{"id":"4ecd7d1e-a9c5-4ff4-8173-f293916febcf","arxiv_id":"2505.06172","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An improved inequality with a correction from Weinberg's F parity gives the minimum operator dimension for each baryon and lepton number combination, and it is shown to be an equality in the νSMEFT up to dimension 25.","lead":"This paper sharpens the known link between the mass dimension of an effective operator and how much baryon and lepton number it can carry. The authors add a correction from a hidden symmetry and show their formula is essentially exact for the Standard Model plus right-handed neutrinos.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The νSMEFT equality claim rests on an unverified extrapolation of derivative operators from d=17 to d=25; a derivative operator above d=17 could add a (ΔB,ΔL) point or lower an exception to the bound.","rationale":"The inequality Eq. (2) follows from F parity and is not threatened by derivative operators. The empirical equality claim, however, is only as strong as the enumeration. The reader's weakest assumption correctly identifies the d=17 derivative cutoff as the main residual gap. I agree with the CONDITIONAL verdict: the paper should either extend the derivative check or soften the 'equality up to d=25' language to 'non-derivative operators.' There is no basis for rejection, and the proposed test is concrete enough to settle the residual doubt.","tokens_in":7220,"tokens_out":35851,"duration_ms":407413,"concrete_test":"Run Sym2Int with derivative operators included up to d=25 for νSMEFT, at least over the grid |ΔB|≤5, |ΔL|≤13, and compare the resulting d_min map with Fig. 3 and Eq. (2). If any new (ΔB,ΔL) point appears below d=25 or any listed exception drops to the bound, the manuscript's exception list and equality claim need revision; if the map is unchanged, the derivative extrapolation is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central new result beyond the inequality is the claim that Eq. (2) is a near-equality in νSMEFT up to d=25. The evidence for this is Sym2Int enumeration of non-derivative operators to d=25 and of derivative operators only to d=17, with the statement 'We also checked derivative operators up to d=17 but found no qualitative difference, as expected since derivatives act similarly to H.' That extrapolation is load-bearing because a derivative and a Higgs have the same F parity and dimension but different hypercharge and weak-isospin structure, so derivative operators are not automatically redundant. A derivative can fix F parity without disturbing hypercharge neutrality in cases where adding H would break it. Consequently, a derivative operator at 18≤d≤25 could either introduce an allowed (ΔB,ΔL) point absent from Fig. 3 or lower one of the claimed exceptions (0,>6) or (1,11) to the Eq. (2) bound, changing the exception list. The inequality itself is unaffected; the concern is the completeness and sharpness of the equality claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper revisits the relation between the mass dimension d of (ν)SMEFT operators and their baryon and lepton numbers. The main analytic result is Eq. (2), which sharpens Kobach's lower bound by an additional +1 term when ΔB·ΔL is negative and odd, based on Weinberg's F parity. The authors test this bound with the Sym2Int program, enumerating non-derivative SMEFT and νSMEFT operators up to d = 25 and derivative operators up to d = 17. They report that in νSMEFT Eq. (2) is an equality up to d = 25 except for (|ΔB|,|ΔL|) = (0, >6) and (1,11), and they argue that these deviations arise from Pauli exclusion among the sterile neutrino fields. The paper concludes that F parity and fermion statistics, in addition to gauge and Lorentz invariance, control the allowed (ΔB,ΔL) landscape.","tokens_in":7450,"tokens_out":20368,"duration_ms":221251,"significance":"If the enumeration is complete, Eq. (2) is a useful sharpening of a classic bound, and the explicit d_min map up to d = 25 for νSMEFT is a valuable reference for model building with baryon- or lepton-number violation. The paper's concrete, falsifiable statements about which (ΔB,ΔL) sectors appear at which dimension, together with its exact Sym2Int-based enumeration rather than a fitted estimate, are strengths. The identification of Pauli exclusion as a limiting mechanism beyond gauge and Lorentz constraints is an interesting and testable observation. The main caveats are that the equality claim for νSMEFT depends on an unproven extrapolation of derivative operators beyond d = 17, and that one step in the derivation of