Introduction
The correlation functions of primordial density perturbations offer one of the most direct windows onto the physics of inflation [1 ] . Surveys of the cosmic microwave background and of the large-scale distribution of galaxies measure these correlations with ever-increasing precision [2 , 3 , 4 ] , and future observations of the 21-cm line of neutral hydrogen could access far more modes [5 , 6 ] . They may also carry the imprint of heavy particles present during inflation, as in the cosmological collider program [7 , 8 , 9 ] . The theoretical challenge is to predict them. At tree level and in the simplest models this is now well understood, but beyond the lowest orders of perturbation theory [10 , 11 ] the relevant integrals have largely resisted computation. At loop level, even the class of functions in which the answer lives remains mostly unknown.
This last question is not merely a technical one. In the study of scattering amplitudes, knowing the function space has proven to be enormously powerful. Once the relevant class of functions is known, an amplitude can often be constructed directly from its singularities and symmetries, without evaluating any integral at all [12 ] . These classes have been mapped out in considerable detail: Feynman integrals evaluate to polylogarithms in the simplest cases [13 ] , and to integrals over elliptic curves or higher-dimensional varieties in others [14 ] . The cosmological bootstrap [15 , 16 ] aims to construct cosmological correlators in the same way, and it requires precisely the same input. Here the natural analog of the scattering amplitude is the wavefunction of the universe [1 , 17 ] . Its coefficients (one for each graph of perturbation theory) encode the correlations of the fields at the end of inflation, and equal-time correlators follow from them. The residue of a coefficient at its total-energy pole is the flat-space scattering amplitude [15 , 18 ] , and in de Sitter space the wavefunction plays the role of the partition function of a possible dual theory on the late-time boundary [1 ] . Over the past decade, the computation of these coefficients has increasingly drawn on tools from the amplitudes program, including canonical differential equations [19 ] , integration-by-parts reduction [20 , 21 ] , and twisted cohomology [22 , 23 ] .
At tree level, the function space is by now well established for conformally coupled scalars in both de Sitter space and power-law cosmologies. The flat-space wavefunction coefficient is the canonical form of a cosmological polytope [18 ] , a rational function with simple poles wherever a partial energy vanishes. The corresponding coefficients in de Sitter space are polylogarithms, while those in power-law cosmologies are integrals of the same form with non-integer powers of the energies. All of them satisfy differential equations of a universal combinatorial form, known as kinematic flow [23 , 24 , 25 , 26 , 27 ] .
Much less is known at loop level. The simplest one-loop graph, with two vertices (or sites), is polylogarithmic [28 , 29 , 30 ] . For the triangle, ref. [28 ] found that a subset of the master integrals obeys an irreducible second-order differential equation, whose solutions are elliptic integrals. This raises the possibility that elliptic functions enter cosmological wavefunction coefficients already at one loop, in contrast to flat space, where they first appear at two loops, in the sunrise graph [31 , 32 , 33 ] .
There is good reason, however, to be cautious about drawing conclusions about an observable from the integrals used to compute it. Master integrals are a convenient basis, but the physical combination can be considerably simpler than its individual pieces, and several earlier results in cosmology are of precisely this kind. The one-loop four-point correlator of a conformally coupled scalar with a quartic interaction in de Sitter space is expressed in terms of single-valued polylogarithms, because an elliptic sector present in anti-de Sitter space cancels [34 ] . More recently, the triangle correlator itself has been computed. The flat-space triangle correlator is a combination of dilogarithms [35 ] , as is the triangle correlator of conformally coupled scalars in de Sitter space [36 ] . These results do not, however, determine the wavefunction coefficient, since correlators lack some of its singularities (those that come from the term of the propagator that imposes the late-time boundary condition) [37 , 38 ] . This leaves a natural question: is the triangle wavefunction coefficient itself elliptic, or do the elliptic pieces cancel?
In this paper, we answer this question by computing the one-loop triangle wavefunction coefficient of a conformally coupled scalar directly. To highlight some of our findings (summarized in Table 1 ):
Table 1. The function spaces of the one-loop polygons of a conformally coupled scalar, as established in this paper. For the triangle, the elliptic curves of the master integrals do not enter the wavefunction coefficient; for the box, they do.
The elliptic periods cancel, and the triangle coefficient is a polylogarithmic function of transcendental weight two. With one external leg at each site, it is the combination of three dilogarithms given in eq. (32) . With several legs per site (the case required by expanding backgrounds), it is given instead by eq. (34) , whose dilogarithms have arguments containing ten square roots, all of which become rational with one leg per site. Its counterparts in de Sitter space and power-law cosmologies follow as integrals of this result over the site energies.
The cancellation has a simple origin. The elliptic curve of the master integrals appears on three poles of the loop integrand. At these poles, the residue of the physical integrand is a flat-space amplitude, a rational function of the loop momentum, which does not produce elliptic periods. Individual master integrals contain odd powers of the edge energies, which are square roots in the loop momentum, and so they do see the curve. In their sum, the physical integrand, the square roots cancel and the periods with them (proposition 1). The maximal cuts of the master integrals therefore suggest a larger function space than the coefficient occupies.
The same three poles come from the term of the propagator that imposes the late-time boundary condition of the wavefunction. As noted above, such poles are absent from cosmological correlators [37 , 38 ] . In § 5 , we show that the integrand of the triangle correlator has no residue on any line where the loop measure defines an elliptic curve, answering, at the level
of the integrand, a question raised in ref. [28 ] . The same holds for the box and the pentagon in d = N d = N d = N d = 3 d = 3 d = 3
The box, by contrast, is genuinely not polylogarithmic, even with one leg per site. Here the residue of the loop integrand at the partial-energy pole of a single site involves two square roots rather than one, and together they define an elliptic curve. We give the discontinuity of the box at threshold in closed form in terms of elliptic integrals, and find that for momenta forming a regular tetrahedron the periods of a K 3 K3 K 3
Our approach combines exact information about the function space with high-precision numerics. This semi-numerical bootstrap approach [39 , 40 ] is well suited to automation, and many of the relevant tools have recently been packaged together in the BootLoops toolkit [41 ] . BootLoops builds upon the work of a community of mathematical physicists but is effectively run by a large language model. We used Claude (Anthropic) with BootLoops for many of the computations in this paper. To bootstrap the triangle coefficient, we derive an exact differential operator for it on a line in kinematic space and show that it factors completely into first-order operators, which fixes the function space. We then determine the remaining rational constants from numerical values of the loop integral (section 4). For general site energies, the symbol fixes the function space and numerical values again fix the constants (section 4.5). Away from the line, our closed forms rest on their agreement with numerical values of the integral, and for one leg per site we also derive the closed form directly from the integral (appendix C). This approach is complementary both to the cosmological bootstrap at loop level [42 , 43 , 44 , 45 , 46 ] and to the non-perturbative de Sitter bootstrap [47 ] , which constrain correlators through symmetries, singularities, and unitarity. Indeed, one of the lessons of this work is that the function space suggested by master integrals can overestimate the true function space of the observable, which is exactly the input a bootstrap requires.
The rest of the paper is organized as follows. In section 2, we define the loop integrand, the loop measure, and the kinematics, and show how coefficients in expanding backgrounds follow from the flat-space wavefunction coefficient. Section 3 describes the elliptic sector of the triangle, and section 4 shows that its periods cancel and gives the closed forms for the full coefficient. We turn to correlators in section 5 and to the box in section 6, and we conclude in section 7. Appendix A details the construction of the closed forms, appendix B collects the derivations for the box, and appendix C sketches a direct derivation of eq. (32) from the loop integral.
Notation and conventions. Following the literature, we refer to the vertices of a graph as sites. We call the one-loop graphs with three and four sites the triangle and the box, and more generally the one-loop graph with N N N
Setup
In this section, we define the wavefunction coefficient V V V
Wavefunction coefficients and correlators
The late-time state of the fields is described by the wavefunction Ψ [ ϕ ] \Psi[\phi] Ψ [ ϕ ] ϕ k ⃗ \phi_{\vec{k}} ϕ k η 0 → 0 \eta_0 \to0 η 0 → 0
ln Ψ [ ϕ ] = − ∑ n 1 n ! ∫ ∏ i = 1 n d 3 k i ( 2 π ) 3 ψ n ( k ⃗ 1 , … , k ⃗ n ) ϕ k ⃗ 1 ⋯ ϕ k ⃗ n ( 2 π ) 3 δ 3 ( ∑ i k ⃗ i ) , (1) \ln\Psi[\phi] = -\sum_n \frac{1}{n!} \int\prod_{i=1}^{n} \frac{\mathrm{d}^{3}k_i}{(2\pi)^3} \, \psi_n(\vec{k}_1,\ldots,\vec{k}_n)\phi_{\vec{k}_1}\cdots\phi_{\vec{k}_n}(2\pi)^3\delta^3\left(\sum_i \vec{k}_i\right), \tag*{(1)} ln Ψ [ ϕ ] = − n ∑ n ! 1 ∫ i = 1 ∏ n ( 2 π ) 3 d 3 k i ψ n ( k 1 , … , k n ) ϕ k 1 ⋯ ϕ k n ( 2 π ) 3 δ 3 ( i ∑ k i ) , (1) defines the wavefunction coefficients ψ n \psi_n ψ n
⟨ ϕ k ⃗ 1 ⋯ ϕ k ⃗ n ⟩ = ∫ D ϕ ∣ Ψ [ ϕ ] ∣ 2 ϕ k ⃗ 1 ⋯ ϕ k ⃗ n ∫ D ϕ ∣ Ψ [ ϕ ] ∣ 2 . (2) \langle\phi_{\vec{k}_1}\cdots\phi_{\vec{k}_n}\rangle= \frac{\int\mathcal{D}\phi\,|\Psi[\phi]|^2\phi_{\vec{k}_1}\cdots\phi_{\vec{k}_n}}{\int\mathcal{D}\phi\,|\Psi[\phi]|^2}. \tag*{(2)} ⟨ ϕ k 1 ⋯ ϕ k n ⟩ = ∫ D ϕ ∣Ψ [ ϕ ] ∣ 2 ∫ D ϕ ∣Ψ [ ϕ ] ∣ 2 ϕ k 1 ⋯ ϕ k n . (2) Although wavefunction coefficients and correlators contain the same information, they are different functions of the momenta. Sections 2 to 4 concern a wavefunction coefficient, while in section 5 we show that the integrand of the corresponding correlator has fewer singularities.
A key simplification is that a wavefunction coefficient in an expanding background is an integral of the flat-space wavefunction coefficient over energies. To see this, consider a conformally coupled scalar with polynomial self-interactions in a spatially flat Friedmann–Robertson–Walker (FRW) background. After rescaling the field by the scale factor, its free action is that of a massless field in flat space, and the scale factor enters only through a factor ( − η ) − ( 1 + σ ) (-\eta)^{-(1+\sigma)} ( − η ) − ( 1 + σ ) σ \sigma σ σ = 0 \sigma=0 σ = 0 η = i t \eta=it η = i t t > 0 t>0 t > 0 X X X e − X t e^{-Xt} e − X t t − ( 1 + σ ) t^{-(1+\sigma)} t − ( 1 + σ )
t − ( 1 + σ ) = 1 Γ ( 1 + σ ) ∫ 0 ∞ d ω ω σ e − ω t , (3) t^{-(1+\sigma)} = \frac{1}{\Gamma(1+\sigma)}\int_0^\infty\mathrm{d}\omega\,\omega^\sigma e^{-\omega t}, \tag*{(3)} t − ( 1 + σ ) = Γ ( 1 + σ ) 1 ∫ 0 ∞ d ω ω σ e − ω t , (3) the factor e − ω t e^{-\omega t} e − ω t e − X t e^{-Xt} e − X t X X X X + ω X+\omega X + ω [18 , 26 ] . A coefficient in the expanding background is therefore an integral of the flat-space wavefunction coefficient over shifted energies,
ψ n ( σ ) ( X ; y ) ∝ ∫ 0 ∞ ∏ v d ω v ω v σ ψ n ( X + ω ; y ) , (4) \psi_n^{(\sigma)}(X;y) \propto\int_0^\infty\prod_v d\omega_v\,\omega_v^\sigma\,\psi_n(X+\omega;y), \tag*{(4)} ψ n ( σ ) ( X ; y ) ∝ ∫ 0 ∞ v ∏ d ω v ω v σ ψ n ( X + ω ; y ) , (4) where ψ n \psi_n ψ n X v X_v X v v v v y e y_e y e e e e σ = − 1 \sigma=-1 σ = − 1 (3) , valid for σ > − 1 \sigma>-1 σ > − 1 ω σ / Γ ( 1 + σ ) → δ ( ω ) \omega^\sigma/\Gamma(1+\sigma)\to\delta(\omega) ω σ /Γ ( 1 + σ ) → δ ( ω ) ψ n ( σ ) \psi_n^{(\sigma)} ψ n ( σ ) ψ n \psi_n ψ n ψ n \psi_n ψ n (4) for de Sitter space in section 4.6.
The loop integrand
The flat-space wavefunction coefficients are integrals over the times t v > 0 t_v>0 t v > 0 e − X v t v e^{-X_vt_v} e − X v t v
G y ( t , t ′ ) = 1 2 y [ e − y ∣ t − t ′ ∣ − e − y ( t + t ′ ) ] . (5) G_y(t,t')=\frac{1}{2y}\left[e^{-y|t-t'|}-e^{-y(t+t')}\right]. \tag*{(5)} G y ( t , t ′ ) = 2 y 1 [ e − y ∣ t − t ′ ∣ − e − y ( t + t ′ ) ] . (5) We omit the coupling constants and an overall phase, so that every ψ n \psi_n ψ n (5) is simply the time-ordered propagator of flat space. The second term, by contrast, is fixed by the boundary condition at t = 0 t=0 t = 0 t t t t ′ t' t ′
At tree level, every factor in the denominator is a partial energy, which we now define. From here on, the subscript of ψ \psi ψ ψ 1 = 1 / X \psi_1=1/X ψ 1 = 1/ X
ψ 2 = ∫ 0 ∞ d t 1 d t 2 e − X 1 t 1 − X 2 t 2 G y ( t 1 , t 2 ) = 1 ( X 1 + X 2 ) ( X 1 + y ) ( X 2 + y ) . (6) \psi_2=\int_0^\infty dt_1\,dt_2\,e^{-X_1t_1-X_2t_2}G_y(t_1,t_2)=\frac{1}{(X_1+X_2)(X_1+y)(X_2+y)}. \tag*{(6)} ψ 2 = ∫ 0 ∞ d t 1 d t 2 e − X 1 t 1 − X 2 t 2 G y ( t 1 , t 2 ) = ( X 1 + X 2 ) ( X 1 + y ) ( X 2 + y ) 1 . (6) Each factor in the denominator is the sum of the X v X_v X v y e y_e y e [18 ] . Because the boundary breaks time translations, energy is not conserved, and the coefficient is instead singular wherever a partial energy vanishes.
We now turn to the one-loop triangle, the graph with three sites and three edges shown in Figure 1 (a). Its sites have energies X 1 X_1 X 1 X 2 X_2 X 2 X 3 X_3 X 3 v v v w w w y v w y_{vw} y v w
Figure 1. (a) The one-loop triangle and (b) the one-loop box. Each site has the energy X v X_v X v y e y_e y e P ⃗ v \vec{P}_v P v ℓ ⃗ \vec{\ell} ℓ ℓ ⃗ \vec{\ell} ℓ
ψ 3 ( X ; y ) = 2 N ( X ; y ) q G ∏ i = 1 9 q i ( X ; y ) , q G = X 1 + X 2 + X 3 , (7) \psi_3(X;y)=\frac{2\mathcal{N}(X;y)}{q_G\prod_{i=1}^{9}q_i(X;y)},\qquad q_G=X_1+X_2+X_3, \tag*{(7)} ψ 3 ( X ; y ) = q G ∏ i = 1 9 q i ( X ; y ) 2 N ( X ; y ) , q G = X 1 + X 2 + X 3 , (7) with ten linear factors in the denominator. In addition to the total energy q G q_G q G q 1 , … , q 9 q_1,\ldots,q_9 q 1 , … , q 9
X 1 + y 12 + y 31 , X 2 + y 12 + y 23 , X 3 + y 23 + y 31 , X 1 + X 2 + y 23 + y 31 , X 2 + X 3 + y 12 + y 31 , X 1 + X 3 + y 12 + y 23 , q G + 2 y 12 , q G + 2 y 23 , q G + 2 y 31 . (8) \begin{aligned}
&X_1+y_{12}+y_{31},\qquad X_2+y_{12}+y_{23},\qquad X_3+y_{23}+y_{31},\\
&X_1+X_2+y_{23}+y_{31},\qquad X_2+X_3+y_{12}+y_{31},\qquad X_1+X_3+y_{12}+y_{23},\\
&q_G+2y_{12},\qquad q_G+2y_{23},\qquad q_G+2y_{31}. \tag*{(8)}
\end{aligned} X 1 + y 12 + y 31 , X 2 + y 12 + y 23 , X 3 + y 23 + y 31 , X 1 + X 2 + y 23 + y 31 , X 2 + X 3 + y 12 + y 31 , X 1 + X 3 + y 12 + y 23 , q G + 2 y 12 , q G + 2 y 23 , q G + 2 y 31 . (8) The numerator N \mathcal{N} N ψ 3 ( X ; y ) \psi_3(X;y) ψ 3 ( X ; y ) [18 ] ,
q G ψ 3 ( X ; y ) = ψ c h a i n ( X 2 + y 12 , X 3 , X 1 + y 12 ; y 23 , y 31 ) + ψ c h a i n ( X 3 + y 23 , X 1 , X 2 + y 23 ; y 31 , y 12 ) + ψ c h a i n ( X 1 + y 31 , X 2 , X 3 + y 31 ; y 12 , y 23 ) . (9) q_G \psi_3(X;y) = \psi_{\mathrm{chain}}(X_2+y_{12},X_3,X_1+y_{12};y_{23},y_{31}) \\
+ \psi_{\mathrm{chain}}(X_3+y_{23},X_1,X_2+y_{23};y_{31},y_{12}) \\
+ \psi_{\mathrm{chain}}(X_1+y_{31},X_2,X_3+y_{31};y_{12},y_{23}) \, . \tag*{(9)} q G ψ 3 ( X ; y ) = ψ chain ( X 2 + y 12 , X 3 , X 1 + y 12 ; y 23 , y 31 ) + ψ chain ( X 3 + y 23 , X 1 , X 2 + y 23 ; y 31 , y 12 ) + ψ chain ( X 1 + y 31 , X 2 , X 3 + y 31 ; y 12 , y 23 ) . (9) where each term is the coefficient of the chain in which one edge e e e y e y_e y e x 1 , x 2 , x 3 x_1,x_2,x_3 x 1 , x 2 , x 3 y , y ′ y,y' y , y ′
ψ c h a i n ( x 1 , x 2 , x 3 ; y , y ′ ) = ( x 1 + x 2 + y ′ ) − 1 + ( x 2 + x 3 + y ) − 1 ( x 1 + x 2 + x 3 ) ( x 1 + y ) ( x 2 + y + y ′ ) ( x 3 + y ′ ) . (10) \psi_{\mathrm{chain}}(x_1,x_2,x_3;y,y') =
\frac{(x_1+x_2+y')^{-1}+(x_2+x_3+y)^{-1}}
{(x_1+x_2+x_3)(x_1+y)(x_2+y+y')(x_3+y')} \, . \tag*{(10)} ψ chain ( x 1 , x 2 , x 3 ; y , y ′ ) = ( x 1 + x 2 + x 3 ) ( x 1 + y ) ( x 2 + y + y ′ ) ( x 3 + y ′ ) ( x 1 + x 2 + y ′ ) − 1 + ( x 2 + x 3 + y ) − 1 . (10) The total energy and the first two rows of eq. (8) are partial energies (of the whole graph, of a single site, and of a pair of sites, respectively), but the factors in the last row are not. Instead, the factor q G + 2 y e q_G+2y_e q G + 2 y e e e e e e e q G + 2 y e q_G+2y_e q G + 2 y e
The loop measure
The edge energies depend on the loop momentum ℓ ⃗ \vec{\ell} ℓ y 12 = ∣ ℓ ⃗ ∣ y_{12}=|\vec{\ell}| y 12 = ∣ ℓ ∣ y 23 = ∣ ℓ ⃗ + P ⃗ 2 ∣ y_{23}=|\vec{\ell}+\vec{P}_2| y 23 = ∣ ℓ + P 2 ∣ y 31 = ∣ ℓ ⃗ − P ⃗ 1 ∣ y_{31}=|\vec{\ell}-\vec{P}_1| y 31 = ∣ ℓ − P 1 ∣ P ⃗ v \vec{P}_v P v v v v P ⃗ 1 + P ⃗ 2 + P ⃗ 3 = 0 \vec{P}_1+\vec{P}_2+\vec{P}_3=0 P 1 + P 2 + P 3 = 0 ℓ ⃗ \vec{\ell} ℓ ℓ ⃗ \vec{\ell} ℓ Figure 1 (c)). In three dimensions, three distances fix a point up to reflection through the plane of the triangle, and for each of the two points the Jacobian is y 12 y 23 y 31 / ( 6 Vol ) y_{12}y_{23}y_{31}/(6\operatorname{Vol}) y 12 y 23 y 31 / ( 6 Vol ) Vol \operatorname{Vol} Vol ℓ ⃗ \vec{\ell} ℓ 288 Vol 2 = B 288\operatorname{Vol}^{2}=B 288 Vol 2 = B
d 3 ℓ = 4 2 B ( y ; P ) − 1 / 2 y 12 y 23 y 31 d y 12 d y 23 d y 31 (11) \mathrm{d}^{3}\ell=4\sqrt{2}\,B(y;P)^{-1/2}y_{12}y_{23}y_{31}\,\mathrm{d}y_{12}\,\mathrm{d}y_{23}\,\mathrm{d}y_{31} \tag*{(11)} d 3 ℓ = 4 2 B ( y ; P ) − 1/2 y 12 y 23 y 31 d y 12 d y 23 d y 31 (11) where B B B
B ( y ; P ) = det ( 0 1 1 1 1 1 0 P 2 2 P 1 2 y 12 2 1 P 2 2 0 P 3 2 y 23 2 1 P 1 2 P 3 2 0 y 31 2 1 y 12 2 y 23 2 y 31 2 0 ) (12) B(y;P)=\det\begin{pmatrix}
0 & 1 & 1 & 1 & 1\\
1 & 0 & P_{2}^{2} & P_{1}^{2} & y_{12}^{2}\\
1 & P_{2}^{2} & 0 & P_{3}^{2} & y_{23}^{2}\\
1 & P_{1}^{2} & P_{3}^{2} & 0 & y_{31}^{2}\\
1 & y_{12}^{2} & y_{23}^{2} & y_{31}^{2} & 0
\end{pmatrix} \tag*{(12)} B ( y ; P ) = det 0 1 1 1 1 1 0 P 2 2 P 1 2 y 12 2 1 P 2 2 0 P 3 2 y 23 2 1 P 1 2 P 3 2 0 y 31 2 1 y 12 2 y 23 2 y 31 2 0 (12) and P v = ∣ P ⃗ v ∣ P_v=\lvert\vec{P}_v\rvert P v = ∣ P v ∣ B > 0 B>0 B > 0 B B B y e 2 y_{e}^{2} y e 2 B B B [28 ] studied the triangle.
The main object of this paper is the integral of the loop integrand over the loop momentum, that is, the one-loop wavefunction coefficient of the triangle. We normalize it as
V ( X ; P ) = ∫ B > 0 B ( y ; P ) − 1 / 2 + ε y 12 y 23 y 31 q G ψ 3 ( X ; y ) d y 12 d y 23 d y 31 → ε → 0 q G 4 2 ∫ d 3 ℓ ψ 3 (13) V(X;P)=\int_{B>0}B(y;P)^{-1/2+\varepsilon}y_{12}y_{23}y_{31}\,q_G\psi_3(X;y)\,\mathrm{d}y_{12}\,\mathrm{d}y_{23}\,\mathrm{d}y_{31}\xrightarrow{\varepsilon\to0}\frac{q_G}{4\sqrt{2}}\int\mathrm{d}^{3}\ell\,\psi_3 \tag*{(13)} V ( X ; P ) = ∫ B > 0 B ( y ; P ) − 1/2 + ε y 12 y 23 y 31 q G ψ 3 ( X ; y ) d y 12 d y 23 d y 31 ε → 0 4 2 q G ∫ d 3 ℓ ψ 3 (13) The factor q G q_G q G ∣ ℓ ⃗ ∣ − 5 \lvert\vec{\ell}\rvert^{-5} ∣ ℓ ∣ − 5 V V V ε \varepsilon ε d = 3 + 2 ε d=3+2\varepsilon d = 3 + 2 ε § 3 have poles at ε = 0 \varepsilon=0 ε = 0 V V V B − 1 / 2 B^{-1/2} B − 1/2 § 3 requires both this square root and the second term of eq. (5) .
Kinematics
The coefficient V V V X v X_v X v P v P_v P v
X v = P v , v = 1 , 2 , 3. (14) X_v=P_v,\qquad v=1,2,3. \tag*{(14)} X v = P v , v = 1 , 2 , 3. (14) With several legs at a site, on the other hand, X v X_v X v P v P_v P v
On a line in kinematic space, the coefficient depends on a single variable and obeys an ordinary differential equation. Following ref. [28 ] , which studies the differential equations along lines of the form X = ( α λ , λ , 1 ) X=(\alpha\lambda,\lambda,1) X = ( α λ , λ , 1 ) α = 2 \alpha=2 α = 2
X = P = ( 2 λ , λ , 1 ) , 1 3 < λ < 1 , (15) X=P=(2\lambda,\lambda,1),\qquad\frac{1}{3}<\lambda<1, \tag*{(15)} X = P = ( 2 λ , λ , 1 ) , 3 1 < λ < 1 , (15) where the range of λ \lambda λ α \alpha α § 3.3 at α = 3 \alpha=3 α = 3 [28 ] . The differential equations of § 3 and of § 4.2 –§ 4.4 are derived on this line; § 4.5 uses a second line with X v > P v X_v>P_v X v > P v (4) shift X v X_v X v P v P_v P v V V V X v > P v X_v>P_v X v > P v § 4.5 .
