Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values

Congkao Wen; Michael B. Green

Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values

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Pith's one-line read Unitarity and analyticity of four-particle superstring tree amplitudes force the Hankel matrices of their low-energy multiple-zeta-value coefficients to be totally positive.

desk verdict Unitarity positivity applied to string tree amplitudes yields new MZV Hankel inequalities; the closed-string gap is cosmetic, not fatal. read the letter →

arxiv 1908.08426 v2 pith:RGVQBPKQ submitted 2019-08-22 hep-th math-phmath.MP

Michael B. Green , Congkao Wen This is my paper · ORCID

classification hep-thmath-phmath.MP MSC 11M3215B4881T30

keywords superstringamplitudesunitarityboundsHankeldeterminantsmultiplezetavaluestotalpositivityStieltjesmomentsequencesno-ghosttheoremlow-energyexpansion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that two very general S-matrix principles, unitarity and analyticity, impose algebraic inequalities on the number-theoretic constants that appear in superstring scattering. For the open-string tree amplitude, each low-energy coefficient is a multiple zeta value, and the principles force Hankel matrices with entries $\zeta(1,\ldots,1,i+j)$ to be totally positive, meaning every minor has the same sign. For the closed string, the full amplitude's coefficients are not moment sequences, but the $s$-channel pole part is; its Hankel determinants therefore constrain rational polynomials of MZVs, including irreducible ones such as $\zeta(2,6)$. The simplest cases reduce to known inequalities on Hankel determinants of ordinary zeta values, now placed on a physical footing.

What carries the argument

The load-bearing objects are Hankel matrices, meaning matrices whose $(i,j)$ entry depends only on $i+j$, built from low-energy expansion coefficients. For the open string the $(p,q)$ coefficient is $\zeta(1,\ldots,1,p+2)$, so the matrix entries are $\zeta(1,\ldots,1,i+j)$; for the closed string the $s$-channel coefficients are $Z(r,q)$, defined by the generating function $\sum_{q\geq 0} Z(p+3,q)\,t^q = \sum_{n\geq 1} n^{-p-1}\,(\Gamma(n+t)/\Gamma(1+t)\Gamma(1+n))^2$, and the matrix entries are $Z(i+j+1,q)$. The mechanism is the Stieltjes half-moment theorem: a positive measure on $[0,\infty)$ whose moments equal these coefficients makes every associated Hankel matrix totally positive. Positivity of the measure follows from partial-wave expansions with positive residues for open strings and, for the closed-string $s$-channel amplitude, from the assumed no-ghost positivity of the Gegenbauer coefficients.

What would settle it

Evaluate the Gegenbauer partial-wave coefficients of the closed-string $s$-channel amplitude at the first few massive poles; a single negative coefficient would break the Stieltjes moment representation. Independently, compute $\det H_{\mathrm{cl}}^{(s,n)}[Z_q]$ to high precision for $n=1,\ldots,10$ and $q=2,3$; any non-positive determinant would contradict the paper's claim.

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Extended reading notes

Core claim

The paper's central claim is that total positivity of these Hankel matrices is a theorem about superstring tree amplitudes, not a numerical accident. More precisely, in the open superstring the coefficient of $s^p t^q$ is $\zeta(1,\ldots,1,p+2)$, so total positivity means every minor of $H_{\mathrm{op}}^{(n)}[\zeta_q]$ with entries $\zeta(1,\ldots,1,i+j)$ is positive; in the closed superstring, the same reasoning applied to the $s$-channel half of the amplitude makes every minor of $H_{\mathrm{cl}}^{(s,n)}[Z_q]$ with entries $Z(i+j+1,q)$ positive, where $Z(r,q)$ is a specified combination of multiple zeta values. Since $Z(r,q)$ contains irreducible MZVs when $q\geq 2$ and $r+q\geq 8$, the positive-minor conditions are inequalities on rational polynomials of single zeta values and irreducible MZVs such as $\zeta(2,6)$. The paper also shows that the irreducible MZVs cancel between the $s$- and $u$-channel parts, so the full closed-string amplitude at fixed $t$ is again a rational polynomial in odd zeta values.

