Prepared for submission to JHEP
Introduction
The observation of gravitational waves from merging black holes [1 ] , and the accuracy that next-generation detectors [2 , 3 ] will require of waveform models [4 ] , have motivated precise analytic calculations of the two-body problem [5 ] . One of these is the post-Minkowskian (PM) expansion of black-hole scattering, in powers of Newton’s constant G G G [6 , 7 ] and enter effective-one-body waveform models [8 , 9 ] . Since its revival a decade ago [8 ] it has been developed with scattering amplitudes [10 , 11 ] , worldline effective field theory [12 ] , and worldline quantum field theory [13 ] , and it now stands at fifth order (5PM), four loops [14 , 15 ] .
As the techniques for computing this expansion have developed, similarities to scattering amplitudes have emerged. Each order in G G G [16 , 17 , 18 , 19 ] , and the same kinds of geometry have recently been found in the PM expansion. K3 surfaces, which govern the three-loop photon self-energy of QED [19 , 20 , 21 ] , also appear in black-hole scattering from 4PM on, although as different families [22 , 23 ] , and the four-loop integrals involve two K3 surfaces and two Calabi–Yau threefolds [22 , 24 , 25 , 26 ] . The space of these functions, however, and whether techniques such as the Landau bootstrap [27 , 28 ] , which constrains an integral through the singularities that the Landau equations [29 , 30 ] allow, can be applied fruitfully, have only begun to be explored.
This paper is an exploration of these connections. We use methods developed for Feynman integrals, in particular ideas from the BootLoops toolkit [31 , 32 ] , and test how they generalize. For example, we generalize AMFlow [33 ] , which computes Feynman integrals numerically, to PMFlow for the integrals of the PM expansion, and use it to obtain the numerical boundary data and linear relations that a bootstrap requires. More generally, the idea of characterizing the analytic structure of an integral from its maximal cut appears to transfer as well. Our main example is the divergence found at 5PM and second order in the mass ratio ν \nu ν γ = 3 \gamma= 3 γ = 3 [15 ] . The Picard–Fuchs differential equation of a period determines where the period can be singular, but not whether the integral over the physical contour is singular there or what kind of singularity it is. Our main result is that γ = 3 \gamma= 3 γ = 3 [34 ] , a pinch at infinite loop momentum that no real on-shell configuration produces [35 ] .
The apparent divergence at γ = 3 \gamma= 3 γ = 3 n ⋅ k = 0 n \cdot k = 0 n ⋅ k = 0 [15 ] . The memory region is named for the memory effect, the permanent displacement a burst of radiation leaves in freely falling test masses, with a nonlinear part sourced by the radiated energy flux [36 , 37 , 38 , 39 ] . In this region two of the four gravitons are radiative, and the impulse depends on the i 0 + i0^+ i 0 + ϵ \epsilon ϵ c M c_M c M [15 ] . The cancellation then holds for any c M c_M c M
The constant c M c_M c M [15 ] . Fixing it requires a rule for the routing, and no such rule has yet been derived from first principles. Driesse et al. fix c M c_M c M c M = 1 c_M = 1 c M = 1 [15 ] . Porto and Riva, analyzing the corresponding sector at fifth post-Newtonian order, reproduce the result of this routing where the two computations overlap, but advocate instead a local-in-time (Poincaré–Bertrand) version of Feynman’s prescription, under which the memory-like contribution at O ( G 5 ν 2 ) O(G^5\nu^2) O ( G 5 ν 2 ) [40 ] . They also observe that the analogous retarded rerouting at O ( G 6 ν 2 ) O(G^6\nu^2) O ( G 6 ν 2 ) [41 ] that all π 2 \pi^2 π 2 [40 ] . Dlapa, Kälin, Liu, and Porto note that the 5PM-2SF results of Driesse et al. are fixed only up to the two boundary integrals [42 ] .
Our goal is not to demonstrate the cancellation, which the routing of Driesse et al. achieves, but to understand where the singularity comes from in the analytic structure of the K3 integrals and whether it singles out a routing. We also begin a study of the 2SF dissipative integrals, which contain the same families with retarded propagators. We work from maximal cuts and boundary limits, without reducing the full four-loop integrand, so none of our results is a 5PM observable.
