{"id":"c93e159b-3d69-4c41-a150-b029955d0496","arxiv_id":"2607.06005","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A method for anisotropic energy-loss functions shows Migdal daily modulation depends only on the ELF quadrupole, applied to silicon and GaAs dark-matter detectors.","lead":"This paper gives a practical way to compute dark-matter event rates in semiconductors when the material’s energy-loss function is anisotropic, not isotropic. The result matters because daily modulation of the Migdal signal is shown to come only from the ELF quadrupole, which can help design and interpret cryogenic Si and GaAs detectors.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"Local-field corrections in the dielectric matrix can imprint angular structure beyond a scalar ELF multipole expansion, potentially reopening daily-modulation channels for Migdal rates even when the macroscopic quadrupole vanishes.","rationale":"The reader’s weakest assumption already identified the continued validity of the ELF/dielectric formalism under full anisotropy and the sufficiency of ELF multipoles as the load-bearing premise. The present concern simply makes that premise concrete: local-field mixing inside the dielectric matrix is a specific higher-order solid-state channel that can reintroduce modulation without an ELF quadrupole. Because the reader already returned UNVERDICTED at low confidence (abstract-level information only) and the structural claim remains plausible for a pure macroscopic ELF, the verdict is left unchanged. The concrete dielectric-matrix test above would settle whether the “solely quadrupole” language holds in a more complete treatment; until that check is performed the claim is interesting but not yet secured. No internal inconsistency is visible—only a completeness gap in the response functions used.","tokens_in":2072,"tokens_out":535,"duration_ms":86677,"concrete_test":"At a representative nuclear-recoil momentum transfer in silicon, evaluate the Migdal rate from the full dielectric matrix (local fields retained) both with the complete response and after the l=2 sector of the macroscopic ELF has been projected out. If a residual daily-modulation amplitude remains at the ≳10 % level of the quadrupole-only result, the selection-rule claim does not survive local-field corrections.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that Migdal daily modulation arises solely from the ELF quadrupole treats the electronic response as a scalar anisotropic energy-loss function that is multipole-expanded and inserted into the rate integrals. In a crystal the microscopic object is the dielectric matrix ε_GG'(q,ω); the macroscopic ELF is recovered only after inversion that mixes local fields (G≠0). Those local-field corrections can generate effective angular dependence that is not equivalent to any finite multipole set of the macroscopic ELF. Because the DM wind already supplies a dipole anisotropy, residual local-field angular structure can couple to produce a time-dependent rate even if every l=2 component of the macroscopic ELF is set to zero. The paper’s practical method and the Si/GaAs applications rest on the scalar-ELF multipoles being a complete and accurate input; nothing in the derivation isolates or suppresses this mixing. This is the least secure condition for the “solely quadrupole” selection rule.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript develops a practical multipole method for computing dark-matter–electron and Migdal rates in semiconductors with a fully anisotropic energy-loss function (ELF). Starting from the dielectric response, the ELF is expanded in spherical harmonics; the daily-modulated rates are then expressed as contractions of ELF multipoles with kinematic moments of the DM velocity distribution. The central structural claim is that the Migdal daily modulation is sourced solely by the quadrupole (ℓ=2) ELF multipoles, whereas every multipole of the ELF contributes to the electron-scattering rate. The formalism is applied to silicon and gallium arsenide, with numerical estimates of the modulation amplitudes.","tokens_in":2235,"tokens_out":793,"duration_ms":12359,"significance":"If the selection rule and the multipole rate formulae hold under the stated dielectric-response assumptions, the paper supplies a clean, reusable computational framework for anisotropic semiconductor targets and a sharp, falsifiable prediction (quadrupole-only Migdal modulation). That prediction is of direct experimental interest for next-generation cryogenic detectors that can resolve daily modulation. The work also clarifies how crystal anisotropy enters the two channels differently, which is a useful conceptual advance beyond the isotropic-ELF literature.","major_comments":[{"comment":"The central claim that Migdal daily modulation arises solely from the ELF quadrupole treats the electronic response as a scalar anisotropic ELF that is multipole-expanded and inserted into the rate integrals. In a crystal the microscopic object is the dielectric matrix ε_GG'(q,ω); the macroscopic ELF is recovered only after inversion that mixes local fields (G≠0). Those local-field corrections can generate effective angular dependence that is not equivalent to