Load-bearing premise

The closed-string result collapses if the residues of the massive $s$-channel poles are not all positive in their Gegenbauer expansion, an assumption the paper invokes from the no-ghost theorem without deriving the partial-wave coefficients.

Editorial extensions

If this is right

All leading principal minors and all minors of $H_{\mathrm{op}}^{(n)}[\zeta_q]$ are strictly positive for every $q\geq 0$ and $n\geq 1$, giving infinitely many inequalities among rational polynomials of single zeta values.
The closed-string Hankel matrices $H_{\mathrm{cl}}^{(s,n)}[Z_q]$ are totally positive, so constraints such as $\det H_{\mathrm{cl}}^{(s,3)}[Z_2]>0$ restrict rational polynomials that include the irreducible multiple zeta value $\zeta(2,6)$.
The irreducible MZVs cancel between the $s$- and $u$-channel parts in the full closed-string amplitude, so the full amplitude's low-energy coefficients remain rational polynomials of odd zeta values while the new MZV inequalities are carried by the $s$-channel split alone.
The known positivity of Hankel determinants of ordinary zeta values becomes a special case of the open-string unitarity constraints, now derived from physical principles rather than observed numerically.
The paper presents these inequalities as necessary conditions for superstring tree amplitudes to be unitary, and notes that proving them by independent number-theoretic means remains open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the paper is right, the large-$n$ decay of $\det H_{\mathrm{cl}}^{(s,n)}[Z_q]$ should be derivable from the high-energy behaviour of the closed-string amplitude; the paper leaves the asymptotic formula open, but deriving it would tie the number theory directly to string dynamics.
The weakest step can be tested independently: expanding the closed-string pole residues at the first few mass levels into Gegenbauer polynomials should show whether all partial-wave coefficients are positive, since a single negative residue would invalidate the Stieltjes moment argument for the closed string.
The same moment-sequence logic applied to $N$-point superstring amplitudes, or to four-point amplitudes with massive external legs, is a natural next test and would exercise the no-ghost theorem more fully than the massless four-point case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

You should know this about arXiv:1908.08426: it is a real result, not a stunt. The authors use the standard unitarity/analyticity positivity machinery on four-particle superstring tree amplitudes and derive total positivity for Hankel matrices whose entries are multiple zeta values, including irreducible MZVs like ζ(2,6). The open-string case is a clean generalization of the single-zeta results of Monien and Haynes–Zudilin, and the closed-string s-channel case gives new inequalities involving irreducible MZVs that cancel in the full amplitude.

What is new: the explicit connection between string theory and Hankel determinants of MZVs. The coefficients g^op_{p,q}=ζ(1,...,1,p+2) arise naturally, and the Z(r,q) coefficients in the closed-string s-channel are exactly the quantities Zagier already found in genus-one amplitudes. The Stieltjes moment argument is sound: the Hankel matrices are totally positive because the low-energy coefficients are moments of a positive discrete measure.

The soft spots are real but minor. The d(0) constant in the quoted asymptotics (3.10)/(3.11) disagrees with the attributed value; that needs checking. Eq. (4.11) has a sign/typo issue. And Section 4.2 asserts the residue positivity of massive closed-string poles from the no-ghost theorem without showing the Gegenbauer coefficients. That last point is the one a referee would press. But the stress-test note is right: the authors' own generating function (4.14) gives a positive discrete measure for the Z(p+3,q) coefficients directly, so the Hankel positivity follows without needing the Gegenbauer expansion. The assertion is under-derived, but the claim is secure.

One limitation worth stating: this only probes four-particle tree amplitudes, so it touches a small corner of the no-ghost theorem. The paper says as much. The new inequalities are interesting but don't reshape the field.

Who is this for? Anyone working on positivity bounds in amplitudes, and number theorists interested in MZV inequalities. It deserves a serious referee. The main theorem stands. With a few small corrections and a short extra paragraph deriving the positivity from (4.14), it should be published without much fuss.

My recommendation: send it to peer review; accept after minor revisions.

