{"id":"2bc43ba3-62fd-4b79-82fe-bf66f2915b3b","arxiv_id":"2507.13070","paper_version":1,"verdict":"UNVERDICTED","confidence":"UNKNOWN","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rychkov argues that identifying operators related by equations of motion, which differ only by contact terms, yields correct CFT scaling dimensions, and that Kousvos and Stergiou's criticism is a terminological difference.","lead":"This comment defends an earlier treatment of CFT scaling dimensions where operators differing by contact terms are identified. It argues that a recent criticism is terminological rather than substantive, and that the disputed issue does not change the computed CFT data.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader judged the paper UNVERDICTED because the subtlety of contact terms in perturbative RG could not be independently adjudicated, and because the comment does not include a worked calculation. On closer inspection, the comment's argument is a direct application of standard CFT facts: correlation functions at non-coincident points are the only data from which conformal scaling dimensions are defined; EOM operators contribute only contact terms; and at the Wilson-Fisher fixed point the equation of motion makes φ^3 proportional to ∂^2φ away from coincident points. Therefore the identification of these operators and the assignment of the common dimension Δ_φ + 2 is not an unproven extra assumption but a consequence of the definitions. K&S's full-mixing-matrix language assigns a dimension to the contact-only EOM direction, which is a formal RG eigenvalue with no CFT significance. The reader's hypothetical failure mode—contact terms leaking into the power-law part of correlators—is excluded by locality: any distribution supported at coincident points can alter only the local part of the correlator, not the non-coincident scaling behavior that defines the CFT dimension. Thus the central claim holds up, and the comment's lack of an explicit perturbative demonstration is a pedagogical shortcoming rather than a load-bearing flaw. A one-loop check of the mixing matrix would settle the issue definitively, but it is not required for the conceptual argument to be correct.","tokens_in":2161,"tokens_out":18070,"duration_ms":237214,"concrete_test":"Compute the one-loop renormalization mixing matrix for the basis {φ^3, ∂^2φ} at the Wilson-Fisher fixed point in d = 4 − ε in the MS scheme, diagonalize at the fixed-point coupling, and verify that the eigenvector with a non-vanishing non-coincident two-point function has scaling dimension Δ_φ + 2 to O(ε), while the orthogonal eigenvector's two-point function is purely contact. If the surviving dimension comes out different, the economical identification used in the comment would be invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The comment's central claim is that in a CFT, operators whose correlation functions differ by contact terms—specifically φ^3 and ∂^2φ at the Wilson-Fisher fixed point—should be identified and assigned equal scaling dimensions. This is sound: CFT scaling dimensions are extracted from non-coincident correlation functions, and the EOM relation □φ + g φ^3 = 0 implies that away from coincident points φ^3 is proportional to ∂^2φ, i.e. to a descendant of φ with dimension Δ_φ + 2. K&S's full-mixing-matrix procedure may assign a formal dimension to the purely contact EOM direction, but that direction is not part of the CFT spectrum. The reader's weakest_assumption worries that contact terms could feed into the renormalization mixing matrix and affect non-coincident correlators; locality excludes this, since contact terms are supported at coincident points and cannot change the power-law scaling that defines CFT dimensions. The comment would be strengthened by an explicit one-loop check, but the absence of such a check is not a correctness gap.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a short comment responding to Kousvos and Stergiou's critique of Rychkov and Tan's epsilon-expansion computation of CFT data. The author defends the identification, modulo contact terms, of the composite operators φ^3 and ∂^2φ at the Wilson-Fisher fixed point, which leads to equal scaling dimensions in the CFT sense. The argument is that CFT correlation functions are defined at non-coincident points, so operators whose correlators differ only by contact terms—such as those proportional to the equations of motion—should be identified; Kousvos and Stergiou's full renormalization-matrix treatment retains these contact directions and assigns them formal \"scaling dimensions\" in a perturbative RG sense, which are not part of the CFT spectrum. The author concludes that the disagreement is terminological and that the more economical scheme is preferable.","tokens_in":2283,"tokens_out":7704,"duration_ms":96871,"significance":"If accepted, the comment clarifies a recurring point of confusion in perturbative CFT: the status of equation-of-motion operators in RG dimension calculations. It does not present new calculations, but it articulates a standard and internally consistent viewpoint, and it explicitly separates the CFT notion of scaling dimension from the formal RG eigenvalue. The argument is not circular: it relies on the definition of CFT correlators at separated points and on locality of contact terms. The main limitation is that the equivalence of the economical and full renormalization schemes is asserted rather than demonstrated with a concrete one-loop example, though the claim is standard and I do not regard the absence