The elliptic sector
In this section, we describe the elliptic curve of the triangle and the integrals in which it appears. After expressing V V V V V V § 4 .
Master integrals
The integrand of eq. (13) is a polynomial divided by a product of linear factors and multiplied by a power of B B B (13) therefore belongs to the family
I [ Q ; n ] = ∫ B > 0 B ( y ; P ) − 1 / 2 + ε Q ( y ) ∏ i = 1 9 q i ( X ; y ) n i d 3 y (16) I[Q;n]=\int_{B>0} B(y;P)^{-1/2+\varepsilon}\frac{Q(y)}{\prod_{i=1}^{9}q_i(X;y)^{n_i}}\,\mathrm{d}^{3}y \tag*{(16)} I [ Q ; n ] = ∫ B > 0 B ( y ; P ) − 1/2 + ε ∏ i = 1 9 q i ( X ; y ) n i Q ( y ) d 3 y (16) where Q Q Q q i q_i q i (8) , and the n i n_i n i B > 0 B>0 B > 0 ε \varepsilon ε q i q_i q i λ \lambda λ
The sectors built only from the first six factors of eq. (8) produce polylogarithms on the line (15) , but the sectors with a factor q G + 2 y e q_G+2y_e q G + 2 y e [28 ] found that the sector with the single denominator q G + 2 y 12 q_G+2y_{12} q G + 2 y 12 q G + 2 y 23 q_G+2y_{23} q G + 2 y 23 q G + 2 y 31 q_G+2y_{31} q G + 2 y 31 § 3.2 to § 3.4 treat the sector with denominator q G + 2 y 12 q_G+2y_{12} q G + 2 y 12
The curve
The elliptic curve can be read off directly from the loop measure. Recall that an integrand containing the square root of a polynomial p ( x ) p(x) p ( x ) w 2 = p ( x ) w^2=p(x) w 2 = p ( x ) p p p p p p
Since the residue of the integrand at the pole of a sector obeys the homogeneous equation of that sector, in the sector with denominator q G + 2 y 12 q_G+2y_{12} q G + 2 y 12 y 12 = − q G / 2 y_{12}=-q_G/2 y 12 = − q G /2 y 23 y_{23} y 23 y 31 y_{31} y 31 B B B z e = y e 2 z_e=y_e^2 z e = y e 2 B B B z 31 = r z 23 z_{31}=r z_{23} z 31 = r z 23 z 23 z_{23} z 23
B = − A ( r ) z 23 2 + O ( z 23 ) , A ( r ) = 2 [ P 2 2 r 2 − ( P 1 2 + P 2 2 − P 3 2 ) r + P 1 2 ] . (17) B=-A(r)z_{23}^{2}+O(z_{23}),\qquad A(r)=2\left[P_2^2r^2-\left(P_1^2+P_2^2-P_3^2\right)r+P_1^2\right]. \tag*{(17)} B = − A ( r ) z 23 2 + O ( z 23 ) , A ( r ) = 2 [ P 2 2 r 2 − ( P 1 2 + P 2 2 − P 3 2 ) r + P 1 2 ] . (17) where A ( r ) A(r) A ( r ) y 12 y_{12} y 12 r r r A ( r ) A(r) A ( r ) y e 2 y_e^2 y e 2 B B B d y 23 d y 31 dy_{23}\,dy_{31} d y 23 d y 31 d z 23 d z 31 dz_{23}\,dz_{31} d z 23 d z 31 d y e = d z e / ( 2 z e ) dy_e=dz_e/(2\sqrt{z_e}) d y e = d z e / ( 2 z e ) z 23 z_{23} z 23
B − 1 / 2 d y 23 d y 31 ⟶ d z 23 z 23 d r 4 − r A ( r ) , d r r A ( r ) = 2 d τ A ( τ 2 ) , τ = y 31 y 23 . (18) B^{-1/2}\,dy_{23}\,dy_{31}\longrightarrow\frac{dz_{23}}{z_{23}}\frac{dr}{4\sqrt{-rA(r)}},\qquad\frac{dr}{\sqrt{rA(r)}}=\frac{2\,d\tau}{\sqrt{A(\tau^2)}},\qquad\tau=\frac{y_{31}}{y_{23}}. \tag*{(18)} B − 1/2 d y 23 d y 31 ⟶ z 23 d z 23 4 − r A ( r ) d r , r A ( r ) d r = A ( τ 2 ) 2 d τ , τ = y 23 y 31 . (18) The residue at large z 23 z_{23} z 23 τ \tau τ A ( τ 2 ) A(\tau^2) A ( τ 2 )
w 2 = P 2 2 τ 4 − ( P 1 2 + P 2 2 − P 3 2 ) τ 2 + P 1 2 . (19) w^2=P_2^2\tau^4-\left(P_1^2+P_2^2-P_3^2\right)\tau^2+P_1^2. \tag*{(19)} w 2 = P 2 2 τ 4 − ( P 1 2 + P 2 2 − P 3 2 ) τ 2 + P 1 2 . (19) Since the discriminant of A ( r ) A(r) A ( r ) P v 2 P_v^2 P v 2 P v P_v P v X v X_v X v q G + 2 y 12 = 0 q_G+2y_{12}=0 q G + 2 y 12 = 0 y 23 y_{23} y 23 y 31 y_{31} y 31 y 12 y_{12} y 12 y 23 y_{23} y 23 y 31 y_{31} y 31
On the line (15) , the curve is isomorphic to w 2 = x ( 1 − x ) ( 1 − m x ) w^2=x(1-x)(1-mx) w 2 = x ( 1 − x ) ( 1 − m x ) m m m
m ( λ ) = λ 2 − 1 9 λ 2 − 1 . (20) m(\lambda)=\frac{\lambda^2-1}{9\lambda^2-1}. \tag*{(20)} m ( λ ) = 9 λ 2 − 1 λ 2 − 1 . (20) in agreement with ref. [28 ] . The sectors with denominators q G + 2 y 23 q_G+2y_{23} q G + 2 y 23 q G + 2 y 31 q_G+2y_{31} q G + 2 y 31
Periods and their differential equation
The homogeneous equation of the elliptic sector is the differential equation obeyed by the periods, whose operator is known as the Picard–Fuchs operator. On the line (15) , it is given by [28 ]
L 2 = d 2 d λ 2 + 45 λ 4 − 30 λ 2 + 1 λ ( λ 2 − 1 ) ( 9 λ 2 − 1 ) d d λ + 27 λ 2 − 10 ( λ 2 − 1 ) ( 9 λ 2 − 1 ) . (21) \mathcal{L}_2=\frac{d^2}{d\lambda^2}+\frac{45\lambda^4-30\lambda^2+1}{\lambda(\lambda^2-1)(9\lambda^2-1)}\frac{d}{d\lambda}+\frac{27\lambda^2-10}{(\lambda^2-1)(9\lambda^2-1)}. \tag*{(21)} L 2 = d λ 2 d 2 + λ ( λ 2 − 1 ) ( 9 λ 2 − 1 ) 45 λ 4 − 30 λ 2 + 1 d λ d + ( λ 2 − 1 ) ( 9 λ 2 − 1 ) 27 λ 2 − 10 . (21) whose solutions are complete elliptic integrals of the first kind, K ( m ) = ∫ 0 π / 2 ( 1 − m sin 2 θ ) − 1 / 2 d θ K(m)=\int_0^{\pi/2}(1-m\sin^2\theta)^{-1/2}\,d\theta K ( m ) = ∫ 0 π /2 ( 1 − m sin 2 θ ) − 1/2 d θ 1 3 < λ < 1 \frac{1}{3}<\lambda<1 3 1 < λ < 1
ϖ 0 ( λ ) = 1 λ K ( 1 − λ 2 8 λ 2 ) , ϖ 1 ( λ ) = 1 λ K ( 9 λ 2 − 1 8 λ 2 ) . (22) \varpi_0(\lambda)=\frac{1}{\lambda}K\left(\frac{1-\lambda^2}{8\lambda^2}\right),\qquad\varpi_1(\lambda)=\frac{1}{\lambda}K\left(\frac{9\lambda^2-1}{8\lambda^2}\right). \tag*{(22)} ϖ 0 ( λ ) = λ 1 K ( 8 λ 2 1 − λ 2 ) , ϖ 1 ( λ ) = λ 1 K ( 8 λ 2 9 λ 2 − 1 ) . (22) where the argument of the first is m / ( m − 1 ) m/(m-1) m / ( m − 1 ) m ( λ ) m(\lambda) m ( λ ) (20) . The first period is analytic at λ = 1 \lambda=1 λ = 1 λ = 1 3 \lambda=\frac{1}{3} λ = 3 1 L 2 \mathcal{L}_2 L 2 λ = 0 , ± 1 3 , ± 1 \lambda=0,\pm\frac{1}{3},\pm1 λ = 0 , ± 3 1 , ± 1 ∞ \infty ∞ 1 3 < λ < 1 \frac{1}{3}<\lambda<1 3 1 < λ < 1 § 4 , since any function that contains the periods (22) inherits their logarithmic singularities.
We note that the operator L 2 \mathcal{L}_2 L 2 q G + 2 y 12 q_G+2y_{12} q G + 2 y 12 q G + y 12 q_G+y_{12} q G + y 12 y 12 y_{12} y 12 [28 ] writes q G + y 12 q_G+y_{12} q G + y 12 q G + 2 y 12 q_G+2y_{12} q G + 2 y 12
The master integrals of the elliptic sector
The master integrals of the elliptic sector can be computed using the methods developed for elliptic Feynman integrals [14 , 48 , 49 ] . On the line (15), the master integrals of the sector with denominator q G + 2 y 12 q_G+2y_{12} q G + 2 y 12 ε \varepsilon ε λ \lambda λ
The master integrals of the three elliptic sectors are therefore genuinely elliptic, and it was from this that ref. [28 ] concluded that the triangle coefficient involves elliptic functions. In the next section, we show that it does not.
The triangle wavefunction coefficient
We now show that the periods of the elliptic curve of § 3 do not, in fact, appear in the coefficient V V V (13) on the factors q G + 2 y e q_G+2y_e q G + 2 y e V V V § 4.4 gives V V V § 4.5 we extend the result to general site energies. Finally, we comment on de Sitter space in § 4.6 .
Why the periods cancel
The key observation is that at the pole q G + 2 y 12 = 0 q_G+2y_{12}=0 q G + 2 y 12 = 0 d τ / w \mathrm{d}\tau/w d τ / w y 12 y_{12} y 12 [18 ] , the residue of the loop integrand (7) is
Res y 12 = − q G / 2 ψ 3 = 1 2 q G ( y 23 2 − E 2 2 ) ( y 31 2 − E 1 2 ) , E 1 = 1 2 ( X 2 + X 3 − X 1 ) , E 2 = 1 2 ( X 1 + X 3 − X 2 ) . (23) \underset{y_{12}=-q_G/2}{\operatorname{Res}}\ \psi_3
=
\frac{1}{2q_G\left(y_{23}^{2}-E_2^{2}\right)\left(y_{31}^{2}-E_1^{2}\right)},
\qquad
E_1=\frac{1}{2}\left(X_2+X_3-X_1\right),
\qquad
E_2=\frac{1}{2}\left(X_1+X_3-X_2\right). \tag*{(23)} y 12 = − q G /2 Res ψ 3 = 2 q G ( y 23 2 − E 2 2 ) ( y 31 2 − E 1 2 ) 1 , E 1 = 2 1 ( X 2 + X 3 − X 1 ) , E 2 = 2 1 ( X 1 + X 3 − X 2 ) . (23) Equation (23) is simply a product of two Feynman propagators with energies E 1 E_1 E 1 E 2 E_2 E 2 y 23 y_{23} y 23 y 31 y_{31} y 31 ℓ \ell ℓ
μ = q G ψ 3 y 12 y 23 y 31 d y 12 d y 23 d y 31 (24) \mu=q_G\psi_3\,y_{12}y_{23}y_{31}\,\mathrm{d}y_{12}\,\mathrm{d}y_{23}\,\mathrm{d}y_{31} \tag*{(24)} μ = q G ψ 3 y 12 y 23 y 31 d y 12 d y 23 d y 31 (24) for the rational part of the integrand of eq. (13) , which has poles on the nine planes (8).
Proposition 1. For arbitrary X v X_v X v P v P_v P v (13) on a plane or on a line involves an elliptic curve. More precisely, the following statements hold.
The residue of μ \mu μ q G + 2 y 12 = 0 q_G+2y_{12}=0 q G + 2 y 12 = 0
− q G 16 d z 23 d z 31 ( z 23 − E 2 2 ) ( z 31 − E 1 2 ) , z e = y e 2 , (25) -\frac{q_G}{16}\frac{\mathrm{d}z_{23}\,\mathrm{d}z_{31}}{(z_{23}-E_2^2)(z_{31}-E_1^2)}, \qquad z_e=y_e^2, \tag*{(25)} − 16 q G ( z 23 − E 2 2 ) ( z 31 − E 1 2 ) d z 23 d z 31 , z e = y e 2 , (25) and the polynomial B B B z 23 z_{23} z 23 z 31 z_{31} z 31 q G + 2 y e = 0 q_G+2y_e=0 q G + 2 y e = 0
The nine planes meet in pairs along 36 lines. Three of the lines lie at infinity. On 12 lines B B B B B B X v X_v X v w 2 = B w^2=B w 2 = B X = P X=P X = P § 4.5 ). Each of these 21 lines lies in one of the planes q G + 2 y e = 0 q_G+2y_e=0 q G + 2 y e = 0
On 9 of the 21 lines the residue of μ \mu μ y y y B = b ( y 2 ) B=b(y^2) B = b ( y 2 ) b b b μ / B \mu/\sqrt{B} μ / B
± q G 16 d Z ( Z − E 2 ) b ( Z ) , Z = y 2 , (26) \pm\frac{q_G}{16}\frac{\mathrm{d}Z}{(Z-E^2)\sqrt{b(Z)}}, \qquad Z=y^2, \tag*{(26)} ± 16 q G ( Z − E 2 ) b ( Z ) d Z , Z = y 2 , (26) with E E E E i E_i E i B − 1 / 2 B^{-1/2} B − 1/2 B − 1 / 2 + ϵ B^{-1/2+\epsilon} B − 1/2 + ϵ
The form (26) is a differential on the curve w 2 = b ( Z ) w^2=b(Z) w 2 = b ( Z ) (13) has no residue, on any plane or line, that involves one of the 21 elliptic curves. By contrast, a master integral with a constant numerator and the same two denominators has the residue d y / b ( y 2 ) \mathrm{d}y/\sqrt{b(y^2)} d y / b ( y 2 ) § 3 , which lies at infinity on the plane q G + 2 y 12 = 0 q_G+2y_{12}=0 q G + 2 y 12 = 0 y 23 y_{23} y 23 y 31 y_{31} y 31 z 23 z_{23} z 23 z 31 z_{31} z 31 z 31 = r z 23 z_{31}=r z_{23} z 31 = r z 23 B − 1 / 2 B^{-1/2} B − 1/2 z 23 z_{23} z 23 r r r z 23 z_{23} z 23 A ( r ) \sqrt{A(r)} A ( r )
Proof. Part 1 is eq. (23) multiplied by the measure, with y 23 y 31 d y 23 d y 31 = 1 4 d z 23 d z 31 y_{23}y_{31}\,\mathrm{d}y_{23}\,\mathrm{d}y_{31}=\frac{1}{4}\,\mathrm{d}z_{23}\,\mathrm{d}z_{31} y 23 y 31 d y 23 d y 31 = 4 1 d z 23 d z 31 B B B B B B
Part 3 follows from part 1. The form (25) has poles only at y 23 = ± E 2 y_{23}=\pm E_2 y 23 = ± E 2 y 31 = ± E 1 y_{31}=\pm E_1 y 31 = ± E 1 q G + 2 y 12 = 0 q_G+2y_{12}=0 q G + 2 y 12 = 0 { 2 , 3 } \{2,3\} { 2 , 3 } { 1 , 3 } \{1,3\} { 1 , 3 } q G + 2 y 12 = 0 q_G+2y_{12}=0 q G + 2 y 12 = 0
It is instructive to see how the holomorphic differential appears in the individual master integrals and cancels in their sum. The decomposition of the residue (23) into partial fractions is
1 ( y 23 2 − E 2 2 ) ( y 31 2 − E 1 2 ) = 1 4 E 1 E 2 ∑ s , s ′ = ± 1 s s ′ ( y 23 − s E 2 ) ( y 31 − s ′ E 1 ) . (27) \frac{1}{(y_{23}^2-E_2^2)(y_{31}^2-E_1^2)}
=
\frac{1}{4E_1E_2}\sum_{s,s'=\pm1}\frac{ss'}{(y_{23}-sE_2)(y_{31}-s'E_1)}. \tag*{(27)} ( y 23 2 − E 2 2 ) ( y 31 2 − E 1 2 ) 1 = 4 E 1 E 2 1 s , s ′ = ± 1 ∑ ( y 23 − s E 2 ) ( y 31 − s ′ E 1 ) s s ′ . (27) Each of the four terms has a non-zero coefficient of the holomorphic differential d r / w \mathrm{d}r/w d r / w s s ′ ss' s s ′ V V V
The partial energies of the sites require a separate argument, which also illustrates where this simplification ends. At the pole where the partial energy of a single site vanishes, the residue of the loop integrand is
Res X 1 = − y 31 − y 12 ψ 3 = 1 4 y 12 y 31 ∑ s , s ′ = ± 1 s s ′ ψ 2 ( X 2 + s y 12 , X 3 + s ′ y 31 ; y 23 ) (28) \underset{X_{1}=-y_{31}-y_{12}}{\operatorname{Res}}\ \psi_{3}=\frac{1}{4y_{12}y_{31}}\sum_{s,s'=\pm1}ss'\,\psi_{2}\left(X_{2}+sy_{12},X_{3}+s'y_{31};y_{23}\right) \tag*{(28)} X 1 = − y 31 − y 12 Res ψ 3 = 4 y 12 y 31 1 s , s ′ = ± 1 ∑ s s ′ ψ 2 ( X 2 + s y 12 , X 3 + s ′ y 31 ; y 23 ) (28) with ψ 2 \psi_{2} ψ 2 (6) . The residue is even in y 12 y_{12} y 12 y 31 y_{31} y 31 y 23 y_{23} y 23 y 12 y_{12} y 12 y 31 y_{31} y 31 y 23 y_{23} y 23 N N N N − 2 N-2 N − 2 Figure 1 ). The box has two, and, as we will see in § 6 , they do define an elliptic curve.
Since proposition 1 holds for arbitrary X v X_v X v P v P_v P v § 4.5 ). It does not by itself determine the function space of V V V B > 0 B>0 B > 0
The differential equation
We reduce the integrand of eq. (13) to master integrals using the integration-by-parts identities of the family (16), as described in appendix § 9 .§ 1 . This expresses V V V λ \lambda λ V V V ϵ \epsilon ϵ ϵ → 0 \epsilon\to0 ϵ → 0
L 0 = ∑ i = 0 20 c i ( λ ) d i d λ i , L 0 V = 0 , (29) \mathcal{L}_{0}=\sum_{i=0}^{20}c_i(\lambda)\frac{\mathrm{d}^{i}}{\mathrm{d}\lambda^{i}},\qquad\mathcal{L}_{0}V=0, \tag*{(29)} L 0 = i = 0 ∑ 20 c i ( λ ) d λ i d i , L 0 V = 0 , (29) has polynomial coefficients c i c_i c i λ = 0 \lambda=0 λ = 0 ± 1 3 \pm\frac{1}{3} ± 3 1 ± 1 \pm1 ± 1 ∞ \infty ∞ L 2 \mathcal{L}_2 L 2
We emphasize that the limit ϵ → 0 \epsilon\to0 ϵ → 0 ϵ = 0 \epsilon=0 ϵ = 0 ϵ \epsilon ϵ § 9 .§ 1 ).
Complete factorization
The operator L 0 \mathcal{L}_0 L 0 λ \lambda λ
L 0 = c 20 ( λ ) ( d d λ − r 20 ′ r 20 ) ( d d λ − r 19 ′ r 19 ) ⋯ ( d d λ − r 1 ′ r 1 ) , (30) \mathcal{L}_0=c_{20}(\lambda)\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}-\frac{r'_{20}}{r_{20}}\right)\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}-\frac{r'_{19}}{r_{19}}\right)\cdots\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}-\frac{r'_1}{r_1}\right), \tag*{(30)} L 0 = c 20 ( λ ) ( d λ d − r 20 r 20 ′ ) ( d λ d − r 19 r 19 ′ ) ⋯ ( d λ d − r 1 r 1 ′ ) , (30) where every r k r_k r k λ \lambda λ d / d λ \mathrm{d}/\mathrm{d}\lambda d / d λ Q ( λ ) \mathbb{Q}(\lambda) Q ( λ )
The solutions of a product of first-order operators follow from a single integration for each factor,
V ( λ ) = r 1 ∫ d λ r 2 r 1 ∫ d λ r 3 r 2 ⋯ ∫ d λ r 20 r 19 , (31) V(\lambda)=r_1\int\mathrm{d}\lambda\,\frac{r_2}{r_1}\int\mathrm{d}\lambda\,\frac{r_3}{r_2}\cdots\int\mathrm{d}\lambda\,\frac{r_{20}}{r_{19}}, \tag*{(31)} V ( λ ) = r 1 ∫ d λ r 1 r 2 ∫ d λ r 2 r 3 ⋯ ∫ d λ r 19 r 20 , (31) with one constant of integration at each step. Since every integrand is a rational function, V ( λ ) V(\lambda) V ( λ ) (30) contains no irreducible factor of second order, which a period of an elliptic curve would require. On the line (15), this excludes the curve (19), its two images under permutations of the sites, and indeed any other elliptic curve. Moreover, the conclusion holds for every right factor of L 0 \mathcal{L}_0 L 0 L 0 \mathcal{L}_0 L 0 V V V
The numerical values of V V V L 2 \mathcal{L}_2 L 2 λ = 1 3 \lambda= \frac{1}{3} λ = 3 1 λ = 1 \lambda= 1 λ = 1 V V V Figure 2 ), as is the closed form of § 4.4 .
Figure 2. Chebyshev coefficients ∣ c n ∣ / ∣ c 0 ∣ |c_n|/|c_0| ∣ c n ∣/∣ c 0 ∣ 0.345 ≤ λ ≤ 0.985 0.345 \leq\lambda\leq0.985 0.345 ≤ λ ≤ 0.985 ϖ 0 \varpi_0 ϖ 0 ϖ 1 \varpi_1 ϖ 1 (22) (gray) have logarithmic singularities at λ = 1 3 \lambda= \frac{1}{3} λ = 3 1 λ = 1 \lambda= 1 λ = 1 V V V 10 − 34 10^{-34} 1 0 − 34 V V V n n n V + 10 − 20 ϖ 0 V + 10^{-20}\varpi_0 V + 1 0 − 20 ϖ 0
The closed form
On the line (15), the solutions of L 0 \mathcal{L}_0 L 0 L 0 g = 0 \mathcal{L}_0 g = 0 L 0 g = 0 G ( l 1 , … , l w ; λ ) G(l_1,\ldots,l_w;\lambda) G ( l 1 , … , l w ; λ ) l i ∈ { 0 , ± 1 , ± 1 3 } l_i \in\{0,\pm1,\pm\frac{1}{3}\} l i ∈ { 0 , ± 1 , ± 3 1 } λ \lambda λ V V V λ = 1 \lambda= 1 λ = 1 V V V § 9 .§ 3 ).
Written in terms of the site energies, the result on the line suggests the following formula for one leg per site and arbitrary momenta:
V = 2 π 8 [ ∑ i X i Li 2 ( u i ) ξ 1 ξ 2 ξ 3 + 8 X 1 X 2 X 3 q G ξ 1 2 ξ 2 2 ξ 3 2 W ] (32) V = \frac{\sqrt{2}\,\pi}{8}\left[\frac{\sum_i X_i\operatorname{Li}_2(u_i)}{\xi_1\xi_2\xi_3}+\frac{8X_1X_2X_3}{q_G\,\xi_1^2\xi_2^2\xi_3^2}\,W\right] \tag*{(32)} V = 8 2 π [ ξ 1 ξ 2 ξ 3 ∑ i X i Li 2 ( u i ) + q G ξ 1 2 ξ 2 2 ξ 3 2 8 X 1 X 2 X 3 W ] (32) W = ∑ i X i ξ i Li 2 ( u i ) + 2 ∑ i < j X i X j ln ( 1 − u i ) ln ( 1 − u j ) − π 2 12 ∑ i < j ξ i ξ j , ξ i = q G − 2 X i , u i = ξ i q G . (33) \mathcal{W} = \sum_i X_i \xi_i \operatorname{Li}_2(u_i) + 2\sum_{i<j} X_i X_j \ln(1-u_i)\ln(1-u_j) - \frac{\pi^2}{12}\sum_{i<j}\xi_i\xi_j,\qquad\xi_i=q_G-2X_i,\quad u_i=\frac{\xi_i}{q_G}. \tag*{(33)} W = i ∑ X i ξ i Li 2 ( u i ) + 2 i < j ∑ X i X j ln ( 1 − u i ) ln ( 1 − u j ) − 12 π 2 i < j ∑ ξ i ξ j , ξ i = q G − 2 X i , u i = q G ξ i . (33) On the line (15), the function space of eq. (32) is exact, while its constants are determined numerically. For other momenta, eq. (32) is an ansatz with 14 rational coefficients. The result on the line determines these 14 coefficients uniquely, and the formula then agrees with numerical values of the loop integral for other triangles of momenta (appendix A.4). Equation (32) can also be derived directly from the integral (13), without the differential equation or numerical values, by carrying out the three integrations over the edge energies, as we sketch in appendix C.
