# Scientific sources - Case40

## References

Pei-Lun He, Di Lao, and Feng He. 2017. *Strong Field Theories beyond Dipole Approximations in Nonrelativistic Regimes*. *Physical Review Letters*, 118, 163203. [Source](https://doi.org/10.1103/PhysRevLett.118.163203).

Direct length-gauge nondipole circular-polarization SFA for five hydrogen states and independent same-branch paired momentum/energy tests. Source fitted alpha/beta/gamma panels support the asymptotic interpretation but are not required reproductions.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | He, Lao and He (2017), article pp. 163203-1–2, Eqs. (1)–(5), especially Eq. (5). Direct length-gauge light-cone SFA amplitude and leading nondipole shift/prefactor. Velocity-gauge Table I is not the commissioned branch. |
| F3 | Article pp. 163203-3–4, Fig. 2(a,b), caption and magnetic-sublevel paragraph. Five hydrogenic state identities, circular handedness and angular-momentum dependence. State-specific binding energies/orbitals follow standard hydrogenic quantum mechanics, not a fitted source ordering. |
| F4, O1, O4, I1 | Article p. 163203-3, Fig. 2(a), Eq. (7); Eq. (5), p. 163203-2. Normalized longitudinal mean and reconstructible one-member/full-five-state response roles. At-least-one valid member and five-of-five family completeness are separate task obligations. |
| O2, O3, I2, I3 | Article p. 163203-3, Eq. (7), Fig. 1(a–c) and adjoining photon-order paragraph. Finite-to-large-order partition structure. The task tests it with independently computed paired means and a transparent discrepancy; fitted coefficient panels are source evidence, not compulsory outputs. |
| M1, M2 | Article pp. 163203-3–4, Fig. 2(a,b), state-dependent partition paragraphs; Eq. (5), p. 163203-2. Orbital-gradient momentum structure and initial angular momentum explain state dependence; opposite magnetic substates need not have identical longitudinal means. |
| M3, M4 | Article p. 163203-3, Eq. (6), tunnelling/under-barrier paragraph, Eq. (7), Fig. 2(b). Continuum energy and under-barrier/bound-state contributions are distinct. The velocity-gauge limiting Eq. (6) does not replace the length-gauge result. |
| M5, M6, M7 | Article p. 163203-3, Eq. (7), Fig. 1(a–c) and large-order limit paragraph; length-gauge qualification on p. 163203-2. Source limits alpha→1, beta→1/3, gamma→0 motivate the tested relation. A same-direct-branch paired-moment discrepancy is the task-defined numerical test, not a printed separate source figure. |
| M8, V1 | Article p. 163203-2, length-gauge paragraph; pp. 163203-3–4, circular versus linear/rescattering branch discussion. Direct circular asymptotic scope and separation from rescattering, spin and absolute-yield claims. |
| V2 | Article p. 163203-3, Eq. (7) and paragraph on unreliable extraction of the small ponderomotive coefficient; Eq. (5), p. 163203-2. Concrete moment-resolution and finite-order inference threats. Conditional validation is task-defined and imposes no particular ATI mesh or residual metric. |

## Formula and convention notes

In atomic units, $W_L^{(1)}(\mathbf p)=-i\int d\eta\,e^{i[S_{\mathbf p}(\eta)+I_p\eta]}(1-p_z/c)\mathbf E(\eta)\cdot i\nabla_{\mathbf k}\widetilde\phi_0(\mathbf k)|_{\mathbf k=(\mathbf p_\perp+\mathbf A,\,p_z-[p^2/2+I_p]/c)}$ (Eq. (5)). Here $\eta=t-z/c$, $\mathbf E=-\partial_\eta\mathbf A$ and $S_{\mathbf p}=\int^\eta[p^2/2+(\mathbf A\cdot\mathbf p+A^2/2)/(1-p_z/c)]d\eta\prime$. Algebraically equivalent leading-$1/c$ forms are accepted in the nonrelativistic nondipole domain; an arbitrary gauge swap in SFA is not automatically equivalent.

For $P(\mathbf p)=|W_L^{(1)}|^2$, $\langle p_z\rangle=\int p_zP\,d^3p/\int P\,d^3p$ and $\langle E_p\rangle=\int(p^2/2)P\,d^3p/\int P\,d^3p$. Both moments must use the same direct-branch state, case, measure and distribution. A peak shift or unnormalized numerator is not the longitudinal mean; overall orbital normalization cancels.

The independently tested asymptotic relation is $\langle p_z\rangle\to[\langle E_p\rangle+I_p/3]/c$. An admissible residual is $D=c\langle p_z\rangle-\langle E_p\rangle-I_p/3$ or a transparently scaled equivalent; it is computed from paired moments rather than used to generate either moment. $N=\lfloor I_p/\omega\rfloor+1$ is the source threshold-order convention, with pulse/channel conventions declared. The five hydrogenic states have $I_p=1/(2n^2)$, so $1s$ and $n=2$ states cannot share an incorrectly fixed binding energy.


## Scope of the source anchors

Source examples verify physical identity and robust structure; their numerical samples, plotting conventions and panel layout are not compulsory candidate deliverables. Reporting, reconstructibility, evidence attribution, conditional validation and Research Closure are examination-defined obligations, not additional results asserted by the article. A derivation or physical entailment is identified as such rather than presented as a directly printed source formula.

Supplemental Material to the article: [official supplement](https://link.aps.org/supplemental/10.1103/PhysRevLett.118.163203). Supplement locators above refer to its own numbered pages and equations.