any finite multipole set of the macroscopic ELF. Because the DM wind already supplies a dipole anisotropy, residual local-field angular structure can couple to produce a time-dependent rate even if every ℓ=2 component of the macroscopic ELF is set to zero. The paper’s practical method and the Si/GaAs applications rest on the scalar-ELF multipoles being a complete and accurate input; nothing in the derivation isolates or suppresses ","section":null},{"comment":"The numerical applications to Si and GaAs are presented as concrete illustrations of the selection rule, yet the manuscript does not quantify how the adopted anisotropic ELF models (or the truncation of the multipole series) affect the reported modulation amplitudes. Without a controlled comparison to a fully isotropic baseline and to a higher-multipole truncation, it is difficult to judge whether the quoted daily-modulation signals are robust predictions or artifacts of the particular ELF parametrization. A short sensitivity study would make the applications load-bearing rather than merely illustrative.","section":null}],"minor_comments":[{"comment":"Notation for the multipole coefficients of the ELF should be standardized early (e.g., a single symbol for Im[−1/ε]_{ℓm}(q,ω)) and used consistently in the rate formulae and figures.","section":null},{"comment":"Figures showing the daily modulation should include error bands or at least a statement of the dominant theoretical uncertainty (ELF model, multipole truncation, velocity-distribution moments).","section":null},{"comment":"A brief comparison table of the isotropic-limit rates against existing literature results for Si/GaAs would help the reader verify the numerical pipeline.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about local-field corrections in the dielectric matrix is genuine and load-bearing for the “solely quadrupole” claim; it is not a mere presentation issue. If the authors can either bound the effect or clearly demarcate the scalar-ELF approximation, the paper becomes a solid contribution. Scope-wise the work fits hep-ph / dark-matter direct detection well."},"author_rebuttal":{"model":"grok-4.5","summary":"We thank the referee for a careful reading and for highlighting both the structural selection rule and the experimental relevance of the Migdal daily-modulation prediction. We address the two major comments below. On the first, we clarify the scope of the dielectric-response assumptions already stated in the manuscript and add an explicit discussion of local-field corrections and the macroscopic ELF; we do not claim that the scalar multipole expansion is exact beyond those assumptions. On the second, we agree that a controlled sensitivity study is needed and will add isotropic-baseline and multipole-truncation comparisons for Si and GaAs so that the reported modulation amplitudes can be judged as robust rather than merely illustrative. We believe these revisions convert the applications into load-bearing results while preserving the central selection-rule claim under the stated framework.","responses":[{"response":"We agree that the microscopic response is the dielectric matrix ε_GG'(q,ω) and that the macroscopic ELF is obtained only after matrix inversion that incorporates local-field corrections (LFCs). Our derivation and selection rule are formulated entirely within the standard macroscopic-ELF framework used in the isotropic literature (and in existing semiconductor DM codes): the rate is written in terms of Im[−1/ε_M(q,ω)], where ε_M is the macroscopic dielectric function after inversion. Under that assumption the multipole expansion of the scalar anisotropic ELF is complete by construction, and the Migdal daily modulation is sourced only by the ℓ=2 multipoles (because the Migdal kinematics contract with a rank-2 tensor built from the nuclear recoil direction). Residual angular structure that cannot be absorbed into any multipole set of the macroscopic ELF would require retaining the full off-diagonal G,G' structure inside the rate integrals themselves—an extension beyond both our work and the isotropic-ELF literature we build on. We do not claim that LFCs are absent; rather, once they have been folded into ε_M(q,ω), any remaining anisotropy is precisely what our multipoles capture. We will revise the manuscript to state this scope explicitly (Introduction and Sec. II), to note that a full microscopic treatment with unsummed LFCs is left for future work, and to emphasize that the quadrupole-only selection rule is a sharp prediction inside the macroscopic-ELF framework. The practical method and the Si/GaAs applications are therefore complete and accurate within the stated assumptions; they are not claimed to be exact beyond them.","revision_made":"yes","referee_comment":"The central claim that Migdal daily modulation arises solely from the ELF quadrupole treats the electronic response as a scalar anisotropic ELF that is multipole-expanded and inserted into the rate integrals. In a crystal the microscopic object is the dielectric matrix ε_GG'(q,ω); the macroscopic ELF is recovered only after inversion that mixes local fields (G≠0). Those local-field corrections can generate effective angular dependence that is not equivalent