of such a check as a correctness gap.","major_comments":[],"minor_comments":[{"comment":"The guarantee that renormalizing modulo contact terms gives correct CFT scaling dimensions would be easier to evaluate with an explicit one-loop comparison for the φ^3/∂^2φ sector, or with a citation to a proof in the RG literature; without this, the reader must rely on the stated locality argument.","section":"Economical-way paragraph"},{"comment":"The premise \"CFT only describes correlation functions away from coincident points\" is an oversimplification when taken literally: contact terms carry Ward-identity and anomaly information in a CFT. Since the argument concerns only the extraction of scaling dimensions, I suggest reformulating the premise accordingly.","section":"Role-of-equations-of-motion paragraph"},{"comment":"There is a typo: \"minimal subtaction\" should read \"minimal subtraction.\"","section":"Item 1 in the introductory list"},{"comment":"There is a typo: \"one one independent operator\" should read \"one independent operator.\"","section":"Third paragraph"},{"comment":"Reference [4] cites the arXiv version; if the comment is published after Kousvos and Stergiou's paper appears in a journal, the reference should be updated.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Slava,\n\nThis is a compact comment, and it does what a good comment should: it states the disagreement crisply and takes a clear position. The substantive claim is that in a CFT, operators whose correlators differ by contact terms are the same operator, because CFT scaling dimensions are read off from non-coincident points. That is the standard understanding, and I think Rychkov is right to push back on framing it as a flaw in the original paper.\n\nWhat is actually new here is the explicit contrast between the two procedures: the \"full way\" of renormalizing the entire mixing matrix, including EOM directions, and the \"economical way\" of identifying operators modulo contact terms before renormalizing. The comment argues these give identical CFT data. That is the heart of the matter, and the logic is sound: contact terms are delta-function supported, so they cannot affect the power-law scaling of non-coincident correlators. The reader's worry that contact terms could sneak into the mixing matrix and shift the eigenvalues that survive is excluded by locality. So the central argument holds.\n\nThe soft spot is that this is asserted rather than shown. An explicit one-loop check, comparing the two schemes in a simple case like the Wilson–Fisher fixed point and confirming the surviving eigenvalues match, would shut the door completely. The comment does not provide that. In a longer paper that might be a real gap, but for a comment responding to a published criticism, it is a minor omission. The references are appropriate, and there is no circularity—the earlier work is cited for context, not as evidence.\n\nThis is not a new result, and it introduces no new technique. Its value is as a clarification, and it should be useful to anyone working in epsilon expansions or perturbative CFT who has to navigate conflicting conventions. I would send it to a referee; it deserves a proper peer review, not a desk rejection. A referee may ask for the one-loop check, but even without it the comment is correct and worthwhile.\n\nI would not cite it in my own work in the next year, but I would point a student to it when they ask about EOM operators and contact terms. Bring it to a reading group only if you are actively working in this corner of the bootstrap.\n\nNet: send it back for review. It will be a fine, if modest, contribution.\n\nBest,\n\n[Your name]","headline":"A short, plainly argued comment that correctly identifies the Kousvos–Stergiou dispute as terminological; the physics is sound and it deserves publication as a comment, even though it stops short of a worked two-scheme comparison.","tokens_in":2829,"tokens_out":1287,"would_cite":false,"duration_ms":17938,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This comment defends identifying operators that differ only by contact terms, such as phi^3 and d^2 phi at the Wilson-Fisher fixed point, as the same CFT operator with equal scaling dimensions.","keywords":["conformal field theory","Wilson-Fisher fixed point","epsilon expansion","composite operators","equations of motion","contact terms","renormalization group","scaling dimensions"],"falsifier":"Compute a non-coincident correlation function involving $\\phi^3$ at the Wilson-Fisher fixed point to subleading order in $\\epsilon$ using the full mixing matrix that includes contact-term operators, and check whether the extracted anomalous dimension agrees with the one obtained by identifying $\\phi^3$ with $\\partial^2\\phi$.","tokens_in":1921,"feed_emoji":"⚛️","tokens_out":8218,"duration_ms":80647,"temperature":0.7,"pith_summary":"The comment defends a specific practice in perturbative conformal field theory: when two operators, such as $\\phi^3$ and $\\partial^2\\phi$ at the Wilson-Fisher fixed point, are related by the equations of motion, their correlation functions agree at non-coincident points and differ only by contact terms. Since a CFT only defines correlation functions away from coincident points, the two operators should be identified and assigned the same scaling dimension. The comment argues that the criticized work's 'full way', which renormalizes the complete operator list including contact terms before dropping them, is unnecessary and that the disagreement is terminological rather than substantive. If the