Nuclear Physics AS18 (1990) 201-206

North-Holland

THE PION-NUCLEON SIGMA TERM AND THRESHOLD

PARAMETERS*

M.P. LOCHER

Paul Schemer Znstitut**, CH-5232 Villigen, Switzerland

M.E. SAINIO

Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20 C, SF-001 70 Helsinki, Finland

Received 7 March 1990

Abstract: We solve a set of integral equations which, exploiting unitarity and analyticity, determine the

behaviour of the S- and P-wave phase shifts near threshold in terms of two subtraction constants

proportional to the scattering lengths ai+ and a:+. These constants also fix the values of the

scattering lengths (volumes) and effective ranges a;+, a:_, a;+, bi+ which in turn can be used to

fix the P-term. Here we investigate, in particular, the role of bl+ in Z, and the ~-p scattering

length a$;” in the light of existing data.

Over a period of many years it has been one of Torleif Ericson’s particular strengths to look at well-studied simple systems from a fresh angle, combining all the available results and getting important and accurate information ‘). A typical example is the rrN system at low energies, where pionic atom level shifts and pion-nucleon scattering experiments are determining fundamental parameters of the strong interaction. One such parameter is the sigma term which measures the strength of chiral symmetry breaking due to the u- and d-quark masses, and thus is intrinsically a small and sensitive quantity. Establishing a discrepancy with the determination of this same quantity from the baryon mass spectrum could have important consequencies for the structure of the nucleon in terms of its strange quark content or for our understanding of chiral symmetry breaking.

In ref. ‘) Ericson has suggested to extrapolate the isosymmetric TN amplitude to the Cheng-Dashen point*** (S = m*, t = 2~’ and q2 = q’2 = p2) by following a line of constant cos 0 = 0. In this way the role of the P-waves, in particular the importance of the P33 amplitude is de-emphasized. In fact this choice is one of infinitely many. Besides some correction terms, the Z-term can be expressed by the low-energy scattering lengths (volumes) al+, UT+, a:_ and the effective range bl+ . As has been shown by Gasser 3), any one of these parameters can be eliminated by means of a sum rule exploiting the one-parameter freedom in the one-loop representation of

l A contribution to the Festschrift in honour of T.E.O. Ericson.

** Formerly Schweizerisches Institut fur Nuklearforschung (SIN).

*** The pion mass is denoted by p and m is the mass of the proton.

0375-9474/90/%03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)

202 M. P. Lecher, M. E. Sainio / TN I-term

the Z-term. Of course, by keeping proper track of all terms in the expansion the resulting sigma term is independent of the particular representation chosen. For a consistent scheme the accuracy, within which this is true, is of the order of 2 MeV (while the Z-term from the analysis of scattering data is of the order of 60 MeV). The merit of Ericson’s scheme is to stress the importance of the effective range parameter bl+ even if its contribution to the amplitude might appear small at first sight. In the present note we shall determine the range of variation of bz+ in the light of old and modern low-energy scattering data. At the same time we shall also determine the range of variation for the 7r-p scattering length, arTP, which is important for experiments with pionic atoms.

When analysing TN scattering amplitudes at low energies, it is crucial to impose the constraints of analyticity, unita~ty and crossing at all times. This programme has been implemented in great detail by the Karlsruhe-Helsinki group for many

years 4*5). For the present analysis we shall use the method of ref. 6), which exploits a set of six forward dispersion relations for the amplitudes D*, B* and 8” (where g is the slope with respect to the momentum transfer at t = 0). We use the notation of ref. “). The bar denotes subtraction of the pseudovector Born term from the amplitudes. We recapitulate the essence of the method 6, without repeating all the details:

The dispersion relations are functions of the pion laboratory energy u =

(k2-tCL ) - 2 1’2 For momenta k > k0 = 185 MeV/c the amplitudes are well determined by the Karlsruhe analysis “) (total cross sections, differential cross sections and polarizations are measured). Hence one can get a first estimate of the real parts of the amplitudes for k < k. by calculating the dispersion integral for k > b from known input. The imaginary part in the low-energy window O< k < k. is then calculated from elastic unitarity. In the next step one re-enters the dispersion relations with the full k-range and iterates until convergence is reached. In this way one obtains unique S and P partial waves as a function of energy in the low-energy window 0 < k < k. assuming that the higher waves, D and F, are known in the same momentum range. The solution depends on only two free parameters, which are the scattering lengths a:+ and a:+. These parameters play the role of subtraction constants in the dispersion relations for ii” and B+.