Eq. (2) is stated too broadly.","major_comments":[{"comment":"The sentence 'any operator with negative odd ΔB·ΔL is odd under F parity' is false as stated. A field string such as q q q \\bar q \\ell^C has ΔB = 1, ΔL = -1 and F parity even, because it contains one anti-quark and one anti-lepton. What is actually needed for the proof is the weaker statement that the minimal fermion content realizing a given (ΔB,ΔL) with negative odd product has odd F parity, and that any F-even realization therefore requires additional F-odd fields. Since the cheapest such field, a Higgs or a derivative, has dimension 1 while a fermion-antifermion pair has dimension 3, the +1 term in Eq. (2) follows. Please revise the proof to state this minimality argument explicitly.","section":"Paragraph containing Eq. (2)"},{"comment":"The equality claim for νSMEFT up to d = 25 is not fully established by the computation described in the paper. The text states that Sym2Int was run to d = 25 only for non-derivative operators, while derivative operators were checked only up to d = 17. Because a derivative has the same F parity and dimension as a Higgs doublet but different hypercharge and SU(2) properties, it is not automatically redundant with H. A derivative operator with 18 ≤ d ≤ 25 could in principle add a new (ΔB,ΔL) point to Fig. 3 or lower the dimension of one of the claimed exceptions, such as (|ΔB|,|ΔL|) = (1,11). Please either extend the derivative enumeration to d = 25, provide a rigorous argument that derivative operators cannot lower d_min, or explicitly restrict the equality claims to the sector and dimension range where derivatives were checked.","section":"Paragraph beginning 'To test Eq. (2)'"}],"minor_comments":[{"comment":"The typesetting of the indicator function in Eq. (2) is garbled: as printed it reads '+11−2N(∆B∆L)' rather than '+1_{ΔBΔL<0, odd}'. Please fix the notation so the condition is unambiguous.","section":"Equation (2)"},{"comment":"There is a typo 'for for' in the sentence 'the equality in (2) only holds up to d = 9 for for ∆L≠ 0 = ∆B νSMEFT'. Also, the notation '∆L≠ 0 = ∆B' is confusing; please write (ΔB = 0, ΔL ≠ 0).","section":"Paragraph after Eq. (2)"},{"comment":"The tuple label '(dνSMEFT, dSMEFT)min min' at the top of the caption is garbled. It should be typeset as (d_min^{νSMEFT}, d_min^{SMEFT}) or a similarly unambiguous notation.","section":"Figure 3 caption"},{"comment":"The statement 'We also checked derivative operators up to d = 17 but found no qualitative difference, as expected since derivatives act similarly to H' would benefit from a precise description of which derivative operator topologies were included and how Sym2Int was configured, because D and H carry different hypercharge and weak-isospin quantum numbers.","section":"Paragraph beginning 'To test Eq. (2)'"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid short contribution and the analytic inequality appears to be correct once the proof is repaired. The main risk is the unsupported extension of the derivative-operator check beyond d = 17; if the authors cannot complete that enumeration or provide a proof, they should soften the equality claim accordingly. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis is a genuinely useful paper. The improved inequality (2) gives a simple way to read off the minimal operator dimension for any baryon/lepton number combination, and the explicit Sym2Int enumeration up to d=25 is a real service, extending Kobach's maps far beyond what existed. The observation that Pauli exclusion, via the finite number of generations, can obstruct saturation in the νSMEFT is new and, as far as I can tell, correct. The computational work is substantial and the presentation is clear.\n\nThe central inequality is almost certainly right, but the proof as sketched has a gap: the statement that \"any operator with negative odd ΔB·ΔL is odd under F parity\" is not true if you add spectator fermion-antifermion pairs, which flip F without changing net baryon or lepton number. The intended argument only needs the claim for operators that saturate Kobach's dimension count, and for those it goes through, so the result stands. I would call that a minor exposition issue, not a fatal one.