3. Low Energy Theory 23
3.3. Comparision with theoretical predictions
MeV
E0+
M1+
neutron
-4 -2 0 2 4
200 300 400 500 600
-4 -2 0 2 4 6
200 300 400 500 600
Figure 3.1.:The integrand of γ = R∞
νthrI(ν)dν3, I(ν) = ∆σ(ν)/4π2ν3 for the pro-ton (above) and neutron (under), where ∆σ(ν) =σ1/2−σ3/2. The dashed line represent the E0+-contribution to γ, while the M1+-contribution to γ is plotted with dotted line.
In Fig. 3.1 the whole integrand in Eq. (3.10) are plotted with solid lines for the proton (left) and neutron (right). The intgrand shows its maximum and minimum at energies ω ' 150 MeV and ω ' 300 MeV, respectively and practically fast vanishes aboveω '500 MeV. It may be due to the damping factor 1/ω3 in the integrand. We have also plotted the contributions from E0+ and M1+ multipoles with dashed and dotted lines, respectively, which are by far the largest contributions toγ. It is noticed that there is large cancellation between the contributions from these two multipoles.
Concretly, γ yields γ =
+1.6(E0+)−2.9(M1+E1+)−0.2(rest) = −1.5 proton,
+2.8(E0+)−3.1(M1+E1+)−0.1(rest) = −0.4 neutron. (3.11) The term of restrefers to involving of all other multipole amplitudes.
3.3. Comparision with theoretical predictions
The nucleon spin polarizabilities calculated within fixed-t dispersion relation by us-ing the SAID solution VPI-SP98K as input for the sus-ingle-pion photoamplitude are
3. Low energy theory
presented in Tables 3.1-3.3 and are compared with the theoretical predictions in the framework of HBChPT [82, 83, 80, 81]. Table 3.1 shows seperately the contributions from one-pion (πN) and two-pion (2πN) intermediate states in thes-channel together with the asymptotic part in the t-channel Aas2 (t). Table 3.2 and Table 3.3 give the forward and backward spin polarizability, γ0 and γπ, as well as the combinations of the spin polarizabilities of Eq. (3.10), which are independent of the amplitude A2, respectively.
In the s-channel, the main contribution to the γi comes from single π-production, whereas two-pion photoproduction gives rise to an only negligible effect to the γi, see Table 3.1. As to the single-pion channel, the result of Drechsel et al. [18] is also represented which is based on the pion multipoles of Hanstein, Drechsel and Tia-tor (HDT) [68]. There are no large differences in magnitiudes of the polarizabilitits with respect to the application of the two various inputs of multipoles. Nevertheless, the spin polarizabilities, especially the forward spin polarizability, from the multipoles by the VIP group show some difference from that obtained by HDT multipole am-plitudes. This may be, as disscused in Ref. [18], due to the fact that in HDT the amplitude A0+(= E0+) for charged pions near the pion threshold is larger than that of SAID. Furthermore, two different upper limits of the integral in Eq. (3.10) such as ωmax = 500 MeV at HDT and ωmax = 1500 MeV at SAID also play a role in this discrepancy.
The chiral perturbation expansion to leading order (LO) O(p3) is determined by the graphs of the effective chiral Lagrangian with one-pion loop (πN) and the t-channelπ0-exchange (π0), where the latter part is in accordance with the high-energy contribution Aas2 in DR (see Table 3.1). The p denotes non-relativistic momenta.
The proton and neutron forward spin polarizability from relativistic ChPT yield-ing γ0p, LO = γ0n, LO = +4.6 disagree dramatically with our values of γ0p = −1.5 and γ0n = −0.4. This disagreement is dimished by including correction terms such as ∆-isobar excitation, which is especially important for the low-energy phenom-ena of the polarizabilities because of the relatively small nucleon-delta mass spliting
∆ = m∆−mN ' 2mπ and a large coupling to the πN-channel. In heavy baryon ChPT the ∆-pole is thus introduced as an explicit degree of freedom, and the chiral expansion is taken to third order in a small energy scale = (mπ,∆), which con-tains anyO(p3) result in Ref. [82] plus additional terms involving the delta resonance.
Not only the ∆-isobar excitation but also the ∆π-loop are kept to the same order in this expansion. Numerically, however, their contribution to the γi is, as shown in Table 3.1, ignorable.
The ∆-pole contribution to the magnetic dipole spin polarizability γM∆1, given by γM1∆ = µ2πN∆
4π∆2, (3.12)
3. Low energy theory
is deduced to +2.4 [5, 81] or to +4.0 [80] depending on the values for the transition magnetic moment µπN∆. In Table 3.1 we give the results with γM1∆ = +4.0 as well as the∆-pole contribution to γE2∆. The sum of the πN-loop in LO and SLO term in HBChPT can be compared with the integral contribution in DR. In most cases, the γi within HBChPT are fairly similar to our results. Nevertheless, our finding of γE1p = −3.7 evidently deviats from γE1p = −5.8 predicted by HBChPT. Indeed, γM2p,n have opposite signs to that of a chiral expansion. These may account for the variousity of the proton forward spin polarizabilty as well as for the discrepancy of γE1 +γM1 andγE1+γE2, see Tables 3.2 and 3.3. The values of the backward spin polarizability following from DR and HBChPT are close to each other by virtue of the dominant contribution of the t-channel with the exception of our somewhat larger non-π0 part of γπn compared with that of HBChPT.
