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CHIRAL CORRECTIONS TO THE S–WAVE

arXiv:hep-ph/9304275v1 20 Apr 1993CHIRAL CORRECTIONS TO THE S–WAVEPION–NUCLEON SCATTERING LENGTHS *V´eronique BernardCentre de Recherches Nucl´eaires et Universit´e Louis Pasteur de StrasbourgPhysique Th´eorique, Bat. 40A, BP 20, 67037 Strasbourg Cedex 2, FranceNorbert KaiserPhysik Department T30, Technische Universit¨at M¨unchenJames Franck Straße, W-8046 Garching, GermanyUlf-G. Meißner†Universit¨at Bern, Institut f¨ur Theoretische PhysikSidlerstr.

5, CH–3012 Bern, SwitzerlandABSTRACTWe calculate the chiral corrections to Weinberg’s prediction for the S–wave πN scat-tering lengths up–to–and–including order O(M 3π) making use of heavy baryon chiralperturbation theory.For the isospin–odd scattering length a−these corrections aresmall and bring the prediction closer to the empirical value. In the case of the isospin–even a+ large cancellations appear so that the O(M 3π) result depends sensitively oncertain resonance parameters which enter the calculation of the contact terms presentat next–to–leading order.BUTP–93/09April 1993CRN 93–13* Work supported in part by Deutsche Forschungsgemeinschaft and by SchweizerischerNationalfonds.† Heisenberg Fellow.0

1. One of the most spectacular successes of current algebra in the sixties was Wein-berg’s prediction [1] for the S–wave pion–nucleon scattering lengths, a1/2 = Mπ/4πF 2π =−2a3/2 = 0.175 M −1π , with Mπ the physical pion mass and Fπ the pion decay constant.Tomozawa [2] also derived the sum rule a1/2 −a3/2 = 3Mπ/8πF 2π = 0.263 M −1π .

Em-pirically, the combination (2a1/2 + a3/2)/3 is best determined. The Karlsruhe–Helsinkigroup gives 0.083 ± 0.004 M −1π[3] consistent with the pionic atom measurement [4] of0.086 ± 0.004 M −1π .

The value of a1/2 −a3/2 is more uncertain. The KH analysis leadsto 0.274 ± 0.005 M −1π[5] whereas the VPI group has recently given a larger value [6].Their analysis was critically reexamined by H¨ohler [7].

In what follows, we will use thecentral values from the work of Koch [3], namely a1/2 = 0.175 M −1πand a3/2 = −0.100M −1π . The agreement of the current algebra predictions with these numbers is ratherspectacular.

However, in the last decade it has become clear that current algebra is onlythe first term in a systematic expansion of the QCD Green functions in powers of (small)external momenta and the (light) quark masses [8,9]. Therefore, one would like to knowwhat the next–to–leading order corrections to the original predictions are.

This is ex-actly the question we will address here. The basic framework to perform the calculationof these corrections is baryon chiral perturbation theory which makes use of an effec-tive Lagrangian of the asymptotically observed fields.

In particular, we use the heavyfermion formulation [10] in which the nucleons are considered as static sources and onehas a one–to–one correspondence between the loop and the small momentum expansion.We will work out one loop and counter term contributions up–to–and–including orderM 3π based on the power counting scheme developed by Weinberg [8] and later extendedto the baryon sector by Gasser et al. [11].

For a review on these methods, the reader isreferred to ref.[12].2. Consider the πN forward scattering amplitude for a nucleon at rest with its four–velocity given by vµ = (1, 0, 0, 0).

* Denoting by b and a the isospin of the outgoing andincoming pion, in order, the scattering amplitude takes the formT ba = T +(ω)δba + T −(ω)iǫbacτ c(1)with q the pion four–momentum and ω = v·q = q0. Under crossing (a ↔b, q →−q) thefunctions T + and T −are even and odd, respectively, T ±(ω) = ±T ±(−ω).

