Sum Rules and Spin-Dependent Polarizabilities of the Deuteron in Effective Field Theory

We construct sum rules for the forward vector and tensor polarizabilities for any spin-$S$ target and apply them to the spin-1 deuteron. We calculate these polarizabilities of the deuteron to the next-to-leading order in the pionless effective field theory.

Low-energy photon scattering on a composite system can be characterized by a host of electromagnetic polarizabilities, many of which depend on the polarization (spin) state of the system. Comparing experimental measurements of these polarizabilities with theoretical predictions allows one to learn about the underlying dynamics of the composite system. In this paper, we are interested in the spin-dependent vector and tensor polarizabilities of the deuteron. The spin-independent electric polarizability α E0 of the nucleus has long been a subject of investigation in the literature, and it has been explored extensively in the potential models [1]. More recently, it has been calculated in effective field theories with and without explicit pion degrees of freedom [2,3]. Experimentally, α E0 has been measured through deuteron scattering off heavy atoms (0.70 ± 0.05 fm 3 ) [4] and also extracted from the photo-production data through a sum rule (0.69 ± 0.04 fm 3 ) [5].
We communicate two sets of results in this paper. First is the sum rules for the spindependent vector and tensor polarizabilities. Because the deuteron binding-energy is 2.2 MeV, extracting the polarizabilities directly from Compton scattering is difficult experimentally. One would need high-intensity, polarized photon beams at energy of order 1 MeV or less, which are not available at the present time. [Compton scattering on the deuteron has been studied in the past and sum rules have been explored [6], and recently it has been investigated in effective field theories [7].] One could scatter a polarized deuteron beam off the Coulomb field of a heavy nucleus, as in [4], observing spin-dependent effects. However, an easier way might be to extract these polarizabilities from the spin-dependent photoproduction cross sections through sum rules. The HIGS facility at Duke could be used for this purpose [8].
The second set of results is on the effective field theory (EFT) calculations of the forward spin polarizabilities. A systematic EFT approach to the deuteron structure and scattering processes has been developed in the last few years [9] and has been applied successfully to many experimental observables (see [10] for a review). Here we use the pionless version of the theory [3] to calculate the vector and tensor polarizabilities up to the next-to-leading order. Since all the counter terms to this order have been fixed from other processes, there are no free parameters in our prediction.
Before specializing to the spin-1 deuteron case, we consider the forward scattering of a circularly polarized photon of positive helicity on a nucleus of spin S and the magnetic quantum number m S (we choose the direction of photon momentum as the quantization axis z). The total number of forward scattering amplitudes is easily found to be 2S + 1 + [S], where [S] denotes the integer part. These amplitudes arise from the initial deuteron and photon states with the total angular momentum projection S + 1, S, (S − 1) 2 , ..., 1 2 , 0 2 for integer nuclear spin, and S + 1, S, (S − 1) 2 , ..., (3/2) 2 , (1/2) 2 for half integer nuclear spin, where the superscripts denote the multiplicity of the amplitudes. Here we are concerned with the forward amplitudes without helicity flip, for which there are exactly 2S + 1.
Let us denote the scattering amplitude for γ(+1) + A(m S ) → γ(+1) + A(m S ) by f (m S ) (θ), where θ is the scattering angle, and the corresponding cross section by σ (m S ) . Then the wellknown optical theorem states where k is the center-of-mass momentum. Since f (m S ) ∼ χ * m S χ m S where χ m S is the spin wave function of the target, we can couple the initial and final spin wave functions into tensors with definite total angular momentum, and a similar relation can be used to defineσ J such that the optical theorem exists between them: Imf J (0) = k 4πσ J . Obviously for J = 0, one has which is just the unpolarized scattering amplitude. On the other hand, a general scattering amplitude f can be expressed in terms of various non-relativistic structures after a non-relativistic reduction. In forward scattering, it is straightforward to write down all tensor structures for a general spin target, where ǫ is the photon polarization, S is the angular momentum operator of the target, and ⊗ indicates a tensor coupling.
[If one considers the spin-flip forward amplitudes as well, one has one more structure ( S ⊗ S... ⊗ S) (J) · (ǫ * ⊗ǫ ⊗k... ⊗k) (J) for every even-J.] It is easy to show that the above amplitudes f i are simply proportional to the ones in the previous section; and and so on. Because of the crossing symmetry under exchanging q µ ↔ −q µ and ǫ ′ * ↔ ǫ, all even-J amplitudes are even functions of the photon energy ω, and all odd-J amplitudes are odd functions of ω. Using the analyticity of the f J , we write down once-subtracted dispersion relations for even-J, Using the optical theorem, one has, which is the basis for various sum rules. Consider the example of J = 0, the low-energy expansion for the amplitude is Substituting the above into the dispersion relation, we recover the well-known Baldin sum rule for the averaged cross section, For the special case of the spin-1 deuteron, the sum rule becomes where we remind the reader that σ (m) denotes the cross section for the deuteron in an m-state.