The variables ξ i \xi_i ξ i ξ i = 0 \xi_i=0 ξ i = 0 q G 2 V q_G^2 V q G 2 V
With one leg per site, the one-loop triangle coefficient is therefore a function of transcendental weight two built from three dilogarithms (here we count the weight of the bracket in eq. (32) , not its overall factor of π \pi π [50 ] , for the configuration in which one external momentum vanishes at arbitrary site energies. In that configuration the integration region reduces to that of the two-site graph, and the result is again a combination of dilogarithms.
General site energies
As discussed in section 2.4, the integrals of eq. (4) require V V V X v > P v X_v>P_v X v > P v
V ( X ; P ) = 2 π 8 Re [ − 1 4 q G R 0 F 0 − 1 2 ∑ e q G R e F e − ∑ i < j ρ i j − T i j F i j − + ∑ i < j ρ i j + T i j F i j + ] . (34) V(X;P)=\frac{\sqrt{2}\pi}{8}\operatorname{Re}\left[-\frac{1}{4}\frac{q_G}{\sqrt{R_0}}F_0-\frac{1}{2}\sum_e\frac{q_G}{\sqrt{R_e}}F_e-\sum_{i<j}\frac{\rho_{ij}^{-}}{\sqrt{T_{ij}}}F_{ij}^{-}+\sum_{i<j}\frac{\rho_{ij}^{+}}{\sqrt{T_{ij}}}F_{ij}^{+}\right]. \tag*{(34)} V ( X ; P ) = 8 2 π Re [ − 4 1 R 0 q G F 0 − 2 1 e ∑ R e q G F e − i < j ∑ T ij ρ ij − F ij − + i < j ∑ T ij ρ ij + F ij + ] . (34) Here R 0 R_0 R 0 − 2 B -2B − 2 B R e R_e R e − 2 B -2B − 2 B q G + 2 y e q_G+2y_e q G + 2 y e e e e
T i j ± = κ ( P ) + 4 P i 2 X j 2 + 4 P j 2 X i 2 ± 4 X i X j ( P i 2 + P j 2 − P k 2 ) , (35) T_{ij}^{\pm}=\kappa(P)+4P_i^2X_j^2+4P_j^2X_i^2\pm4X_iX_j(P_i^2+P_j^2-P_k^2), \tag*{(35)} T ij ± = κ ( P ) + 4 P i 2 X j 2 + 4 P j 2 X i 2 ± 4 X i X j ( P i 2 + P j 2 − P k 2 ) , (35) ρ i j + = X k X i + X j − X k , ρ i j − = X k q G ( X i − X j ) 2 − X k 2 . \rho_{ij}^{+}=\frac{X_k}{X_i+X_j-X_k},\qquad\rho_{ij}^{-}=\frac{X_kq_G}{(X_i-X_j)^2-X_k^2}. ρ ij + = X i + X j − X k X k , ρ ij − = ( X i − X j ) 2 − X k 2 X k q G . where κ ( P ) \kappa(P) κ ( P ) P v 2 P_v^2 P v 2 k k k F F F
D ( x , x ˉ ) = Li 2 ( x ) − Li 2 ( x ˉ ) + 1 2 ln ( x x ˉ ) ln ( 1 − x 1 − x ˉ ) , (36) \mathcal{D}(x,\bar{x})=\operatorname{Li}_2(x)-\operatorname{Li}_2(\bar{x})+\frac{1}{2}\ln(x\bar{x})\ln\left(\frac{1-x}{1-\bar{x}}\right), \tag*{(36)} D ( x , x ˉ ) = Li 2 ( x ) − Li 2 ( x ˉ ) + 2 1 ln ( x x ˉ ) ln ( 1 − x ˉ 1 − x ) , (36) where x ˉ \bar{x} x ˉ x x x x x x x ˉ \bar{x} x ˉ x ˉ \bar{x} x ˉ x x x D \mathcal{D} D 2 i 2i 2 i (34) is real. Since permutations of the sites relate the three F e F_e F e F i j + F^+_{ij} F ij + F i j − F^-_{ij} F ij −
The function F 0 F_0 F 0 P v P_v P v − y e -y_e − y e f k f_k f k c j c_j c j
P ( h ) = ∏ k = 1 4 ( h − f k ) − h ∏ j = 1 3 ( h − c j ) (37) \mathcal{P}(h) = \prod_{k=1}^{4}(h-f_k)-h\prod_{j=1}^{3}(h-c_j) \tag*{(37)} P ( h ) = k = 1 ∏ 4 ( h − f k ) − h j = 1 ∏ 3 ( h − c j ) (37) is a quadratic polynomial in the auxiliary variable h h h R 0 R_0 R 0 h ± h_\pm h ± P \mathcal{P} P D a = D ( 1 − a / h − , 1 − a / h + ) D_a=D(1-a/h_-,1-a/h_+) D a = D ( 1 − a / h − , 1 − a / h + )
F 0 = − 2 [ ∑ k = 1 4 D f k − ∑ j = 1 3 D c j ] . (38) F_0=-2\left[\sum_{k=1}^{4}D_{f_k}-\sum_{j=1}^{3}D_{c_j}\right]. \tag*{(38)} F 0 = − 2 [ k = 1 ∑ 4 D f k − j = 1 ∑ 3 D c j ] . (38) Strikingly, when R 0 < 0 R_0<0 R 0 < 0 (38) is proportional to the volume of an ideal hyperbolic tetrahedron. The function F e F_e F e F i j + F^+_{ij} F ij + F i j − F^-_{ij} F ij −
We now turn to how eq. (34) was obtained. On the second line, X = P + 1 3 ( 1 , 1 , 1 ) X=P+\frac{1}{3}(1,1,1) X = P + 3 1 ( 1 , 1 , 1 ) P = ( 2 λ , λ , 1 ) P=(2\lambda,\lambda,1) P = ( 2 λ , λ , 1 ) B B B X = P X=P X = P B B B
Once again, the periods of these curves do not enter the coefficient: on this line, its numerical values are reproduced by a sum of ten terms, each a polylogarithm of weight two divided by a square root. The ten square roots are those of eq. (34) , and all ten become perfect squares at X = P X=P X = P
To move to general kinematics, we use the symbol [13 ] . Recall that the symbol of a function of weight two is a sum of ordered pairs of letters, in which the first entry of each pair determines where the function has branch points; moreover, a sum of pairs is the symbol of a function only if it satisfies an integrability condition. We consider symbols whose first entries are the positive sums of energies that occur on the second line, with i i i j j j k k k
q G , X i + P i , X i + X j + P k , X i + P j + P k , P 1 + P 2 + P 3 . (39) q_{\mathcal{G}},\quad X_i+P_i,\quad X_i+X_j+P_k,\quad X_i+P_j+P_k,\quad P_1+P_2+P_3. \tag*{(39)} q G , X i + P i , X i + X j + P k , X i + P j + P k , P 1 + P 2 + P 3 . (39) The second entries are letters of the form ( p + R ) / ( p − R ) (p+\sqrt{R})/(p-\sqrt{R}) ( p + R ) / ( p − R ) R R R p p p p 2 − R p^2-R p 2 − R q G / R q_{\mathcal{G}}/\sqrt{R} q G / R ρ i j + / T i j + \rho^+_{ij}/\sqrt{T^+_{ij}} ρ ij + / T ij + − 1 4 -\frac{1}{4} − 4 1 − 1 2 -\frac{1}{2} − 2 1 − 1 -1 − 1 + 1 +1 + 1 (34) . These numbers also follow from the integrand itself: the part of the symbol whose first entry is X 1 + P 1 X_1+P_1 X 1 + P 1 (34) . The coefficients are residues of the integrand at the points and on the lines where its partial energies vanish together (appendix C.5). With these values, eq. (34) agrees with numerical values of the loop integral throughout X v ≥ P v X_v\ge P_v X v ≥ P v X = P X=P X = P
With the definition (36), eq. (34) holds for X v ≥ P v > 0 X_v\ge P_v>0 X v ≥ P v > 0 P k = ∣ X i − X j ∣ P_k=\lvert X_i-X_j\rvert P k = ∣ X i − X j ∣ P P P X = P X=P X = P X v ≥ P v X_v\ge P_v X v ≥ P v X v > 0 X_v>0 X v > 0 X v < P v X_v<P_v X v < P v π 2 / 2 \pi^{2}/2 π 2 /2 P → t P P\to tP P → tP t = 0 t=0 t = 0 X v = P v X_v=P_v X v = P v X v < P v X_v<P_v X v < P v
To summarize, the one-loop triangle coefficient is a polylogarithmic function of weight two. This statement is exact on the line (15), where the operator L 0 \mathcal{L}_0 L 0 X = P X=P X = P X v > P v X_v>P_v X v > P v
De Sitter space
For a cubic interaction in de Sitter space, the integrals (4) have σ = 0 \sigma= 0 σ = 0 ∣ ℓ ⃗ ∣ − 5 |\vec{\ell}|^{-5} ∣ ℓ ∣ − 5
∫ 0 ∞ d 3 ω ψ 3 ( X + ω ; y ) ⟶ ζ ( 3 ) 8 ∣ ℓ ⃗ ∣ 3 , (40) \int_{0}^{\infty} \mathrm{d}^{3}\omega\ \psi_{3}(X+\omega;y) \longrightarrow\frac{\zeta(3)}{8|\vec{\ell}|^{3}}, \tag*{(40)} ∫ 0 ∞ d 3 ω ψ 3 ( X + ω ; y ) ⟶ 8∣ ℓ ∣ 3 ζ ( 3 ) , (40) so the loop integral of the de Sitter coefficient diverges logarithmically. With a cutoff Λ \Lambda Λ ln Λ \ln\Lambda ln Λ π ζ ( 3 ) / 2 \pi\zeta(3)/2 π ζ ( 3 ) /2 Δ = 2 \Delta= 2 Δ = 2 [51 ] . Assuming that the cutoff breaks the symmetry only through the logarithm, we have
ψ 3 d S = ∫ Λ d 3 ℓ ∫ 0 ∞ d 3 ω ψ 3 ( X + ω ; y ) = − π ζ ( 3 ) 2 ln P 1 + P 2 + P 3 Λ + b 0 , (41) \psi_{3}^{\mathrm{dS}}=\int^{\Lambda}\mathrm{d}^{3}\ell\int_{0}^{\infty}\mathrm{d}^{3}\omega\ \psi_{3}(X+\omega;y)=-\frac{\pi\zeta(3)}{2}\ln\frac{P_{1}+P_{2}+P_{3}}{\Lambda}+b_{0}, \tag*{(41)} ψ 3 dS = ∫ Λ d 3 ℓ ∫ 0 ∞ d 3 ω ψ 3 ( X + ω ; y ) = − 2 π ζ ( 3 ) ln Λ P 1 + P 2 + P 3 + b 0 , (41) where b 0 b_{0} b 0 − 3 π 2 ζ ( 3 ) -3\pi^{2}\zeta(3) − 3 π 2 ζ ( 3 )
With one leg per site, the de Sitter triangle is therefore determined by just two numbers (the coefficient of the logarithm and b 0 b_{0} b 0 V V V V V V
Correlators
Having established that the elliptic periods cancel in the wavefunction coefficient, we now turn to the equal-time correlator. In this section, we show that the integrand of the triangle correlator has no pole on the planes q G + 2 y e = 0 q_G+2y_e=0 q G + 2 y e = 0 § 3 . The absence of the factors q G + 2 y e q_G+2y_e q G + 2 y e [37 , 38 ] , and the relation between correlators and wavefunction coefficients has been studied from several directions [52 , 53 , 54 ] . In particular, ref. [28 ] noted this absence in connection with the elliptic sector and asked whether the correlator requires elliptic functions at all. Here we derive the cancellation from the recursion relations and then consider the other singular loci of the correlator integrand. Throughout this section, all statements concern integrands at fixed loop momentum.
The correlator integrand
The Born rule (2) expresses a correlator through products of wavefunction coefficients. At one loop, the integral in eq. (2) also includes the profile of the field running around the loop; with the Gaussian weight ∣ Ψ ∣ 2 |\Psi|^{2} ∣Ψ ∣ 2 y e y_e y e 1 / ( 2 y e ) 1/(2y_e) 1/ ( 2 y e ) G G G ψ ^ G = ∏ e ( 2 y e ) ψ G \widehat{\psi}_{G}=\prod_{e}(2y_{e})\psi_{G} ψ G = ∏ e ( 2 y e ) ψ G [37 ] and appendix A of ref. [38 ] ) gives
C G = ∑ S ⊆ e d g e s 2 n ( S ) − 1 ∏ j = 1 n ( S ) ψ ^ H j , (42) C_{G}=\sum_{S\subseteq\mathrm{edges}}2^{n(S)-1}\prod_{j=1}^{n(S)}\widehat{\psi}_{H_j}, \tag*{(42)} C G = S ⊆ edges ∑ 2 n ( S ) − 1 j = 1 ∏ n ( S ) ψ H j , (42) where the graph with the edges in S S S n ( S ) n(S) n ( S ) H j H_j H j y e y_e y e C G C_G C G ∏ e 2 y e \prod_e 2y_e ∏ e 2 y e (42) , namely the loop and the chains in which one edge is cut, appear with the same coefficient.
Cancellation of the factors q G + 2 y e q_G + 2y_e q G + 2 y e
Consider the polygon with N N N G e G_e G e e e e (42) with poles at q G + 2 y e = 0 q_G + 2y_e = 0 q G + 2 y e = 0
Φ G = ψ ^ G + ∑ e ψ ^ G e . (43) \Phi_G = \hat{\psi}_G + \sum_e \hat{\psi}_{G_e}. \tag*{(43)} Φ G = ψ ^ G + e ∑ ψ ^ G e . (43) The recursion of ref. [18 ] , which is eq. (9) for the triangle, gives
q G ψ ^ G = ∑ e 2 y e ψ ^ G e , ( q G + 2 y e ) ψ ^ G e = ∑ f ≠ e 2 y f ψ ^ G e ∖ f . (44) q_G \hat{\psi}_G = \sum_e 2y_e \hat{\psi}_{G_e}, \qquad(q_G + 2y_e)\hat{\psi}_{G_e} = \sum_{f \ne e} 2y_f \hat{\psi}_{G_e\setminus f}. \tag*{(44)} q G ψ ^ G = e ∑ 2 y e ψ ^ G e , ( q G + 2 y e ) ψ ^ G e = f = e ∑ 2 y f ψ ^ G e ∖ f . (44) where G e ∖ f G_e \setminus f G e ∖ f f f f G e G_e G e q G + 2 y e q_G + 2y_e q G + 2 y e e e e 2 y e ψ ^ G e 2y_e\hat{\psi}_{G_e} 2 y e ψ ^ G e
q G Φ G = ∑ e ≠ f 2 y f ψ ^ G e ∖ f . (45) q_G \Phi_G = \sum_{e \ne f} 2y_f \hat{\psi}_{G_e\setminus f}. \tag*{(45)} q G Φ G = e = f ∑ 2 y f ψ ^ G e ∖ f . (45) On the right-hand side, ψ ^ G e ∖ f \hat{\psi}_{G_e\setminus f} ψ ^ G e ∖ f e e e f f f G G G ψ ^ G \hat{\psi}_G ψ ^ G ψ ^ G e \hat{\psi}_{G_e} ψ ^ G e q G + 2 y e = 0 q_G + 2y_e = 0 q G + 2 y e = 0 Φ G \Phi_G Φ G (42) have two or more edges cut, and their poles are likewise partial energies of G G G Φ G \Phi_G Φ G q G q_G q G (8) . The argument extends by induction to polygons with trees attached, and ref. [37 ] derives the cancellation more generally, for any number of loops and any time dependence of the background, as a cosmological analog of the Kinoshita–Lee–Nauenberg theorem.
Physically, the factors q G + 2 y e q_G + 2y_e q G + 2 y e (5) is the sum of a time-ordered term and a factorized term, e − y ( t + t ′ ) e^{-y(t+t')} e − y ( t + t ′ ) y e y_e y e + y e +y_e + y e + y e +y_e + y e q G + 2 y e q_G + 2y_e q G + 2 y e G e G_e G e
Other singular loci of the correlator integrand
The cancellation removes the curve of section 3 from the correlator integrand of the triangle, and it turns out that the other singular loci do not contain elliptic curves either. By proposition 1, every line on which B B B q G + 2 y e = 0 q_G + 2y_e = 0 q G + 2 y e = 0 (13) , on the other hand, has residues on 12 of them, but by proposition 1 these residues do not involve the periods of an elliptic curve. For the triangle, then, neither integrand has a residue. involving an elliptic curve, though for different reasons: the correlator integrand has no pole on the lines with an elliptic curve, while the residues of the wavefunction integrand on these lines are differentials on curves of genus zero.
For the box and the pentagon, the result depends on the dimension. In d = N d=N d = N N N N N + 1 N+1 N + 1 d = N d=N d = N d = 3 d=3 d = 3 § 6 , the correlator of the box in d = 3 d=3 d = 3
Finally, the cancellation also holds for time-dependent vertex factors and at two loops. With vertex factors ( − η ) − ( 1 + σ ) (-\eta)^{-(1+\sigma)} ( − η ) − ( 1 + σ ) (42) contain a phase for each connected component that depends on its number of sites. Since the loop and the chains G e G_e G e Φ G \Phi_G Φ G (4) apply to Φ G \Phi_G Φ G [37 ] . Whether the integrand of a two-loop correlator has residues that involve an elliptic curve remains an open question.
The box
Unlike the triangle, the one-loop box is not polylogarithmic. In this section, we show that the residue of its loop integrand at the partial-energy pole of a single site has two square roots, which together define an elliptic curve. We give the discontinuity of the box at threshold in closed form in terms of elliptic integrals, describe the K3 surface that appears above the threshold, and compute the box and its correlator on a line. In the last subsection, we summarize what we know for general momenta. We use the notation of section 2, with sites v = 1 , … , 4 v=1,\ldots,4 v = 1 , … , 4 y 12 y_{12} y 12 y 23 y_{23} y 23 y 34 y_{34} y 34 y 41 y_{41} y 41 Figure 1 (b)), and we take the wavefunction coefficient to be V 4 = ∫ d 3 ℓ ψ 4 V_4=\int\mathrm{d}^3\ell\,\psi_4 V 4 = ∫ d 3 ℓ ψ 4
The partial-energy pole of a single site
For the box, the residue on the factors q G + 2 y e q_G+2y_e q G + 2 y e X 1 + y 41 + y 12 = 0 X_1+y_{41}+y_{12}=0 X 1 + y 41 + y 12 = 0
Res X 1 = − y 41 − y 12 ψ 4 = 1 4 y 12 y 41 ∑ s , s ′ = ± 1 s s ′ ψ c h a i n ( X 2 + s y 12 , X 3 , X 4 + s ′ y 41 ; y 23 , y 34 ) , (46) \underset{X_1=-y_{41}-y_{12}}{\operatorname{Res}}\ \psi_4
=
\frac{1}{4y_{12}y_{41}}
\sum_{s,s'=\pm1}ss'\,\psi_{\mathrm{chain}}(X_2+sy_{12},X_3,X_4+s'y_{41};y_{23},y_{34}), \tag*{(46)} X 1 = − y 41 − y 12 Res ψ 4 = 4 y 12 y 41 1 s , s ′ = ± 1 ∑ s s ′ ψ chain ( X 2 + s y 12 , X 3 , X 4 + s ′ y 41 ; y 23 , y 34 ) , (46) where ψ c h a i n \psi_{\mathrm{chain}} ψ chain (28) . The residue is even in y 12 y_{12} y 12 y 41 y_{41} y 41 y 23 y_{23} y 23 y 34 y_{34} y 34 (10) .
At the pole, the loop momentum lies on a spheroid with foci at the two points where y 12 y_{12} y 12 y 41 y_{41} y 41 § 4.1 ); the box, however, has two. Along a line on the spheroid, the two square roots define the curve Y 2 = z 23 z 34 Y^2 = z_{23}z_{34} Y 2 = z 23 z 34 z e = y e 2 z_e = y_e^2 z e = y e 2
The threshold discontinuity
With one leg per site (X v = P v X_v = P_v X v = P v V 4 V_4 V 4 V 4 V_4 V 4 X 1 = − P 1 X_1 = -P_1 X 1 = − P 1 y 41 + y 12 = − X 1 y_{41} + y_{12} = -X_1 y 41 + y 12 = − X 1 χ = y 41 − y 12 \chi= y_{41} - y_{12} χ = y 41 − y 12 − P 1 ≤ χ ≤ P 1 -P_1 \le\chi\le P_1 − P 1 ≤ χ ≤ P 1
y 23 2 = ( P 1 + χ ) P 2 2 + ( P 1 − χ ) P 12 2 2 P 1 − P 1 2 − χ 2 4 , y 34 2 = ( P 1 + χ ) P 41 2 + ( P 1 − χ ) P 4 2 2 P 1 − P 1 2 − χ 2 4 , (47) y_{23}^2 = \frac{(P_1+\chi)P_{2}^{2}+(P_1-\chi)P_{12}^{2}}{2P_1}-\frac{P_1^2-\chi^2}{4}, \qquad y_{34}^2 = \frac{(P_1+\chi)P_{41}^{2}+(P_1-\chi)P_{4}^{2}}{2P_1}-\frac{P_1^2-\chi^2}{4}, \tag*{(47)} y 23 2 = 2 P 1 ( P 1 + χ ) P 2 2 + ( P 1 − χ ) P 12 2 − 4 P 1 2 − χ 2 , y 34 2 = 2 P 1 ( P 1 + χ ) P 41 2 + ( P 1 − χ ) P 4 2 − 4 P 1 2 − χ 2 , (47) on the segment, with P 12 = ∣ P ⃗ 1 + P ⃗ 2 ∣ P_{12} = \lvert\vec{P}_1+\vec{P}_2\rvert P 12 = ∣ P 1 + P 2 ∣ P 41 = ∣ P ⃗ 4 + P ⃗ 1 ∣ P_{41} = \lvert\vec{P}_4+\vec{P}_1\rvert P 41 = ∣ P 4 + P 1 ∣
D 0 = π 4 P 1 ∫ − P 1 P 1 d χ ∑ s , s ′ = ± 1 s s ′ ψ c h a i n ( P 2 + s P 1 − χ 2 , P 3 , P 4 + s ′ P 1 + χ 2 ; y 23 , y 34 ) . (48) D_0 = \frac{\pi}{4P_1}\int_{-P_1}^{P_1}\mathrm{d}\chi\sum_{s,s'=\pm1}ss'\,\psi_{\mathrm{chain}}\left(P_2+s\frac{P_1-\chi}{2},\,P_3,\,P_4+s'\frac{P_1+\chi}{2};\,y_{23},y_{34}\right). \tag*{(48)} D 0 = 4 P 1 π ∫ − P 1 P 1 d χ s , s ′ = ± 1 ∑ s s ′ ψ chain ( P 2 + s 2 P 1 − χ , P 3 , P 4 + s ′ 2 P 1 + χ ; y 23 , y 34 ) . (48) For generic momenta, the four zeros of z 23 z 34 z_{23}z_{34} z 23 z 34 (48) is an integral on an elliptic curve whose j j j
Importantly, the threshold discontinuity is not merely a feature of the integrand, but part of the wavefunction coefficient itself. When the momentum of the leg at site 1 becomes soft, the coefficient with one leg per site takes the form
V 4 = − D 0 ln 2 P 1 + terms analytic at P 1 = 0 , (49) V_4=-D_0\ln2P_1+\text{terms analytic at }P_1=0, \tag*{(49)} V 4 = − D 0 ln 2 P 1 + terms analytic at P 1 = 0 , (49) and numerical values of V 4 V_4 V 4 (49) .
The integral (48) can be evaluated in closed form. Its integrand is a rational function of χ \chi χ y 23 y_{23} y 23 y 34 y_{34} y 34 Y 2 = z 23 z 34 Y^2=z_{23}z_{34} Y 2 = z 23 z 34 § 6.1 . The result is
D 0 = π 4 P 1 [ 1 4 ∑ p o l e s ρ ( L 0 + η L 23 + η ′ L 34 + η η ′ L Π ) + 4 p 0 J F ] , J F = ∫ − P 1 P 1 d χ 4 Y = 2 R F , (50) D_0=\frac{\pi}{4P_1}\left[\frac{1}{4}\sum_{\mathrm{poles}}\rho\left(L_0+\eta L_{23}+\eta' L_{34}+\eta\eta' L_{\Pi}\right)+4p_0J_F\right], \qquad J_F=\int_{-P_1}^{P_1}\frac{\mathrm{d}\chi}{4Y}=2R_F, \tag*{(50)} D 0 = 4 P 1 π 4 1 poles ∑ ρ ( L 0 + η L 23 + η ′ L 34 + η η ′ L Π ) + 4 p 0 J F , J F = ∫ − P 1 P 1 4 Y d χ = 2 R F , (50) where L 0 L_0 L 0 L 23 L_{23} L 23 L 34 L_{34} L 34 L Π L_{\Pi} L Π R F R_F R F R J R_J R J [55 ] . The sum runs over the poles of the integrand on the four sheets ( η , η ′ ) = ( ± y 23 , ± y 34 ) (\eta,\eta')=(\pm y_{23},\pm y_{34}) ( η , η ′ ) = ( ± y 23 , ± y 34 ) ρ \rho ρ p 0 p_0 p 0 (47) and (48) ; we derive the complete expressions in appendix § 10 .§ 3 . That these elliptic integrals do not combine into logarithms is shown in § 6.6 . Figure 4 compares eq. (50) with numerical values of the integral (48).