The geometry of the PM integrals is known in detail through four loops. The conservative functions are polylogarithmic at 3PM [43 , 44 , 45 ] , and at 4PM, including radiation-reaction tail terms [46 , 47 , 48 , 49 ] , they contain products of complete elliptic integrals, the periods of another K3 surface [22 , 25 , 46 , 48 , 50 , 51 ] . At 5PM the conservative sector is known at first [14 ] and second [15 ] order in ν \nu ν [52 , 53 , 54 ] , and Bern et al. gave its potential-graviton part as a velocity expansion [55 ] . Frellesvig, Morales, and Wilhelm identified the first Calabi–Yau threefold of its four-loop integrals in the conservative sector at 5PM-2SF [24 ] , see also Refs. [23 , 56 ] , and the periods of a second govern the 5PM-1SF dissipative integrals and enter the radiated energy and recoil [57 ] . Klemm et al. found the second threefold and, at 2SF, the K3 surface of the divergence, governed by the Apéry operator, and gave their Picard–Fuchs operators [22 ] . Brammer et al. completed the loop-by-loop Baikov classification of Ref. [25 ] and showed that no other geometry occurs in the four-loop families, conservative or dissipative, at any self-force order [26 ] . In the conservative 5PM-2SF results [15 , 55 ] the Apéry K3 period remains while the threefolds drop out, and Bern et al. derive their absence from the ultraviolet structure of the integrand [58 ] . The divergence at γ = 3 \gamma= 3 γ = 3
To find where the singularity comes from, we use the Baikov polynomial of Brammer et al. to map the surface of the maximal cut of the planar K3 sector, topology 40 of Ref. [26 ] , onto the Apéry–Fermi K3 [59 ] , obtain the torus period of the cut as a series, and prove that the Apéry operator annihilates it. Klemm et al. found this operator from differential equations [22 ] , and Duhr et al. from the maximal cut in other variables [23 ] . We then show that the known node [59 , 60 , 61 ] of the Apéry K3 surface lies on the integration torus in the scattering region, where the torus period is pinched. The divergence is therefore a second-type Landau singularity of the maximal cut. The routing of the graviton propagators does not follow from the pinch, and solving the Landau equations in Feynman parameters [62 , 63 ] , we find nothing that singles out γ = 3 \gamma= 3 γ = 3
The dissipative sector at 2SF is much less developed. The radiated energy and the radiation-reacted impulse are known at 3PM [64 , 65 , 66 , 67 , 68 , 69 , 70 , 71 ] and 4PM [72 , 73 , 74 ] , where nonlinear memory terms in the radiation-reaction force contribute to the impulse [75 ] , but at 5PM only at 1SF [57 ] ; see also Refs. [76 , 77 , 78 ] . In the post-Newtonian expansion two gauge-invariant observables at 5PM-2SF have been predicted [41 , 79 ] , and the even-in-velocity relative impulse with nonlinear radiation-reaction and memory effects at O ( G 5 ν 2 ) O(G^5\nu^2) O ( G 5 ν 2 ) [40 ] . To our knowledge no 5PM-2SF results for radiation reaction, radiated energy, or recoil with full velocity dependence have appeared, and the 2SF dissipative function space, its boundary data, and its continuation to bound orbits have not been determined.
We determine the constants relating the maximal cut to the holomorphic period for all four geometries (Table 1), recovering that of the 2SF threefold [22 ] , and obtain a closed formula for the Frobenius coefficients of its period, and a similar formula follows from tabulated Calabi–Yau operators [22 , 80 , 81 ] . We continue the Frobenius bases of the four Picard–Fuchs operators to bound-orbit energies [6 , 7 ] and find the expected monodromies. For the four dissipative families of Ref. [15 ] , with both worldline routings retained, we give first-stage master-integral counts. From the response matrices of the auxiliary-mass flow [33 , 82 ] , which we compute with PMFlow, we find ten linear relations among master integrals on three of the four leading top sectors of PX2, the smallest family, and all ten are exact reduction identities. As boundary data, we evaluate twelve PX2 master integrals at one velocity with Feynman graviton propagators.