At thresholdone has ⃗q = 0 and the pertinent scattering lengths are defined bya± = 14π1 + Mπm−1 T ±(Mπ)(2)* Remember that we treat the nucleons as very heavy fields.1

with m the nucleon mass. The S–wave scattering lengths for the total πN isospin 1/2and 3/2 are related to a± viaa1/2 = a+ + 2a−,a3/2 = a+ −a−(3)The abovementioned central empirical values translate into a+ = −0.83 · 10−2 M −1πand a+ = 9.17 · 10−2 M −1π .

In what follows, we will not exhibit the canonical units of10−2 M −1π . The benchmark values are therefore a+ = −0.83±0.38 and a−= 9.17±0.17*compared to the current algebra predictions of a+ = 0 and a−= 8.76 (using Mπ = 138MeV and Fπ = 93 MeV).3.

To calculate the scattering lengths, we use the effective pion–nucleon Lagrangian.We work in flavor SU(2) and in the isospin limit mu = md = ˆm. The pion fields arecollected in the matrix U(x) = exp[i⃗τ · ⃗π(x)/Fπ] = u2(x).

The effective Lagrangian toorder O(q3), where q denotes a genuine small momentum or a quark mass, readsLπN = L(1)πN + L(2)πN + L(3)πNL(1)πN = ¯H(iv · D + gAS · u)HL(2)πN = c1 ¯HH Tr (χ+) +c2 −g2A8m ¯Hv · u v · uH + c3 ¯Hu · uHL(3)πN =b1 −g2A32m2vµvλvρ ¯HKµλρH + b2 vρ ¯HKµµρH + b3 vρ ¯HKµρµH(4)withuµ = iu†∇µUu†uµλ = iu†∇µ∇λUu†Kµλρ = i[uµλ, uρ](5)where H denotes the heavy nucleon field, Sµ the covariant spin–operator subject to theconstraint v·S = 0, ∇µ the covariant derivative acting on the pions and we adhere to thenotations of ref.[13]. The superscripts (1,2,3) denote the chiral dimension.

The lowestorder effective Lagrangian is of order O(q). The one loop contribution is suppressedwith respect to the tree level by q2 thus contributing at O(q3).In addition, thereare contact terms of order q2 and q3 with coefficients not fixed by chiral symmetry.Notice furthermore that one in addition has to go further in the 1/m expansion of therelativistic tree level graphs as can be seen from the terms which come together with* We have added the errors obtained for a1/2 and a3/2 in quadrature.2

the ones proportional to c2 and b1. At next–to–leading order, all these terms have tobe retained.

Due to crossing symmetry, L(2)πN contributes only to T +(ω) whereas L(3)πNsolely enters T −(ω). We have not exhibited the standard meson Lagrangian L(2)ππ +L(4)ππ.The contact terms appearing in L(4)ππ only contribute to the shift of the pion mass, thepseudoscalar coupling Gπ and the pion decay constant from their chiral limit to thephysical values.

For the following, we defineL =Mπ8πF 2π,µ = Mπm(6)The calculation of the scattering lengths is straightforward. For the isospin–odd a−onearrives ata−= a−(Mπ) + a−(M 2π) + a−(M 3π)a−(Mπ) = L ,a−(M 2π) = −Lµa−(M 3π) = Lµ21 + g2A4+ L2Mππ1 −2 ln Mπλ−64πL2MπF 2πbr1(λ) + b2 + b3(7)with λ the scale of dimensional regularization.

While the constants b2 and b3 are finite,b1 has to be renormalized as follows to render the isospin–odd scattering amplitudeT −(ω) finite,b1 = br1(λ) −K2F 2K = λd−416π21d −4 + 12γE −1 −ln 4π(8)Here, F is the pion decay constant in the chiral limit. In what follows, we will useλ = m∆= 1.232 GeV, motivated by the resonance saturation principle.

Of course,physical observables do not depend on this particular choice. Notice that the contactterm contributions are suppressed by a factor M 2π with respect to the leading currentalgebra term.