Figure 4. The threshold discontinuity D 0 D_0 D 0 P 1 = 1 P_1 = 1 P 1 = 1 P 3 = 2 P_3 = 2 P 3 = 2 P 4 = 7 4 P_4 = \frac{7}{4} P 4 = 4 7 P 12 = 9 5 P_{12} = \frac{9}{5} P 12 = 5 9 P 41 = 2 P_{41} = 2 P 41 = 2 P 2 P_2 P 2 P 12 P_{12} P 12 P 41 P_{41} P 41 (50) and the points are numerical values of the integral (48) .
For momenta in a plane, one of the poles becomes a double pole, and eq. (50) then contains the additional term of eq. (91) . With this term, the formula holds for planar quadrilaterals in general position. In the planar case all poles are real, and the result does not depend on the branches of the logarithms. For momenta that do not lie in a plane, the branches at the complex poles follow from an analytic continuation, which agrees with numerical values but which we have not proved.
The K3 surface above the threshold
Above the threshold, the discontinuity becomes an integral over the double cover of the spheroid defined by the two roots, which is a K3 surface fibered by elliptic curves. Its periods obey a differential equation in y 41 + y 12 y_{41} + y_{12} y 41 + y 12 (99) in appendix B.6). It is not a symmetric power of a second-order operator, so its solutions are not simply products of periods of an elliptic curve. Instead, its five solutions obey the quadratic relation (100) , a characteristic property of the periods of a K3 surface.
The loop integrand has a component along these periods. With several legs per site, the leading term of the residue at large site energies generates all five solutions of the fifth-order operator, so the periods of the K3 surface enter the loop integrand of the box. With one leg per site, the evidence is as follows. The solutions of the fifth-order operator have square-root branch points at complex values of y 41 + y 12 y_{41} + y_{12} y 41 + y 12 V 4 V_4 V 4
The box on a line
Just as for the triangle in § 4 , we can obtain exact differential equations for the box on a line in kinematic space. We take the momenta to form a regular tetrahedron of unit side, with site energies
X v = X = 1 + t , v = 1 , … , 4 , (51) X_v = X = 1 + t,\qquad v = 1,\ldots,4, \tag*{(51)} X v = X = 1 + t , v = 1 , … , 4 , (51) so that t = 0 t = 0 t = 0 C 4 ( t ) C_4(t) C 4 ( t ) V 4 V_4 V 4 X X X X − 4 X^{-4} X − 4 ∫ d 3 ℓ / ( y 12 y 23 y 34 y 41 ) \int\mathrm{d}^{3}\ell/(y_{12}y_{23}y_{34}y_{41}) ∫ d 3 ℓ / ( y 12 y 23 y 34 y 41 ) (53) . An integration-by-parts reduction with the regulator of eq. (13) gives exact first-order differential systems in t t t V 4 V_{4} V 4 C 4 C_{4} C 4 ε → 0 \varepsilon\to0 ε → 0 ε \varepsilon ε
These systems determine the function space on the line. An exact rational change of basis brings the system for C 4 C_{4} C 4 X X X (101) ), and the system has no other singular points at finite X X X § 6.3 , the second-order operators of three elliptic curves, and first-order operators with rational and square-root letters. The three curves are those on which two partial energies vanish, while the curve of § 6.2 degenerates on this line and does not appear. The periods of the three curves are hypergeometric functions of their j j j (93) ), and they enter both V 4 V_{4} V 4 C 4 C_{4} C 4 t = 0 t=0 t = 0 § 10 .§ 7 ). The ancillary files contain the two systems along with a program that evaluates their solutions.
The correlator on the line
Remarkably, on this line the differential system determines the correlator completely. The result is
C 4 ( t ) = u ( t ) + Ω v ( t ) , C_{4}(t)=u(t)+\Omega v(t), C 4 ( t ) = u ( t ) + Ω v ( t ) , Ω = ∫ d 3 ℓ y 12 y 23 y 34 y 41 = 21.4444295139 … , (52) \Omega=\int\frac{\mathrm{d}^{3}\ell}{y_{12}y_{23}y_{34}y_{41}}=21.4444295139\ldots, \tag*{(52)} Ω = ∫ y 12 y 23 y 34 y 41 d 3 ℓ = 21.4444295139 … , (52) where u u u v v v C 4 C_{4} C 4 § 10 .§ 7 , and in the integral the four edge energies are the distances from ℓ ⃗ \vec{\ell} ℓ Figure 5 .
Figure 5. The correlator of the box on the line of section 6.4, multiplied by X 4 X^4 X 4 (52) and the points are numerical values of the loop integral. The dotted line is the constant Ω \Omega Ω X X X
Notably, the functions u u u v v v C 4 C_{4} C 4 t > − 1 t>-1 t > − 1 C 4 C_{4} C 4 § 10 .§ 7 form a space of dimension 4. At large X X X
C 4 = ∑ n ≥ 4 X − n ( α n + β n ln X ) , α 4 = Ω , β 5 = − 24 π , β 7 = − 158 3 π , (53) C_{4}=\sum_{n\geq4}X^{-n}(\alpha_{n}+\beta_{n}\ln X),\qquad\alpha_{4}=\Omega,\qquad\beta_{5}=-24\pi,\qquad\beta_{7}=-\frac{158}{3}\pi, \tag*{(53)} C 4 = n ≥ 4 ∑ X − n ( α n + β n ln X ) , α 4 = Ω , β 5 = − 24 π , β 7 = − 3 158 π , (53) with β n = 0 \beta_{n}=0 β n = 0 n n n β n \beta_{n} β n X X X v v v X − 4 X^{-4} X − 4
v = 4 10 X ( 10 X 2 − 1 ) ( 16 X 4 − 10 X 2 + 3 ) . (54) v=\frac{4\sqrt{10}}{X\sqrt{(10X^{2}-1)(16X^{4}-10X^{2}+3)}}. \tag*{(54)} v = X ( 10 X 2 − 1 ) ( 16 X 4 − 10 X 2 + 3 ) 4 10 . (54) The coefficients β n \beta_{n} β n α 5 \alpha_{5} α 5 ε \varepsilon ε § 10 .§ 7 ), then determine u u u v v v v v v X 2 = 1 10 X^{2}=\frac{1}{10} X 2 = 10 1 u + Ω v u+\Omega v u + Ω v (52) (appendix § 10 .§ 7 ). The differential system therefore determines Ω \Omega Ω Ω \Omega Ω § 10 .8 writes it in terms of hypergeometric functions at rational arguments, eq. (117) .
The expression u + Ω v u+\Omega v u + Ω v C 4 C_4 C 4 (34) , since u u u V 4 V_4 V 4
The box is not polylogarithmic
As the triangle shows, an elliptic curve among the master integrals does not by itself exclude a polylogarithmic coefficient. For the box, however, two independent arguments exclude it. The first concerns the function D 0 D_0 D 0 D 0 D_0 D 0 λ ± 1 \lambda^{\pm1} λ ± 1 λ + λ − 1 \lambda+\lambda^{-1} λ + λ − 1 λ \lambda λ D 0 D_0 D 0
The second argument is exact and concerns the loop integrand. As the shape of the quadrilateral varies, the residue (46) and its derivatives obey a linear differential system. With one leg per site, the solutions of this system include the periods of the curve Y 2 = z 23 z 34 Y^2=z_{23}z_{34} Y 2 = z 23 z 34
Combined with eq. (49) , the two arguments show that the periods of an elliptic curve enter the soft limit of the box with one leg per site. Furthermore, on the line of section 6.4 the periods of three elliptic curves enter both V 4 V_4 V 4 C 4 C_4 C 4
General momenta
The results of § 6.4 and § 6.5 hold on a single line, for the regular tetrahedron. They are, however, restrictions of a structure in all variables of the box, namely the four site energies and the six lengths of the quadrilateral of momenta. Here we state its main properties, leaving a fuller treatment to a separate paper. The singular points of the master integrals lie on the zeros of 1333 irreducible polynomials in the site energies and the squares of the lengths, and every singular point in this section lies on one of them. The curves on which two partial energies vanish form two families of elliptic curves, one for pairs of partial energies that share an edge and one for pairs that share none. Their invariants are known in all variables, and on the line they reduce to the j j j § 6.4 . The K3 surfaces of § 6.3 form a single family with six moduli, whose transcendental lattice has rank 5 for the regular tetrahedron, 6 for tetrahedra whose opposite edges have equal lengths, and 8 for generic momenta. In each case, the rank equals the order of the differential equation of the periods, so the operator (99) and the form of the relation (100) are special to the regular tetrahedron.
The elliptic structure at the partial-energy pole of a single site can also be made explicit for arbitrary momenta. In the coordinates σ 1 = y 41 + y 12 \sigma_1 = y_{41} + y_{12} σ 1 = y 41 + y 12 σ 2 = y 12 + y 23 \sigma_2 = y_{12} + y_{23} σ 2 = y 12 + y 23 σ 3 = y 23 + y 34 \sigma_3 = y_{23} + y_{34} σ 3 = y 23 + y 34 d 3 ℓ / ( y 12 y 23 y 34 y 41 ) \mathrm{d}^3\ell/(y_{12}y_{23}y_{34}y_{41}) d 3 ℓ / ( y 12 y 23 y 34 y 41 ) g d σ 1 d σ 2 d σ 3 / Δ \sqrt{g}\,\mathrm{d}\sigma_1\,\mathrm{d}\sigma_2\,\mathrm{d}\sigma_3/\sqrt{\Delta} g d σ 1 d σ 2 d σ 3 / Δ g g g Δ \Delta Δ σ 2 \sigma_2 σ 2 σ 1 \sigma_1 σ 1 σ 3 \sigma_3 σ 3 § 10 .§ 5 ). The loop integral is therefore an integral over the plane of σ 1 \sigma_1 σ 1 σ 3 \sigma_3 σ 3 w 2 = Δ ( σ 2 ) w^2 = \Delta(\sigma_2) w 2 = Δ ( σ 2 )
For one quadrilateral in general position, we have also reduced the correlator on the line X v = P v + t X_v = P_v + t X v = P v + t v v v (54) is a rational function divided by the square root of a polynomial, and on this line the system has at most 16 solutions of this type. Writing Ω \Omega Ω t t t (52) with the edge energies of the given quadrilateral, we have not yet determined which combination of the 16 solutions multiplies Ω \Omega Ω C 4 C_4 C 4 Ω / ( X 1 X 2 X 3 X 4 ) \Omega/(X_1X_2X_3X_4) Ω/ ( X 1 X 2 X 3 X 4 )
Conclusion
Bootstrapping loop corrections to cosmological correlators requires, first and foremost, knowing the class of functions in which they live. In flat space, decades of work on scattering amplitudes have mapped out this function space in considerable detail: one-loop amplitudes are polylogarithmic, and elliptic functions first appear at two loops. In cosmology the situation has been much less clear. The master integrals of the one-loop triangle involve an elliptic curve, which suggested that the triangle itself is elliptic, and that cosmological loops leave the space of polylogarithms one loop earlier than their flat-space counterparts.
In this paper, we have shown that this is not the case for the triangle. The elliptic curve appears in its master integrals, but the periods cancel in the physical combination, and the wavefunction coefficient of a conformally coupled scalar is a polylogarithmic function of transcendental weight two. We have given it in closed form both with one leg per site, eq. (32) , and for general site energies, eq. (34) , the latter being the input required for de Sitter space and power-law cosmologies. Along the way, we identified the origin of the cancellation: at the poles of the elliptic sector, the residue of the loop integrand is a flat-space amplitude, rational in the loop momentum (proposition 1). We also showed that the correlator integrand of the triangle has no residue involving an elliptic curve, though for a different reason, namely that it has no pole on the relevant planes at all.
The box, by contrast, is the first polygon of this theory that is not polylogarithmic, so cosmological loops do leave the space of polylogarithms at one loop, one graph later than expected. Its residue at the partial-energy pole of a single site involves two square roots rather than one. Its periods enter the wavefunction coefficient even with one leg per site, and in the correlator the elliptic periods do not cancel (§ 6.5 ). For momenta forming a regular tetrahedron, the periods of a K 3 K3 K 3 § 6 ). Taken together, the triangle and the box illustrate a broader lesson: the maximal cuts of the master integrals are not a reliable guide to the function space of the physical observable.
The triangle closed forms were bootstrapped rather than derived. The function space is fixed exactly on one line and by the symbol elsewhere, and the remaining rational coefficients were determined from numerical values of the integral. For one leg per site, eq. (32) can also be derived by carrying out the integrals over the edge energies analytically (appendix § 11 ). In this derivation the elliptic curve never appears, since the residue of proposition 1 reduces it to logarithms at the first integration, and with one leg per site the poles of the remaining integrals lie on lines along which every square root is rational. With several legs per site the poles move off these lines, and the radicands of eq. (34) are the values of the square roots at the points where they meet. For general site energies we have not carried the derivation through, but within the function space of § 4.5 the integrand fixes the ten coefficients of eq. (34) as residues (appendix § 11 .§ 5 ). This confirms the bootstrap and explains why the coefficients are as simple as − 1 4 -\frac{1}{4} − 4 1 − 1 2 -\frac{1}{2} − 2 1 ± 1 \pm1 ± 1
It would also be valuable to carry these results into expanding backgrounds in full. With one leg per site, symmetry determines the de Sitter triangle up to two numbers, the coefficient of the logarithm and a constant (§ 4.6 ). With several legs at a site, however, the de Sitter and power-law coefficients are integrals of eq. (34) over the site energies. Whether these integrals remain polylogarithmic (or whether the ten square roots can be rationalized simultaneously) is an open question.
For the box, the open problems concern the functions themselves. On one line, exact conditions on a differential system determine both the correlator and the coefficient with no fitted numbers. However, one of the two functions in the correlator is an iterated integral, of depth at most ten, over kernels that include the periods of three elliptic curves and of a K 3 K3 K 3 Ω \Omega Ω § 10 .8), but has not been related to known constants. For generic momenta, the differential equation obeyed by the periods of the K 3 K3 K 3 § 6.7 ), we have not constructed its solutions. Further afield, the residue of the pentagon at the partial-energy pole of a single site has three square roots and defines a curve of genus five. Beyond one loop, we have examined only integrands, and in the two-loop graphs of § 5.3 , the correlator integrand contains partial energies only.
Finally, all fields in this paper are conformally coupled, which keeps the loop integrand rational. For fields of general mass, which are of particular interest for the cosmological collider program, the singular loci of the integrand are the same. However, the integrand is no longer rational, and our methods do not apply directly. The leading cosmological collider signals of a triangle loop of massive fields have recently been obtained using cutting rules [56 ] , and one-loop signals of this kind can be integrated numerically [57 ] . The function space of such loops, however, remains open.
Acknowledgments
Many of the computations in this paper were performed by AI: mostly Claude Fable 5.1 (Anthropic) with checks and a few additions by GPT Astra 6 (OpenAI). Much of the text was drafted by AI, then edited and rewritten by the authors. The authors take full responsibility for the validity and accuracy of the results and presentation.
Construction of the closed form
In this appendix, we describe the construction of the closed form (32) in detail. We begin with the operator L 0 \mathcal{L}_0 L 0 V V V
The operator and its singular points
The integration-by-parts identities of the family (16) follow from the vector fields that preserve B B B n e n_e n e n 0 n_0 n 0 ∑ e n e ∂ B / ∂ y e = n 0 B \sum_e n_e \, \partial B/\partial y_e = n_0 B ∑ e n e ∂ B / ∂ y e = n 0 B [28 ] . For a rational function g g g
∫ d y 12 d y 23 d y 31 B − 1 / 2 + ε [ ∑ e ∂ ( n e g ) ∂ y e + ( ε − 1 2 ) n 0 g ] = 0. (55) \int\mathrm{d}y_{12}\,\mathrm{d}y_{23}\,\mathrm{d}y_{31}\,B^{-1/2+\varepsilon}\left[\sum_e \frac{\partial(n_e g)}{\partial y_e}+\left(\varepsilon-\frac{1}{2}\right)n_0 g\right]=0. \tag*{(55)} ∫ d y 12 d y 23 d y 31 B − 1/2 + ε [ e ∑ ∂ y e ∂ ( n e g ) + ( ε − 2 1 ) n 0 g ] = 0. (55) We solve these identities over finite fields at numerical values of λ \lambda λ ε \varepsilon ε V V V λ \lambda λ λ \lambda λ ε \varepsilon ε ε \varepsilon ε λ \lambda λ ε = 0 \varepsilon=0 ε = 0 L 0 \mathcal{L}_0 L 0 c i ( λ ) c_i(\lambda) c i ( λ ) λ \lambda λ
The operator L 0 \mathcal{L}_0 L 0 λ = 0 \lambda=0 λ = 0 ± 1 3 \pm\frac{1}{3} ± 3 1 ± 1 \pm1 ± 1 q G q_G q G ξ i \xi_i ξ i X 2 X_2 X 2
q G = 3 λ + 1 , ξ 1 = 1 − λ , ξ 2 = 1 + λ , ξ 3 = 3 λ − 1 , X 2 = λ . (56) q_G=3\lambda+1,\qquad\xi_1=1-\lambda,\qquad\xi_2=1+\lambda,\qquad\xi_3=3\lambda-1,\qquad X_2=\lambda. \tag*{(56)} q G = 3 λ + 1 , ξ 1 = 1 − λ , ξ 2 = 1 + λ , ξ 3 = 3 λ − 1 , X 2 = λ . (56) The points λ = 1 \lambda=1 λ = 1 λ = 1 3 \lambda=\frac{1}{3} λ = 3 1 λ = 0 \lambda=0 λ = 0 λ = − 1 3 \lambda=-\frac{1}{3} λ = − 3 1 λ = − 1 \lambda=-1 λ = − 1
We note that if the limit ε → 0 \varepsilon\to0 ε → 0 L 0 \mathcal{L}_0 L 0 V V V L 2 L_2 L 2 § 3.1 , gives an operator of order 9 that divides L 0 L_0 L 0 V V V (32) indicates that the operator of lowest order annihilating V ( λ ) V(\lambda) V ( λ )
To test L 0 L_0 L 0 V V V (13) at ε = 0 \varepsilon= 0 ε = 0 y 12 y_{12} y 12 § 9 .§ 4 and the points of Figure 3 , which use an independent evaluation to 11 digits, all comparisons with numerical values in § 4 and in this appendix hold to this accuracy. Let φ \varphi φ a ≤ λ ≤ b a \le\lambda\le b a ≤ λ ≤ b V V V φ \varphi φ
Figure 3. The combination q G 2 V q_{\mathrm{G}}^{2}V q G 2 V P = ( 7 5 , 7 10 , 1 ) P=\left(\frac{7}{5},\frac{7}{10},1\right) P = ( 5 7 , 10 7 , 1 ) X v = P v + δ X_v=P_v+\delta X v = P v + δ δ = 0 \delta=0 δ = 0 δ < 0 \delta<0 δ < 0 X v < P v X_v<P_v X v < P v δ = 0 \delta=0 δ = 0
∫ a b φ L 0 V d λ = ∫ a b V L 0 † φ d λ , L 0 † φ = ∑ i = 0 20 ( − 1 ) i d i d λ i ( c i φ ) . (57) \int_a^b \varphi L_0 V\,\mathrm{d}\lambda= \int_a^b V L_0^\dagger\varphi\,\mathrm{d}\lambda,\qquad L_0^\dagger\varphi= \sum_{i=0}^{20}(-1)^i\frac{\mathrm{d}^i}{\mathrm{d}\lambda^i}(c_i\varphi). \tag*{(57)} ∫ a b φ L 0 V d λ = ∫ a b V L 0 † φ d λ , L 0 † φ = i = 0 ∑ 20 ( − 1 ) i d λ i d i ( c i φ ) . (57) The right-hand side contains only the values of V V V φ \varphi φ L 0 V = 0 L_0V = 0 L 0 V = 0 φ m = [ ( λ − a ) ( b − λ ) ] 20 Leg m \varphi_m = [(\lambda-a)(b-\lambda)]^{20}\operatorname{Leg}_m φ m = [( λ − a ) ( b − λ ) ] 20 Leg m Leg m \operatorname{Leg}_m Leg m m m m (57) vanishes to the accuracy of the numerical values for L 0 L_0 L 0
The space of solutions
We now construct all solutions of L 0 L_0 L 0 ± 1 3 \pm\frac{1}{3} ± 3 1 ± 1 \pm1 ± 1 G ( ; λ ) = 1 G(; \lambda) = 1 G ( ; λ ) = 1
d d λ G ( a , w ; λ ) = G ( w ; λ ) λ − a . (58) \frac{\mathrm{d}}{\mathrm{d}\lambda}G(a,w;\lambda) = \frac{G(w;\lambda)}{\lambda-a}. \tag*{(58)} d λ d G ( a , w ; λ ) = λ − a G ( w ; λ ) . (58) where a a a w w w
g ( λ ) = ∑ w R w ( λ ) G ( w ; λ ) . (59) g(\lambda) = \sum_w R_w(\lambda)G(w;\lambda). \tag*{(59)} g ( λ ) = w ∑ R w ( λ ) G ( w ; λ ) . (59) where the R w R_w R w λ \lambda λ L 0 L_0 L 0
G ( w ; λ + τ ) = ∑ w = w ′ w ′ ′ h w ′ ( τ ) G ( w ′ ′ ; λ ) . (60) G(w;\lambda+\tau) = \sum_{w=w'w''} h_{w'}(\tau)G(w'';\lambda). \tag*{(60)} G ( w ; λ + τ ) = w = w ′ w ′′ ∑ h w ′ ( τ ) G ( w ′′ ; λ ) . (60) where the sum runs over all ways to split the word w w w w ′ w' w ′ w ′ ′ w'' w ′′ h w ′ ( τ ) h_{w'}(\tau) h w ′ ( τ ) w ′ w' w ′ λ \lambda λ λ + τ \lambda+\tau λ + τ τ \tau τ λ \lambda λ i i i g g g i ! i! i ! τ i \tau^i τ i g ( λ + τ ) g(\lambda+\tau) g ( λ + τ ) [ τ i ] [\tau^i] [ τ i ]
L 0 g = ∑ w ′ ′ G ( w ′ ′ ; λ ) ∑ i = 0 20 c i i ! [ τ i ] ∑ w ′ R w ′ w ′ ′ ( λ + τ ) h w ′ ( τ ) . (61) L_0 g = \sum_{w''} G(w'';\lambda)\sum_{i=0}^{20}c_i\,i!\,[\tau^i]\sum_{w'}R_{w'w''}(\lambda+\tau)h_{w'}(\tau). \tag*{(61)} L 0 g = w ′′ ∑ G ( w ′′ ; λ ) i = 0 ∑ 20 c i i ! [ τ i ] w ′ ∑ R w ′ w ′′ ( λ + τ ) h w ′ ( τ ) . (61) Since the multiple polylogarithms are linearly independent over the rational functions, the coefficient of each G ( w ′ ′ ; λ ) G(w'';\lambda) G ( w ′′ ; λ ) w ′ w' w ′ L 0 R w ′ ′ L_0R_{w''} L 0 R w ′′ w ′ ′ w^{\prime\prime} w ′′ L 0 \mathcal{L}_0 L 0 L 0 \mathcal{L}_0 L 0
The coefficient on the line
The coefficient V V V λ = 1 \lambda= 1 λ = 1 L 0 \mathcal{L}_0 L 0 V V V [40 ] . Up to an overall factor of 2 π \sqrt{2}\,\pi 2 π ln 2 \ln2 ln 2 ln 2 2 \ln^2 2 ln 2 2 π 2 \pi^2 π 2
All multiple polylogarithms of weight two in the result combine into the three dilogarithms Li 2 ( u i ) \operatorname{Li}_2(u_i) Li 2 ( u i ) L i = ln ( 1 − u i ) = ln ( 2 X i / q G ) L_i = \ln(1-u_i) = \ln(2X_i/q_G) L i = ln ( 1 − u i ) = ln ( 2 X i / q G ) (56) we obtain
4 2 q G ξ 1 2 ξ 2 2 ξ 3 2 π V = 2 λ ( 1 − λ ) p 1 Li 2 ( u 1 ) + λ ( 1 + λ ) p 2 Li 2 ( u 2 ) + ( 3 λ − 1 ) p 3 Li 2 ( u 3 ) + 4 π 2 3 λ 2 ( λ 2 − 6 λ + 1 ) + 32 λ 3 [ 2 λ L 1 L 2 + 2 L 1 L 3 + L 2 L 3 ] , (62) \frac{4\sqrt{2}\,q_G\,\xi_1^2\xi_2^2\xi_3^2}{\pi}V = 2\lambda(1-\lambda)p_1\operatorname{Li}_2(u_1)+\lambda(1+\lambda)p_2\operatorname{Li}_2(u_2)
+ (3\lambda-1)p_3\operatorname{Li}_2(u_3)+\frac{4\pi^2}{3}\lambda^2(\lambda^2-6\lambda+1)
+32\lambda^3\left[2\lambda L_1L_2+2L_1L_3+L_2L_3\right], \tag*{(62)} π 4 2 q G ξ 1 2 ξ 2 2 ξ 3 2 V = 2 λ ( 1 − λ ) p 1 Li 2 ( u 1 ) + λ ( 1 + λ ) p 2 Li 2 ( u 2 ) + ( 3 λ − 1 ) p 3 Li 2 ( u 3 ) + 3 4 π 2 λ 2 ( λ 2 − 6 λ + 1 ) + 32 λ 3 [ 2 λ L 1 L 2 + 2 L 1 L 3 + L 2 L 3 ] , (62) where the three polynomials are
p 1 = 9 λ 3 + 25 λ 2 − λ − 1 , p 2 = − 9 λ 3 + 25 λ 2 + λ − 1 , p 3 = − 3 λ 3 + 15 λ 2 + 3 λ + 1. (63) p_1=9\lambda^3+25\lambda^2-\lambda-1,\qquad p_2=-9\lambda^3+25\lambda^2+\lambda-1,\qquad p_3=-3\lambda^3+15\lambda^2+3\lambda+1. \tag*{(63)} p 1 = 9 λ 3 + 25 λ 2 − λ − 1 , p 2 = − 9 λ 3 + 25 λ 2 + λ − 1 , p 3 = − 3 λ 3 + 15 λ 2 + 3 λ + 1. (63) The right-hand side of eq. (62) vanishes to second order at λ = 1 \lambda=1 λ = 1 λ = 1 3 \lambda=\frac{1}{3} λ = 3 1 V V V (62) .