For the memory region we derive in closed form the Laurent coefficients of the boundary integral I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M ) ϵ \epsilon ϵ I 2 ( M ) I_2^{(M)} I 2 ( M ) [15 ] reports the ϵ − 4 \epsilon^{-4} ϵ − 4 c M = 1 c_M=1 c M = 1 [15 ] and with Porto and Riva’s fifth-post-Newtonian implementation of the same routing [40 ] .
Section 2 sets up the integral families, the four geometries and the memory-region integrals. Section 3 contains the analysis of the maximal cut of topology 40 and the constants of the other three geometries, Section 4 the dissipative master integrals and boundary constants of PX2, Section 5 the continuation to bound kinematics, and Section 6 the memory-region integrals and c M c_M c M
Post-Minkowskian integrals at fifth order
In this section we fix the single-variable kinematics and the four integral families of 5PM scattering at second self-force order. We then list the four non-polylogarithmic geometries that these integrals and the 4PM integrals involve, and we recall the two boundary integrals to which the memory-region contribution of Driesse et al. reduces at the static point of zero relative velocity.
Kinematics and integral families
We use the single-scale kinematics of Refs. [26 , 48 , 50 , 83 ] , in which the two bodies scatter with four-velocities u 1 u_1 u 1 u 2 u_2 u 2 u i 2 = 1 u_i^2 = 1 u i 2 = 1 q q q u i ⋅ q = 0 u_i \cdot q = 0 u i ⋅ q = 0 q 2 = − 1 q^2 = -1 q 2 = − 1
γ = u 1 ⋅ u 2 = 1 + x 2 2 x , (1) \gamma= u_1 \cdot u_2 = \frac{1+x^2}{2x}, \tag*{(1)} γ = u 1 ⋅ u 2 = 2 x 1 + x 2 , (1) the relative Lorentz factor, where we use x = γ − γ 2 − 1 x = \gamma- \sqrt{\gamma^2 - 1} x = γ − γ 2 − 1 γ 2 − 1 \sqrt{\gamma^2 - 1} γ 2 − 1 0 < x < 1 0 < x < 1 0 < x < 1
Besides quadratic graviton propagators 1 / k 2 1/k^2 1/ k 2 1 / ( 2 u i ⋅ k ) 1/(2u_i \cdot k) 1/ ( 2 u i ⋅ k ) [12 , 13 , 67 ] , L L L L L L δ ( 2 k ⋅ u i ) \delta(2k \cdot u_i) δ ( 2 k ⋅ u i ) min ( n 1 , n 2 ) S F \min(n_1,n_2)\mathrm{SF} min ( n 1 , n 2 ) SF n i n_i n i i i i u i → − u i u_i \to-u_i u i → − u i [22 , 25 ] .
