Sigma-term update

Volume 253, number 1,2 PHYSICS LETTERS B 3 January 1991

Sigma-term update J. Gasser, H. Leutwyler Institute for Theoretical Physics, University of Bern, Sidlerstrafle 5, CH-3012 Bern, Switzerland

and

M.E. Sainio Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20C, SF-O0170 Helsinki, Finland

Received 24 September 1990

The determination of the a-term from nN scattering is critically examined. The currently available data indicate a= 45 MeV. Low-energy precision measurements are needed to clarify discrepancies in experimental data and to reduce the uncertainty in the strangeness content for which we obtain y-~ 0.2.

1. A venerable low-energy theorem of current algebra [ 1 ] relates the value of the isospin even nN scattering ampli tude at the Cheng-Dashen point ~

~ = F 2 / ) + (2/12) ( 1 )

to the a-term matrix element

1 a= ~ (Pl rh(au+dd) IP) ,

rh= ~ (rnu +rod) . (2)

The theorem states that if the quark masses mu, mo are sent to zero, the ratio .S/a tends to one. This prediction has received continued interest during the last twenty years ( [ 3-10] ) mainly because the analysis of the nN data ~2 leads to 27= 64 + 8 MeV [ 11 ], while naive estimates of the a-term based on the baryon

¢r Work supported in part by Schweizerischer Nationalfonds. ~t Notation:/~ and m are the masses of pion and proton respec-

tively. The function/9 + (t) is the scattering amplitude at v = 0 with the Born term removed, D +(t)=A + (s, t)ls=u-g2/m. As pointed out in ref. [2] a critical examination of the radia- tive corrections occurring in the determination of the pion de- cay constant leads to F. = 92.4 MeV.

~2 A comprehensive analysis of the extrapolation required to reach the Cheng-Dashen point is given in the handbook arti- cle by H6hler [ 3 ], which also contains a review of the literature on the a-term up to 1982.

mass spectrum give a-~ 25 MeV: either, one of these two values is wrong, or the low-energy theorem a=27 is of fby a factor of two or more.

2. Rather radical theoretical conclusions were in- ferred from this disagreement. They are based on the observation that approximate SU (3) symmetry only allows one to estimate the proton matrix element of the SU ( 3 )-breaking piece of the hamil tonian, not the a-term itself. The two quantit ies are related by

ms-rh (Plt2u+dd-2gslp) = ( - ~ - l )

(3)

where y=2(pl:~slp)/(plau+aYdlp) is the share contr ibuted by the operator gs. If y were substantial or if the ratio mJrh were smaller than the standard value [ 12 ] m~/th ~_ 26, then a should indeed be larger. Both of these possibilities are extensively discussed in the literature, but neither of them is plausible, for the following reasons: ( i) If (p]~slp) were the cul- prit, then a large fraction of the proton mass would originate in the piece msgs of the hamil tonian (the matrix element of m~.~s is larger than the matrix element of 2rhYs by the factor mJ2rh-~ 13 - if the latter amounts to 30 MeV, the former is of order 400 MeV). The proton is the lightest state with nonzero baryon

252 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )


number - why does it choose a quark configuration for which (plYslp) is large if this costs that much energy? (ii) I f instead the standard value of the ratio mJrh were off by a factor of the order of two, one would have to conclude that the success of the GeU- Mann-Okubo formula for the pseudoscalar mesons is a queer accident and that, for unknown reasons, the chiral order parameter of lowest dimension, ( 01 qql 0 ) , happens to be suppressed. In our opinion, this "solution" of the puzzle is even more hard to swallow than "solution" (i).

3. The purpose of the present paper is to critically examine the ingredients involved in a determination of the a-term from ~N data. In particular, we analyze the modification of the current algebra result generated by the quark masses, discuss the uncertainties involved in the extraction of _r from the available data, determine the corresponding range in the value of a and compare the result with the information ex- tracted from the baryon mass spectrum. Our main conclusion is that speculations based on the disagreement mentioned above are without foundation: the r~N data are consistent with a small strangeness content and with the standard value ms/rh~_ 26. Addi- tional low-energy data would be very useful to fur- ther reduce the uncertainty in the value ofy.