General triangles
For general momenta we have no differential operator, so we instead infer the result from the structure of eq. (62) . In terms of the site energies X = ( 2 λ , λ , 1 ) X=(2\lambda,\lambda,1) X = ( 2 λ , λ , 1 ) X i ξ i X_i\xi_i X i ξ i
p i = q G ξ j ξ k + 8 X 1 X 2 X 3 , (64) p_i=q_G\xi_j\xi_k+8X_1X_2X_3, \tag*{(64)} p i = q G ξ j ξ k + 8 X 1 X 2 X 3 , (64) where i i i j j j k k k i = 1 i=1 i = 1 ( 3 λ + 1 ) ( 1 + λ ) ( 3 λ − 1 ) + 16 λ 2 (3\lambda+1)(1+\lambda)(3\lambda-1)+16\lambda^2 ( 3 λ + 1 ) ( 1 + λ ) ( 3 λ − 1 ) + 16 λ 2 p 1 p_1 p 1 L i L j L_iL_j L i L j 16 X 1 X 2 X 3 X i X j 16X_1X_2X_3X_iX_j 16 X 1 X 2 X 3 X i X j
4 2 q G ξ 1 2 ξ 2 2 ξ 3 2 π V = − ∑ i ξ i b ( X i ; X j , X k ) Li 2 ( u i ) + π 2 c ( X ) + 16 X 1 X 2 X 3 ∑ i < j X i X j L i L j (65) \frac{4\sqrt{2}\,q_G\xi_1^2\xi_2^2\xi_3^2}{\pi}V
=
-\sum_i \xi_i b(X_i;X_j,X_k)\operatorname{Li}_2(u_i)+\pi^2c(X)
+16X_1X_2X_3\sum_{i<j}X_iX_jL_iL_j \tag*{(65)} π 4 2 q G ξ 1 2 ξ 2 2 ξ 3 2 V = − i ∑ ξ i b ( X i ; X j , X k ) Li 2 ( u i ) + π 2 c ( X ) + 16 X 1 X 2 X 3 i < j ∑ X i X j L i L j (65) for an arbitrary triangle. Here b ( x ; y , z ) b(x;y,z) b ( x ; y , z ) y y y z z z c c c b b b c c c (65) to the line and compare it with eq. (62) . The three dilogarithms are linearly independent over the rational functions of λ \lambda λ λ \lambda λ b b b c c c
b ( x ; y , z ) = − x [ ( x + y + z ) ( x − y + z ) ( x + y − z ) + 8 x y z ] , c ( X ) = − 2 3 X 1 X 2 X 3 ∑ i < j ξ i ξ j . (66) b(x;y,z)=-x\left[(x+y+z)(x-y+z)(x+y-z)+8xyz\right],\qquad c(X)=-\frac{2}{3}X_{1}X_{2}X_{3}\sum_{i<j}\xi_i\xi_j . \tag*{(66)} b ( x ; y , z ) = − x [ ( x + y + z ) ( x − y + z ) ( x + y − z ) + 8 x y z ] , c ( X ) = − 3 2 X 1 X 2 X 3 i < j ∑ ξ i ξ j . (66) With these polynomials eq. (65) becomes eq. (32) .
While the comparison on the line determines the 14 coefficients, it does not test the form of the ansatz itself. We therefore compare eq. (32) with numerical values of the loop integral at 25 other triangles, which include two nearly folded and two soft configurations. With the coefficients fixed on the line, the formula agrees with the numerical values at all 25 triangles, to 11 digits, the precision of the evaluation used for this check. An independent fit of the 14 coefficients to numerical values returns the same polynomials, eq. (66) . (34) agrees with the numerical values on the line X = P + 1 3 ( 1 , 1 , 1 ) X=P+\frac{1}{3}(1,1,1) X = P + 3 1 ( 1 , 1 , 1 ) X v ≥ P v X_v\ge P_v X v ≥ P v
Limits
We give three limits of eq. (32) . The first two are the smallest and the largest value of q G 2 V q_G^2V q G 2 V X v = X X_v=X X v = X u i = 1 3 u_i=\frac{1}{3} u i = 3 1
V = 2 π 8 X 2 [ 11 Li 2 ( 1 3 ) + 16 ln 2 2 3 − 2 π 2 3 ] . (67) V=\frac{\sqrt{2}\pi}{8X^2}\left[11\operatorname{Li}_2\left(\frac{1}{3}\right)+16\ln^2\frac{2}{3}-\frac{2\pi^2}{3}\right]. \tag*{(67)} V = 8 X 2 2 π [ 11 Li 2 ( 3 1 ) + 16 ln 2 3 2 − 3 2 π 2 ] . (67) When the momentum at site 1 becomes soft, the other two site energies approach a common value X X X X 1 X_1 X 1
V ⟶ 3 2 π 128 X 2 , X 1 ⟶ 0. (68) V\longrightarrow\frac{3\sqrt{2}\pi}{128X^2},\qquad X_1\longrightarrow0. \tag*{(68)} V ⟶ 128 X 2 3 2 π , X 1 ⟶ 0. (68) For a folded configuration with X 1 = X 2 + X 3 X_1=X_2+X_3 X 1 = X 2 + X 3 (32) vanishes to second order in ξ 1 \xi_1 ξ 1 Li 2 ( x ) + Li 2 ( 1 − x ) = π 2 6 − ln x ln ( 1 − x ) \operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)=\frac{\pi^2}{6}-\ln x\ln(1-x) Li 2 ( x ) + Li 2 ( 1 − x ) = 6 π 2 − ln x ln ( 1 − x ) X 23 = X 2 + X 3 X_{23}=X_2+X_3 X 23 = X 2 + X 3 x = X 3 / X 23 x=X_3/X_{23} x = X 3 / X 23
V ⟶ 2 π 384 X 23 2 x 2 ( 1 − x ) 2 N ( x ) , ξ 1 ⟶ 0 , (69) V\longrightarrow\frac{\sqrt{2}\pi}{384X_{23}^2x^2(1-x)^2}N(x),\qquad\xi_1\longrightarrow0, \tag*{(69)} V ⟶ 384 X 23 2 x 2 ( 1 − x ) 2 2 π N ( x ) , ξ 1 ⟶ 0 , (69) N ( x ) = 6 ( 2 x 3 − 3 x 2 + 3 x − 1 ) Li 2 ( x ) + 6 x ( x 2 − x + 1 ) ln x ln ( 1 − x ) − 3 x 2 ( x − 2 ) ln x + 3 ( x 3 − x 2 − x + 1 ) ln ( 1 − x ) − π 2 x 3 − 9 x 2 + 9 x . (70) N(x)=6(2x^3-3x^2+3x-1)\operatorname{Li}_2(x)+6x(x^2-x+1)\ln x\ln(1-x)-3x^2(x-2)\ln x
+3(x^3-x^2-x+1)\ln(1-x)-\pi^2x^3-9x^2+9x. \tag*{(70)} N ( x ) = 6 ( 2 x 3 − 3 x 2 + 3 x − 1 ) Li 2 ( x ) + 6 x ( x 2 − x + 1 ) ln x ln ( 1 − x ) − 3 x 2 ( x − 2 ) ln x + 3 ( x 3 − x 2 − x + 1 ) ln ( 1 − x ) − π 2 x 3 − 9 x 2 + 9 x . (70) The functions of the general formula
Here we give the four functions of eq. (34) . For a square root w w w x x x y y y
D [ x ; y ] = D ( x − w y − w , x + w y + w ) . (71) \mathcal{D}[x;y]=\mathcal{D}\left(\frac{x-w}{y-w},\frac{x+w}{y+w}\right). \tag*{(71)} D [ x ; y ] = D ( y − w x − w , y + w x + w ) . (71) with D \mathcal{D} D (36) . The functions F 0 F_0 F 0 F e F_e F e ℓ 12 , ℓ 13 , ℓ 23 , ℓ 14 , ℓ 24 , ℓ 34 \ell_{12},\ell_{13},\ell_{23},\ell_{14},\ell_{24},\ell_{34} ℓ 12 , ℓ 13 , ℓ 23 , ℓ 14 , ℓ 24 , ℓ 34
f 1 = ℓ 23 + ℓ 24 + ℓ 34 , f 2 = ℓ 13 + ℓ 14 + ℓ 34 , f 3 = ℓ 12 + ℓ 14 + ℓ 24 , f 4 = ℓ 12 + ℓ 13 + ℓ 23 , c 1 = ℓ 12 + ℓ 13 + ℓ 24 + ℓ 34 , c 2 = ℓ 12 + ℓ 14 + ℓ 23 + ℓ 34 , c 3 = ℓ 13 + ℓ 14 + ℓ 23 + ℓ 24 . (72) \begin{aligned}
f_1 &= \ell_{23}+\ell_{24}+\ell_{34}, & f_2 &= \ell_{13}+\ell_{14}+\ell_{34},\\
f_3 &= \ell_{12}+\ell_{14}+\ell_{24}, & f_4 &= \ell_{12}+\ell_{13}+\ell_{23},\\
c_1 &= \ell_{12}+\ell_{13}+\ell_{24}+\ell_{34}, & c_2 &= \ell_{12}+\ell_{14}+\ell_{23}+\ell_{34},\\
c_3 &= \ell_{13}+\ell_{14}+\ell_{23}+\ell_{24}. \tag*{(72)}
\end{aligned} f 1 f 3 c 1 c 3 = ℓ 23 + ℓ 24 + ℓ 34 , = ℓ 12 + ℓ 14 + ℓ 24 , = ℓ 12 + ℓ 13 + ℓ 24 + ℓ 34 , = ℓ 13 + ℓ 14 + ℓ 23 + ℓ 24 . f 2 f 4 c 2 = ℓ 13 + ℓ 14 + ℓ 34 , = ℓ 12 + ℓ 13 + ℓ 23 , = ℓ 12 + ℓ 14 + ℓ 23 + ℓ 34 , (72) The polynomial P ( h ) \mathcal{P}(h) P ( h ) (37) is quadratic, its discriminant is the radicand R R R w = R w=\sqrt{R} w = R p a = − P ′ ( a ) p_a=-\mathcal{P}'(a) p a = − P ′ ( a ) D [ p a ; p 0 ] \mathcal{D}[p_a;p_0] D [ p a ; p 0 ] D a \mathcal{D}_a D a F 0 F_0 F 0
( ℓ 12 , ℓ 13 , ℓ 23 ) = ( P 2 , P 1 , P 3 ) , ( ℓ 14 , ℓ 24 , ℓ 34 ) = 1 2 ( X 1 + X 2 − X 3 , X 2 + X 3 − X 1 , X 1 + X 3 − X 2 ) . (73) (\ell_{12},\ell_{13},\ell_{23})=(P_2,P_1,P_3),\qquad
(\ell_{14},\ell_{24},\ell_{34})=\frac{1}{2}(X_1+X_2-X_3,X_2+X_3-X_1,X_1+X_3-X_2). \tag*{(73)} ( ℓ 12 , ℓ 13 , ℓ 23 ) = ( P 2 , P 1 , P 3 ) , ( ℓ 14 , ℓ 24 , ℓ 34 ) = 2 1 ( X 1 + X 2 − X 3 , X 2 + X 3 − X 1 , X 1 + X 3 − X 2 ) . (73) and F 0 F_0 F 0 (38) . For the edge e = 12 e=12 e = 12
( ℓ 12 , ℓ 13 , ℓ 23 ) = ( P 2 , P 1 , P 3 ) , ( ℓ 14 , ℓ 24 , ℓ 34 ) = ( 1 2 q G , 1 2 q G − X 2 , 1 2 q G − X 1 ) . (74) (\ell_{12},\ell_{13},\ell_{23})=(P_2,P_1,P_3),\qquad
(\ell_{14},\ell_{24},\ell_{34})=\left(\frac{1}{2}q_G,\frac{1}{2}q_G-X_2,\frac{1}{2}q_G-X_1\right). \tag*{(74)} ( ℓ 12 , ℓ 13 , ℓ 23 ) = ( P 2 , P 1 , P 3 ) , ( ℓ 14 , ℓ 24 , ℓ 34 ) = ( 2 1 q G , 2 1 q G − X 2 , 2 1 q G − X 1 ) . (74) A prime denotes the same quantities with ℓ 24 → − ℓ 24 \ell_{24}\to-\ell_{24} ℓ 24 → − ℓ 24 ℓ 34 → − ℓ 34 \ell_{34}\to-\ell_{34} ℓ 34 → − ℓ 34
F 12 = − D [ p f 1 ; p 0 ] + D [ p c 1 ; p 0 ] + ∑ k = 2 4 D [ p f k ; p 0 ′ ] − D [ p c 2 ′ ; p 0 ′ ] − D [ p c 3 ′ ; p 0 ′ ] + D [ p 0 ′ ; p f 4 ] − D [ p f 4 ′ ; p 0 ] . (75) \begin{aligned}
F_{12}={}&-\mathcal{D}[p_{f_1};p_0]+\mathcal{D}[p_{c_1};p_0]+\sum_{k=2}^{4}\mathcal{D}[p_{f_k};p'_0]-\mathcal{D}[p'_{c_2};p'_0]-\mathcal{D}[p'_{c_3};p'_0]\\
&+\mathcal{D}[p'_0;p_{f_4}]-\mathcal{D}[p'_{f_4};p_0]. \tag*{(75)}
\end{aligned} F 12 = − D [ p f 1 ; p 0 ] + D [ p c 1 ; p 0 ] + k = 2 ∑ 4 D [ p f k ; p 0 ′ ] − D [ p c 2 ′ ; p 0 ′ ] − D [ p c 3 ′ ; p 0 ′ ] + D [ p 0 ′ ; p f 4 ] − D [ p f 4 ′ ; p 0 ] . (75) The functions F 12 ± F_{12}^{\pm} F 12 ± w = T 12 ± w=\sqrt{T_{12}^{\pm}} w = T 12 ±
F 12 − = D [ a 1 ; a 2 ] + D [ a 3 ; a 7 ] + D [ a 4 ; a 7 ] − D [ a 5 ; a 6 ] , F 12 + = − D [ b 1 ; b 2 ] + D [ b 1 ; b 5 ] + D [ b 1 ; b 6 ] + D [ b 3 ; b 4 ] − D [ b 3 ; b 5 ] . (76) \begin{aligned}
F_{12}^{-}&=\mathcal{D}[a_1;a_2]+\mathcal{D}[a_3;a_7]+\mathcal{D}[a_4;a_7]-\mathcal{D}[a_5;a_6],\\
F_{12}^{+}&=-\mathcal{D}[b_1;b_2]+\mathcal{D}[b_1;b_5]+\mathcal{D}[b_1;b_6]+\mathcal{D}[b_3;b_4]-\mathcal{D}[b_3;b_5]. \tag*{(76)}
\end{aligned} F 12 − F 12 + = D [ a 1 ; a 2 ] + D [ a 3 ; a 7 ] + D [ a 4 ; a 7 ] − D [ a 5 ; a 6 ] , = − D [ b 1 ; b 2 ] + D [ b 1 ; b 5 ] + D [ b 1 ; b 6 ] + D [ b 3 ; b 4 ] − D [ b 3 ; b 5 ] . (76) with the polynomials
a 1 = P 2 2 − ( P 1 + P 3 ) 2 + 2 X 1 ( P 1 + P 3 ) − 2 P 1 X 2 , a 2 = P 2 2 − P 1 2 − P 3 2 + 2 X 1 ( X 1 − X 2 ) , a 3 = P 3 2 − ( P 1 − P 2 ) 2 + 2 P 1 X 2 − 2 P 2 X 1 , a 4 = P 2 2 + P 3 2 − P 1 2 + 2 X 2 ( X 1 − X 2 ) , a 6 = P 3 2 − ( P 1 − P 2 ) 2 − 2 P 1 X 2 + 2 P 2 X 1 , a 5 = ( P 2 + P 3 ) 2 − P 1 2 + 2 X 1 P 2 − 2 X 2 ( P 2 + P 3 ) , a 7 = P 2 2 − ( P 1 + P 3 ) 2 − 2 X 1 ( P 1 + P 3 ) + 2 P 1 X 2 , b 1 = P 3 2 − P 1 2 − P 2 2 − 2 X 1 X 2 , b 2 = P 3 2 − ( P 1 − P 2 ) 2 − 2 P 1 X 2 − 2 P 2 X 1 , b 3 = P 2 2 − ( P 1 + P 3 ) 2 + 2 X 1 ( P 1 + P 3 ) + 2 P 1 X 2 , b 4 = P 2 2 − P 1 2 − P 3 2 + 2 X 1 ( X 1 + X 2 ) , b 5 = P 2 2 + P 3 2 − P 1 2 − 2 X 2 ( X 1 + X 2 ) , b 6 = ( P 2 + P 3 ) 2 − P 1 2 + 2 X 1 P 2 + 2 X 2 ( P 2 + P 3 ) . (77) \begin{aligned}
a_1&=P_2^2-(P_1+P_3)^2+2X_1(P_1+P_3)-2P_1X_2,\\
a_2&=P_2^2-P_1^2-P_3^2+2X_1(X_1-X_2), & a_3&=P_3^2-(P_1-P_2)^2+2P_1X_2-2P_2X_1,\\
a_4&=P_2^2+P_3^2-P_1^2+2X_2(X_1-X_2), & a_6&=P_3^2-(P_1-P_2)^2-2P_1X_2+2P_2X_1,\\
a_5&=(P_2+P_3)^2-P_1^2+2X_1P_2-2X_2(P_2+P_3),\\
a_7&=P_2^2-(P_1+P_3)^2-2X_1(P_1+P_3)+2P_1X_2,\\
b_1&=P_3^2-P_1^2-P_2^2-2X_1X_2, & b_2&=P_3^2-(P_1-P_2)^2-2P_1X_2-2P_2X_1,\\
b_3&=P_2^2-(P_1+P_3)^2+2X_1(P_1+P_3)+2P_1X_2,\\
b_4&=P_2^2-P_1^2-P_3^2+2X_1(X_1+X_2), & b_5&=P_2^2+P_3^2-P_1^2-2X_2(X_1+X_2),\\
b_6&=(P_2+P_3)^2-P_1^2+2X_1P_2+2X_2(P_2+P_3). \tag*{(77)}
\end{aligned} a 1 a 2 a 4 a 5 a 7 b 1 b 3 b 4 b 6 = P 2 2 − ( P 1 + P 3 ) 2 + 2 X 1 ( P 1 + P 3 ) − 2 P 1 X 2 , = P 2 2 − P 1 2 − P 3 2 + 2 X 1 ( X 1 − X 2 ) , = P 2 2 + P 3 2 − P 1 2 + 2 X 2 ( X 1 − X 2 ) , = ( P 2 + P 3 ) 2 − P 1 2 + 2 X 1 P 2 − 2 X 2 ( P 2 + P 3 ) , = P 2 2 − ( P 1 + P 3 ) 2 − 2 X 1 ( P 1 + P 3 ) + 2 P 1 X 2 , = P 3 2 − P 1 2 − P 2 2 − 2 X 1 X 2 , = P 2 2 − ( P 1 + P 3 ) 2 + 2 X 1 ( P 1 + P 3 ) + 2 P 1 X 2 , = P 2 2 − P 1 2 − P 3 2 + 2 X 1 ( X 1 + X 2 ) , = ( P 2 + P 3 ) 2 − P 1 2 + 2 X 1 P 2 + 2 X 2 ( P 2 + P 3 ) . a 3 a 6 b 2 b 5 = P 3 2 − ( P 1 − P 2 ) 2 + 2 P 1 X 2 − 2 P 2 X 1 , = P 3 2 − ( P 1 − P 2 ) 2 − 2 P 1 X 2 + 2 P 2 X 1 , = P 3 2 − ( P 1 − P 2 ) 2 − 2 P 1 X 2 − 2 P 2 X 1 , = P 2 2 + P 3 2 − P 1 2 − 2 X 2 ( X 1 + X 2 ) , (77) The other functions follow from relabelings of the sites. The functions F 23 F_{23} F 23 F 23 ± F_{23}^{\pm} F 23 ± F 12 F_{12} F 12 F 12 ± F_{12}^{\pm} F 12 ± ( 1 , 2 , 3 ) → ( 2 , 3 , 1 ) (1,2,3)\to(2,3,1) ( 1 , 2 , 3 ) → ( 2 , 3 , 1 ) X v X_v X v P v P_v P v F 31 F_{31} F 31 F 13 ± F_{13}^{\pm} F 13 ± ( 1 , 2 , 3 ) → ( 1 , 3 , 2 ) (1,2,3)\to(1,3,2) ( 1 , 2 , 3 ) → ( 1 , 3 , 2 )
Derivations for the box
In this appendix, we derive the results of § 6 . We first derive the residue at the partial-energy pole of a single site and the geometry of the threshold, and then reduce D 0 D_0 D 0 § 6.4 , an elliptic fibration of the loop measure for general momenta, the fifth-order operator, the conditions that determine u u u v v v C 4 C_4 C 4 V 4 V_4 V 4
The residue at the partial-energy pole of a single site
The pole at X 1 + y 41 + y 12 = 0 X_1+y_{41}+y_{12}=0 X 1 + y 41 + y 12 = 0 t 1 t_1 t 1 t 1 > t 2 t_1>t_2 t 1 > t 2 (5) factorizes,
G y 12 ( t 1 , t 2 ) = e − y 12 t 1 2 y 12 ( e y 12 t 2 − e − y 12 t 2 ) , (78) G_{y_{12}}(t_1,t_2)=\frac{e^{-y_{12}t_1}}{2y_{12}}\left(e^{y_{12}t_2}-e^{-y_{12}t_2}\right), \tag*{(78)} G y 12 ( t 1 , t 2 ) = 2 y 12 e − y 12 t 1 ( e y 12 t 2 − e − y 12 t 2 ) , (78) and the same holds for G y 41 ( t 1 , t 4 ) G_{y_{41}}(t_1,t_4) G y 41 ( t 1 , t 4 ) t 1 t_1 t 1 t 2 t_2 t 2 t 4 t_4 t 4
∫ max ( t 2 , t 4 ) ∞ d t 1 e − ( X 1 + y 41 + y 12 ) t 1 = e − ( X 1 + y 41 + y 12 ) max ( t 2 , t 4 ) X 1 + y 41 + y 12 . (79) \int_{\max(t_2,t_4)}^\infty\mathrm{d}t_1\,e^{-(X_1+y_{41}+y_{12})t_1}=\frac{e^{-(X_1+y_{41}+y_{12})\max(t_2,t_4)}}{X_1+y_{41}+y_{12}}. \tag*{(79)} ∫ m a x ( t 2 , t 4 ) ∞ d t 1 e − ( X 1 + y 41 + y 12 ) t 1 = X 1 + y 41 + y 12 e − ( X 1 + y 41 + y 12 ) m a x ( t 2 , t 4 ) . (79) At the pole the exponential on the right-hand side equals one, and the residue is an integral over the times of the other three sites. Each of the two brackets of the form (78) adds ± y \pm y ± y (46) . The recursion for ψ 4 \psi_4 ψ 4 (28) .