There have been two recent determinations of the proton backward spin polarizability of γπ,expp = −36.8± 0.6stat.+syst. ± 2.0mod. obtained from the LARA experiment of Ref. [29] using the data between 200 MeV . ω . 800 MeV and 300 . θlab . 1500 andγπ,expp =−36.1±2.1stat.∓0.4syst.±0.8mod. from the TAPS measurement [20] with energies between 55 MeV and 165 MeV and angles 590 .θlab.1550. The indicesstat and sys denote the statistical and systematic error, respectively. And mod refers to the model-dependent one. These extracted γπ,expp are in quite good accordance with our result presented here. On the other hand, γπ,expp =−27.1±2.2stat.+syst.±2.6mod deduced from the analysis of the LEGS group [2, 53] using data up to the 2πthreshold (309 MeV) is appreciably different to our calculation forγπp, and in particular, the non-π0 part ofγπ,expp yieldingγπ,expp, non−π0 = 17.9±3.4 is approximately twice as large as the theoretical prediction γπ,thep, non−π0 '+8.
3. Low energy theory
Table 3.1.: Separate contributions to the spin polarizabilities calculated within DR using SAID solution [73, 74] in comparision with the results of Ref. [18]
and the predictions of HBChPT [81]. The contribution from one-pion photoproduction (πN) in DR corresponds to the sum ofπN-loop (πN) in leading order (LO) and the ∆-pole term (∆) in subleading order (SLO) of HBChPT. On the other hand, the contribution from two-pion photopro-duction (2πN) in DR can be compared with the term of ∆π-loop (∆π) in HBChPT. The asymptotic parts in the t-channel (Aas2 (t)) of DR are com-parable to the t-channelπ0-exchange (π0) of HBChPT. (All results are in units of 10−4fm4).
Dispersion Relation HBChPT
Aas2 (t) non-π0 contributions LO SLO πN
πN 2πN sum π0 πN ∆ ∆π +SLO
SAID HDT
γE1p +11.2 −4.1 −4.5 +0.4 −3.7 +11.0 −5.7 0.0 +0.6 −5.1 γM1p −11.3 +3.3 +3.4 −0.2 +2.9 −11.0 −1.1 +4.0 +0.2 +3.1 γE2p −11.2 +2.4 +2.3 −0.3 +2.1 −11.0 +1.1 +0.75 −0.3 +1.6 γM2p +11.3 −0.3 −0.6 +0.2 −0.1 +11.0 +1.1 0.0 −0.2 +0.9 γE1n +11.2 −5.9 −5.5 +0.4 −5.5 +11.0 −5.7 0.0 +0.6 −5.1 γM1n −11.3 +4.1 +3.4 −0.2 +3.9 −11.0 −1.1 +4.0 +0.2 +3.1 γE2n −11.2 +3.2 +2.6 −0.3 +2.9 −11.0 +1.1 +0.75 −0.3 +1.6 γM2n +11.3 −1.0 −0.6 +0.2 −0.8 +11.0 +1.1 0.0 −0.2 +0.9 γ0p 0.0 −1.3 −0.6 −0.2 −1.5 0.0 +4.6 −4.75 −0.3 −0.1 γ0n 0.0 −0.3 −0.1 −0.1 −0.4 0.0 +4.6 −4.75 −0.3 −0.1 γπp −45.0 +10.0 +10.8 −2.1 +7.9 −44.0 +4.6 +4.75 −0.5 +8.9 γπn +45.0 +14.2 +12.1 −1.4 +12.8 +44.0 +4.6 +4.75 −0.5 +8.9
3. Low energy theory
Table 3.2.: The results for the forwardγ0 and backward γπ spin polarizabilities of the nucleon in the framework of DR (SAID), DR (HDT) and HBChPT. (all results are in units of 10−4fm4).
Proton Neutron
SAID HDT HBChPT SAID HDT HBChPT
γ0 −1.5 −0.6 −0.5 −0.4 −0.1 −0.5 γπ −37.1 −34.2 −35.1 +57.8 +57.1 +52.9
Table 3.3.: Linear combinations of the nucleon spin polarizabilities which are not af-fected by the asymptotic part of the amplitude A2. (all results are in units of 10−4fm4).
SAID HDT HBChPT
γi proton neutron proton neutron
γE1+γM1 −0.7 −1.8 −1.1 −2.1 −4.0 γE2+γM2 +2.2 +2.2 +1.7 +2.0 +2.0 γE1+γE2 −1.5 −2.7 −2.2 −2.9 −4.4 γM1+γM2 +3.0 +3.1 +2.8 +2.8 +2.4