4. Before we embark upon the details of our analysis, we wish to make a remark concerning the range of applicability of chiral perturbation theory in the present context. Chiral perturbation theory provides an expansion of the scattering amplitude in powers of pion momenta and quark masses which controls the vicinity of the Cheng-Dashen point. The leading term of the expansion is the Born term. Since the corresponding predictions for the scattering lengths [ 1 ] are in good agreement with experiment, the leading term does provide an approximate representation of the scattering amplitude at the threshold of the physical region. In the context of the a-term, we are, however, not dealing with the leading, chirally symmetric contribution, but with the symmetry breaking effects generated by the quark masses. For these intrinsi- cally small effects, the chiral representation does not provide an accurate prediction of the momentum dependence, even if carried out to one loop. In our analysis, we will therefore make use of chiral perturba-

tion theory only at the Cheng-Dashen point, relying on analyticity, unitarity and phase-shift analysis to extrapolate the amplitude from the physical region to this point.

5. In the chiral limit, the relation Y,/a= 1 is exact. To analyze the corrections generated by the quark masses, we expand both a and _r in powers of rh. The leading terms in this expansion are proportional to rh and the low-energy theorem requires these leading terms to be the same. The next term in the expansion is a nonanalytic contribution proportional to rh 3/2 and the coefficients are not the same. The difference is removed, however, if one writes the low-energy theorem in the form proposed by Brown, Pardee and Peccei [ 1 ],

• ~'----- a (2 ]22) + d R , (4)

where a(t) is the matrix element of the operator rh (au + rid) at momentum transfer t. In this form, the nonanalytic terms of order rh 3/2 drop out such that the chiral expansion of dR starts with a contribution of order rh 2, possibly containing a chiral logarithm [ 13,14 ]. The numerical value of BR is not known - it represents the inherent limitation of our analysis. We, however, expect dR to be small. Evaluation of chiral perturbation theory to one-loop yields dR = 0.35 MeV [ 14 ]. This evaluation does not account for all contributions of order rh 2, but it indicates the order of magnitude of these terms - in view of the accuracy to which the value of 2; can be measured in the foresee- able future, the uncertainty associated with the numerical value of dR represents an academic problem.

6. Since the low-energy theorem (4) concerns the value of a ( t ) at t=2/z 2 while the a-term represents the value at t= 0, we need to analyze the difference,

A a - a ( 2 / 1 2 ) - a ( 0 ) . (5)

A similar term arises in the extrapolation of the scattering amplitude from the physical region t ~ 0 to the Cheng-Dashen point. The first two terms in the Taylor series D ÷ (t) = d~o + td~l +... generate the contribution

Xd =F~ ( d& + 2p2d~ ) . (6)

We denote the remainder by/[D,

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S=/~d +AD • (7)

The point of this decomposition is that the Taylor coefficients d~o and d~-i can be expressed in terms of the scattering amplitude in the physical region by means of forward dispersion relations [ 3,15 ]. The quantity Se is, therefore, accessible through phase- shift analysis - we will discuss the results of this analysis below. In terms of Z'd, the low-energy theorem takes the form

tr=Z'd +A, A=AD-A~-AR. (8)

7. In previous work, the correction A D w a s calcu- lated by means of dispersion relations (a comprehensive analysis, including a discussion of earlier work is given in ref. [ 3 ] ), while A, was estimated on the ba- sis of ehiral perturbation theory [ 13,16,14 ]. Both of these evaluations need to be revised.