The threshold
At the pole the sum of the distances from ℓ ⃗ \vec{\ell} ℓ y 12 y_{12} y 12 y 41 y_{41} y 41 P 1 P_1 P 1 y 41 + y 12 = P 1 y_{41}+y_{12}=P_1 y 41 + y 12 = P 1 y 12 = 1 2 ( P 1 − χ ) y_{12}=\frac{1}{2}(P_1-\chi) y 12 = 2 1 ( P 1 − χ ) y 41 = 1 2 ( P 1 + χ ) y_{41}=\frac{1}{2}(P_1+\chi) y 41 = 2 1 ( P 1 + χ ) χ = y 41 − y 12 \chi=y_{41}-y_{12} χ = y 41 − y 12 C C C a a a y 12 y_{12} y 12 b b b
∣ ℓ ⃗ − C ⃗ ∣ 2 = ( P 1 + χ ) a 2 + ( P 1 − χ ) b 2 2 P 1 − P 1 2 − χ 2 4 . (80) \left|\vec{\ell}-\vec{C}\right|^2=\frac{(P_1+\chi)a^2+(P_1-\chi)b^2}{2P_1}-\frac{P_1^2-\chi^2}{4}. \tag*{(80)} ℓ − C 2 = 2 P 1 ( P 1 + χ ) a 2 + ( P 1 − χ ) b 2 − 4 P 1 2 − χ 2 . (80) With the two remaining vertices of the quadrilateral in place of C C C (47) . Both z 23 = y 23 2 z_{23}=y_{23}^2 z 23 = y 23 2 z 34 = y 34 2 z_{34}=y_{34}^2 z 34 = y 34 2 χ \chi χ 1 4 \frac{1}{4} 4 1
z 23 = 1 4 ( χ − e 1 ) ( χ − e 2 ) , z 34 = 1 4 ( χ − e 3 ) ( χ − e 4 ) , 16 Y 2 = ∏ i = 1 4 ( χ − e i ) , (81) z_{23}=\frac{1}{4}(\chi-e_1)(\chi-e_2),\qquad z_{34}=\frac{1}{4}(\chi-e_3)(\chi-e_4),\qquad16Y^2=\prod_{i=1}^{4}(\chi-e_i), \tag*{(81)} z 23 = 4 1 ( χ − e 1 ) ( χ − e 2 ) , z 34 = 4 1 ( χ − e 3 ) ( χ − e 4 ) , 16 Y 2 = i = 1 ∏ 4 ( χ − e i ) , (81) so that Y 2 = z 23 z 34 Y^2=z_{23}z_{34} Y 2 = z 23 z 34
e 1 , 2 = − m ± m 2 − n , m = P 2 2 − P 12 2 P 1 , n = 2 ( P 2 2 + P 12 2 ) − P 1 2 , (82) e_{1,2}=-m\pm\sqrt{m^2-n},\qquad m=\frac{P_2^2-P_{12}^2}{P_1},\qquad n=2(P_2^2+P_{12}^2)-P_1^2, \tag*{(82)} e 1 , 2 = − m ± m 2 − n , m = P 1 P 2 2 − P 12 2 , n = 2 ( P 2 2 + P 12 2 ) − P 1 2 , (82) and e 3 , 4 e_{3,4} e 3 , 4 P 2 → P 41 P_2\to P_{41} P 2 → P 41 P 12 → P 4 P_{12}\to P_4 P 12 → P 4
Reduction of D 0 D_0 D 0
The integrand of eq. (48) is a rational function of χ \chi χ y 23 y_{23} y 23 y 34 y_{34} y 34 y 23 → − y 23 y_{23} \to-y_{23} y 23 → − y 23 y 34 → − y 34 y_{34} \to-y_{34} y 34 → − y 34 χ \chi χ χ \chi χ Y Y Y
The chain coefficient (10) has poles where one of six linear factors vanishes. Each zero lies at a point χ = α \chi=\alpha χ = α η = ± y 23 ( α ) \eta=\pm y_{23}(\alpha) η = ± y 23 ( α ) η ′ = ± y 34 ( α ) \eta'=\pm y_{34}(\alpha) η ′ = ± y 34 ( α ) ρ \rho ρ (50) ,
D 0 = π 4 P 1 [ 1 4 ∑ ( α , η , η ′ ) ρ ( L 0 + η L 23 + η ′ L 34 + η η ′ L Π ) + 4 p 0 J F ] . (83) D_0=\frac{\pi}{4P_1}\left[\frac{1}{4}\sum_{(\alpha,\eta,\eta')}\rho\left(L_0+\eta L_{23}+\eta' L_{34}+\eta\eta' L_{\Pi}\right)+4p_0J_F\right]. \tag*{(83)} D 0 = 4 P 1 π 4 1 ( α , η , η ′ ) ∑ ρ ( L 0 + η L 23 + η ′ L 34 + η η ′ L Π ) + 4 p 0 J F . (83) with the integrals
L 0 = ∫ − P 1 P 1 d χ χ − α , L 23 = ∫ − P 1 P 1 d χ ( χ − α ) y 23 , L Π = ∫ − P 1 P 1 d χ ( χ − α ) y 23 y 34 , (84) L_0=\int_{-P_1}^{P_1}\frac{\mathrm{d}\chi}{\chi-\alpha},\qquad L_{23}=\int_{-P_1}^{P_1}\frac{\mathrm{d}\chi}{(\chi-\alpha)y_{23}},\qquad L_{\Pi}=\int_{-P_1}^{P_1}\frac{\mathrm{d}\chi}{(\chi-\alpha)y_{23}y_{34}}, \tag*{(84)} L 0 = ∫ − P 1 P 1 χ − α d χ , L 23 = ∫ − P 1 P 1 ( χ − α ) y 23 d χ , L Π = ∫ − P 1 P 1 ( χ − α ) y 23 y 34 d χ , (84) and L 34 L_{34} L 34 L 23 L_{23} L 23 y 34 y_{34} y 34 y 23 y_{23} y 23 Y Y Y J F = ∫ d χ / ( 4 Y ) J_F=\int\mathrm{d}\chi/(4Y) J F = ∫ d χ / ( 4 Y ) J F J_F J F χ = e 1 \chi=e_1 χ = e 1
p 0 = 1 4 ∑ ( α , η , η ′ ) ρ η η ′ α − e 1 . (85) p_0=\frac{1}{4}\sum_{(\alpha,\eta,\eta')}\frac{\rho\eta\eta'}{\alpha-e_1}. \tag*{(85)} p 0 = 4 1 ( α , η , η ′ ) ∑ α − e 1 ρ η η ′ . (85) The first two integrals in eq. (84) are elementary,
L 0 = ln P 1 − α − P 1 − α , L_0=\ln\frac{P_1-\alpha}{-P_1-\alpha}, L 0 = ln − P 1 − α P 1 − α , L 23 = − 1 z 23 ( α ) [ ln 2 z 23 ( α ) + z 23 ′ ( α ) ( χ − α ) + 2 z 23 ( α ) z 23 ( χ ) χ − α ] χ = − P 1 χ = P 1 . (86) L_{23}=-\frac{1}{\sqrt{z_{23}(\alpha)}}\left[\ln\frac{2z_{23}(\alpha)+z'_{23}(\alpha)(\chi-\alpha)+2\sqrt{z_{23}(\alpha)z_{23}(\chi)}}{\chi-\alpha}\right]_{\chi=-P_1}^{\chi=P_1}. \tag*{(86)} L 23 = − z 23 ( α ) 1 [ ln χ − α 2 z 23 ( α ) + z 23 ′ ( α ) ( χ − α ) + 2 z 23 ( α ) z 23 ( χ ) ] χ = − P 1 χ = P 1 . (86) For the elliptic integrals we use the symmetric integrals of Carlson [55 ] , which are R F ( a , b , c ) = 1 2 ∫ 0 ∞ d τ [ ( τ + a ) ( τ + b ) ( τ + c ) ] − 1 / 2 R_F(a,b,c)=\frac{1}{2}\int_0^\infty\mathrm{d}\tau\left[(\tau+a)(\tau+b)(\tau+c)\right]^{-1/2} R F ( a , b , c ) = 2 1 ∫ 0 ∞ d τ [ ( τ + a ) ( τ + b ) ( τ + c ) ] − 1/2 R J ( a , b , c , p ) = 3 2 ∫ 0 ∞ d τ ( τ + p ) − 1 [ ( τ + a ) ( τ + b ) ( τ + c ) ] − 1 / 2 R_J(a,b,c,p)=\frac{3}{2}\int_0^\infty\mathrm{d}\tau(\tau+p)^{-1}\left[(\tau+a)(\tau+b)(\tau+c)\right]^{-1/2} R J ( a , b , c , p ) = 2 3 ∫ 0 ∞ d τ ( τ + p ) − 1 [ ( τ + a ) ( τ + b ) ( τ + c ) ] − 1/2 r i = P 1 − e i r_i=\sqrt{P_1-e_i} r i = P 1 − e i r ˉ i = − P 1 − e i \bar{r}_i=\sqrt{-P_1-e_i} r ˉ i = − P 1 − e i
U i j = r i r j r ˉ k r ˉ l + r ˉ i r ˉ j r k r l 2 P 1 , (87) U_{ij}=\frac{r_i r_j\bar{r}_k\bar{r}_l+\bar{r}_i\bar{r}_j r_k r_l}{2P_1}, \tag*{(87)} U ij = 2 P 1 r i r j r ˉ k r ˉ l + r ˉ i r ˉ j r k r l , (87) where i i i j j j k k k l l l
J F = ∫ − P 1 P 1 d χ 4 Y = 2 R F ( U 12 2 , U 13 2 , U 14 2 ) . (88) J_F=\int_{-P_1}^{P_1}\frac{\mathrm{d}\chi}{4Y}=2R_F\left(U_{12}^{2},U_{13}^{2},U_{14}^{2}\right). \tag*{(88)} J F = ∫ − P 1 P 1 4 Y d χ = 2 R F ( U 12 2 , U 13 2 , U 14 2 ) . (88) Since Y = y 23 y 34 Y=y_{23}y_{34} Y = y 23 y 34 L Π = 4 ( J Π − J F ) / ( α − e 1 ) L_{\Pi}=4(J_{\Pi}-J_F)/(\alpha-e_1) L Π = 4 ( J Π − J F ) / ( α − e 1 )
J Π = ∫ − P 1 P 1 ( χ − e 1 ) d χ 4 ( χ − α ) Y = 2 d 12 d 13 d 14 3 d 15 R J ( U 12 2 , U 13 2 , U 14 2 , U 15 2 ) + 1 w ln S 5 + w S 5 − w . (89) J_{\Pi}=\int_{-P_1}^{P_1}\frac{(\chi-e_1)\mathrm{d}\chi}{4(\chi-\alpha)Y}=\frac{2d_{12}d_{13}d_{14}}{3d_{15}}R_J\left(U_{12}^{2},U_{13}^{2},U_{14}^{2},U_{15}^{2}\right)+\frac{1}{w}\ln\frac{S_5+w}{S_5-w}. \tag*{(89)} J Π = ∫ − P 1 P 1 4 ( χ − α ) Y ( χ − e 1 ) d χ = 3 d 15 2 d 12 d 13 d 14 R J ( U 12 2 , U 13 2 , U 14 2 , U 15 2 ) + w 1 ln S 5 − w S 5 + w . (89) where d i j = e j − e i d_{ij}=e_j-e_i d ij = e j − e i e 5 = α e_5=\alpha e 5 = α
U 15 2 = U 12 2 − d 13 d 14 d 25 d 15 , w 2 = S 5 2 + ( P 1 2 − α 2 ) U 15 2 r 1 2 r ˉ 1 2 , S 5 = 1 2 P 1 [ r 2 r 3 r 4 r 1 ( − P 1 − α ) + r ˉ 2 r ˉ 3 r ˉ 4 r ˉ 1 ( P 1 − α ) ] . (90) \begin{aligned}
U_{15}^{2} &= U_{12}^{2}-\frac{d_{13}d_{14}d_{25}}{d_{15}}, \qquad
w^{2}=S_{5}^{2}+\frac{(P_{1}^{2}-\alpha^{2})U_{15}^{2}}{r_{1}^{2}\bar{r}_{1}^{2}},\\
S_{5} &= \frac{1}{2P_{1}}\left[\frac{r_{2}r_{3}r_{4}}{r_{1}}(-P_{1}-\alpha)+\frac{\bar{r}_{2}\bar{r}_{3}\bar{r}_{4}}{\bar{r}_{1}}(P_{1}-\alpha)\right]. \tag*{(90)}
\end{aligned} U 15 2 S 5 = U 12 2 − d 15 d 13 d 14 d 25 , w 2 = S 5 2 + r 1 2 r ˉ 1 2 ( P 1 2 − α 2 ) U 15 2 , = 2 P 1 1 [ r 1 r 2 r 3 r 4 ( − P 1 − α ) + r ˉ 1 r ˉ 2 r ˉ 3 r ˉ 4 ( P 1 − α ) ] . (90) Equations (83) to (90) are the complete result. We determine the branches of the logarithms and of the Carlson integrals by continuation from χ = − P 1 \chi=-P_{1} χ = − P 1 R F R_{F} R F R J R_{J} R J U i j U_{ij} U ij α = ± P 1 \alpha=\pm P_{1} α = ± P 1
For momenta in a plane the factor P 3 + y 23 + y 34 P_{3}+y_{23}+y_{34} P 3 + y 23 + y 34 χ = α ∗ \chi=\alpha_{*} χ = α ∗ a ∗ / ( χ − α ∗ ) 2 + b ∗ / ( χ − α ∗ ) a_{*}/(\chi-\alpha_{*})^{2}+b_{*}/(\chi-\alpha_{*}) a ∗ / ( χ − α ∗ ) 2 + b ∗ / ( χ − α ∗ ) F ( α ) = L 0 + η L 23 + η ′ L 34 + η η ′ L Π F(\alpha)=L_{0}+\eta L_{23}+\eta^{\prime}L_{34}+\eta\eta^{\prime}L_{\Pi} F ( α ) = L 0 + η L 23 + η ′ L 34 + η η ′ L Π (83) , where η \eta η η ′ \eta^{\prime} η ′ χ = α \chi=\alpha χ = α ∫ d χ g ( χ ) / ( χ − α ) 2 \int\mathrm{d}\chi\,g(\chi)/(\chi-\alpha)^{2} ∫ d χ g ( χ ) / ( χ − α ) 2 ∫ d χ g ( χ ) / ( χ − α ) \int\mathrm{d}\chi\,g(\chi)/(\chi-\alpha) ∫ d χ g ( χ ) / ( χ − α ) α \alpha α
ρ F ( α ) ⟶ a ∗ d F d α ( α ∗ ) + b ∗ F ( α ∗ ) , ρ η η ′ α − e 1 ⟶ a ∗ d d α [ η η ′ α − e 1 ] α ∗ + b ∗ η η ′ α ∗ − e 1 . (91) \rho F(\alpha)\longrightarrow a_{*}\frac{\mathrm{d}F}{\mathrm{d}\alpha}(\alpha_{*})+b_{*}F(\alpha_{*}),\qquad
\frac{\rho\eta\eta^{\prime}}{\alpha-e_{1}}\longrightarrow a_{*}\frac{\mathrm{d}}{\mathrm{d}\alpha}\left[\frac{\eta\eta^{\prime}}{\alpha-e_{1}}\right]_{\alpha_{*}}+b_{*}\frac{\eta\eta^{\prime}}{\alpha_{*}-e_{1}}. \tag*{(91)} ρF ( α ) ⟶ a ∗ d α d F ( α ∗ ) + b ∗ F ( α ∗ ) , α − e 1 ρ η η ′ ⟶ a ∗ d α d [ α − e 1 η η ′ ] α ∗ + b ∗ α ∗ − e 1 η η ′ . (91) in eqs. (83) and (85) . The derivatives are taken along the sheet. The ancillary implementation includes this term.
We obtain the monodromy of D 0 D_{0} D 0 § 6.6 by numerical continuation in one of the lengths of the quadrilateral. We restrict it to the minimal local system of D 0 D_{0} D 0 λ ± 1 \lambda^{\pm1} λ ± 1 λ + λ − 1 = 4 \lambda+\lambda^{-1}=4 λ + λ − 1 = 4 λ + λ − 1 = 18 \lambda+\lambda^{-1}=18 λ + λ − 1 = 18 λ = 9 + 4 5 \lambda=9+4\sqrt{5} λ = 9 + 4 5 ∣ λ ∣ > 1 \lvert\lambda\rvert>1 ∣ λ ∣ > 1 (46) has rank 6 on one family of quadrilaterals and rank 4 on a second family. In both cases its solutions include the two periods of the curve. The system of a logarithmic form on the same curve has rank 1.
The elliptic curves on the line
Three elliptic curves appear on the line of § 6.4 , namely the curves on which two partial energies vanish. The two partial energies may be those of two adjacent sites, of two opposite sites, or of one site and the pair of adjacent sites that contains it. We denote the curves of these three cases by E a d j E_{\mathrm{adj}} E adj E o p p E_{\mathrm{opp}} E opp E n e s t E_{\mathrm{nest}} E nest
The periods of all three are hypergeometric functions of the j j j Y E 2 = 4 X E 3 − g 2 X E − g 3 Y_{E}^{2}=4X_{E}^{3}-g_{2}X_{E}-g_{3} Y E 2 = 4 X E 3 − g 2 X E − g 3 g 2 → ν 4 g 2 g_{2}\to\nu^{4}g_{2} g 2 → ν 4 g 2 g 3 → ν 6 g 3 g_{3}\to\nu^{6}g_{3} g 3 → ν 6 g 3 ν − 1 \nu^{-1} ν − 1 g 2 − 1 / 4 g_{2}^{-1/4} g 2 − 1/4
J = 1728 j = 1 − 27 g 3 2 g 2 3 . (92) J=\frac{1728}{j}=1-\frac{27g_{3}^{2}}{g_{2}^{3}}. \tag*{(92)} J = j 1728 = 1 − g 2 3 27 g 3 2 . (92) This function obeys the hypergeometric equation with parameters 1 12 \frac{1}{12} 12 1 5 12 \frac{5}{12} 12 5 0 < J < 1 0<J<1 0 < J < 1 g 3 > 0 g_3>0 g 3 > 0
ϖ E = 2 π ( 12 g 2 ) 1 / 4 2 F 1 ( 1 12 , 5 12 ; 1 ; J ) . (93) \varpi_E=\frac{2\pi}{(12g_2)^{1/4}}\,{}_2F_1\left(\frac{1}{12},\frac{5}{12};1;J\right). \tag*{(93)} ϖ E = ( 12 g 2 ) 1/4 2 π 2 F 1 ( 12 1 , 12 5 ; 1 ; J ) . (93) For g 3 < 0 g_3<0 g 3 < 0
On the line j j j t t t c 4 c_4 c 4 j j j g 2 g_2 g 2 c 4 c_4 c 4 L E \mathcal{L}_E L E J ( t ) J(t) J ( t ) c 4 1 / 4 c_4^{1/4} c 4 1/4
L E = c 4 − 1 / 4 [ J ( 1 − J ) d 2 d J 2 + ( 1 − 3 J 2 ) d d J − 5 144 ] c 4 1 / 4 , (94) \mathcal{L}_E=c_4^{-1/4}\left[J(1-J)\frac{\mathrm{d}^2}{\mathrm{d}J^2}+\left(1-\frac{3J}{2}\right)\frac{\mathrm{d}}{\mathrm{d}J}-\frac{5}{144}\right]c_4^{1/4}, \tag*{(94)} L E = c 4 − 1/4 [ J ( 1 − J ) d J 2 d 2 + ( 1 − 2 3 J ) d J d − 144 5 ] c 4 1/4 , (94) with both sides normalized to unit leading coefficient in d / d t \mathrm{d}/\mathrm{d}t d / d t (94) holds exactly for the three operators of the reduction. The point J = 0 J=0 J = 0 j = ∞ j=\infty j = ∞ t = 0 t=0 t = 0 t = − 1 + 3 / 2 t=-1+\sqrt{3}/2 t = − 1 + 3 /2 E o p p E_{\mathrm{opp}} E opp t t t t ↦ j t\mapsto j t ↦ j j = 0 j=0 j = 0 ∞ \infty ∞ L E \mathcal{L}_E L E j j j
An elliptic fibration for general momenta
For arbitrary momenta, the measure of the correlator can be written as an elliptic fibration in coordinates built from the edge energies. Let A ⃗ e \vec{A}_e A e y e = ∣ ℓ ⃗ − A ⃗ e ∣ y_e=\lvert\vec{\ell}-\vec{A}_e\rvert y e = ∣ ℓ − A e ∣ e ⃗ 1 = A ⃗ 41 − A ⃗ 12 \vec{e}_1=\vec{A}_{41}-\vec{A}_{12} e 1 = A 41 − A 12 e ⃗ 2 = A ⃗ 23 − A ⃗ 12 \vec{e}_2=\vec{A}_{23}-\vec{A}_{12} e 2 = A 23 − A 12 e ⃗ 3 = A ⃗ 34 − A ⃗ 12 \vec{e}_3=\vec{A}_{34}-\vec{A}_{12} e 3 = A 34 − A 12 G i j = e ⃗ i ⋅ e ⃗ j G_{ij}=\vec{e}_i\cdot\vec{e}_j G ij = e i ⋅ e j g = det G g=\det G g = det G P 1 2 P_1^2 P 1 2 P 2 2 P_2^2 P 2 2 A ⃗ 12 A ⃗ 34 \vec{A}_{12}\vec{A}_{34} A 12 A 34 g g g
σ 1 = y 41 + y 12 , σ 2 = y 12 + y 23 , σ 3 = y 23 + y 34 , (95) \sigma_1=y_{41}+y_{12},\qquad\sigma_2=y_{12}+y_{23},\qquad\sigma_3=y_{23}+y_{34}, \tag*{(95)} σ 1 = y 41 + y 12 , σ 2 = y 12 + y 23 , σ 3 = y 23 + y 34 , (95) and the vectors L = ( σ 1 , σ 2 , σ 2 − σ 3 ) L=(\sigma_1,\sigma_2,\sigma_2-\sigma_3) L = ( σ 1 , σ 2 , σ 2 − σ 3 ) m = 1 2 ( G 11 − σ 1 2 , G 22 − σ 2 2 , G 33 − ( σ 2 − σ 3 ) 2 ) m=\frac{1}{2}\left(G_{11}-\sigma_1^2,G_{22}-\sigma_2^2,G_{33}-(\sigma_2-\sigma_3)^2\right) m = 2 1 ( G 11 − σ 1 2 , G 22 − σ 2 2 , G 33 − ( σ 2 − σ 3 ) 2 ) η i = ℓ ⃗ ⋅ e ⃗ i \eta_i=\vec{\ell}\cdot\vec{e}_i η i = ℓ ⋅ e i A ⃗ 12 \vec{A}_{12} A 12 η = L y 12 + m \eta=Ly_{12}+m η = L y 12 + m y 41 2 = y 12 2 − 2 η 1 + G 11 y_{41}^2=y_{12}^2-2\eta_1+G_{11} y 41 2 = y 12 2 − 2 η 1 + G 11 y 41 = σ 1 − y 12 y_{41}=\sigma_1-y_{12} y 41 = σ 1 − y 12 ∣ ℓ ⃗ ∣ 2 = y 12 2 \lvert\vec{\ell}\rvert^2=y_{12}^2 ∣ ℓ ∣ 2 = y 12 2 η T G − 1 η = y 12 2 \eta^{\mathsf{T}}G^{-1}\eta=y_{12}^2 η T G − 1 η = y 12 2 y 12 y_{12} y 12
A y 12 2 + 2 B y 12 + C = 0 , A = L T adj ( G ) L − g , B = L T adj ( G ) m , C = m T adj ( G ) m , (96) \mathcal{A}y_{12}^2+2\mathcal{B}y_{12}+\mathcal{C}=0,\qquad\mathcal{A}=L^{\mathsf{T}}\operatorname{adj}(G)L-g,\qquad\mathcal{B}=L^{\mathsf{T}}\operatorname{adj}(G)m,\qquad\mathcal{C}=m^{\mathsf{T}}\operatorname{adj}(G)m, \tag*{(96)} A y 12 2 + 2 B y 12 + C = 0 , A = L T adj ( G ) L − g , B = L T adj ( G ) m , C = m T adj ( G ) m , (96) so that y 12 = ( − B ± Δ ) / A y_{12}=(-\mathcal{B}\pm\sqrt{\Delta})/\mathcal{A} y 12 = ( − B ± Δ ) / A
Δ ( σ 2 ; σ 1 , σ 3 ) = B 2 − A C . (97) \Delta(\sigma_2;\sigma_1,\sigma_3)=\mathcal{B}^2-\mathcal{A}\mathcal{C}. \tag*{(97)} Δ ( σ 2 ; σ 1 , σ 3 ) = B 2 − A C . (97) Although A \mathcal{A} A B \mathcal{B} B C \mathcal{C} C σ 2 \sigma_2 σ 2 Δ \Delta Δ σ 2 \sigma_2 σ 2 σ 1 \sigma_1 σ 1 σ 3 \sigma_3 σ 3 G G G σ 1 \sigma_1 σ 1 σ 3 \sigma_3 σ 3 w 2 = Δ ( σ 2 ) w^2=\Delta(\sigma_2) w 2 = Δ ( σ 2 ) y 23 = σ 2 − y 12 y_{23}=\sigma_2-y_{12} y 23 = σ 2 − y 12 y 34 = σ 3 − σ 2 + y 12 y_{34}=\sigma_3-\sigma_2+y_{12} y 34 = σ 3 − σ 2 + y 12 y 41 = σ 1 − y 12 y_{41}=\sigma_1-y_{12} y 41 = σ 1 − y 12 σ 2 \sigma_2 σ 2 w w w ℓ ⃗ ↦ ( σ 1 , σ 2 , σ 3 ) \vec{\ell}\mapsto(\sigma_1,\sigma_2,\sigma_3) ℓ ↦ ( σ 1 , σ 2 , σ 3 )
d 3 ℓ y 12 y 23 y 34 y 41 = g Δ d σ 1 d σ 2 d σ 3 . (98) \frac{\mathrm{d}^3\ell}{y_{12}\,y_{23}\,y_{34}\,y_{41}}=\frac{\sqrt{g}}{\sqrt{\Delta}}\,\mathrm{d}\sigma_1\,\mathrm{d}\sigma_2\,\mathrm{d}\sigma_3. \tag*{(98)} y 12 y 23 y 34 y 41 d 3 ℓ = Δ g d σ 1 d σ 2 d σ 3 . (98) Both statements hold as identities for independent Gram entries; we have checked them symbolically, and eq. (98) numerically at generic points. The correlator integrand is this measure times a rational function of the site energies and the edge energies, with poles at the partial energies: X 1 + σ 1 X_1+\sigma_1 X 1 + σ 1 X 2 + σ 2 X_2+\sigma_2 X 2 + σ 2 X 3 + σ 3 X_3+\sigma_3 X 3 + σ 3 X 4 + σ 1 − σ 2 + σ 3 X_4+\sigma_1-\sigma_2+\sigma_3 X 4 + σ 1 − σ 2 + σ 3 y 12 y_{12} y 12 σ 2 \sigma_2 σ 2 w w w σ 1 \sigma_1 σ 1 σ 3 \sigma_3 σ 3 w 2 = Δ ( σ 2 ; σ 1 , σ 3 ) w^2=\Delta(\sigma_2;\sigma_1,\sigma_3) w 2 = Δ ( σ 2 ; σ 1 , σ 3 ) Ω \Omega Ω (52) is an integral of elliptic integrals over the plane of σ 1 \sigma_1 σ 1 σ 3 \sigma_3 σ 3 σ 2 \sigma_2 σ 2