Since all low-energy phase shifts are then known, the threshold parameters are all determined as well. These are a;+, b,l, a:- and a;+ (while a$+ and a:+ are free parameters). Of course, for any pair (a:+, a :+) the corresponding low-energy phase shifts can be compared to existing low-energy data sets in the window 0 < k < ko,

which at this stage have not been yet used. Minimizing x2 one obtains (in favourable cases) a unique solution and hence a unique pair (a:+, a:+). For such a solution the sigma-term for on-shell pions can be calculated e.g. from the formula 6,

~=4&,[(1+~x+~x2)u~+-/1*b~++3x$‘a~+]++,,, (I)

where x = p/m. Chiral perturbation theory to one loop predicts -X0 = -12.6 MeV

M.P. Locher, M.E. Sainio / TN X-term 203

[ref. “)I. Eq. (1) corresponds to the special case T = 0, a parameter to be discussed

below, of the more general expression based on the one-loop representation of the pion-nucleon amplitude ‘**)

D+( Y, t) = L)~~I, + O;fv + 0;: + O:,i , (2)

where v = (S - u)/4m, and similarly for B+( V, t). The polynomial part is

II&,( V, t) = d&+ v2d&+ td,: , (3)

and the pseudovector Born term is denoted by D&. The A-contribution and the loop corrections (D&i) are normalized in such a way that their Taylor expansions around v = t = 0 start with terms higher than the ones appearing in D&,,. The definition of the X-term

ZZ=F2,~+(V=O,t=2$) (4)

yields from eqs. (2) and (3) at the one-loop level

~=F2,(do+o+2~2do:)+-r;d+~uni. (5)

A general expression of the 2 in terms of the threshold parameters nl+, a:+, a:_ and bi+ can be reached as follows 3): (i) the contributions from I&,, Di and O:ni in eq. (2) for the threshold parameters are calculated, appropriately expanding all the kinematical factors, (ii) a linear combination of the parameters is taken in such a way that the contributions from the polynomial part add up to the leading first term in eq. (5). A term propo~ional to dTO can be kept with an arbitrary coefficient. Terms proportional to b&,, the first term in the expansion for I?+( V, t)/ V, should not appear since eq. (5) does not contain b;tb. These requirements yield for the Df amplitude at the Cheng-Dashen point the contributions

1(~)=4~[x,u~++~~(x~b~++~~a:++x,a:_)], (6)

where

x1 =:(x2+2x+4-2x7)) x2=-(1+7)

X3=3X-27, X,= -7,

and r is an arbitrary real parameter representing the freedom in the coefficient of d&. The polynomial part then adds up to

g+(T) = G+2EL2G* - r(l +X)P2G, (7)

where the third term can be compensated using eq. (2) and the integral over the TN total cross section

J+=D+(Y=/L, t=O)-ij+(v=o, t=0)

204 MI? Lmher, M.E. Sainio / ?rN Z-term

The final result for JZ at the one-loop level is then

~==F2,[1(7)+(1+X)7J+]+~R, (9)

where -CR contains contributions from the pseudovector Born term, the A and the one-loop corrections. Expressions like the one in eq. (9) have been discussed before (apart from loop corrections). Also, the treatment of the A may differ, but by concentrating on the part involving low-energy parameters, one could say that the approach of refs. 9*1o) corresponds to the case with T = - 1. Similarly, the case T = 0 has been discussed in ref. ‘I), and the result of Ericson “) is close to the value T = X. It is, in fact, also possible to suppress both P-wave scattering volumes in eq. (6), if their ratio is known, by choosing T = 3x/(2+ (~:_/a:+)) x0.2837.

So far our results are for on-shell pions. For off-mass-shell pions X is reduced by 5 MeV at the one-loop level ‘)

cr=Z-5MeV. (10)

The correction due to the higher loops is expected to be somewhat larger. In fig. 1 we illustrate the sensitivity of the effective range parameter bl+ to the

low-energy data set (k < !q,) chosen. Solution A uses the Bertin data ‘*) for GT+P scattering. These data are the main constraint in the Karlsruhe analysis at low energies (solution C; corresponding to KH.80 as quoted in ref. “)). Our method of finding analytic solutions imposes strong constraints between the amplitudes at different energies. In this way the consistency of data for all channels in the

0.136

0.128 -0.02 -0.03

a',+b.?)

Fig. 1. The value of the effective range bz+ as a function of the parameters oz+ and a:+ is shown by the solid lines. On the left are the curves for ,&“+ 1 (solid line) and ,&“+4 (dotted line) for the Bertin ef af. data (minimum A) and for the Frank et ai. data (minimum B). The box C is the area corresponding

to the KH.80 solution for a:+ and a:+, see table 2.4.7.2. of ref. ‘?.