At 5PM-2SF the integrals form four families, which differ in how the four graviton lines cross and which we denote P, PX1, PX2, and NP (P, NP1, NP2, and NPM in the notation of Driesse et al. [15 ] ). For the crossed family PX2 we write the integrals as
I n 5 ⋯ n 22 = ∫ k 1 ⋯ k 4 δ ( D 1 ) δ ( D 2 ) δ ( D 3 ) δ ( D 4 ) D 5 n 5 D 6 n 6 ⋯ D 22 n 22 , (2) I_{n_5\cdots n_{22}}=\int_{k_1\cdots k_4}\frac{\delta(D_1)\delta(D_2)\delta(D_3)\delta(D_4)}{D_5^{n_5}D_6^{n_6}\cdots D_{22}^{n_{22}}}, \tag*{(2)} I n 5 ⋯ n 22 = ∫ k 1 ⋯ k 4 D 5 n 5 D 6 n 6 ⋯ D 22 n 22 δ ( D 1 ) δ ( D 2 ) δ ( D 3 ) δ ( D 4 ) , (2) D 1 = 2 k 1 ⋅ u 1 , D 2 = 2 k 2 ⋅ u 1 , D 3 = 2 k 3 ⋅ u 2 , D 4 = 2 k 4 ⋅ u 2 , D 5 = 2 k 1 ⋅ u 2 , D 6 = 2 k 1 ⋅ u 2 − 2 k 2 ⋅ u 2 , D 7 = 2 k 3 ⋅ u 1 , D 8 = 2 k 3 ⋅ u 1 − 2 k 4 ⋅ u 1 , D 9 = ( k 1 − k 2 ) 2 , D 10 = ( k 1 − k 3 ) 2 , D 11 = ( k 1 − k 4 ) 2 , D 12 = ( k 2 − k 3 ) 2 , D 13 = ( k 2 − k 4 ) 2 , D 14 = ( k 3 − k 4 ) 2 , D 14 + j = ( k j + q ) 2 , D 18 + j = k j 2 . \begin{aligned}
D_1&=2k_1\cdot u_1, & D_2&=2k_2\cdot u_1, & D_3&=2k_3\cdot u_2, & D_4&=2k_4\cdot u_2,\\
D_5&=2k_1\cdot u_2, & D_6&=2k_1\cdot u_2-2k_2\cdot u_2, & D_7&=2k_3\cdot u_1, & D_8&=2k_3\cdot u_1-2k_4\cdot u_1,\\
D_9&=(k_1-k_2)^2, & D_{10}&=(k_1-k_3)^2, & D_{11}&=(k_1-k_4)^2, & D_{12}&=(k_2-k_3)^2,\\
D_{13}&=(k_2-k_4)^2, & D_{14}&=(k_3-k_4)^2, & D_{14+j}&=(k_j+q)^2, & D_{18+j}&=k_j^2.
\end{aligned} D 1 D 5 D 9 D 13 = 2 k 1 ⋅ u 1 , = 2 k 1 ⋅ u 2 , = ( k 1 − k 2 ) 2 , = ( k 2 − k 4 ) 2 , D 2 D 6 D 10 D 14 = 2 k 2 ⋅ u 1 , = 2 k 1 ⋅ u 2 − 2 k 2 ⋅ u 2 , = ( k 1 − k 3 ) 2 , = ( k 3 − k 4 ) 2 , D 3 D 7 D 11 D 14 + j = 2 k 3 ⋅ u 2 , = 2 k 3 ⋅ u 1 , = ( k 1 − k 4 ) 2 , = ( k j + q ) 2 , D 4 D 8 D 12 D 18 + j = 2 k 4 ⋅ u 2 , = 2 k 3 ⋅ u 1 − 2 k 4 ⋅ u 1 , = ( k 2 − k 3 ) 2 , = k j 2 . with loop momenta k 1 , … , k 4 k_1,\ldots,k_4 k 1 , … , k 4 j = 1 , … , 4 j=1,\ldots,4 j = 1 , … , 4 D 1 , … , D 4 D_1,\ldots,D_4 D 1 , … , D 4 D 5 , … , D 8 D_5,\ldots,D_8 D 5 , … , D 8 1 / ( D i + i σ 0 ) 1/(D_i+i\sigma0) 1/ ( D i + iσ 0 ) σ = ± 1 \sigma=\pm1 σ = ± 1 D 9 , … , D 22 D_9,\ldots,D_{22} D 9 , … , D 22 i 0 i0 i 0
A top sector of these families has thirteen of the D i D_i D i D i D_i D i § 4 ). PX2 has four leading top sectors, whose denominators are D 1 , … , D 9 D_1,\ldots,D_9 D 1 , … , D 9 { D 13 , D 14 , D 15 , D 21 } \{D_{13},D_{14},D_{15},D_{21}\} { D 13 , D 14 , D 15 , D 21 } S 1 S_1 S 1 { D 11 , D 14 , D 16 , D 21 } \{D_{11},D_{14},D_{16},D_{21}\} { D 11 , D 14 , D 16 , D 21 } S 2 S_2 S 2 { D 11 , D 14 , D 17 , D 20 } \{D_{11},D_{14},D_{17},D_{20}\} { D 11 , D 14 , D 17 , D 20 } S 3 S_3 S 3 { D 13 , D 14 , D 17 , D 19 } \{D_{13},D_{14},D_{17},D_{19}\} { D 13 , D 14 , D 17 , D 19 } families_dissipative.json.