The fifth-order operator
The order of the differential equation of the periods is 5 for the regular tetrahedron, 6 for tetrahedra whose opposite edges have equal lengths, and 8 for generic momenta. The fifth-order operator of § 6.3 arises in three ways. It is the operator of the periods of the surface as a function of Σ = y 41 + y 12 \Sigma=y_{41}+y_{12} Σ = y 41 + y 12 X v = 1 + t X_v=1+t X v = 1 + t Σ = − ( 1 + t ) \Sigma=-(1+t) Σ = − ( 1 + t ) Σ → 1 + t \Sigma\to1+t Σ → 1 + t Σ \Sigma Σ
The operator is L 5 = ∑ i = 0 5 q i ( Σ ) d i / d Σ i \mathcal{L}_5=\sum_{i=0}^{5}q_i(\Sigma)\mathrm{d}^i/\mathrm{d}\Sigma^i L 5 = ∑ i = 0 5 q i ( Σ ) d i / d Σ i
q 5 = Σ 2 ( Σ 2 − 1 ) 2 ( 3 Σ 2 + 1 ) ( Σ 4 − 4 Σ 2 + 12 ) ( 26 Σ 4 − 41 Σ 2 + 47 ) , q 4 = Σ ( Σ 2 − 1 ) ( 2184 Σ 12 − 10788 Σ 10 + 30503 Σ 8 − 38330 Σ 6 + 33139 Σ 4 − 2272 Σ 2 + 1692 ) , q 3 = 19578 Σ 14 − 93867 Σ 12 + 232539 Σ 10 − 318461 Σ 8 + 274863 Σ 6 − 70908 Σ 4 − 3964 Σ 2 + 1692 , q 2 = 3 Σ 3 ( 21918 Σ 10 − 83741 Σ 8 + 168795 Σ 6 − 189111 Σ 4 + 115763 Σ 2 − 7896 ) , q 1 = 18 Σ 2 ( 4082 Σ 10 − 12389 Σ 8 + 22570 Σ 6 − 19379 Σ 4 + 5392 Σ 2 + 1316 ) , q 0 = 90 Σ 5 ( 182 Σ 6 − 447 Σ 4 + 846 Σ 2 − 509 ) . (99) \begin{aligned}
q_5&=\Sigma^2(\Sigma^2-1)^2(3\Sigma^2+1)(\Sigma^4-4\Sigma^2+12)(26\Sigma^4-41\Sigma^2+47),\\
q_4&=\Sigma(\Sigma^2-1)(2184\Sigma^{12}-10788\Sigma^{10}+30503\Sigma^8-38330\Sigma^6+33139\Sigma^4-2272\Sigma^2+1692),\\
q_3&=19578\Sigma^{14}-93867\Sigma^{12}+232539\Sigma^{10}-318461\Sigma^8+274863\Sigma^6-70908\Sigma^4-3964\Sigma^2+1692,\\
q_2&=3\Sigma^3(21918\Sigma^{10}-83741\Sigma^8+168795\Sigma^6-189111\Sigma^4+115763\Sigma^2-7896),\\
q_1&=18\Sigma^2(4082\Sigma^{10}-12389\Sigma^8+22570\Sigma^6-19379\Sigma^4+5392\Sigma^2+1316),\\
q_0&=90\Sigma^5(182\Sigma^6-447\Sigma^4+846\Sigma^2-509). \tag*{(99)}
\end{aligned} q 5 q 4 q 3 q 2 q 1 q 0 = Σ 2 ( Σ 2 − 1 ) 2 ( 3 Σ 2 + 1 ) ( Σ 4 − 4 Σ 2 + 12 ) ( 26 Σ 4 − 41 Σ 2 + 47 ) , = Σ ( Σ 2 − 1 ) ( 2184 Σ 12 − 10788 Σ 10 + 30503 Σ 8 − 38330 Σ 6 + 33139 Σ 4 − 2272 Σ 2 + 1692 ) , = 19578 Σ 14 − 93867 Σ 12 + 232539 Σ 10 − 318461 Σ 8 + 274863 Σ 6 − 70908 Σ 4 − 3964 Σ 2 + 1692 , = 3 Σ 3 ( 21918 Σ 10 − 83741 Σ 8 + 168795 Σ 6 − 189111 Σ 4 + 115763 Σ 2 − 7896 ) , = 18 Σ 2 ( 4082 Σ 10 − 12389 Σ 8 + 22570 Σ 6 − 19379 Σ 4 + 5392 Σ 2 + 1316 ) , = 90 Σ 5 ( 182 Σ 6 − 447 Σ 4 + 846 Σ 2 − 509 ) . (99) In a suitable basis the five solutions Π 1 , … , Π 5 \Pi_1,\ldots,\Pi_5 Π 1 , … , Π 5
Π 1 Π 2 + Π 3 Π 4 + Π 5 2 = 0. (100) \Pi_1\Pi_2+\Pi_3\Pi_4+\Pi_5^2=0. \tag*{(100)} Π 1 Π 2 + Π 3 Π 4 + Π 5 2 = 0. (100) The symmetric square of L 5 \mathcal{L}_5 L 5 V 4 V_4 V 4 Σ c \Sigma_c Σ c Σ 4 − 4 Σ 2 + 12 \Sigma^4-4\Sigma^2+12 Σ 4 − 4 Σ 2 + 12 ( Σ − Σ c ) 1 / 2 (\Sigma-\Sigma_c)^{1/2} ( Σ − Σ c ) 1/2 ( Σ − Σ c ) 5 / 2 (\Sigma-\Sigma_c)^{5/2} ( Σ − Σ c ) 5/2 V 4 V_4 V 4 V 4 V_4 V 4
The solutions on the line
The reduction on the line gives first-order systems for the master integrals and their coefficients in the expansion in ε \varepsilon ε ε \varepsilon ε V 4 V_4 V 4 C 4 C_4 C 4 − π / 8 -\pi/8 − π /8 − π / 16 -\pi/16 − π /16 V 4 V_4 V 4 C 4 C_4 C 4 ε \varepsilon ε V 4 V_4 V 4 C 4 C_4 C 4
We compare the systems with numerical values of both functions. Some care is needed here, since a fit of all boundary constants to values on a short interval would reproduce the values of any smooth function, and so would not test the systems. We therefore use values close to the singular points and at large t t t
An exact rational change of basis brings the system of dimension 28 to a form with simple poles only. The poles lie at the zeros of the 19 polynomials
3 X 2 + 1 , 10 X 2 − 1 , X 4 − 4 X 2 + 12 , 16 X 4 − 10 X 2 + 3 , X , X + 2 , X − 2 , 3 X + 1 , 3 X − 1 , 2 X 2 + 1 , 13 X 2 − 4 , 12 X 4 − 20 X 2 + 9 , X + 1 , X − 1 , 2 X + 1 , 2 X − 1 , 2 X 2 − 1 , 2 X 2 − 3 , 4 X 4 − 4 X 2 + 3 , (101) \begin{aligned}
&3X^2+1,\quad10X^2-1,\quad X^4-4X^2+12,\quad16X^4-10X^2+3,\\
&X,\quad X+2,\quad X-2,\quad3X+1,\quad3X-1,\quad2X^2+1,\quad13X^2-4,\quad12X^4-20X^2+9,\\
&X+1,\quad X-1,\quad2X+1,\quad2X-1,\quad2X^2-1,\quad2X^2-3,\quad4X^4-4X^2+3, \tag*{(101)}
\end{aligned} 3 X 2 + 1 , 10 X 2 − 1 , X 4 − 4 X 2 + 12 , 16 X 4 − 10 X 2 + 3 , X , X + 2 , X − 2 , 3 X + 1 , 3 X − 1 , 2 X 2 + 1 , 13 X 2 − 4 , 12 X 4 − 20 X 2 + 9 , X + 1 , X − 1 , 2 X + 1 , 2 X − 1 , 2 X 2 − 1 , 2 X 2 − 3 , 4 X 4 − 4 X 2 + 3 , (101) and the system has no other singular points at finite X X X
The loop integral converges for t > − 1 t>-1 t > − 1 C 4 C_4 C 4
t = 1 , − 1 + 3 2 , 0 , − 1 + 1 2 , − 1 + 2 13 , − 1 2 (102) t=1,\quad-1+\sqrt{\frac{3}{2}},\quad0,\quad-1+\frac{1}{\sqrt{2}},\quad-1+\frac{2}{\sqrt{13}},\quad-\frac{1}{2} \tag*{(102)} t = 1 , − 1 + 2 3 , 0 , − 1 + 2 1 , − 1 + 13 2 , − 2 1 (102) form a space of dimension 4. Those that are also single-valued around the logarithmic point t = − 2 3 t=-\frac{2}{3} t = − 3 2 V 4 V_4 V 4 t = − 1 + 1 / 10 t=-1+1/\sqrt{10} t = − 1 + 1/ 10 X X X v v v X − 4 X^{-4} X − 4 1 / X 1/X 1/ X
v = X − 4 + 29 80 X − 6 + 923 12800 X − 8 − 1531 204800 X − 10 + O ( X − 12 ) . (103) v=X^{-4}+\frac{29}{80}X^{-6}+\frac{923}{12800}X^{-8}-\frac{1531}{204800}X^{-10}+O(X^{-12}). \tag*{(103)} v = X − 4 + 80 29 X − 6 + 12800 923 X − 8 − 204800 1531 X − 10 + O ( X − 12 ) . (103) The function u u u α 4 = 0 \alpha_4=0 α 4 = 0 β 4 \beta_4 β 4 β 8 \beta_8 β 8 α 5 \alpha_5 α 5 ε \varepsilon ε − π / 8 -\pi/8 − π /8 v v v t = − 2 3 t=-\frac{2}{3} t = − 3 2
The condition at t = − 1 + 1 / 10 t=-1+1/\sqrt{10} t = − 1 + 1/ 10 X 2 = 1 10 X^2=\frac{1}{10} X 2 = 10 1 v v v v → − v v\to-v v → − v u → u + 2 c v v u\to u+2c_vv u → u + 2 c v v c v c_v c v u + Ω v u+\Omega v u + Ω v c v = Ω c_v=\Omega c v = Ω c v c_v c v 16 X 4 − 10 X 2 + 3 16X^4-10X^2+3 16 X 4 − 10 X 2 + 3
The expansion at large site energy
We now derive the leading coefficients of eq. (53). At large X X X d = 3 − 2 ε d=3-2\varepsilon d = 3 − 2 ε ε ˉ = − ε \bar{\varepsilon}=-\varepsilon ε ˉ = − ε (13) . In the soft region the loop momentum is of order one. We expand the integrand of the correlator in y e / X y_e/X y e / X
1 y 12 y 23 y 34 y 41 ∑ m ≥ 0 X − 4 − m κ m ( y ) , κ 0 = 1 , κ 1 = − 3 2 ( y 12 + y 23 + y 34 + y 41 ) , (104) \frac{1}{y_{12}y_{23}y_{34}y_{41}}\sum_{m\ge0}X^{-4-m}\kappa_m(y),\qquad\kappa_0=1,\qquad\kappa_1=-\frac{3}{2}(y_{12}+y_{23}+y_{34}+y_{41}), \tag*{(104)} y 12 y 23 y 34 y 41 1 m ≥ 0 ∑ X − 4 − m κ m ( y ) , κ 0 = 1 , κ 1 = − 2 3 ( y 12 + y 23 + y 34 + y 41 ) , (104) where κ m \kappa_m κ m m m m X X X − 8 -8 − 8 X − 5 − 2 ε ˉ X^{-5-2\bar{\varepsilon}} X − 5 − 2 ε ˉ ε ˉ \bar{\varepsilon} ε ˉ X X X
The leading term comes from the soft region alone, and it is Ω X − 4 \Omega X^{-4} Ω X − 4 Ω \Omega Ω (52) . Its numerical value is Ω = 21.4444295139357745195031667653426556 \Omega=21.4444295139357745195031667653426556 Ω = 21.4444295139357745195031667653426556 Ω \Omega Ω x e x_e x e 1 / y e = 1 / ∣ ℓ ⃗ − A ⃗ e ∣ 1/y_e=1/\lvert\vec{\ell}-\vec{A}_e\rvert 1/ y e = 1/ ∣ ℓ − A e ∣ A ⃗ e \vec{A}_e A e
∑ e x e ( ℓ ⃗ − A ⃗ e ) 2 = ( ℓ ⃗ − ∑ e x e A ⃗ e ) 2 + ∑ e < f x e x f (105) \sum_e x_e(\vec{\ell}-\vec{A}_e)^2=\left(\vec{\ell}-\sum_e x_e\vec{A}_e\right)^2+\sum_{e<f}x_ex_f \tag*{(105)} e ∑ x e ( ℓ − A e ) 2 = ( ℓ − e ∑ x e A e ) 2 + e < f ∑ x e x f (105) on the simplex ∑ e x e = 1 \sum_e x_e=1 ∑ e x e = 1 1 / π 2 1/\pi^2 1/ π 2 ∫ d 3 ℓ ( ℓ 2 + Δ ) − 2 = π 2 Δ − 1 / 2 \int\mathrm{d}^3\ell\,(\ell^2+\Delta)^{-2}=\pi^2\Delta^{-1/2} ∫ d 3 ℓ ( ℓ 2 + Δ ) − 2 = π 2 Δ − 1/2 ℓ ⃗ \vec{\ell} ℓ
Ω = ∫ ∑ x = 1 d 3 x x 1 x 2 x 3 x 4 E 2 ( x ) , E 2 ( x ) = ∑ e < f x e x f . (106) \Omega=\int_{\sum x=1}\frac{\mathrm{d}^3x}{\sqrt{x_1x_2x_3x_4}\sqrt{E_2(x)}},\qquad E_2(x)=\sum_{e<f}x_ex_f. \tag*{(106)} Ω = ∫ ∑ x = 1 x 1 x 2 x 3 x 4 E 2 ( x ) d 3 x , E 2 ( x ) = e < f ∑ x e x f . (106) The integral over the simplex can also be reduced to one dimension. With ϱ = x ( 1 − x ) \varrho=x(1-x) ϱ = x ( 1 − x ) h = 1 + 6 ϱ h=1+6\varrho h = 1 + 6 ϱ
m ± = 1 2 ( 1 ± x 4 − ( 1 − x ) 4 h 2 − 8 ϱ ( 1 + 2 ϱ ) ( 4 + 9 ϱ ) h 2 ) , (107) m_{\pm}=\frac{1}{2}\left(1\pm\frac{x^4-(1-x)^4}{h^2}-\frac{8\varrho\sqrt{(1+2\varrho)(4+9\varrho)}}{h^2}\right), \tag*{(107)} m ± = 2 1 ( 1 ± h 2 x 4 − ( 1 − x ) 4 − h 2 8 ϱ ( 1 + 2 ϱ ) ( 4 + 9 ϱ ) ) , (107) and find
Ω = 2 2 π 2 ∫ 0 1 d x h 2 F 1 ( 1 4 , 3 4 ; 1 ; m + ) 2 F 1 ( 1 4 , 3 4 ; 1 ; m − ) . (108) \Omega=2\sqrt{2}\pi^2\int_0^1\frac{\mathrm{d}x}{\sqrt{h}}\,{}_2F_1\left(\frac{1}{4},\frac{3}{4};1;m_+\right){}_2F_1\left(\frac{1}{4},\frac{3}{4};1;m_-\right). \tag*{(108)} Ω = 2 2 π 2 ∫ 0 1 h d x 2 F 1 ( 4 1 , 4 3 ; 1 ; m + ) 2 F 1 ( 4 1 , 4 3 ; 1 ; m − ) . (108) Each hypergeometric function is a complete elliptic integral of the first kind. A simpler form is given in eq. (117) below. To derive it, we write x 1 = x θ x_1=x\theta x 1 = x θ x 2 = x ( 1 − θ ) x_2=x(1-\theta) x 2 = x ( 1 − θ ) x 3 = ( 1 − x ) θ ′ x_3=(1-x)\theta' x 3 = ( 1 − x ) θ ′ x 4 = ( 1 − x ) ( 1 − θ ′ ) x_4=(1-x)(1-\theta') x 4 = ( 1 − x ) ( 1 − θ ′ ) θ \theta θ θ ′ \theta' θ ′ x x x
∫ 0 1 ∫ 0 1 d θ d θ ′ θ ( 1 − θ ) θ ′ ( 1 − θ ′ ) E 2 = π 2 ϱ F 2 ( 1 2 ; 1 2 , 1 2 ; 1 , 1 ; − x 4 ( 1 − x ) , 1 − x 4 x ) . (109) \int_0^1\int_0^1\frac{\mathrm{d}\theta\,\mathrm{d}\theta'}{\sqrt{\theta(1-\theta)\theta'(1-\theta')}E_2}
=\frac{\pi^2}{\sqrt{\varrho}}F_2\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};1,1;-\frac{x}{4(1-x)},\frac{1-x}{4x}\right). \tag*{(109)} ∫ 0 1 ∫ 0 1 θ ( 1 − θ ) θ ′ ( 1 − θ ′ ) E 2 d θ d θ ′ = ϱ π 2 F 2 ( 2 1 ; 2 1 , 2 1 ; 1 , 1 ; − 4 ( 1 − x ) x , 4 x 1 − x ) . (109) Two identities for Appell functions [58 ] then give eq. (108) . The first is the quadratic transformation
F 2 ( a ; b , b ′ ; 2 b , 2 b ′ ; ξ , ζ ) = ( 1 − ξ + ζ 2 ) − a F 4 ( a 2 , a + 1 2 ; b + 1 2 , b ′ + 1 2 ; ξ 2 ( 2 − ξ − ζ ) 2 , ζ 2 ( 2 − ξ − ζ ) 2 ) . (110) F_2(a;b,b';2b,2b';\xi,\zeta)=\left(1-\frac{\xi+\zeta}{2}\right)^{-a}F_4\left(\frac{a}{2},\frac{a+1}{2};b+\frac{1}{2},b'+\frac{1}{2};\frac{\xi^2}{(2-\xi-\zeta)^2},\frac{\zeta^2}{(2-\xi-\zeta)^2}\right). \tag*{(110)} F 2 ( a ; b , b ′ ; 2 b , 2 b ′ ; ξ , ζ ) = ( 1 − 2 ξ + ζ ) − a F 4 ( 2 a , 2 a + 1 ; b + 2 1 , b ′ + 2 1 ; ( 2 − ξ − ζ ) 2 ξ 2 , ( 2 − ξ − ζ ) 2 ζ 2 ) . (110) and the second is the product formula
F 4 ( a , b ; c , a + b − c + 1 ; m + ( 1 − m − ) , m − ( 1 − m + ) ) = 2 F 1 ( a , b ; c ; m + ) 2 F 1 ( a , b ; a + b − c + 1 ; m − ) . (111) F_4(a,b;c,a+b-c+1;m_+(1-m_-),m_-(1-m_+))={}_2F_1(a,b;c;m_+){}_2F_1(a,b;a+b-c+1;m_-). \tag*{(111)} F 4 ( a , b ; c , a + b − c + 1 ; m + ( 1 − m − ) , m − ( 1 − m + )) = 2 F 1 ( a , b ; c ; m + ) 2 F 1 ( a , b ; a + b − c + 1 ; m − ) . (111) At order X − 5 X^{-5} X − 5 − 6 T 3 -6T_3 − 6 T 3 T 3 T_3 T 3 (52) without the factor 1 / y 41 1/y_{41} 1/ y 41 (106) now give
T 3 = ∫ d ℓ x 1 y 12 y 23 y 34 = π − ε ˉ Γ ( ε ˉ ) ∫ Σ x = 1 d 2 x x 1 x 2 x 3 E 2 ( x ) − ε ˉ = 2 π ε ˉ − 2 π ( γ E + ln π ) − J 1 + O ( ε ˉ ) , (112) T_3=\int d^\ell x\,\frac{1}{y_{12}y_{23}y_{34}}=\pi^{-\bar{\varepsilon}}\Gamma(\bar{\varepsilon})\int_{\Sigma_{x=1}}\frac{d^2x}{\sqrt{x_1x_2x_3}}E_2(x)^{-\bar{\varepsilon}}=\frac{2\pi}{\bar{\varepsilon}}-2\pi(\gamma_E+\ln\pi)-J_1+O(\bar{\varepsilon}), \tag*{(112)} T 3 = ∫ d ℓ x y 12 y 23 y 34 1 = π − ε ˉ Γ ( ε ˉ ) ∫ Σ x = 1 x 1 x 2 x 3 d 2 x E 2 ( x ) − ε ˉ = ε ˉ 2 π − 2 π ( γ E + ln π ) − J 1 + O ( ε ˉ ) , (112) with
∫ Σ x = 1 d 2 x x 1 x 2 x 3 = 2 π , J 1 = ∫ Σ x = 1 d 2 x x 1 x 2 x 3 ln E 2 ( x ) = 4 π ln 3 − 8 π . (113) \int_{\Sigma_{x=1}}\frac{d^2x}{\sqrt{x_1x_2x_3}}=2\pi,\qquad J_1=\int_{\Sigma_{x=1}}\frac{d^2x}{\sqrt{x_1x_2x_3}}\ln E_2(x)=4\pi\ln3-8\pi. \tag*{(113)} ∫ Σ x = 1 x 1 x 2 x 3 d 2 x = 2 π , J 1 = ∫ Σ x = 1 x 1 x 2 x 3 d 2 x ln E 2 ( x ) = 4 π ln 3 − 8 π . (113) The pole of the soft region is therefore − 12 π / ε ˉ -12\pi/\bar{\varepsilon} − 12 π / ε ˉ X − 2 ε ˉ X^{-2\bar{\varepsilon}} X − 2 ε ˉ − 24 π X − 5 ln X -24\pi X^{-5}\ln X − 24 π X − 5 ln X β 5 \beta_5 β 5 (53) . The finite parts combine to
α 5 = π 12 ( 22 + 291 ln 3 − 108 ln 2 ) . (114) \alpha_5=\frac{\pi}{12}\left(22+291\ln3-108\ln2\right). \tag*{(114)} α 5 = 12 π ( 22 + 291 ln 3 − 108 ln 2 ) . (114) The higher orders follow in the same way. Each soft term is a one-loop integral with simple poles only, so that no higher power of the logarithm appears. A logarithm requires an ultraviolet pole of a soft term, which exists only at odd orders, so that β n = 0 \beta_n=0 β n = 0 n n n
α 6 = − 95 3 π + 8 Ω ~ , β 7 = − 158 3 π , α 7 = π ( 22519 360 + 423 8 ln 3 − 16 ln 2 ) , (115) \alpha_6=-\frac{95}{3}\pi+8\widetilde{\Omega},\qquad\beta_7=-\frac{158}{3}\pi,\qquad\alpha_7=\pi\left(\frac{22519}{360}+\frac{423}{8}\ln3-16\ln2\right), \tag*{(115)} α 6 = − 3 95 π + 8 Ω , β 7 = − 3 158 π , α 7 = π ( 360 22519 + 8 423 ln 3 − 16 ln 2 ) , (115) where Ω ~ = − 3.3142572208468856411918120617 \widetilde{\Omega}=-3.3142572208468856411918120617 Ω = − 3.3142572208468856411918120617 (52) with y 12 − 1 y_{12}^{-1} y 12 − 1 y 12 y_{12} y 12 (103) the coefficients are α n = a n Ω + u n \alpha_n=a_n\Omega+u_n α n = a n Ω + u n a n a_n a n u n u_n u n X − n X^{-n} X − n v v v u u u β n \beta_n β n u u u β 9 = − 627 10 π \beta_9=-\frac{627}{10}\pi β 9 = − 10 627 π β 11 = 10247 840 π \beta_{11}=\frac{10247}{840}\pi β 11 = 840 10247 π ∣ X ∣ > 2 \lvert X\rvert>2 ∣ X ∣ > 2 Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω
S j ( a , b ) = ∑ m , n ≥ 0 ( 1 ) m + 2 n ( j + 1 ) m + n ( 2 j + 1 ) m ( 1 2 ) n [ ( j + 3 2 ) m + n ] 2 ( 1 ) n a m b n m ! n ! , (116) S_j(a,b)=\sum_{m,n\geq0}\frac{(1)_{m+2n}(j+1)_{m+n}(2j+1)_m\left(\frac{1}{2}\right)_n}{\left[\left(j+\frac{3}{2}\right)_{m+n}\right]^2(1)_n}\frac{a^m b^n}{m!\,n!}, \tag*{(116)} S j ( a , b ) = m , n ≥ 0 ∑ [ ( j + 2 3 ) m + n ] 2 ( 1 ) n ( 1 ) m + 2 n ( j + 1 ) m + n ( 2 j + 1 ) m ( 2 1 ) n m ! n ! a m b n , (116) one finds
Ω = 2 π 3 3 [ 2 F 1 ( 1 8 , 3 8 ; 1 ; 1 9 ) ] 2 − 4 π S 0 ( 1 4 , 1 16 ) , Ω ~ = 3 4 Ω − 4 E , (117) \Omega=\frac{2\pi^3}{\sqrt{3}}\left[{}_2F_1\left(\frac{1}{8},\frac{3}{8};1;\frac{1}{9}\right)\right]^2-4\pi S_0\left(\frac{1}{4},\frac{1}{16}\right),\qquad\widetilde{\Omega}=\frac{3}{4}\Omega-4E, \tag*{(117)} Ω = 3 2 π 3 [ 2 F 1 ( 8 1 , 8 3 ; 1 ; 9 1 ) ] 2 − 4 π S 0 ( 4 1 , 16 1 ) , Ω = 4 3 Ω − 4 E , (117) where E E E
E = ∫ 0 1 d x ∫ 0 1 d θ ∫ 0 1 d η Q 1 / 2 θ ( 1 − θ ) η ( 1 − η ) , Q = x ( 1 − x ) + x 2 θ ( 1 − θ ) + ( 1 − x ) 2 η ( 1 − η ) , (118) E=\int_0^1dx\int_0^1d\theta\int_0^1d\eta\,\frac{Q^{1/2}}{\sqrt{\theta(1-\theta)\eta(1-\eta)}},\qquad Q=x(1-x)+x^2\theta(1-\theta)+(1-x)^2\eta(1-\eta), \tag*{(118)} E = ∫ 0 1 d x ∫ 0 1 d θ ∫ 0 1 d η θ ( 1 − θ ) η ( 1 − η ) Q 1/2 , Q = x ( 1 − x ) + x 2 θ ( 1 − θ ) + ( 1 − x ) 2 η ( 1 − η ) , (118) and Ω \Omega Ω Q − 1 / 2 Q^{-1/2} Q − 1/2 E = 4.8493948390746791327 … E=4.8493948390746791327\ldots E = 4.8493948390746791327 … F 2 F_2 F 2 ( 1 4 , 1 4 ) \left(\frac{1}{4},\frac{1}{4}\right) ( 4 1 , 4 1 ) S 1 ( 1 4 , 1 16 ) S_1\left(\frac{1}{4},\frac{1}{16}\right) S 1 ( 4 1 , 16 1 ) Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω
In fact, every coefficient of eq. (53) is a rational combination of π \pi π π ln 2 \pi\ln2 π ln 2 π ln 3 \pi\ln3 π ln 3 Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω X = ∞ X=\infty X = ∞ π \pi π π ln 2 \pi\ln2 π ln 2 π ln 3 \pi\ln3 π ln 3 Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω α n \alpha_n α n π \pi π π ln 2 \pi\ln2 π ln 2 π ln 3 \pi\ln3 π ln 3 n n n π \pi π Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω n n n β n \beta_n β n π \pi π n n n n n n