Let us review the calculations of α E0 and β M 0 in the pionless effective field theory [2,3,11]. We remind the reader that in the pionless theory, the leading order effective lagrangian is where N is the nucleon field, P i = τ 2 σ 2 σ i / √ 8 and P i = σ 2 τ 2 τ i / √ 8 are the triplet S 1 and singlet S 0 two-nucleon projection operators, respectively. The covariant derivative is D = ∂ + ieQ A with Q = (1 + τ 3 )/2 as the charge operator and A the photon vector potential. µ (0) = (µ p + µ n )/2 and µ (1) = (µ p − µ n )/2 are the isoscalar and isovector nucleon magnetic moments in nuclear magnetons. The two-body coupling constants are where µ is a renormalization scale, a To the N 3 LO order, the scalar electric polarizability is [3,11] where FIG. 2: The next-to-leading order contribution to the electric tensor polarizability of the deuteron. The black box represents the triplet C is suppressed by two orders of Q-counting relative to α E0 because of the non-relativistic dynamics of the deuteron [2], where A is the leading-order singlet S 0 scattering amplitude at energy E = −B. [Note that the above result differs from [2] by a term proportional to (µ (0) ) 2 , which is cancelled by an omitted diagram with the triplet S 1 bubble chain between the two photon insertions.] Numerically, β M 0 = 0.067 fm 3 , about 10% of α E0 .
Turning to the case of J = 2, we have the low-energy expansion, which allows us to write a J = 2 sum rule, Specializing to the spin-1 deuteron, the sum rule becomes, In the effective theory with pions, the leading contribution comes at the NLO in Q-counting from the potential pion exchange [2,7]. On the other hand, in the pionless theory, there is a leading contribution coming from the two-body operator that couples the single S 0 and triplet S 1 channels [12], where T
The leading-order Feynman diagrams are shown in Fig. 1, and a straightforward calculation yields, At the NLO, there are contributions from Fig. 2 plus the correction for wave function renormalization. The result is that where in the second line we have introduced Z d = 1/(1 − γρ d ) = 1.69 [11]. Numerically, we have α E2 = −0.068 fm 3 at this order, which is very close to the result from the potential pion contribution [2]. For completeness, we quote the magnetic tensor polarizability β M 2 which formally comes at N 2 LO [2], However, it is very large numerically, 0.195 fm 3 , because of the large isovector magnetic moment. [Again, the (µ (0) ) 2 dependence in Ref. [2] should be absent.] For odd-J, one can write down a dispersion relation without subtraction, Again using the optical theorem, one has For J = 1, the scattering amplitude has a low-energy expansion, where the first term corresponds to the famous low-energy theorem [13] with the anomalous magnetic moment κ defined as µ − 2S [14], where µ is the magnetic moment in unit of eh/2Mc. The next term defines the forward spin-polarizability γ. Substituting the above into Eq. (25), the first term yields the famous Drell-Hearn Gerasimov (DHG) sum rule, now extended to target of any spin S, where σ 1 = [3/S(S + 1)(2S + 1)] ms m S σ m S . For a discussion about the DHG sum rule for the deuteron, see Ref. [15]. The sum rule for the forward spin polarizability reads Specializing to the deuteron, we have This is the sum rule potentially useful for extracting the forward spin polarizability from the polarized photo-production cross section. The relevant Feynman diagrams for the forward Compton scattering are shown in Fig. 3. Taking into account the crossing symmetry, the result for the leading-order spin polarizability is At the next-to-leading order, there are contributions from the C 2 coupling in the singlet and triplet channels, where with r 0 = 2.73 fm. There are also contributions from the following electromagnetic counter term [16] The relevant next-to-leading order Feynman diagrams for the Compton amplitude are shown in Fig. 2. The calculated vector polarizability is The last term in the first square bracket can be added to the leading-order result if one replaces A . It is easy to verify that if the following equation holds, the above result is independent of the renormalization scale µ. This is consistent with the np capture calculation [16]. Using the counter term determined from the neutron capture, L 1 = 3.26 fm 4 , we find γ NLO = 0.324 fm 4 which is 10% of the leading-order result. Therefore the effective field theory expansion seems to converge well.
In summary, we have presented sum rules for the vector and tensor polarizabilities of the deuteron. We have evaluated both quantities in the pionless effective field theory to the next-to-leading order. The Q-expansion seems to converge well for the vector polarizability.
We thank Jiunn-Wei Chen and Barry Holstein for discussions related to the subject of this paper. This work was supported by the U. S. Department of Energy via grant DE-FG02-93ER-40762.