Referee Report

0 major / 5 minor

Summary. The paper derives positivity constraints on Hankel determinants of multiple zeta values (MZVs) from unitarity and analyticity of massless four-particle superstring tree amplitudes. After reviewing the Stieltjes half-moment theorem, it shows that for the open superstring the low-energy coefficients g^op_{p,q}=ζ(1,...,1,p+2) form moment sequences, so the Hankel matrices H^op_n[ζ_q] are totally positive; this generalizes known results for single zeta values. For the closed superstring, the full amplitude has u-channel poles that spoil naive positivity, but the s-channel contribution A^(s)_cl has coefficients Z(p+3,q), and the paper argues that the associated Hankel matrices Hcl_n^{(s)}[Z_q] are totally positive, yielding new inequalities on rational polynomials containing irreducible MZVs such as ζ(2,6). The paper also explains the cancellation of even zeta values and irreducible MZVs in the full closed-string amplitude and connects the Z(r,q) quantities to the genus-one setup via Zagier's relation.

Significance. If the claims hold, the paper supplies a physical derivation of previously known Hankel positivity for single zeta values and an infinite family of new positivity constraints on MZV polynomials, including cases with irreducible MZVs. A notable strength is that the key positivity is not fitted or conjectural: for fixed q, Eq. (4.14) expresses Z(p+3,q) as an explicit positive discrete Stieltjes measure, so the Hankel determinants are positive by the theorem quoted from [19]. The explicit low-energy expansions, the connection to the single-valued projection, and the link to Zagier's genus-one results make the paper valuable for both the amplitudes and the number-theory communities. The scope is appropriately modest, being limited to four-particle tree amplitudes, and the authors state clearly which aspects would require higher-point or higher-genus generalizations.

minor comments (5)

[Section 2.2, Eq. (2.15)] The equality g^(s)_p,0 = g^(u)_p,0 is not correct for odd p; from Eq. (2.12) one obtains g^(u)_p,0 = (-1)^p g^(s)_p,0, so for an amplitude with both s- and u-channel poles the t=0 coefficients satisfy g_{2n+1,0}=0 and the full sequence is not a Stieltjes half-moment sequence. The general total-positivity statement in this subsection should be restricted to the s-channel (or colour-ordered) amplitude or to the even-p subsequence, since the subsequent explicit string results do not rely on the incorrect equality.
[Section 4.2, after Eq. (4.7) and around Eq. (4.14)] The text asserts that the closed-string s-channel residues have positive Gegenbauer partial-wave coefficients via the no-ghost theorem, but it does not demonstrate this for the squared residues. The conclusion follows more directly and rigorously from Eq. (4.14): for fixed q, Z(p+3,q)=∑_n c_{n,q} n^{-(p+3)} with c_{n,q}=[t^q]∏_{m<n}(1+t/m)^2>0, which displays the required Stieltjes measure dμ_q(y)=∑_n c_{n,q} n^{-3}δ(y-1/n)dy. This explicit one-line proof should be stated in the text.
[Section 3.1, Eqs. (3.10)-(3.11)] The quoted asymptotic constant d(0) is reported as 0.66367, whereas the value attributed to Zagier in [23] is 0.35147; the authors should verify the transcription and reconcile the discrepancy, since both values appear in the same formulas.
[General notation] There are several typographical inconsistencies, e.g., 'Euler–Mascharoni' should be 'Mascheroni', 'rˆole' should be 'role', and the Gegenbauer index is written as (D-2)/2 in Eqs. (2.11), (2.15), and (2.16) but as (D-3)/2 elsewhere; these should be harmonized.
[Section 4.2, Eq. (4.15)] The summation variable q is used both for the fixed order and for the vector being summed; using a boldface symbol, e.g. \mathbf{q}, would remove the ambiguity.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation chain: Hankel positivity follows from explicit positive Stieltjes measures in (3.6) and (4.14); the only soft spot is a terse Gegenbauer-positivity assertion that (4.14) independently secures.