(i) In the dispersive analysis of AD the cusp associated with nn intermediate states plays an important role. The nn phase-shift representation used in ref. [ 3 ] does, however, not obey the constraints imposed by chiral symmetry which fix both the scattering length a ° and the effective range b ° rather accurately [ 17 ]. We have repeated the analysis with a representation which incorporates these constraints at threshold, relying on experimental results at higher energies. It turns out that this change in the nn phase shift affects the value Of AD very little. Numerically, we find

AD = 12 MeV (9)

(details of the analysis are given in ref. [ 18 ] ). (ii) In the case of A~, the problem is that only the

leading, nonanalytic term in the chiral expansion is under control; the slope of the form factor a ( t ) also receives a contribution from one of the low-energy constants in the effective lagrangian of order p4 whose value is not known. In the accompanying paper [ 18 ], we show that chiral perturbation theory is not needed here: as is the case with AD, the quantity A, can be worked out by means of dispersion relations. Em- ploying methods similar to those used in the analysis of the electromagnetic form factors [ 3 ], we find (for details, see ref. [ 18 ] )

A~= 15 MeV. (10)

8. The results quoted in eqs. (9) and (10) differ considerably from our old estimates [16,15,19] which were based on chiral perturbation theory. To identify the origin of this disagreement we first stress that we are dealing with very small quantities here. At leading order in chiral perturbation theory, the scattering amplitude at u = 0 is given by D ÷ (t) = g2/ rn - this does provide an approximate description of the physical amplitude ~3 in the range 0 ~< t~< 2/~ 2. At the next order in the chiral expansion, one-loop contributions as well as a polynomial of the type ddo + dd-l t show up. If these contributions are taken into account, the chiral representation of D + (t) is accurate in the above range to better than 1%. In units of the total amplitude, the curvature term given in (9), however, represents an effect of less than 1% and, here, even one-loop accuracy is not good enough. The dispersive analysis shows that the magnitude of AD, A~ is driven by the t-channel S-wave amplitude f ° ( t ) . In the one-loop representation, this partial wave is replaced by the corresponding Born approximation. It turns out that this approximation provides an adequate description only in the immediate vicinity of the point t = p2. In the range 4p 2 < t < (500 MeV): which generates the main contribution to the dispersion integrals for AD, A~, the Born approximation is, however, offby about a factor of two. We have checked that in this region the one-loop contributions are indeed of the same sign and of the same order of magnitude as the Born term - no wonder that at leading order, chiral perturbation theory does not give a reliable estimate, neither for Ao, nor for A,.

We take this occasion to comment on the criticism raised in ref. [ 20 ], where it is claimed that chiral perturbation theory omits important physics connected to the A-resonance. By definition, the low-energy constants dd-o and d~-~ occurring in the chiral representation must account for all analytic terms of order p2. In particular, they include the contributions of order p2 generated by the A-singularity. In fact, the main contribution to ddo and dd-~ stems from the A [3].

#3 In regions where the Born term is small, higher order contributions may be of comparable magnitude - in particular, these contributions significantly affect the zeros of the amplitude [ 3 ]. The problem does not concern us here, because we limit ourselves to g---0 and to the interval 0~< t~< 2/z 2 where zeros do not occur.

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What the chiral representation for D + ( t ) at first nonleading order does not account for is only what remains of the d-term after the piece of order p2 is removed. In the range 0~<t~2p 2, this remainder amounts to about 0.2%. As illustrated by the example of riD, d~, chiral perturbation theory does have its limitations, but it certainly does not miss gross fea- tures such as the singularities generated by resonances.

9. The problem does not afflict the dispersive analysis which leads to the values (9), (10). Here the essential ingredients are the I = 0 S-wave ~Tt phase shift and the behaviour of the partial w a v e f ° (t) at small spacelike momentum transfers. The results for riD, do are not sensitive to the uncertainties associated with these quantities [ 18 ]. Moreover, in connection with the a-term, we are interested only in the difference d D - d ~ which is even less sensitive to the details of the input than the individual terms. Neglecting the contribution from the term rig, we conclude that the net correction is

d = - 3 MeV (11)

to within an uncertainty of order 1 MeV. The main point to note here is that the same physical effect which strongly curves the t-dependence of the scattering ampli tude/3 +( t ) also generates a large slope in the form factor a( t ) - in the difference, the effect almost cancels out.