\n\nThe bigger caveat is the νSMEFT \"equality\" claim. The enumeration is restricted to non-derivative operators for d up to 25; derivatives are only checked to d=17. The authors argue that derivatives behave like H, and that is plausible for F parity and dimension, but derivatives and H differ in hypercharge and in how they can be contracted, so the extrapolation is not automatic. A derivative operator between d=18 and d=25 could in principle saturate a point that the non-derivative map misses, or lower one of the claimed exceptions like (1,11). I don't think this undermines the inequality, which is a lower bound and stands regardless, but the equality claim is stronger than the evidence strictly supports. The authors do phrase it with \"as far as we can tell,\" which is honest, but the paper would be stronger with an explicit statement about the derivative-operator completeness assumption.\n\nThe d_min maps to d=25 will be useful to anyone working on nucleon decay or lepton-number violation. This paper deserves peer review; a careful referee can ask the authors to tighten the F-parity statement and to spell out the status of derivative operators.\n\nBest.","headline":"Useful sharpening of Kobach's bound, backed by serious enumeration to d=25; the νSMEFT equality claim rests on an unproven derivative extrapolation, but the inequality itself holds.","tokens_in":7976,"tokens_out":7978,"would_cite":true,"duration_ms":78090,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The smallest dimension for an operator with charges $(\\Delta B,\\Delta L)$ is at least $\\frac{9}{2}|\\Delta B|+\\frac{3}{2}|\\Delta L|$, plus one if $\\Delta B\\Delta L$ is negative and odd, and in the neutrino-extended SMEFT the bound is…","keywords":["baryon number violation","lepton number violation","SMEFT","νSMEFT","operator dimension","F parity","identical-fermion exclusion","effective field theory"],"falsifier":"Run an independent enumeration that drops the no-derivative restriction and searches all of $(\\nu)$SMEFT through dimension 25: if any operator with a given $(\\Delta B,\\Delta L)$ appears at a dimension below the values shown in the paper's map, the equality claim for that point is wrong. The decisive targets are a $d\\le20$ operator for $(|\\Delta B|,|\\Delta L|)=(1,11)$ in $\\nu$SMEFT and any SMEFT operator at $d<19$ for $(1,7)$.","tokens_in":7016,"feed_emoji":"⚛️","tokens_out":11010,"duration_ms":102075,"temperature":0.7,"pith_summary":"The paper tries to pin down the smallest mass dimension $d_{\\min}$ at which an effective operator can carry baryon number $\\Delta B$ and lepton number $\\Delta L$. It derives the bound $d_{\\min}\\ge \\frac{9}{2}|\\Delta B|+\\frac{3}{2}|\\Delta L|+1$ when $\\Delta B\\Delta L$ is negative and odd, improving the earlier inequality by one unit in that case. Using an automated enumeration of operators up to dimension 25, it finds that in the version of the effective field theory that includes right-handed neutrinos ($\\nu$SMEFT) the bound is an equality for essentially every allowed charge pair, with the only exceptions at $(|\\Delta B|,|\\Delta L|)=(0,>6)$ and $(1,11)$. If this is right, the operator dimension needed for baryon- or lepton-number-violating signals is fixed by the charges alone, without enumerating every operator.","feed_headline":"Operator dimension bound is exact up to d=25 in νSMEFT","feed_subtitle":"A new +1 parity term matches every explicit operator except pure-lepton cases at high dimension.","key_machinery":"The load-bearing object is the multiplicative F parity, under which SM fermions are even while antifermions, bosons, and derivatives are odd; Lorentz and $SU(2)_L$ invariance conserve it. Adding F parity to the counting of quarks (which cost dimension $9/2$ per unit of $\\Delta B$) and leptons (dimension $3/2$ per unit of $\\Delta L$) produces the corrected bound in Eq. (2). The second mechanism is the anticommutation of spinor fields: an identical fermion field can appear at most twice in a non-derivative product, capping the accessible pure-lepton number and lifting $d_{\\min}$ for $(0,\\Delta L>6)$.","core_discovery":"On its own terms, the central claim is that the earlier inequality $d_{\\min}\\ge \\frac{9}{2}|\\Delta B|+\\frac{3}{2}|\\Delta L|$ is not the full story: conservation of the multiplicative F parity forces an extra unit of mass dimension whenever $\\Delta B\\Delta L$ is negative and odd, yielding the corrected lower bound. The paper further claims that in $\\nu$SMEFT this improved inequality is saturated up to $d=25$ except for the two identified families, while in the SMEFT it is essentially an equality whenever $\\Delta B\\neq0$. The departures from equality are caused by hypercharge invariance at low dimension and, for pure-lepton $\\nu$SMEFT operators, by the anticommutation of fermionic fields, which prevents more than six neutrino fields from appearing in a non-derivative operator.","pith_inferences":["If derivative operators continue to respect the