full rationale

The paper's claims are derived from fixed string amplitudes, not from fitted parameters or self-referential normalizations. For open strings, Eq. (3.6) expresses g_{p,0}=zeta(p+2) as sum_n n^{-(p+2)}, which is a Stieltjes moment sequence with positive discrete measure; the general q-coefficients zeta(1,...,1,p+2) have the same structure because the nested sums have positive terms. For closed strings, Eq. (4.14) gives Z(p+3,q)=sum_n n^{-(p+3)} [t^q] prod_{m<n}(1+t/m)^2, and the coefficients of the product are positive for t>=0, so {Z(p+3,q)}_p is again a Stieltjes moment sequence with positive measure; total positivity then follows from the external theorem [19]. The no-ghost/Gegenbauer statement in Section 4.2 after Eq. (4.7) is terse and not proven in detail, but it is not needed in the explicit measure argument, and the measure argument does not reduce to the conclusion. Self-citations [16], [35], and [38] occur, but none is load-bearing: [35]'s relevant identity is proved by Zagier in an appendix, and [23,24] provide external mathematical benchmarks. The mild caveat is that the paper uses tree-level unitarity as an input, which is a physical assumption rather than a circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No physical free parameters are fitted; the coefficients are fixed by the given string amplitude expressions. The main axioms are the Stieltjes moment theorem, Gegenbauer positivity, the D=10 no-ghost theorem, the MZV coefficient identities from [26,27], and the Regge asymptotic drop of the boundary term. No new entities such as particles, mediators, or forces are introduced.

assumptions (5)

standard math Stieltjes half-moment theorem: a sequence is a moment sequence with positive measure on [0, infinity) iff its Hankel matrices are totally positive (Fallat, Johnson, Sokal [19], Theorem 2.8).
Used in Section 2.1 to convert coefficient positivity into total positivity of Hankel matrices and all minors.
standard math Gegenbauer polynomial derivatives at y=1 are positive: partial_y^q G_l^{(D-3)/2}(y)|_{y=1} > 0 for all q and l.
Used in Section 2.2, eq. (2.22), to conclude positivity of the t^q coefficients of the s-channel pole sum.
domain assumption No-ghost theorem in D=10 critical superstring theory: the physical spectrum has positive norm, so the residues p_a in the partial wave expansion are positive.
Invoked in Section 3 before eq. (3.3) and in Section 4.2 to justify positivity of s-channel pole residues for open and closed superstrings.
standard math The low-energy coefficients of the open string are zeta(1,...,1,p+2) (from Zagier and Zerbini [26]) and the closed-string s-channel coefficients are Z(p+3,q) with Z(r,q) defined in eq. (4.15); the reduction of these MZVs to rational polynomials in zeta values uses standard MZV identities.
Used in Sections 3.1 and 4.2 to write explicit Hankel entries; the MZV reduction identities are cited from [26,27].
domain assumption The superstring amplitudes are Regge behaved, so the contour integral at |s| to infinity in the dispersion relation (2.9) can be dropped for the subtracted amplitudes.
Used in Sections 2, 3, 4 to justify the dispersion relation giving coefficients as sums over poles.

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Cite this review

Pith. "Pith review of Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values." pith.science (2026). https://pith.science/paper/RGVQBPKQ

@misc{pith2026190808426,
  author       = {Pith},
  title        = {Pith review of: Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGVQBPKQ}},
  note         = {Machine review of arXiv:1908.08426}
}

read the original abstract

The interplay of unitarity and analyticity has long been known to impose strong constraints on scattering amplitudes in quantum field theory and string theory. This has been highlighted in recent times in a number of papers and lecture notes. Here we examine such conditions in the context of superstring tree-level scattering amplitudes, leading to positivity constraints on determinants of Hankel matrices involving polynomials of multiple zeta values. These generalise certain constraints on polynomials of single zeta values in the mathematics literature.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Shift symmetries, soft limits, and the double copy beyond leading order
hep-th 2019-08 conditional novelty 6.0 of 10

For higher-derivative corrections, shift symmetry no longer guarantees double-copy compatibility; even-point amplitudes can be made compatible by tuning coefficients, odd-point ones cannot.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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