Soft polarized scattering

Summary: Retaining the photon spatial phase and the target's complete polarization multiplet gives the low-frequency forward spin response in the conditionally gauged scalar Dirac model.

Keep the incoming wave's spatial structure

Use the normalized full spatial profile and conditional current coupling derived in Soft spatial current response. The particle occupies its stated $d$ ordinary spatial directions in the one nested manifold. Choose propagation along $x_3$ and transverse polarizations along $x_1,x_2$. This selects three measurement directions; it does not split the particle's space into different kinds of dimension.

The incident photon amplitude is a complex transverse vector $\varepsilon=(\varepsilon_1,\varepsilon_2)$, with $\varepsilon^\dagger\varepsilon=1$. Its source operator includes

$$\alpha_i e^{ikx_3} =V_i^{(0)}+ikV_i^{(1)} -\frac{k^2}{2}V_i^{(2)}+O(k^3), \qquad V_i^{(n)}=x_3^n\alpha_i,\quad i=1,2. \tag{1}$$

The current readout includes the opposite spatial phase. Thus a uniform electric-field calculation cannot simply be called the real-photon amplitude. Set $k=\omega$ for the conditional case where photon speed equals the underlying Dirac speed, both set to one.

The target also retains its complete complex polarization $z$ in

$$\phi_z(x)=W(x)z,\qquad \int W^\dagger W\,dx=I_N,\qquad z^\dagger z=1. \tag{2}$$

These coefficients describe the field's spatial pattern. The complete tensor description explains which relative amplitudes and phases a single circulation tensor can omit.

The complete zero-frequency multiplet contributes

Let $P$ project on the full resting multiplet (2), and let $B_{ij}$ denote $-i\alpha_i\alpha_j$ restricted to its constant upper polarization frame. Spatial parity and direct radial integration give

$$PV_i^{(0)}P=0,\qquad PV_i^{(1)}P=-F_0B_{i3}. \tag{3}$$

The coefficient $F_0=(1-2G/d)/(2M)$ is the full field's current moment, including its lower radial component.

A polarization variation within $W$ orthogonal to the occupied ray changes neither scalar density nor scalar source at first order. These are exact degenerate stationary directions, as in Dirac polarization waves. The ordinary and conjugate zero-frequency resolvents therefore give

$$\begin{aligned} \Pi^{(11)}_{ij}(\omega) &=\frac1\omega z^\dagger[PV_i^{(1)}P,PV_j^{(1)}P]z+O(1)\\ &=\frac{2iF_0^2}{\omega}J_{ij}+O(1). \end{aligned} \tag{4}$$

Here $J_{ij}=z^\dagger B_{ij}z$. The commutator identity $[B_{i3},B_{j3}]=2iB_{ij}$ gives the second line. Projection only onto the occupied ray would miss this term. The global phase direction cancels between the two signed responses; the full multiplet commutator retains the physical polarization changes.

The phase coefficients in the incoming and outgoing vertices make (4) contribute $+k^2\Pi^{(11)}$. At $k=\omega$, this is an order-$\omega$ spin response, despite being second order in spatial phase.

Include the other spatial-phase terms and shape feedback

For Hermitian source labels $A,B$, use

$$\Pi_{AB}= \langle A\phi,(\omega-H_b+M)^{-1}B\phi\rangle +\langle B\phi,(-\omega-H_b+M)^{-1}A\phi\rangle. \tag{5}$$

Let $C$ contain all scalar first moments and all scalar second moments, including breathing. With their curvature matrix $\mathcal K$, the complete source response is

$$\chi_{VV} =\Pi_{VV} +\Pi_{VC}\mathcal K (I-\Pi_{CC}\mathcal K)^{-1}\Pi_{CV}. \tag{6}$$

This is the actual scalar source variation, not a fixed-target prescription. The connection is sourced reciprocally by that same current in the conditional action. A physical scattering rate additionally needs its derived radiation operator and incident/outgoing flux normalization.

Parity separates $V^{(0)},V^{(2)}$ from $V^{(1)}$. It removes the single phase cross term even with all shape feedback retained. The $V^{(0)}$–$V^{(2)}$ response must also be checked: multiplied by $k^2$, a spurious inverse-frequency pole could contaminate the soft limit.

The complete angular calculation below retains that cross response. In the tested radial branches and transverse preparations it remains bounded, so it cannot change the order-$\omega$ term. The only singular even-source contribution needed at that order is (4). Higher phase terms are suppressed by additional powers of $k$, with odd cross terms removed by parity. This expansion assumes an isolated stationary multiplet and translation sector, with no additional zero-frequency shape instability.