10. We now turn to the relation between Sd = F 2 (dd-o + 2/~2d~-1 ) and the scattering amplitude in the physical region. As emphasized by HiShler [ 3,4 ], analyticity is an essential ingredient in the data analysis - parametrizations in terms of scattering lengths and effective ranges may be useful as an in- tuitive guide [6 ], but are useless on a quantitative level (for a detailed discussion, see ref. [4] ). The constraints imposed by analyticity and unitarity are very strong, due to the wealth of data at higher energies. As pointed out in ref. [ 15 ], these constraints allow one to parametrize the scattering amplitude in the low-energy region in terms of only two constants which originate in the subtractions required in the relevant dispersion relations. It is convenient to sub- tract at threshold, i.e. to identify the two subtraction constants with the scattering lengths ad-+ and a ~-+. If these two parameters are accurately known, the

available high-energy data allow one to fix the two S- wave and four P-waves in the low-energy region and, at the same time to calculate the value of Zd. In this sense, the algorithm described in ref. [ 15 ] provides an adequate substitute for the effective range formu- lae: the energy dependence of the relevant phase shifts is specified in terms of two constants. We emphasize that the choice of ad~+ and a ~-+ as basic low-energy parameters is not essential - one may equally well use, say a~-+ and b~-÷ (for the S-problem this choice may appear to be more natural [21,6,19]). The constraints imposed on the scattering amplitude by the high-energy data very strongly correlate the various threshold parameters. One such correlation stems from the familiar sum rules [22] for the S-wave effective ranges; the correlations between a~-÷, a ~-+, the remaining S- and P-wave threshold parameters and the coefficients dd-o, dd-~ are given explicitly in ref. [ 15 ]. Within these constraints, one choice is physi- cally equivalent to the other.

An accurate determination of the two constants requires precise data at low energies. We illustrate the accuracy needed with the recent measurement of the differential n-+p cross sections in the Coulomb interference region [23]. This experiment provides a determination of Re D ÷ in the forward direction, at a pion kinetic energy of 54.3 MeV. Since the energy dependence of the amplitude is fixed through forward dispersion relations in terms of total cross sections, the result 16.8 + 1.3 G e V - t obtained for the real part of the nuclear forward amplitude [23] amounts to a measurement of the scattering length ad-+, or, equivalently, of the coefficient ddo (the uncertainties involved in this determination o fa d-+, or, equivalently, of the coefficient dd~o (the uncertainties involved in this determination of ad'+ are discussed in ref. [4] ). I f the coefficient dJ-~ were known accurately, the quoted error in the real part of the forward amplitude would correspond to an uncertainty in the value of Sd of F 2 × ( 1.3 GeV - i ) ,,, 11 MeV - in reality the uncertainty is larger because of the error associated with d~l.

11. The result of the Karlsruhe analysis relies on the data of Bertin et al. [ 24 ] at low energies; the corresponding values of dd'o, d~-~ given in ref. [3 ] imply 2~d = 51 + 7 MeV. New data have been published since then [23,25-28 ], but unfortunately large discrepan-

255


cies appear between different data sets for n+p scattering [ 4,29 ]. For this reason, we have analyzed the results separately for each experiment, using the algorithm described in ref. [ 15 ] which also allows us to estimate the uncertainty in the value of Xd due to experimental errors. This algorithm assumes that the imaginary parts of the invariant amplitudes are known for k~,b>~ko; we choose ko= 185 MeV/c and use the KH.80 solution [ 30 ]. In addition, a representation of the D- and F-waves for klab ~ ko is needed as an input; here we rely on ref. [ 31 ]. The forward dispersion relations listed in ref. [ 15 ] then determine the scattering amplitude in the low-energy region in terms of two constants a~'+, a +÷. We apply the electromagnetic corrections according to the prescrip- tion of Tromborg et al. [ 32 ], calculate the corresponding differential cross sections and compare the result with the low-energy data. Finally, we search for the best fit, the parameters in the search being ad~+, a ~-+ and an overall normalization parameter for each