For the dissipative problem we retain both routings σ = ± 1 \sigma=\pm1 σ = ± 1 [67 , 70 ] . Since the integration-by-parts (IBP) system is the same for either sign, σ \sigma σ [23 , 67 ] .
In the first-stage reductions we use the automatic sector symmetries of Kira [84 , 85 , 86 ] , 52 for PX2, each of which maps the linearized propagators D 1 , … , D 8 D_1,\ldots,D_8 D 1 , … , D 8 σ \sigma σ [23 ] . We do not impose relations that hold only for time-symmetric worldline propagators, such as those under u i → − u i u_i \to-u_i u i → − u i
The four geometries
On the maximal cut of a top sector in the loop-by-loop Baikov representation [25 , 26 , 87 ] , a contour integral over the remaining n n n
ϖ ( x ) = 1 ( 2 π i ) n ∮ T n d n t Q ( t ; x ) (3) \varpi(x)=\frac{1}{(2\pi i)^n}\oint_{\mathbb{T}_n}\frac{\mathrm{d}^n t}{\sqrt{Q(t;x)}} \tag*{(3)} ϖ ( x ) = ( 2 π i ) n 1 ∮ T n Q ( t ; x ) d n t (3) and call the maximal-cut leading singularity. It is the holomorphic period of the cut surface of the sector, which is the double cover y 2 = Q ( t ; x ) y^2=Q(t;x) y 2 = Q ( t ; x ) t t t [22 , 50 ] .
The periods of the cut surface solve the sector’s homogeneous differential equation in x x x z z z z = 0 z=0 z = 0 ϖ 0 , … , ϖ n \varpi_0,\ldots,\varpi_n ϖ 0 , … , ϖ n ϖ 0 ( z ) = 1 + O ( z ) \varpi_0(z)=1+O(z) ϖ 0 ( z ) = 1 + O ( z ) z z z ϖ 1 , … , ϖ n \varpi_1,\ldots,\varpi_n ϖ 1 , … , ϖ n ϖ k \varpi_k ϖ k k k k ln z \ln z ln z (3) is then proportional to ϖ 0 \varpi_0 ϖ 0
Through 3PM the scattering observables are multiple polylogarithms with the alphabet { x , 1 ± x } \{x,1\pm x\} { x , 1 ± x } [83 ] (§ 3.4 ). At 4PM the cut surface is a K3 surface of Sym 2 \operatorname{Sym}^2 Sym 2 [22 , 25 , 50 , 51 ] , and at 5PM three more non-polylogarithmic geometries appear [22 , 24 ] , none of banana type. By the classification of Brammer et al. [26 ] these four are all that occur through 5PM. With θ = z d / d z \theta=z\,\mathrm{d}/\mathrm{d}z θ = z d / d z
L K 3 = ( 2 θ − 1 ) 3 + z 2 ( 2 θ + 1 ) 3 − 4 z θ ( 4 θ 2 + 1 ) , z = x 2 (4) L_{\mathrm{K3}}=(2\theta-1)^3+z^2(2\theta+1)^3-4z\theta(4\theta^2+1),\qquad z=x^2 \tag*{(4)} L K3 = ( 2 θ − 1 ) 3 + z 2 ( 2 θ + 1 ) 3 − 4 z θ ( 4 θ 2 + 1 ) , z = x 2 (4) L C Y 3 = θ 4 − 2 8 z ( θ + 1 2 ) 4 , z = 2 − 8 x 4 (5) L_{\mathrm{CY}_3}=\theta^4-2^8z\left(\theta+\frac{1}{2}\right)^4,\qquad z=2^{-8}x^4 \tag*{(5)} L CY 3 = θ 4 − 2 8 z ( θ + 2 1 ) 4 , z = 2 − 8 x 4 (5) L A = θ 3 − z ( 2 θ + 1 ) ( 17 θ 2 + 17 θ + 5 ) + z 2 ( θ + 1 ) 3 , z = x 2 (6) L_A=\theta^3-z(2\theta+1)(17\theta^2+17\theta+5)+z^2(\theta+1)^3,\qquad z=x^2 \tag*{(6)} L A = θ 3 − z ( 2 θ + 1 ) ( 17 θ 2 + 17 θ + 5 ) + z 2 ( θ + 1 ) 3 , z = x 2 (6) L C Y 3 ′ = θ 4 − 2 4 z ( 192 θ 4 + 128 θ 3 + 112 θ 2 + 48 θ + 7 ) + 2 14 z 2 ( 192 θ 4 + 256 θ 3 + 208 θ 2 + 64 θ + 7 ) − 2 30 z 3 ( θ + 1 2 ) 4 , z = 1 − γ 2 2 10 . (7) \begin{aligned}