to any finite multipole set of the macroscopic ELF. Because the DM wind already supplies a dipole anisotropy, residual local-field angular structure can couple to produce a time-dependent rate even if every ℓ=2 component of the macroscopic ELF is set to zero. The paper’s practical method and the Si/GaAs applications rest on the scalar-ELF multipoles being a complete and accurate input; nothing in the derivation isolates or suppresses"},{"response":"We agree. The applications were intended as illustrations of the selection rule, but without a controlled sensitivity study the quoted modulation amplitudes cannot be judged as robust. In the revised manuscript we will add: (i) a fully isotropic baseline obtained by retaining only the monopole of the same ELF models, so that the pure anisotropic contribution to the daily modulation is isolated; (ii) a multipole-truncation study (e.g. ℓ_max = 2 vs. 4 vs. 6) for both Si and GaAs, quantifying the residual change in the modulation amplitude; and (iii) a brief discussion of how the adopted anisotropic ELF parametrizations (and any model uncertainties they carry) propagate into the reported amplitudes. These additions will make the numerical results load-bearing rather than merely illustrative, while leaving the analytic selection rule unchanged.","revision_made":"yes","referee_comment":"The numerical applications to Si and GaAs are presented as concrete illustrations of the selection rule, yet the manuscript does not quantify how the adopted anisotropic ELF models (or the truncation of the multipole series) affect the reported modulation amplitudes. Without a controlled comparison to a fully isotropic baseline and to a higher-multipole truncation, it is difficult to judge whether the quoted daily-modulation signals are robust predictions or artifacts of the particular ELF parametrization. A short sensitivity study would make the applications load-bearing rather than merely illustrative."}],"tokens_in":1664,"tokens_out":1031,"duration_ms":14885,"standing_objections":[]},"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: they drop the isotropic ELF approximation everyone uses, give a practical multipole method for a general anisotropic energy-loss function, and get a structural result—daily modulation of the Migdal rate comes only from the ELF quadrupole, while DM–electron scattering feels every spherical-harmonic component. They apply it to silicon and gallium arsenide. That is the paper.\n\nWhat is actually new is the method plus that selection rule. The isotropic literature does not have the multipole split, and a clean “only quadrupole for Migdal modulation” statement is useful for experiment design if it holds. The framing is calculation-first: expand the anisotropic ELF, insert into the rate integrals, read off which pieces survive. No free parameters, no invented entities. Credit where due—this is concrete subfield progress for cryogenic semiconductor DM searches, not a rebrand of the isotropic case.\n\nThe soft spot is real but proportionate. The whole construction treats the electronic response as a scalar anisotropic ELF that you multipole-expand and feed into the rates. In a crystal the microscopic object is the dielectric matrix ε_GG'(q,ω); the macroscopic ELF only appears after inversion that mixes local fields (G≠0). Those local-field corrections can leave angular structure that is not equivalent to any finite multipole set of the macroscopic ELF. The DM wind already supplies a dipole, so residual local-field angular dependence can couple into a time-dependent Migdal rate even if every l=2 piece of the macroscopic ELF is zero. Nothing in the abstract isolates or suppresses that mixing. So the “solely quadrupole” rule is only as solid as the scalar-ELF approximation. That is the place a referee should push. Everything else—method, Si/GaAs application—looks like careful phenomenology once you grant the premise. I have only the abstract, so I cannot audit the integrals or the numerics; the logic as stated is coherent.\n\nWho it is for: people building or interpreting cryogenic Si and GaAs detectors, especially anyone chasing daily modulation. A serious referee should see it. I would not desk-reject; the method is concrete and the selection rule is sharp enough to be useful even if local fields force a revision. Worth a careful read in the subfield.","headline":"Usable anisotropic-ELF multipole method plus a clean Migdal-quadrupole selection rule for daily modulation in Si/GaAs; the scalar-ELF premise is the soft spot.","tokens_in":2900,"tokens_out":562,"would_cite":false,"duration_ms":23737,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Daily Migdal modulation in anisotropic semiconductors arises solely from the quadrupole of the energy-loss function.","keywords":["Migdal effect","energy loss function","anisotropic ELF","dark matter detection","semiconductor detectors","daily modulation","dielectric function","electron scattering"],"falsifier":"Extract the multipole moments of the ELF for silicon or GaAs from first-principles dielectric calculations, recompute the Migdal daily-modulation amplitude with only the quadrupole retained, and compare with a full anisotropic numerical integration or with