non-derivative pattern beyond $d=17$, then the charge-pair formula likely remains exact to arbitrarily high dimension in $\\nu$SMEFT; this is an extrapolation, not a claim in the paper.","The fermion-exclusion obstruction suggests that searches for $\\Delta L=8$ processes mediated by right-handed neutrinos will be suppressed by two additional powers of the new-physics scale relative to the naive dimension bound, a shift that could be visible in same-sign dilepton searches.","The same F-parity-plus-exclusion logic could sharpen dimension bounds for other accidental global symmetries in effective field theories with identical chiral fermions, though the paper does not explore that application."],"forward_implications":["The minimal dimension for any baryon-number-violating operator follows from charges alone, so bottom-up searches for nucleon decay can be organized by $(\\Delta B,\\Delta L)$ rather than by model.","In $\\nu$SMEFT, every charge pair up to $d=25$ has a known minimal operator dimension, so the scale suppression of a given new-physics signal can be read off without enumerating operators.","Because F parity forces operators with the same charges to appear only at $d_{\\min}+2N$, an operator and its $|H|^2$ or $D_\\mu D^\\mu$ descendants exhaust the possible dimensions for a fixed charge pair.","For $\\Delta B\\neq0$ SMEFT operators, the simple formula is a practical equality for $|\\Delta L|<5$, giving a quick test of whether a proposed ultraviolet model produces the leading low-dimensional operator."],"supporting_citations":[{"why":"introduces the dimension-five operator that is the benchmark $\\Delta L=2$ case.","marker":"[5]"},{"why":"defines the multiplicative F parity and its conservation, the new ingredient in the corrected bound.","marker":"[6]"},{"why":"gives the earlier inequality and the angular-momentum parity constraint that the corrected bound sharpens.","marker":"[12]"},{"why":"supplies the operator-enumeration program used to construct explicit operators.","marker":"[13]"},{"why":"documents the enumeration method that underpins the $d\\le25$ operator lists.","marker":"[14]"},{"why":"is the cited source for the exclusion of identical fermions in a non-derivative operator.","marker":"[16]"},{"why":"supports the vanishing of operators with repeated neutrino fields, the obstruction behind the $(0,\\Delta L>6)$ exceptions.","marker":"[17]"}],"fun_headline_variants":["Improved bound ties baryon, lepton, and operator dimension","Parity term sharpens B,L-operator dimension bound","Almost exact B,L-dimension rule in νSMEFT","New inequality makes baryon-lepton dimension bound tight in νSMEFT","B and L dimension link nearly exact in νSMEFT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equality claims depend on the operator enumeration being complete through dimension 25 and on derivative operators, checked only through dimension 17, never producing a lower-dimensional operator than the non-derivative enumeration at higher dimension.","fun_headline_variants_meta":{"raw":{"variants":["Improved bound ties baryon, lepton, and operator dimension","Parity term sharpens B,L-operator dimension bound","Almost exact B,L-dimension rule in νSMEFT","New inequality makes baryon-lepton dimension bound tight in νSMEFT","B and L dimension link nearly exact in νSMEFT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001087,"raw_usage":{"total_tokens":4466,"prompt_tokens":789,"completion_tokens":3677,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":3592}},"tokens_in":405,"tokens_out":3677,"duration_ms":32667,"temperature":1.0,"reasoning_tokens":3592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:48:01.073929+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent enumeration that drops the no-derivative restriction and searches all of $(\\nu)$SMEFT through dimension 25: if any operator with a given $(\\Delta B,\\Delta L)$ appears at a dimension below the values shown in the paper's map, the equality claim for that point is wrong. The decisive targets are a $d\\le20$ operator for $(|\\Delta B|,|\\Delta L|)=(1,11)$ in $\\nu$SMEFT and any SMEFT operator at $d<19$ for $(1,7)$.","supporting_citations":[{"cited_title":"Varieties of Baryon and Lepton Nonconservation,","cited_arxiv_id":null,"evidence_quote":"defines the multiplicative F parity and its conservation, the new ingredient in the corrected bound."},{"cited_title":"¨Uber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren,","cited_arxiv_id":null,"evidence_quote":"is the cited source for the exclusion of identical fermions in a non-derivative operator."}],"review_version":1}