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Abstract

Stieberger (arXiv:0907.2211) writes the disk amplitude of three closed strings as a combination of six-point open-string partial amplitudes, for massless external states. We continue the six kinematic invariants of that relation to closed-string tachyon kinematics and check it numerically at eight kinematic points inside the region where all integrals converge. Three routes (a direct two-dimensional closed-string integral, the six-contour combination of the appendix of that paper, and real six-point open-string integrals) agree to relative differences between 1e-9 and 1e-6. The check holds when two of the six open-string partial amplitudes, whose integration region in the source reads η>1 only, are taken over both η>1 and η<-1; with the half region as written the relative differences are 9e-2 to 3e-1. We read this as the source intending the invariant partial amplitude, but we have not established that reading. A second, pre-specified sample of 40 points drawn at random (Latin hypercube) from a wider part of the convergent region gives the same picture: at the default quadrature 33 of 40 points agree to better than 1e-5 (median 1.3e-7), five more converge to 2.0e-7 to 3.5e-7 when the nodes are refined, and for two points the closed-string integral is unstable under refinement so they remain unresolved; the half-region version is off by 1.7e-2 to 0.60 at every one of the 40 points. The claim is limited to the convergent region and a single analytic continuation s_i = c_i + 2; it is not a proof and it does not reach the cubic coupling that motivated it. The main finding is not the agreement itself but that the printed integration region of two partial amplitudes is incomplete under the reading that they are invariant partial amplitudes.

1. Motivation and scope

Garousi [G1] conjectured that the coefficient function of the D-brane Born–Infeld action, in its dependence on the closed-string tachyon T, is $e^{-\Phi}(1-T/2)^{-1/2}$, and checked it to second order in T. Both [G1] and [G2] leave the cubic term, whose conjectured coefficient is 5/128, to future work; it requires the three-tachyon disk amplitude. A route to that amplitude is the relation of Stieberger [S] between the three-closed-string disk amplitude and six-point open-string partial amplitudes. That paper treats massless states. This note only asks whether the relation survives the continuation to tachyon kinematics, numerically. It does not extract the cubic coupling. A later paper by Bischof, Haack and Stieberger [B] treats three closed strings on the disk directly; we have read only its introduction, so the relation of the present check to it is not established here.

2. Setup

We use the notation of [S]. Closed-string vertex positions are fixed to $z_1=i$, $z_2=ix$, $z_3=z$ with $0<x<1$, and the amplitude is

$$\mathcal A(1,2,3)=2\int_{-1}^{1}dx,(1-x^2)\int_{\mathbb C}d^2z,\langle V_c(-i,i),V_c(-ix,ix),V_c(\bar z,z)\rangle+\text{h.c.}$$

with six invariants $s_1=q_1Dq_1,\ s_2=q_1Dq_2,\ s_3=q_2Dq_2,\ s_4=q_2Dq_3,\ s_5=q_3Dq_3,\ s_6=q_1Dq_3$, where $D$ is diagonal with +1 along and −1 transverse to the brane. Writing $c_i=q_iDq_i$, $a_{ij}=q_iq_j$ and $b_{ij}=q_iDq_j$, momentum conservation along the brane gives $a_{12}=\tfrac12(-c_1-c_2+c_3)-1-b_{12}$ and its cyclic images. For tachyons $q_i^2=2$, so $s_i=c_i+2$ for the diagonal invariants (for massless states $s_i=c_i$); this shift is the only change we make relative to [S]. All integer shifts $m_i,n_i,\tilde\kappa$ of [S] are set to zero.

The relation to be checked is

$$\begin{aligned}\mathcal A(1,2,3)=\Big{&\sin(\pi s_4)A(1,3,6,4,5,2)+\sin!\big[\pi\big(\tfrac{s_1-s_3-s_5}{2}-s_4\big)\big]A(1,4,6,3,5,2)\&+\sin!\big[\pi\big(\tfrac{-s_1+s_3-s_5}{2}-s_6\big)\big]\big[A(1,3,4,6,2,5)+A(1,4,3,6,2,5)\big]\&+\big(\sin[\tfrac{\pi}{2}(s_1-s_3-s_5)]-\sin[\tfrac{\pi}{2}(s_1-s_3+s_5)]\big)\big[A(1,3,4,6,5,2)+A(1,4,3,6,5,2)\big]\Big},\end{aligned}$$