angular distribution. Preliminary results of the present analysis are re-

ported in ref. [ 33 ] and a more complete account will be published elsewhere. The main results are as fol- lows. If data below 170 MeV/c are considered, only those of Bertin [ 24 ] and Frank [ 27 ] yield consistent solutions (provided Benin data at 153 MeV/c and Frank ~+p data at 95 MeV/c are removed from the data base). Single energy measurements [ 23,25 ] do not provide enough information to arrive at a stable solution, Ritchie [28 ] data concentrate to the upper end of the allowed momentum range which again gives trouble with stability, and for the Brack [26 ] experiment the ~+p and ~ - p results appear incon- sistent in the present framework. (The same would be true for the Frank data, but there the quoted normalization uncertainties are large. ) The result for the Benin data is I d = 4 8 MeV. In fig. 1 the curves

2 Y(min "l- 1 and Z2i. + 4, corresponding to one and two standard deviations for Id respectively, are displayed in the (a6F+, a i~+ )-plane with a solid dot denoting the minimum (solution A). For the Frank data the result i s / d = 5 0 MeV (solution B in fig. 1 ). As far as the value o f l d goes, the two data sets thus lead to essentially the same result. To estimate the error bars to be attached to these results, a number of effects were considered. These considerations yield

0.137

0.136

0.135

O.13&

0.133 ':a~

+~ 0.132

0.131

0.130

0.129

0.128

I '

! : / .... 'B

• /

*0.01 0.00 -0.01 -0.02 a + -I o.(~ )

Fig. 1. Value of the quantity Id as a function of the scattering lengths ad-÷ and a ~-+. Solution A results from a fit to the data of Bertin et al. [24] while B corresponds to the data of Frank et al. [ 27 ]. The error ellipses represent ~ 2 i n + 1 (full line) and Z~i, + 4 (dotted line ). The rectangle C indicates the KH. 80 values of the two scattering lengths [ 3 ]. The vertical bar on the right of the figure represents the preliminary value ( 2 a , + a 3 ) = ( 0 . 2 5 8 + 0 .012) / t - ' found in a recent measurement of the level shift in pionic hydrogen [34]. The vertical lines shown on the left indicate the result of the Coulomb interference experiment [ 23 ] which according to ref. [ 4 ] implies a +÷ = (0.012 + 0.013 )# - i.

/ a = 4 8 + 4 + 4 + 4 M e V (solution A) ,

/ d = 5 0 + 3 + 7 + 4 M e V (so lu t ionB) . (12)

The first error is related to the statistics and can be read directly from fig. 1, the second shows the sensitivity to modifications of the data base (change in Id occurring if data above 170 MeV/c are included in the fit) and the third is an estimate of the uncertainty associated with the input of our algorithm (at the precision needed to extract the small quantity Id, the D-wave contributions to the real parts play a significant role; random variations of this part of the input by 15% generate an uncertainty in the result for Id of about 2.5 MeV; comparing different phase-shift solutions we infer that the uncertainties associated with the imaginary parts for klab >I ko are of about the same size). The value used for the coupling constant g, of course, also enters in our input and it plays a central role. Since it is very strongly correlated with the phase shift information used, we are not in a position to discuss the sensitivity of the result to uncertainties in the value of g, but refer to ref. [4]. The numerical

256


value used for F~, on the other hand, does not play a significant role here. We did investigate the isospin breaking effects generated by the mass difference A ÷ + - A °, supplementing the electromagnetic corrections with a term which accounts for the corresponding shift of the peak in a Breit-Wigner formula. The effect slightly improves the quality of our fits, but it affects the result for Xd only at the 1 MeV level.

In fig. 1 the solution C refers to the KH.80 range of the parameters ad-+ and a~-+ [3]. The dash-dotted line on the left of the figure corresponds to the ReD + ( t = 0 ) value at 54.3 MeV of ref. [23] and the dashed line is the respective error band. The recent measurements of the integrated n+p cross section [35] are consistent with the Karlsruhe results, and, therefore, lend support for the solutions A and B, which are not too far from the Karlsruhe solution. Some low-energy information is also available for the charge exchange reaction n - p ~ n ° n [ 36-38 ]. How- ever, the main issue here is the isospin even amplitude at the Cheng-Dashen point, which is not sensitive to the isospin odd part probed by the charge exchange process. Generally the experiment is in reasonable agreement with the Karlsruhe amplitudes which are close to the solutions A and B obtained here. As indicated in the figure, the recent data on the level shift of pionic hydrogen [ 34] favour solution A.