a future experimental modulation measurement; a nonzero modulation arising from the isotropic or dipole parts alone would falsify the claim.","tokens_in":2946,"feed_emoji":"⚛️","tokens_out":847,"duration_ms":30475,"temperature":0.7,"pith_summary":"Cryogenic semiconductor detectors can see light dark matter either through direct electron recoils or through nuclear recoils that shake electrons free via the Migdal effect. Both rates are controlled by the material's energy-loss function (ELF), which is usually treated as isotropic. This paper supplies a practical method that keeps the full angular dependence of the ELF and decomposes it into spherical harmonics. The central result is that the daily modulation of the Migdal rate is generated only by the quadrupole piece of that decomposition, while every multipole of the ELF contributes to the dark-matter–electron scattering rate. The method is applied to silicon and gallium arsenide, the two materials most relevant to existing cryogenic detectors. A sympathetic reader cares because a clean multipole separation turns daily modulation into a diagnostic that can help separate signal channels and reduce background confusion.","feed_headline":"Only the ELF quadrupole drives daily Migdal modulation","feed_subtitle":"In semiconductors every multipole of the energy-loss function enters electron scattering, but not the Migdal effect.","key_machinery":"The spherical-harmonic multipole expansion of the anisotropic energy-loss function (the imaginary part of the inverse dielectric function). Once the ELF is written as a sum of multipoles, the angular integrals that produce the daily modulation isolate the quadrupole for Migdal scattering while leaving all multipoles active for electron scattering.","core_discovery":"For a fully anisotropic energy-loss function the daily modulation of the Migdal event rate is produced exclusively by the quadrupole (l = 2) spherical-harmonic component of the ELF, whereas the dark-matter–electron scattering rate receives contributions from all multipoles. A practical multipole decomposition of the anisotropic ELF is given and used to compute both rates in silicon and gallium arsenide.","pith_inferences":["Materials whose ELF quadrupole is unusually large could exhibit a Migdal daily modulation large enough to serve as a directional signature without a dedicated directional detector.","The multipole isolation suggests that dielectric anisotropy could be deliberately engineered or selected to optimize daily-modulation sensitivity for light dark matter.","Similar multipole filtering may apply to other inelastic processes mediated by the dielectric response, such as plasmon-assisted or phonon-assisted scattering."],"forward_implications":["Daily modulation amplitudes for Migdal events in Si and GaAs can be predicted from the ELF quadrupole alone, without recomputing the full anisotropic rate each sidereal day.","The different multipole content of Migdal versus electron-scattering rates supplies a diagnostic that can help separate the two channels in the same detector.","Isotropic-ELF calculations miss a modulation signal that appears once material anisotropy is restored.","The same multipole method can be applied immediately to other cubic or anisotropic semiconductor targets used in direct detection."],"fun_headline_variants":["ELF quadrupole alone drives daily Migdal modulation","Only ELF quadrupole causes Migdal daily modulation","Migdal daily modulation stems solely from ELF quadrupole","Electron scattering uses all ELF multipoles; Migdal only l=2","Solely ELF quadrupole yields daily Migdal rate modulation"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"That the ordinary dielectric energy-loss-function formalism continues to control both rates once the ELF is allowed to be fully anisotropic, with no extra daily-modulation channels opened by higher solid-state or many-body corrections beyond the multipoles.","fun_headline_variants_meta":{"raw":{"variants":["ELF quadrupole alone drives daily Migdal modulation","Only ELF quadrupole causes Migdal daily modulation","Migdal daily modulation stems solely from ELF quadrupole","Electron scattering uses all ELF multipoles; Migdal only l=2","Solely ELF quadrupole yields daily Migdal rate modulation"]},"model":"grok-4.5","cost_usd":0.011044,"raw_usage":{"total_tokens":2378,"prompt_tokens":710,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":110440000,"prompt_tokens_details":{"text_tokens":710,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1610,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":710,"tokens_out":58,"duration_ms":121349,"temperature":1.0,"reasoning_tokens":1610,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T20:13:52.753993+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Extract the multipole moments of the ELF for silicon or GaAs from first-principles dielectric calculations, recompute the Migdal daily-modulation amplitude with only the quadrupole retained, and compare with a full anisotropic numerical integration or with a future experimental modulation measurement; a nonzero modulation arising from the isotropic or dipole parts alone would falsify the claim.","supporting_citations":[],"review_version":1}