where the $A$ are real three-fold integrals over $x,\xi,\eta$ with open-string vertex positions $z=(-1,1,-x,x,\eta,-\xi)$ (Eq. ISTO of [S]); each integrand is $x^{\alpha_3}(1+x)^{1+\alpha_1}(1-x)^{1+\alpha_2}$ times a product of $|\cdot|^{\text{exponent}}$ factors in $\xi$ and $\eta$. The exponents are $\lambda=b_{13}$, $\gamma=a_{13}$, $\beta=b_{23}$, $\epsilon=a_{23}$, $\kappa=c_3$ and $\alpha_1=2b_{12}$, $\alpha_2=2a_{12}$, $\alpha_3=c_2$.

3. Checks

We take eight kinematic points with $c_i\in[-0.7,-0.4]$ and $b_{ij}\in[-0.38,-0.18]$. They were chosen by hand as an exploratory sample of the convergent region (a random sample follows below), and every candidate that satisfied the convergence condition (all kinematic exponents above −1, and a bound on the large-$v$ power of the closed-string integrand) was kept (only eight candidates were generated, none discarded). The points are therefore not a random or stratified sample, and they may favour regions where the relation holds easily. We compare three quantities, each as a relative difference.

1. The inversion of the six-point orderings, $v'=Cv$, from the monodromy relation matrix (60 orderings, rank 54 at thresholds 1e-6, 1e-9, 1e-12 alike; the singular values drop from 0.28 to 4e-16 across the gap, so the rank does not depend on the threshold within that range).
2. $2^{c_2}\times$ the right-hand side of the relation above against the direct two-dimensional closed-string integral $\mathcal A$.
3. The left-hand side $(1+e^{i\pi s_5})\mathcal A$ of the intermediate relation of [S] against $W_1+W_2$, the combination of the six contours of its appendix.

Table 1. Relative differences at the eight points ($u_{ij}=b_{ij}-a_{ij}$).

The relative differences of check (1) range from 5.5e-11 to 2.5e-7. The largest value in (2) (8.9e-7) is due to truncation of the closed-string quadrature at its default node count; with more nodes (three levels on four points) the open-string side converges and the open–closed difference is at most 5.0e-9 at the finest level.

A random sample (added in this revision). The eight points above were chosen by hand, and a review of the first version pointed out the resulting selection bias. We therefore drew 40 points by Latin hypercube sampling (seed 20261003) from $c_i\in[-0.8,-0.3]$, $b_{ij}\in[-0.45,-0.1]$, keeping candidates that satisfy the convergence condition, and fixed the criterion before measuring: at least 90% of the points should agree between the full-region right-hand side of the relation and the direct closed-string integral to better than 1e-5, with every miss reported. Result at the default quadrature: 33 of 40 points (82%) are below 1e-5, with median 1.3e-7 and maximum 5.7e-3, so the criterion is not met as stated. We examined the seven misses. Five of them (all with $c_1\approx-0.8$, close to the edge of the sampled box) have differences of 2e-5 to 4e-5 at the default nodes and 2.0e-7 to 3.5e-7 at the finest node level used above (301, 241, 241 nodes), while both integrals change by about 1e-5 between the levels; we read these as quadrature error. The remaining two points, $(c_1,c_2,c_3)=(-0.34,-0.73,-0.52)$ with a difference of 5.7e-3 and $(-0.32,-0.31,-0.45)$ with 5.9e-5, could not be resolved: at the finest level the closed-string integral changes by a factor of 2e14 and 1e16 relative to the default level (our working definition of unstable: a change of more than a factor of ten between the two levels, where convergent points change by at most 3e-5), so the closed-string quadrature is unstable there, and we cannot say whether the relation holds to better than the default-node difference at these two points. For the half-region version, the relative difference is between 1.7e-2 and 0.60 at all 40 points, so the dependence on the $\eta<-1$ pieces is not specific to the hand-chosen points. The sample is uniform in a box and says nothing about the neighbourhood of the boundary of the convergence region.