L_{\mathrm{CY}'_3}={}&\theta^4-2^4z(192\theta^4+128\theta^3+112\theta^2+48\theta+7)\\
&+2^{14}z^2(192\theta^4+256\theta^3+208\theta^2+64\theta+7)-2^{30}z^3\left(\theta+\frac{1}{2}\right)^4,
\qquad z=\frac{1-\gamma^2}{2^{10}}. \tag*{(7)}
\end{aligned} L CY 3 ′ = θ 4 − 2 4 z ( 192 θ 4 + 128 θ 3 + 112 θ 2 + 48 θ + 7 ) + 2 14 z 2 ( 192 θ 4 + 256 θ 3 + 208 θ 2 + 64 θ + 7 ) − 2 30 z 3 ( θ + 2 1 ) 4 , z = 2 10 1 − γ 2 . (7) The 4PM operator L K 3 L_{\mathrm{K3}} L K3 θ 2 − z ( θ + 1 2 ) 2 \theta^2-z\left(\theta+\frac{1}{2}\right)^2 θ 2 − z ( θ + 2 1 ) 2 z \sqrt{z} z L C Y 3 L_{\mathrm{CY}_3} L CY 3
ϖ 0 = 4 F 3 ( 1 2 , 1 2 , 1 2 , 1 2 ; 1 , 1 , 1 ; x 4 ) (8) \varpi_0={}_4F_3\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1,1;x^4\right) \tag*{(8)} ϖ 0 = 4 F 3 ( 2 1 , 2 1 , 2 1 , 2 1 ; 1 , 1 , 1 ; x 4 ) (8) caption: The four non-polylogarithmic geometries of post-Minkowskian two-body scattering through 5PM [22 , 24 , 26 ] , with the topology labels of Ref. [26 ] . The singular points listed are those at finite z ≠ 0 z\ne0 z = 0 Q Q Q (3) , with Q 6 Q_6 Q 6 P 8 P_8 P 8 Q 8 Q_8 Q 8 [26 ] , and ( 2 π i ) − n (2\pi i)^{-n} ( 2 π i ) − n T n T^n T n L S LS L S ϖ 0 \varpi_0 ϖ 0 [22 ] (Section 3). The parities in parentheses are those of the sector under u i → − u i u_i\to-u_i u i → − u i ϖ 0 ( z ) = ( 2 / π ) 2 K ( z ) 2 \varpi_0(z)=(2/\pi)^2K(z)^2 ϖ 0 ( z ) = ( 2/ π ) 2 K ( z ) 2 K K K A n A_n A n (9) , and ϖ m a x c u t \varpi_{\mathrm{maxcut}} ϖ maxcut (3) of the CY3 ′ _3^{\prime} 3 ′
:::
a threefold period [22 ] . The Apéry operator L A L_{\mathrm{A}} L A ϖ 0 = ∑ n ≥ 0 A n z n \varpi_0=\sum_{n\geq0}A_nz^n ϖ 0 = ∑ n ≥ 0 A n z n A n A_n A n
A n = ∑ k = 0 n ( n k ) 2 ( n + k k ) 2 = 1 , 5 , 73 , 1445 , … (9) A_n=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}^2=1,\ 5,\ 73,\ 1445,\ \ldots \tag*{(9)} A n = k = 0 ∑ n ( k n ) 2 ( k n + k ) 2 = 1 , 5 , 73 , 1445 , … (9) and the associated surface is the K3 of Beukers and Peters [22 , 60 ] . Finally, the threefold of L C Y 3 ′ L_{\mathrm{CY}_3^{\prime}} L CY 3 ′ χ = 80 \chi=80 χ = 80 [22 , 81 ] . For the coefficients a n = 1 , 112 , 47376 , … a_n=1,\ 112,\ 47376,\ldots a n = 1 , 112 , 47376 , … ϖ 0 = ∑ n a n z n \varpi_0=\sum_n a_nz^n ϖ 0 = ∑ n a n z n L C Y 3 ′ L_{\mathrm{CY}_3^{\prime}} L CY 3 ′ (25) in Section 3.6.