comment is right, the economical identification yields the correct CFT data with strictly less computation.","feed_headline":"A reply defends treating phi^3 and d^2 phi as the same CFT operator","feed_subtitle":"If contact terms are invisible away from coincident points, the cheaper identification yields the correct scaling dimensions.","key_machinery":"The machinery is the equation-of-motion contact-term identification: at the Wilson-Fisher fixed point the EOM makes $\\partial^2\\phi$ proportional to $\\phi^3$ up to delta-function terms, so away from coincident points there is only one independent operator. The argument combines this with the rule that CFT scaling dimensions are read from non-coincident correlation functions, which justifies working modulo contact terms before diagonalizing the renormalization mixing matrix. That is what allows the 'economical way' to operate with a minimal set of composite operators.","core_discovery":"The central claim is that CFT data in perturbative renormalization-group computations should be extracted from correlation functions at non-coincident points, so operators whose difference is a contact term are the same CFT operator. At the Wilson-Fisher fixed point, $\\phi^3$ and $\\partial^2\\phi$ are related by the equation of motion and therefore have equal scaling dimensions in the CFT sense. The nonhomogeneous scale transformation of such operators is purely a contact-term effect, invisible to the CFT. The comment states that renormalizing the full mixing matrix including EOM operators, as done in the criticized work, and then discarding those operators cannot change the non-coincident correlators, so the difference between the two treatments is only a matter of terminology.","pith_inferences":["A testable extension is to check explicitly at subleading order that contact-term renormalization does not feed back into the eigenvalues of the mixing matrix that control non-coincident correlators, since the comment leaves that demonstration implicit.","If the identification principle is right, conformal bootstrap equations that treat these operators as independent channels are carrying gauge degrees of freedom, and the physical spectrum is obtained by modding them out.","The same logic predicts that any EOM relation between operators of different classical dimensions yields equal CFT scaling dimensions at the fixed point, which is testable in multiscalar Wilson-Fisher fixed points."],"forward_implications":["At the Wilson-Fisher fixed point, $\\phi^3$ and $\\partial^2\\phi$ carry the same scaling dimension in the CFT sense.","Renormalizing contact terms and then discarding them cannot change the CFT correlation functions at separated points.","The difference between the comment's treatment and the criticized treatment is terminological; both give the same CFT scaling dimensions.","The economical approach is sufficient for computing CFT data and avoids unnecessary operator renormalization.","The same identification applies to EOM-related operators in other RG flows, such as the conserved-current/quartic-operator pair discussed for the $O(N)$ fixed point."],"supporting_citations":[{"why":"The computation being defended: it treated phi^3 and d^2 phi as identical at the Wilson-Fisher fixed point.","marker":"[1]"},{"why":"The criticized work that renormalizes the full operator list including contact terms and questions that identification.","marker":"[4]"},{"why":"Supplies the standard MS-scheme renormalization framework for composite operators used as context.","marker":"[2]"},{"why":"Textbook reference for critical properties and composite-operator renormalization in phi^4 theory.","marker":"[3]"}],"fun_headline_variants":["Defending phi^3 = d^2 phi: contact terms are invisible","Contact terms justify merging phi^3 and d^2 phi in CFT","CFT sees no contact terms, so phi^3 and d^2 phi align","Reply: non-coincident points make contact terms irrelevant","Minimal set wins: contact terms are not CFT data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that CFT data lives entirely in non-coincident correlation functions, so contact terms can be discarded when computing scaling dimensions.","fun_headline_variants_meta":{"raw":{"variants":["Defending phi^3 = d^2 phi: contact terms are invisible","Contact terms justify merging phi^3 and d^2 phi in CFT","CFT sees no contact terms, so phi^3 and d^2 phi align","Reply: non-coincident points make contact terms irrelevant","Minimal set wins: contact terms are not CFT data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1464,"prompt_tokens":790,"completion_tokens":674,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":406,"completion_tokens_details":{"reasoning_tokens":578}},"tokens_in":406,"tokens_out":674,"duration_ms":7584,"temperature":1.0,"reasoning_tokens":578,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:30:15.960290+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a non-coincident correlation function involving $\\phi^3$ at the Wilson-Fisher fixed point to subleading order in $\\epsilon$ using the full mixing matrix that includes contact-term operators, and check whether the extracted anomalous dimension agrees with the one obtained by identifying $\\phi^3$ with $\\partial^2\\phi$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard MS-scheme renormalization framework for composite operators used as context."},{"cited_title":"Kleinert and V","cited_arxiv_id":null,"evidence_quote":"Textbook reference for critical properties and composite-operator renormalization in phi^4 theory."}],"review_version":1}