12. The data indicate that Xd is approximately equal to 50 MeV. None of the data sets published in the last decade allows one to reduce the uncertainties in this value and it does not make sense to combine them as they are contradictory. We have detailed our error estimates above; the reader is invited to add errors in quadrature and, more urgently, to resolve the experimental discrepancies. Using the results for A and A D given above, solution A implies

am 45MeV, Xm60MeV. (13)

The value for a is lower than the previous estimates by about 10 MeV, while the value for X essentially confirms the conclusion H6hler and his collaborators reached in 1982 [3,11 ].

The uncertainties in the above numbers could be reduced substantially if the experimental situation were to improve - in our opinion both the procedure used to determine the value of Xa from the experimental information and the corrections which relate

this value to the a-term matrix element are now understood (note that the 10 MeV shift mentioned above originates in the t-dependence of the a-term form factor which turns out to be considerably more pronounced than what is indicated by chiral perturbation theory at leading order).

13. Let us now confront this result with the theoretical predictions based on approximate SU(3) symmetry. Expanding the masses of the baryon octet in powers of ms - rh around a common value mu = md = ms = rh of the quark masses, retaining only the terms of first order in m s - rh and assuming that the strange quark content y of the proton vanishes, one arrives at the result am 25 MeV mentioned above. The corrections generated by the terms of order (ms - rh ) 2 are expected to modify this result at the (20-30) % level characteristic of SU (3) breaking effects. They were analyzed in ref. [39] where it was shown that the self-energies associated with virtual mesons generate a remarkably large, positive contribution, enhancing the value of a by about 10 MeV #4. The result is [ 16 ]

a=ao/(1-y) , a o = 3 5 + 5 MeV. (14)

In view of ( 13 ), this implies a strange quark constant of ym 0.2. This value appears to be quite reasonable as the corresponding contribution of the term ms~s to the proton mass is of order (ms/2rh) X 10 M e V - 130 MeV. There is no evidence for a large strange quark content of the proton, but the uncertainties are con- siderable - accurate new data are required to pin them down. Ignoring these uncertainties, the conclusions concerning the discrepancy mentioned in the first paragraph are the following: (i) The value of the amplitude at the Cheng-Dashen point, 27___ 60 MeV, is consistent with the old estimate [3]. (ii) The magnitude of the a-term matrix element, am45 MeV, neither agrees with the naive estimate, a-~ 25 MeV nor with the value am_r suggested by the low-energy theorem. (iii) About half of the difference between 25 and 45 MeV is attributed to SU(3)-breaking effects in the matrix elements of the scalar currents, the

a4 In that analysis a cutoffin momentum space was used to tame the ultraviolet divergences. The calculation is presently being repeated [40] using modern techniques ofchiral perturbation theory where a cutoff is not needed.

257


o t h e r ha l f to the con t r i bu t i on gene ra t ed by the oper -

a to r ~s. ( i v ) T h e d i f fe rence be tween a and X s tems

f r o m the t - dependence o f the m a t r i x e l e m e n t

(p ' l~u+ddlp) . T h e a n o m a l o u s th re sho ld associ-

a ted wi th nn i n t e rmed ia t e states causes the scalar fo rm

fac tor a(t) to rise m u c h m o r e rap id ly wi th t t han the

f o r m factors o f the e l ec t romagne t i c current : on the

in te rva l 0~< t~< 2/t 2 the scalar f o r m fac tor grows f r o m

a ( 0 ) - ~ 4 5 M e V t o a ( 2 # 2 ) - 60 M e V [18] .

We thank Professor G. H 6 h l e r for generous sup-

por t o f ou r work and for useful c o m m e n t s conce rn ing

the p resen t paper .

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