The angular construction uses the complete monogenic families of degrees zero, one and two and their opposite radial partners. For example, a harmonic spinor polynomial $p_\ell(x)w$ decomposes as

$$p_\ell(x)w=M_\ell(x)+\alpha(x)N_{\ell-1}(x),\qquad N_{\ell-1}=\frac{\alpha_i\partial_i(p_\ell w)} {d+2\ell-2}. \tag{7}$$

Both $M_\ell$ and $N_{\ell-1}$ are monogenic. Their radial Dirac operators retain the full upper and lower profile functions. Formal radial dimension shifts account for the radial weight; they add no physical spatial directions. The unused cubic angular family is orthogonal to every $V^{(0)}$ and scalar dipole source, so it does not enter the $V^{(0)}$–$V^{(2)}$ cross term.

Combine the two spin contributions

The spatially uniform current kernel has the independently derived term

$$\chi^{(00)}_{ij} =\frac{\delta_{ij}}{\mathcal I} +i\omega\left(\frac1{2\mathcal I^2} -\frac{2F_0}{\mathcal I}\right)J_{ij} +O(\omega^2). \tag{8}$$

Adding (4) on shell gives

$$\boxed{ \chi^{\rm forward}_{ij} =\frac{\delta_{ij}}{\mathcal I} +i\omega\,\frac{(g_{\rm mag}-2)^2}{8\mathcal I^2}J_{ij} +O(\omega^2),\qquad g_{\rm mag}=4\mathcal I F_0.} \tag{9}$$

The cancellation is

$$\frac1{2\mathcal I^2}-\frac{2F_0}{\mathcal I}+2F_0^2 =2\left(F_0-\frac1{2\mathcal I}\right)^2. \tag{10}$$

For an outgoing polarization $\varepsilon'$, the coherent response is $\varepsilon'^\dagger\chi^{\rm forward}\varepsilon$. Both photon phases and both complex polarization components remain present, together with the target's spatial circulation.

Compare the form with the low-energy scattering theorem

For the conventional normalized electromagnetic radiation equation, a forward amplitude has factor $-q^2/(4\pi)$ times the current kernel used here. Put $\kappa_{\rm mag}=(g_{\rm mag}-2)/2$. Equations (9)–(10) then give the spin-amplitude coefficient

$$-\frac{q^2\kappa_{\rm mag}^2}{8\pi\mathcal I^2}\,\omega. \tag{11}$$

This is the anomalous-moment-squared structure of the electromagnetic low-energy theorem. Equation (4) of Gorchtein and Zhang states the corresponding forward expansion with their explicit electromagnetic normalization. Here it is a comparison of response formulas; their separate weak-current analysis is not a V2 interaction mechanism.

For arbitrary outgoing direction $\widehat n'$, the same leading uniform current would give, with that radiation normalization,

$$\frac{d\sigma_{\varepsilon'\varepsilon}}{d\Omega} =r_q^2|\varepsilon'^\dagger\varepsilon|^2,\qquad r_q=\frac{q^2}{4\pi\mathcal I}, \tag{12}$$

at sufficiently low energy and negligible recoil. Summing the two outgoing transverse polarizations gives $r_q^2(1-|\widehat n'\cdot\varepsilon|^2)$. For a normalized incident polarization the angular integral is $8\pi r_q^2/3$. Consequently a chosen polarization can eliminate a particular output channel without making a freely moving charged target transparent to every output. This complements the complete target-transition test.

Equations (11)–(12) require a correctly normalized physical photon action. The present calculation derives the target current kernel under the declared coupling hypothesis; it does not derive that radiation action. The six-dimensional scalar profile's magnetic moment also disagrees with the single-electron benchmark, as quantified in the companion note. Matching the structural identity (11) is therefore a useful consistency test, not a completed observational interaction law.

The photon speed matters as well. If its speed in the same units is $c_\gamma$, then $k=\omega/c_\gamma$ and the two terms combine as

$$\frac1{2\mathcal I^2}-\frac{2F_0}{\mathcal I} +\frac{2F_0^2}{c_\gamma^2}. \tag{13}$$

The cancellation in (10) requires $c_\gamma=1$ in the stated model. The depth-two scalar translation wave has $c_{\rm tr}=1/2$ in these Dirac units. Identifying it as the photon would therefore require a different, fully consistent completion of its propagation and coupling.

Verification

soft_photon_response_checks.py constructs the angular source functions directly on the actual spatial sphere, orthonormalizes their complete monogenic spans, and integrates their radial profiles independently of the soft formula. It retains every scalar first and second moment in (6), both frequency signs, and all three displayed phase vertices.

Checks compare the low-frequency coefficients for aligned and arbitrary complex target polarizations and tilted incident planes at depths 3, 5 and 6, with two radial resolutions. Direct full-spatial integrals and separate radial circular-channel resolvents check the ingredients of (8) up to depth 10. The executable reports the conditional electron magnetic-moment discrepancy rather than treating the benchmark as fitted.