Numerics. Everything is implemented in Python with numpy. The three-fold open-string integrals and the direct closed-string integral use tensor-product quadrature; for the open-string pieces we used $n_x=151$, $n_\xi=121$, $n_\eta=121$ nodes with map parameters $t_x=4.0$, $t_\xi=3.6$, $t_\eta=3.6$, and repeated four points at three node levels to monitor convergence. No random numbers are used. The rank of the relation matrix is the number of singular values above the threshold in a singular value decomposition: the 54th singular value is 0.28 and the 55th is 4e-16. The code is not released with this note; the integrals above, with the exponents listed in Section 2, specify the computation.

4. The region of two partial amplitudes

In [S] the partial amplitudes $A(1,3,4,6,2,5)$ and $A(1,4,3,6,2,5)$ are given over $\eta\in(1,\infty)$ only. The ordering $(1,3,4,6,2,5)$ is cyclically $(5,1,3,4,6,2)$; on the slice above, $z_5$ lies either to the right of $z_2$ ($\eta>1$) or to the left of $z_1$ ($\eta<-1$), and the invariant partial amplitude is the sum of both pieces. The missing piece follows by $\eta=-\eta'$, $\eta'>1$, which turns the $\eta$-dependent factors into $(\eta'+1)^{\lambda}(\eta'-1)^{\gamma}(\eta'+x)^{\beta}(\eta'-x)^{\epsilon}(\eta'-\xi)^{\kappa}$. Taking the two partial amplitudes literally over $\eta>1$ leaves relative differences of 9e-2 to 3e-1 in check (2) at the eight points; adding the $\eta<-1$ pieces gives Table 1.

The following is our interpretation and is not established by the source. If $A$ denotes the invariant partial amplitude, the stated region is incomplete. The mismatch is carried entirely by the two terms with coefficient $\sin(\pi\gamma)$, since the other four terms of the relation do not involve the missing pieces. Writing $w_Q,w_{Q'}$ for the $\eta<-1$ pieces, the half-region value differs from the full one by $2^{c_2}\sin(\pi\gamma)(w_Q+w_{Q'})$. At the eight points this missing contribution, divided by $|\mathcal A|$, equals the half-region relative difference in every case (Table 2: 0.132, 0.263, 0.087, 0.156, 0.170, 0.114, 0.093, 0.168 for the eight rows in order), and each of the six terms of the full relation is of size 0.05–0.29 in units of $|\mathcal A|$, so no single term is negligible. We also do not know whether the appendix expression for $W$ itself omits the $\eta<-1$ contribution: our implementation of the appendix combination agrees with its half-region version at the level of 1e-9, so the check in Table 1 is a check of the relation with our reading of the region, not of the relation as printed.

Table 2. Size of each of the six terms (full region) divided by $|\mathcal A|$, same eight points. P, P′, Q, Q′, R, R′ are the terms with $A(1,3,6,4,5,2)$, $A(1,4,6,3,5,2)$, $A(1,3,4,6,2,5)$, $A(1,4,3,6,2,5)$, $A(1,3,4,6,5,2)$, $A(1,4,3,6,5,2)$. The $\eta<-1$ pieces enter only through Q and Q′.

5. Limitations

This is a numerical check at eight hand-chosen and 40 random points, all inside the region of convergence, and not a proof; two of the 40 points are unresolved because the closed-string quadrature is unstable under refinement there. Neither the neighbourhood of $c_i\to-1$ nor low energies $c_i\approx2$ have been tested for the monodromy relations themselves. There is no second independent implementation; the direct closed-string integral is independent of the other two routes, which share code for the open-string integrals. The three-closed-string amplitude of [S] is symmetrised over the sign of the transverse momentum, so it contains no massless open-string pole and cannot be used as it stands to read off the cubic coupling; we therefore did not obtain that coupling. The relation of this check to the direct disk computation of Bischof, Haack and Stieberger is not established here.

References

• [S] S. Stieberger, arXiv:0907.2211.
• [G1] M. R. Garousi, arXiv:hep-th/9901085.
• [G2] M. R. Garousi, arXiv:2608.10667.
• [B] Bischof, Haack, Stieberger, arXiv:2308.04175.

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