Matters are different for the isospin–even scattering length a+. It consistsof contributions of order M 2π and M 3π,a+ = a+(M 2π) + a+(M 3π)a+(M 2π) = 32πF 2πL2c2 + c3 −2c1 −g2A8ma+(M 3π) = 34g2AL2Mπ −32πF 2πL2µc2 + c3 −2c1 −g2A8m(9)The coefficients c1,2,3 are all finite.

From the form of eq. (9) it is obvious that the contactterms play a more important role in the determination of a+ than for a−.3

4. The most difficult task is to pin down the various low-energy constants appearingin eqs.

(7) and (9). Let us first consider c1,2,3.

The coefficient c1 can be unambigouslyfixed from the pion–nucleon σ–term [13],c1 = −14M 2πσπN(0) + 9g2AM 3π64πF 2π= −0.87 ± 0.11 GeV−1(10)using gA = 1.26 and σπN(0) = 45 ± 8 MeV [14]. To estimate the remaining constants,we make use of the principle of resonance saturation [15].

It states that to a high degreeof accuracy the low–energy constants can be calculated from resonance exchanges byintegrating out the heavy fields from an effective Lagrangian of the pions chirally coupledto the various resonances. In the meson sector, this has been shown to work very well.We extend this method to the baryon sector since it is essentially the only method ofestimating the unknown coefficients.

From the meson sector, scalar meson exchange cancontribute to c1 and c3,c1 −12c3S = c1 −c1cdcm(11)with 2cd/cm = L5/L8 [15]. The central values for the parameters cd and cm given inref.

[15] are cm = 42 MeV and cd = 32 MeV, i.e. 2cd/cm = 1.56.

However, withinthe uncertainty of L5 and L8, this ratio can vary between 0.75 and 2.25. In addition,intermediate ∆(1232) states give a contribution to c2 + c3.

The general π∆N–vertexcan be written asLπ∆N = gπ∆N2m¯∆µagµν −(Z + 12)γµγν∂νπa N + h.c.(12)where ∆µ denotes the Rarita–Schwinger field and Z parametrizes the off–shell behaviourof the spin–3/2 field.⋆While Peccei [16] fixed Z = −1/4, a more recent phenomenologicalanalysis of Benmerrouche et al. [17] gives a rather wide range, −0.8 ≤Z ≤0.3.

Using theempirically well–fulfilled SU(4) coupling constant relation gπ∆N = 3gπN/√2 togetherwith the Goldberger–Treiman relation, we can cast the ∆(1232) contribution to c2 + c3into the formc2 + c3∆= −g2A2m2∆(12 −Z)2m∆(1 + Z) + m(12 −Z)(13)⋆It is mandatory to use here the relativistic formulation of the spin–3/2 field sinceotherwise one would miss a contribution of order M 2π to a+.4

Clearly, this dependence on Z is one of the major sources of uncertainty in fixing thevalues of the contact terms c2 and c3. Furthermore, there is also a contribution fromthe Roper N ∗(1440) resonance to c2 + c3,c2 + c3N∗= −g2AR16(m + m∗)(14)with R = 1 .

. .1/4 for gπNN∗= (1/2 .

. .1/4)gπN [18].

In complete analogy, one also hasa ∆and N ∗contribution to the low–energy constants br1, b2 and b3. At λ = m∆, it canbe written asbr1(λ) + b2 + b3λ=m∆= −g2A(Z −12)28m2∆+R32(m + m∗)2(15)Finally, other baryon resonances have been neglected since their couplings are eithervery small or poorly (not) known.

* It is interesting to note that for Z = −1/4 andR = 1, the N ∗(1440) contribution to c2 + c3 and to br1(λ) + b2 + b3 is 4 and 12 per centof the ∆–contribution, respectively. However, in the latter case the contact terms playa much less pronounced role as already discussed.5.

We now present our numerical results. Consider first the scattering length a−.