α 8 = − 11783 81 π − 3 Ω + 22 Ω ~ , α 10 = − 339187 720 π − 7 Ω + 466 15 Ω ~ , α 9 = π ( 34688681 129600 − 244 15 ln 2 + 60317 960 ln 3 ) . (119) \begin{aligned}
\alpha_8 &= -\frac{11783}{81}\pi- 3\Omega+ 22\widetilde{\Omega}, \qquad
\alpha_{10} = -\frac{339187}{720}\pi- 7\Omega+ \frac{466}{15}\widetilde{\Omega},\\
\alpha_9 &= \pi\left(\frac{34688681}{129600} - \frac{244}{15}\ln2 + \frac{60317}{960}\ln3\right). \tag*{(119)}
\end{aligned} α 8 α 9 = − 81 11783 π − 3Ω + 22 Ω , α 10 = − 720 339187 π − 7Ω + 15 466 Ω , = π ( 129600 34688681 − 15 244 ln 2 + 960 60317 ln 3 ) . (119) The structure of the system also fixes the class of functions. The matrix of the system of dimension 28 is block triangular with 19 diagonal blocks (25 for the system of V 4 V_4 V 4 L 5 \mathcal{L}_5 L 5 (99) , three of size 2, which are the Picard–Fuchs systems of the three elliptic curves of appendix § 10 .§ 4 , and blocks of size 1 whose solutions are rational functions times the square roots of the letters. The longest chain of blocks has length 11 (12 for V 4 V_4 V 4 u u u C 4 C_4 C 4 V 4 V_4 V 4
The wavefunction coefficient on the line
For the coefficient V 4 V_4 V 4 1 / ∏ e y e 1/\prod_e y_e 1/ ∏ e y e X X X X − 5 X^{-5} X − 5
V 4 = ∑ n ≥ 5 X − n ( α n V + β n V ln X ) . (120) V_4 = \sum_{n\geq5} X^{-n}\left(\alpha_n^V+\beta_n^V\ln X\right). \tag*{(120)} V 4 = n ≥ 5 ∑ X − n ( α n V + β n V ln X ) . (120) Through X − 11 X^{-11} X − 11 π \pi π π ln 2 \pi\ln2 π ln 2 π ln 3 \pi\ln3 π ln 3 Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω X − 12 X^{-12} X − 12 X − 11 X^{-11} X − 11 β 11 V = − 11 21 π \beta_{11}^V=-\frac{11}{21}\pi β 11 V = − 21 11 π X X X X − 12 X^{-12} X − 12 t = − 1 + 1 / 10 t=-1+1/\sqrt{10} t = − 1 + 1/ 10 Ω \Omega Ω t = 0 t=0 t = 0 V 4 V_4 V 4 α 12 V \alpha_{12}^V α 12 V α 14 V \alpha_{14}^V α 14 V
α 12 V = 16 ∫ d d ℓ y 12 y 23 y 34 y 41 − 11596 5103 π , α 14 V = 256 ∫ d d ℓ y 12 3 y 23 y 34 y 41 − 2718675343 220449600 π . (121) \alpha_{12}^V=16\int\mathrm{d}^d\ell\, y_{12}y_{23}y_{34}y_{41}-\frac{11596}{5103}\pi,\qquad
\alpha_{14}^V=256\int\mathrm{d}^d\ell\, y_{12}^{3}y_{23}y_{34}y_{41}-\frac{2718675343}{220449600}\pi. \tag*{(121)} α 12 V = 16 ∫ d d ℓ y 12 y 23 y 34 y 41 − 5103 11596 π , α 14 V = 256 ∫ d d ℓ y 12 3 y 23 y 34 y 41 − 220449600 2718675343 π . (121) and the two determinations agree to 36 digits. Integration by parts reduces the two integrals to Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω π \pi π
α 12 V = Ω 56 + 19 Ω ~ 105 − 58466 25515 π , α 14 V = − 4 Ω 105 + 2038 Ω ~ 1575 − 115599847 8817984 π . (122) \alpha_{12}^V=\frac{\Omega}{56}+\frac{19\widetilde{\Omega}}{105}-\frac{58466}{25515}\pi,\qquad
\alpha_{14}^V=-\frac{4\Omega}{105}+\frac{2038\widetilde{\Omega}}{1575}-\frac{115599847}{8817984}\pi. \tag*{(122)} α 12 V = 56 Ω + 105 19 Ω − 25515 58466 π , α 14 V = − 105 4Ω + 1575 2038 Ω − 8817984 115599847 π . (122) and these expressions agree with the coefficients determined by the system to 32 digits. As for the correlator, the soft and hard regions determine all 34 boundary constants of the system exactly, and every α n V \alpha_n^V α n V π \pi π π ln 2 \pi\ln2 π ln 2 π ln 3 \pi\ln3 π ln 3 n n n π \pi π Ω \Omega Ω Ω ~ \widetilde{\Omega} Ω n n n β n V \beta_n^V β n V π \pi π n n n n n n
α 11 V π = 760664581 137168640 − 1605383 209952 ln 2 + 69 64 ln 3 , α 13 V π = 67959374957557 9777380659200 − 1160027389 340122240 ln 2 + 2506681 645120 ln 3 , α 15 V π = 26196178644661273 1048624075699200 − 490944815353 314272949760 ln 2 + 24289259 2027520 ln 3 , β 13 V π = − 197 56 , β 15 V π = − 1297433 110880 . (123) \begin{aligned}
\frac{\alpha_{11}^V}{\pi} &= \frac{760664581}{137168640} - \frac{1605383}{209952}\ln2 + \frac{69}{64}\ln3,\\
\frac{\alpha_{13}^V}{\pi} &= \frac{67959374957557}{9777380659200} - \frac{1160027389}{340122240}\ln2 + \frac{2506681}{645120}\ln3,\\
\frac{\alpha_{15}^V}{\pi} &= \frac{26196178644661273}{1048624075699200} - \frac{490944815353}{314272949760}\ln2 + \frac{24289259}{2027520}\ln3,\\
\frac{\beta_{13}^V}{\pi} &= -\frac{197}{56},\qquad\frac{\beta_{15}^V}{\pi} = -\frac{1297433}{110880}. \tag*{(123)}
\end{aligned} π α 11 V π α 13 V π α 15 V π β 13 V = 137168640 760664581 − 209952 1605383 ln 2 + 64 69 ln 3 , = 9777380659200 67959374957557 − 340122240 1160027389 ln 2 + 645120 2506681 ln 3 , = 1048624075699200 26196178644661273 − 314272949760 490944815353 ln 2 + 2027520 24289259 ln 3 , = − 56 197 , π β 15 V = − 110880 1297433 . (123) and the series agrees with the numerical values of V 4 V_4 V 4
Direct derivation of the closed form with one leg per site
In this appendix, we sketch a derivation of eq. (32) from the integral (13) that uses neither the differential equation of section 4.2 nor numerical values. We set X v = P v X_v = P_v X v = P v ξ i = q G − 2 X i \xi_i = qG - 2X_i ξ i = q G − 2 X i u i = ξ i / q G u_i = \xi_i/qG u i = ξ i / q G (33) . Two of the three integrations can be carried out in closed form, after which every remaining integral is the integral of a rational function times a logarithm in one variable. The symbol of these integrals and the cancellation of the terms of lower weight involve large expressions (about 6 × 10 4 6 \times10^4 6 × 1 0 4
Site sums and the first integration
As coordinates we use the sums of the two edge energies at each site,
s 1 = y 12 + y 31 , s 2 = y 12 + y 23 , s 3 = y 23 + y 31 , (124) s_1 = y_{12} + y_{31},\qquad s_2 = y_{12} + y_{23},\qquad s_3 = y_{23} + y_{31}, \tag*{(124)} s 1 = y 12 + y 31 , s 2 = y 12 + y 23 , s 3 = y 23 + y 31 , (124) so that the partial energy of site v v v X v + s v X_v + s_v X v + s v d 3 y = 1 2 d 3 s \mathrm{d}^3y = \frac{1}{2}\mathrm{d}^3s d 3 y = 2 1 d 3 s B B B (12) is quadratic in each s v s_v s v V = U 12 + U 23 + U 31 V = U_{12} + U_{23} + U_{31} V = U 12 + U 23 + U 31 U 12 U_{12} U 12 (13) of the first term of the recursion (9), in which the edge y 12 y_{12} y 12 U 23 U_{23} U 23 U 31 U_{31} U 31 (10) , the integrand of U 12 U_{12} U 12
B − 1 / 2 y 12 y 23 y 31 ( X 1 + s 1 ) ( X 2 + s 2 ) ( X 3 + s 3 ) ( q G + 2 y 12 ) [ 1 X 2 + X 3 + s 1 + 1 X 1 + X 3 + s 2 ] . (125) \frac{B^{-1/2}y_{12}y_{23}y_{31}}{(X_1+s_1)(X_2+s_2)(X_3+s_3)(qG+2y_{12})}
\left[\frac{1}{X_2+X_3+s_1}+\frac{1}{X_1+X_3+s_2}\right]. \tag*{(125)} ( X 1 + s 1 ) ( X 2 + s 2 ) ( X 3 + s 3 ) ( q G + 2 y 12 ) B − 1/2 y 12 y 23 y 31 [ X 2 + X 3 + s 1 1 + X 1 + X 3 + s 2 1 ] . (125) We first integrate over s 3 s_3 s 3 B = c 2 s 3 2 + c 1 s 3 + c 0 B = c_2s_3^2 + c_1s_3 + c_0 B = c 2 s 3 2 + c 1 s 3 + c 0 s 1 s_1 s 1 s 2 s_2 s 2 B > 0 B > 0 B > 0 B B B c 2 < 0 c_2 < 0 c 2 < 0 s 3 s_3 s 3 (125) is a polynomial of degree one plus two simple poles, so the integral gives three terms. The polynomial part gives a rational function of s 1 s_1 s 1 s 2 s_2 s 2 − c 2 \sqrt{-c_2} − c 2
− 2 c 2 = κ ( P ) + 4 P 1 2 s 2 2 + 4 P 2 2 s 1 2 − 4 ( P 1 2 + P 2 2 − P 3 2 ) s 1 s 2 , (126) -2c_2 = \kappa(P) + 4P_1^2s_2^2 + 4P_2^2s_1^2 - 4(P_1^2 + P_2^2 - P_3^2)s_1s_2, \tag*{(126)} − 2 c 2 = κ ( P ) + 4 P 1 2 s 2 2 + 4 P 2 2 s 1 2 − 4 ( P 1 2 + P 2 2 − P 3 2 ) s 1 s 2 , (126) with κ \kappa κ (35) . The pole X 3 + s 3 = 0 X_3+s_3=0 X 3 + s 3 = 0 − B \sqrt{-B} − B s 3 = − P 3 s_3=-P_3 s 3 = − P 3 − B -B − B s 1 s_1 s 1 s 2 s_2 s 2 X 3 = P 3 X_3=P_3 X 3 = P 3 q G + 2 y 12 = 0 qG+2y_{12}=0 q G + 2 y 12 = 0 (25) , which is a rational function of z 23 = y 23 2 z_{23}=y_{23}^2 z 23 = y 23 2 z 31 = y 31 2 z_{31}=y_{31}^2 z 31 = y 31 2 B B B (19) therefore never appears: the remaining integrals over this plane give logarithms.
Tangent lines and the second integration
After the second integration, the square roots of − c 2 -c_2 − c 2 − B -B − B B B B s 3 s_3 s 3
c 1 2 − 4 c 0 c 2 = − 4 q G ξ 1 ξ 2 ξ 3 ( s 1 2 − P 1 2 ) ( s 2 2 − P 2 2 ) ( ( s 1 − s 2 ) 2 − P 3 2 ) (127) c_1^2-4c_0c_2=-4q_G\xi_1\xi_2\xi_3\left(s_1^2-P_1^2\right)\left(s_2^2-P_2^2\right)\left(\left(s_1-s_2\right)^2-P_3^2\right) \tag*{(127)} c 1 2 − 4 c 0 c 2 = − 4 q G ξ 1 ξ 2 ξ 3 ( s 1 2 − P 1 2 ) ( s 2 2 − P 2 2 ) ( ( s 1 − s 2 ) 2 − P 3 2 ) (127) and the six lines on which it vanishes, s 1 = ± P 1 s_1=\pm P_1 s 1 = ± P 1 s 2 = ± P 2 s_2=\pm P_2 s 2 = ± P 2 s 1 − s 2 = ± P 3 s_1-s_2=\pm P_3 s 1 − s 2 = ± P 3 c 2 = 0 c_2=0 c 2 = 0 − 2 c 2 -2c_2 − 2 c 2 − 2 c 2 = ( P 1 2 + 2 P 1 s 2 + P 2 2 − P 3 2 ) 2 -2c_2=\left(P_1^2+2P_1s_2+P_2^2-P_3^2\right)^2 − 2 c 2 = ( P 1 2 + 2 P 1 s 2 + P 2 2 − P 3 2 ) 2 s 1 = − P 1 s_1=-P_1 s 1 = − P 1 s 1 ≥ P 1 s_1\ge P_1 s 1 ≥ P 1 s 2 ≥ P 2 s_2\ge P_2 s 2 ≥ P 2 ∣ s 1 − s 2 ∣ ≤ P 3 \lvert s_1-s_2\rvert\le P_3 ∣ s 1 − s 2 ∣ ≤ P 3 X 1 + s 1 = 0 X_1+s_1=0 X 1 + s 1 = 0 X 2 + s 2 = 0 X_2+s_2=0 X 2 + s 2 = 0 (125) . Each term of the bracket contains one of them, and we integrate each term first along the direction of its own pole: the term with X 1 + X 3 + s 2 X_1+X_3+s_2 X 1 + X 3 + s 2 s 1 s_1 s 1 X 2 + X 3 + s 1 X_2+X_3+s_1 X 2 + X 3 + s 1 s 2 s_2 s 2 s 3 = − P 3 s_3=-P_3 s 3 = − P 3 q G + 2 y 12 = 0 q_G+2y_{12}=0 q G + 2 y 12 = 0 z 23 z_{23} z 23 − B -B − B
The result is that each U e U_e U e
∫ a b d v [ ∑ g K g ( v ) ln ∣ A g ( v ) ∣ + R 0 ( v ) ] (128) \int_a^b\mathrm{d}v\left[\sum_g K_g(v)\ln\lvert A_g(v)\rvert+R_0(v)\right] \tag*{(128)} ∫ a b d v [ g ∑ K g ( v ) ln ∣ A g ( v )∣ + R 0 ( v ) ] (128) where K g K_g K g R 0 R_0 R 0 A g A_g A g v v v
The symbol
The terms of weight two in eq. (128) come from the simple poles of the K g K_g K g
∫ ln ∣ v − c ∣ v − p d v = ln ∣ p − c ∣ ln ∣ v − p ∣ − Re Li 2 ( v − p c − p ) (129) \int\frac{\ln\lvert v-c\rvert}{v-p}\,\mathrm{d}v=\ln\lvert p-c\rvert\ln\lvert v-p\rvert-\operatorname{Re}\operatorname{Li}_2\left(\frac{v-p}{c-p}\right) \tag*{(129)} ∫ v − p ln ∣ v − c ∣ d v = ln ∣ p − c ∣ ln ∣ v − p ∣ − Re Li 2 ( c − p v − p ) (129) which gives the symbol of each integral in closed form. The only poles that are not rational functions of the momenta are a pair ± ρ \pm\rho ± ρ (32) , such as X 1 + X 3 X_1+X_3 X 1 + X 3 X 2 + X 3 X_2+X_3 X 2 + X 3 U e U_e U e X v X_v X v U 12 U_{12} U 12
U 12 2 π ≃ a 1 Li 2 ( u 1 ) + a 2 Li 2 ( u 2 ) + a 3 Li 2 ( u 3 ) + b 1 l 1 l 3 + b 2 l 2 l 3 , l i = ln ( 1 − u i ) (130) \frac{U_{12}}{\sqrt{2}\,\pi}\simeq a_1\operatorname{Li}_2(u_1)+a_2\operatorname{Li}_2(u_2)+a_3\operatorname{Li}_2(u_3)+b_1l_1l_3+b_2l_2l_3,\qquad l_i=\ln(1-u_i) \tag*{(130)} 2 π U 12 ≃ a 1 Li 2 ( u 1 ) + a 2 Li 2 ( u 2 ) + a 3 Li 2 ( u 3 ) + b 1 l 1 l 3 + b 2 l 2 l 3 , l i = ln ( 1 − u i ) (130) where ≃ \simeq ≃
a 1 = 2 q G ξ 2 + ξ 1 ξ 3 16 q G ξ 1 ξ 2 2 , a 2 = 2 q G ξ 1 + ξ 2 ξ 3 16 q G ξ 1 2 ξ 2 , a 3 = X 3 ( ξ 1 ξ 2 + X 3 ξ 3 ) 4 q G ξ 1 2 ξ 2 2 , a_1=\frac{2q_G\xi_2+\xi_1\xi_3}{16q_G\xi_1\xi_2^2},\qquad
a_2=\frac{2q_G\xi_1+\xi_2\xi_3}{16q_G\xi_1^2\xi_2},\qquad
a_3=\frac{X_3\left(\xi_1\xi_2+X_3\xi_3\right)}{4q_G\xi_1^2\xi_2^2}, a 1 = 16 q G ξ 1 ξ 2 2 2 q G ξ 2 + ξ 1 ξ 3 , a 2 = 16 q G ξ 1 2 ξ 2 2 q G ξ 1 + ξ 2 ξ 3 , a 3 = 4 q G ξ 1 2 ξ 2 2 X 3 ( ξ 1 ξ 2 + X 3 ξ 3 ) , b i = X i 2 X 3 2 q G ξ 1 2 ξ 2 2 ξ 3 (131) b_i=\frac{X_i^2X_3^2}{q_G\xi_1^2\xi_2^2\xi_3} \tag*{(131)} b i = q G ξ 1 2 ξ 2 2 ξ 3 X i 2 X 3 2 (131) In U 23 U_{23} U 23 U 31 U_{31} U 31 Li 2 ( u 1 ) \operatorname{Li}_2(u_1) Li 2 ( u 1 ) V / 2 π V/\sqrt{2\pi} V / 2 π a 1 + a 3 + a 2 a_1+a_3+a_2 a 1 + a 3 + a 2
X 1 8 ξ 1 ξ 2 ξ 3 + X 1 2 X 2 X 3 q G ξ 1 2 ξ 2 2 ξ 3 2 (132) \frac{X_1}{8\xi_1\xi_2\xi_3}+\frac{X_1^2X_2X_3}{q_G\xi_1^2\xi_2^2\xi_3^2} \tag*{(132)} 8 ξ 1 ξ 2 ξ 3 X 1 + q G ξ 1 2 ξ 2 2 ξ 3 2 X 1 2 X 2 X 3 (132) its coefficient in eq. (32) . Similarly, the coefficient of l 1 l 3 l_1l_3 l 1 l 3 b 1 b_1 b 1 U 12 U_{12} U 12 b 2 b_2 b 2 U 23 U_{23} U 23
X 1 2 X 2 3 ( ξ 1 + ξ 3 ) q G ξ 1 2 ξ 2 2 ξ 3 2 = 2 X 1 2 X 2 X 3 2 q G ξ 1 2 ξ 2 2 ξ 3 2 (133) \frac{X_1^2X_2^3(\xi_1+\xi_3)}{q_G\xi_1^2\xi_2^2\xi_3^2}
=
\frac{2X_1^2X_2X_3^2}{q_G\xi_1^2\xi_2^2\xi_3^2} \tag*{(133)} q G ξ 1 2 ξ 2 2 ξ 3 2 X 1 2 X 2 3 ( ξ 1 + ξ 3 ) = q G ξ 1 2 ξ 2 2 ξ 3 2 2 X 1 2 X 2 X 3 2 (133) using ξ 1 + ξ 3 = 2 X 2 \xi_1+\xi_3=2X_2 ξ 1 + ξ 3 = 2 X 2 V V V (32) .
Terms of lower weight and the constants
The same computation gives the terms of lower weight. No U e U_e U e U 12 U_{12} U 12
U 12 2 π ∣ weight one = − X 1 l 1 + X 2 l 2 8 q G ξ 1 ξ 2 + X 3 2 l 3 4 q G ξ 1 ξ 2 ξ 3 . (134) \left.\frac{U_{12}}{\sqrt{2\pi}}\right|_{\text{weight one}}
=
-\frac{X_1l_1+X_2l_2}{8q_G\xi_1\xi_2}
+\frac{X_3^2l_3}{4q_G\xi_1\xi_2\xi_3}. \tag*{(134)} 2 π U 12 weight one = − 8 q G ξ 1 ξ 2 X 1 l 1 + X 2 l 2 + 4 q G ξ 1 ξ 2 ξ 3 X 3 2 l 3 . (134) These do not vanish, but they cancel in the sum over the three terms: the coefficient of l 1 l_1 l 1 V V V − X 1 ( ξ 2 + ξ 3 − 2 X 1 ) / ( 8 q G ξ 1 ξ 2 ξ 3 ) = 0 -X_1(\xi_2+\xi_3-2X_1)/(8q_G\xi_1\xi_2\xi_3)=0 − X 1 ( ξ 2 + ξ 3 − 2 X 1 ) / ( 8 q G ξ 1 ξ 2 ξ 3 ) = 0 ξ 2 + ξ 3 = 2 X 1 \xi_2+\xi_3=2X_1 ξ 2 + ξ 3 = 2 X 1 l 2 l_2 l 2 l 3 l_3 l 3
The difference between V V V (32) therefore has a vanishing symbol and no terms of lower weight, so it is a combination, with constant coefficients, of the rational prefactors that multiply the functions of weight two in eq. (32) and in the U e U_e U e q G ξ 1 2 ξ 2 2 ξ 3 2 q_G\xi_1^2\xi_2^2\xi_3^2 q G ξ 1 2 ξ 2 2 ξ 3 2 ξ i = 0 \xi_i=0 ξ i = 0 V V V (32) are finite there. For the closed form, this is because the bracket vanishes to second order (§ 4.4 ). For V V V ( X v + s v ≥ 2 P v (X_v+s_v\geq2P_v ( X v + s v ≥ 2 P v q G + 2 y e q_G+2y_e q G + 2 y e q G ) q_G) q G ) π 2 \pi^2 π 2 (33) . As a check, evaluating the antiderivatives at the ends of each range gives V V V (32) to a relative precision of 10 − 45 10^{-45} 1 0 − 45
Several legs per site
With several legs per site, X v > P v X_v>P_v X v > P v X v + s v = 0 X_v+s_v=0 X v + s v = 0 (34) . R 0 R_0 R 0 R e R_e R e − 2 B -2B − 2 B § 4.5 ), and the other six are values of eq. (126) : T 12 ± T_{12}^{\pm} T 12 ± − 2 c 2 -2c_2 − 2 c 2 ( s 1 , s 2 ) = ( − X 1 , ± X 2 ) (s_1,s_2)=(-X_1,\pm X_2) ( s 1 , s 2 ) = ( − X 1 , ± X 2 ) T i j ± T_{ij}^{\pm} T ij ± X v > P v X_v>P_v X v > P v
The ten rational coefficients of eq. (34) can nevertheless be obtained from the integrand. By the construction of § 4.5 , each square root has exactly one integrable symbol, so it suffices to compute one nonzero part of the symbol of V V V X 1 + P 1 X_1+P_1 X 1 + P 1 V V V X 1 = − P 1 X_1=-P_1 X 1 = − P 1 X 1 + s 1 = 0 X_1+s_1=0 X 1 + s 1 = 0 s 3 s_3 s 3 ± π / − B \pm\pi/\sqrt{-B} ± π / − B s 3 s_3 s 3 π / − c 2 \pi/\sqrt{-c_2} π / − c 2 s 2 s_{2} s 2 q G + 2 y e = 0 q_{G}+2y_{e}=0 q G + 2 y e = 0 s 2 s_{2} s 2 s 2 s_{2} s 2 (34) , up to a common sign (the orientation of the discontinuity) that the limit X → P X \to P X → P (13) has the residue − q G / 16 -q_{G}/16 − q G /16 R 0 R_{0} R 0 + q G / 16 +q_{G}/16 + q G /16 e e e q G + 2 y e q_{G}+2y_{e} q G + 2 y e R e R_{e} R e T i j ± T_{ij}^{\pm} T ij ± s 3 s_{3} s 3 − ρ i j − / 8 -\rho_{ij}^{-}/8 − ρ ij − /8 i i i j j j − ρ i j + / 8 -\rho_{ij}^{+}/8 − ρ ij + /8 i i i j j j { i , j } \{i,j\} { i , j } § 4.5 , but it still assumes the form of the function space found there.