M.P. Lecher, M.E. Sainio / aN X-term 205

low-energy window can be tested. For Bertin et al. ‘*) only the data at k = 153 MeV/ c

have unreasonably large x2 p.d.f. (bigger than two). If we float the overall normali- zation of the data at this momentum we find that the problem is not a question of normalization. The more recent data of Frank et al. 13) (not available in the Karlsruhe analysis ‘*‘)) lead to the solution marked B. In this case the r+p data for the momentum k = 95 MeV/c would require renormalization of about +30% to be consistent with the remaining data on the basis of the dispersion relations. Also, for higher momenta the r’+p data tend to be lower than the data of ref. ‘*), but the quoted normalization uncertainties are large. The 7r-p data on the other hand are consistent. Very recent results from TRIUMF 14) are between the Bertin and Frank data for 7r+p scattering. It is, however, not possible within the present dispersion analysis to find a solution which is consistent with their 7r-p data at the same time. In our scheme it is always assumed that the amplitudes for k> ko are correct. For these momenta total cross section measurements are available and they have been used as input for the Karlsruhe analysis.

As has been discussed in ref. “) the X-term from eq. (1) is about 60 MeV for the Karlsruhe solution “). On the other hand, about 39 MeV was obtained “) from the then available mesic hydrogen atom energy shift. The Z for the case T = x is only slightly larger than the r = 0 value. From eq. (9) we obtain for 7 =x

2 =4&[(1+;X-+x’)n,=- ~‘((1 +x)@++xd:_ -xti:+}]+corrections ,

(11)

where the pseudovector Born contributions have been directly subtracted from the low-energy constants. The corrections in eq. (11) turn out to be about 5 MeV in the one-loop calculation. For the solution C the individual contributions in the order of eq. (11) are

t: = [0.5 -{ -51.4+5.7 -9.2}] MeV+corrections

(the pseudovector Born contribution to bl+ . IS about 0.0127~~~). Here it can be seen that the term proportional to Fi+ makes about 85% of the total B value. The solutions A and B, corresponding to the Bertin and Frank data, respectively, yield from eq. (11) the Z-values 58 and 60 MeV. The contributions of the Kz+ terms are in these two cases 47 and 52 MeV, i.e. 82-87% of the total Z:

From pionic hydrogen one obtains the r-p scattering length from the energy level shift. For the appropriate isospin combination, atiP = al+ + a;+, we suggest the use refs. 5*6) of the antisymmetric scattering length a;+, which is known very accurately from a sum rule. In this way we avoid any interpretational problems with the mesic deuteron data which could also be used for this purpose. In fig. 2 we show the results for ulJP corresponding to the solutions discussed in fig. 1. The discrepancy between the solutions based on the low-energy scattering data and the atomic measurement 15) is quite pronounced. It will be interesting to see where the recent runs of the W- hydrogen atomic level-shift experiment 16) will place this

206 M.P. Luther, ME. Sainio I TN Z-term

I ,I,, LL ” 0.01 -0.02

a*,+bi')

i i i i i

i

Fig. 2. The value of the n’-p scattering length a&* as a function of the parameters a:+ and a:+ is shown by the solid lines. Solutions A, B and C as in fig. 1. The error band shown by the chain lines reflects

the result of the n-p atomic measurement t&p = 0.059 * 0.006 p-’ [ref. “)I.

interesting quantity. In order to get stable results our experience shows that experi- mental data should be analysed comprehensively. High-precision scattering data should be taken extensively in one expe~mental set-up for all charge channels and for large energy and angular ranges.

We are grateful to Juerg Gasser and Heinrich Leutwyler from the University of Bern for numerous discussions and constant encouragement.

References

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Nefkens, Los Alamos report LA-11184-C (1987), p. 266 4) G. Hiihier, in ~doIt-B~mstein, ed. H. Schopper, vol. 9 b2 (Springer, Berlin, 1983) 5) R. Koch and E. Pietarinen, Nucl. Phys. A336 (1980) 331;

R. Koch, Z. Phys. Cl5 (1982) 161; C29 (1985) 597; Nucl. Phys. A448 (1986) 707 6) J. Gasser, H. Leutwyler, M.P. Locher and M.E. Sainio, Phys. Lett. B213 (1988) 85 7) J. Gasser, Nucl. Phys. B279 (1987) 65 8) J. Gasser, M.E. Sainio and A. &arc, Nucl. Phys. B307 (1988) 779 9) G. Altarelli, N. Cabibbo and L. Maiani, Phys. Lett. B35 (1971) 415; Nucl. Phys. 834 (1971) 621

10) M.G. Olsson and E.T. Osypowski, J. of Phys. G6 (1980) 423

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