UsingMπ = 138 MeV, Fπ = 93 MeV, m = 938.9 MeV, Z = −1/4 and R = 1, we finda−= (8.76 −1.29 + 1.69) · 10−2 M −1π= 9.16 · 10−2 M −1π(16)where we have explicitely shown the contributions of order Mπ, M 2π and M 3π (and mo-mentarily reinstated the units). The total result is in good agreement with the empiricalvalue.

The largest part of the M 3π term comes from the pion loop diagrams, it amountsto 1.31 for λ = m∆. Varying Z within its band of allowed values, a−varies by ±0.15.The dependence on R shows up only in the third digit and is thus irrelevant.Theone–loop corrections bring the the lowest order value closer to the empirical one.

Onemight argue that since the M 3π contribution is even larger than the M 2π one, the chiralseries has no chance of converging. However, the important one loop effect can onlyshow up at order M 3π due to the chiral counting.

The two–loop contribution carries an* A remark on the ρ–meson is in order.The chiral power counting enforces a ρππvertex of order q2 of the form L(2)ρππ = gρππ Tr (ρµν[uµ, uν]) [15].In forward directionthe contraction of the ρ–meson propagator with the corresponding ρππ matrix elementvanishes. Therefore, one has no explicit ρ–meson induced contributions to a−of order q2and q3.5

explicit factor M 5π and is therefore expected to be much smaller. Clearly, one would liketo perform a calculation beyond O(q3) as done here, but this is beyond the scope ofthis paper.

In the case of the isospin–even scattering length a+, the situation is muchless satisfactory. There are large cancellations between the loop contribution and the1/m suppressed kinematical terms of order M 2π and M 3π.

For Mπ = 138 MeV, these twoamount to 0.91−0.87 = 0.04 in the conventional units. Therefore, the role of the contactterms is even further magnified.

In fig.1, we show a+ as a function of Z for our standardinput and R = 1. The empirical value of a+ can be obtained for a small and positivevalue of Z, Z ≃0.15.

As shown in fig.2, a+ varies also strongly as the ratio 2cd/cmchanges. Setting Z = 0, the empirical value results for 2cd/cm = 1.32, not far from itscentral value of 1.56.

Clearly, a better understanding of these resonance parameters isnecessary before one can draw a final conclusion on the accuracy of the chiral expansionfor a+. It is, however, gratifying that slight variations of these resonance parametersallow one to obtain the empirical value.6.

To summarize, we have used heavy baryon chiral perturbation theory to calculate thecorrections up–to–and–including order M 3π to Weinberg’s lowest order (current algebra)predictions for the two S–wave pion–nucleon scattering lengths, a±. To estimate thestrength of the various contact terms, we have made use of the principle of resonancesaturation which is known to work very accurately in the meson sector.

The main resultsof this investigation are:• The chiral corrections to the isospin–odd scattering length a−are small and positiveand move the lowest order prediction closer to the empirical value. The main effectcomes form the pion loop diagrams.

The contact term contribution is relativelysmall, thus masking the uncertainty in estimating the coefficients which appeartogether with these terms.• The situation is very different for the isospin–even scattering length a+. The contactterm contribution completely dominates the chiral expansion since there is a largecancellation between the one–loop and the kinematical corrections.

The total resultfor a+ is very sensitive to some of the resonance parameters, the empirical value ofa+ can, however, be obtained by reasonable choices of these.Evidently, further investigations have to go into two directions.First, a calculationbeyond O(q3) has to be performed to find out how fast the chiral expansion of thescattering lengths converges. Second, a better understanding of the coefficients of thecontact terms appearing at order q2 and higher is necessary to further pin down theprediction for a+.

We hope to come back to these topics in the near future.6

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T. Ericson and W. Weise, ”Pions and Nuclei”, Clarendon Press, Oxford, 1988.Figure CaptionsFig.1 The scattering length a+ as a function of Z in units of 10−2 M −1π . The input isspecified in the text.

The empirical range is also shown (the solid line gives thecentral value).Fig.2 The scattering length a+ as a function of 2cd/cm in units of 10−2 M −1π . The inputis specified in the text.

For notations, see fig.1.7

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