The Apéry operator is singular at a velocity inside the scattering region [15 ] , since the smaller root of z 2 − 34 z + 1 z^2-34z+1 z 2 − 34 z + 1
z + = 17 − 12 2 = x 2 ∣ γ = 3 , (10) z_+=17-12\sqrt{2}=x^2\big|_{\gamma=3}, \tag*{(10)} z + = 17 − 12 2 = x 2 γ = 3 , (10) that is, x = 3 − 2 2 x=3-2\sqrt{2} x = 3 − 2 2 γ = 3 \gamma=3 γ = 3 v / c = 8 / 3 v/c=\sqrt{8}/3 v / c = 8 /3
The memory-region integrals
Driesse et al. identified in the four-loop radiative integrals a distinct integration region, the memory region, besides the potential and tail regions [15 ] . In it two of the four gravitons are radiative (RPPR and PRRP in their notation; R radiative, P potential), and at 5PM-2SF it gives rise to a memory back-reaction on the impulse. They named the region for gravitational memory [37 , 88 ] , the linear memory of Zel’dovich and Polnarev and the nonlinear memory of Christodoulou and of Blanchet and Damour [36 , 38 , 39 ] . The nonlinear memory of the emitted radiation is known at leading PM order [89 ] and in a multipole expansion through O ( G 5 ) O(G^5) O ( G 5 ) [90 ] .
We call the two radiative gravitons of the memory region the memory gravitons. They can have retarded, advanced, or Feynman propagators, and the impulse depends on this choice even in dimensional regularization [15 ] . In the conservative calculation of Driesse et al. the memory-region contribution reduces, at the static point γ = 1 \gamma= 1 γ = 1
I 1 ( M ) = ∫ ℓ 1 … 4 δ ^ ( ℓ 1 ⋅ v 1 ) δ ^ ( ℓ 2 ⋅ v 1 ) δ ^ ( ℓ 3 ⋅ v 2 ) δ ^ ( ℓ 4 ⋅ v 2 ) D q 3 D 34 D 04 D 31 D 24 , (11) I_1^{(M)} = \int_{\ell_1\ldots4} \frac{\hat{\delta}(\ell_1 \cdot v_1)\hat{\delta}(\ell_2 \cdot v_1)\hat{\delta}(\ell_3 \cdot v_2)\hat{\delta}(\ell_4 \cdot v_2)}{D_{q3}D_{34}D_{04}D_{31}D_{24}}, \tag*{(11)} I 1 ( M ) = ∫ ℓ 1 … 4 D q 3 D 34 D 04 D 31 D 24 δ ^ ( ℓ 1 ⋅ v 1 ) δ ^ ( ℓ 2 ⋅ v 1 ) δ ^ ( ℓ 3 ⋅ v 2 ) δ ^ ( ℓ 4 ⋅ v 2 ) , (11) and I 2 ( M ) I_2^{(M)} I 2 ( M ) D 31 D_{31} D 31 D 24 D_{24} D 24 v 1 , 2 v_{1,2} v 1 , 2 ℓ i \ell_i ℓ i ∫ ℓ = ∫ d 4 − 2 ϵ ℓ / ( 2 π ) 4 − 2 ϵ \int_\ell= \int d^{4-2\epsilon}\ell/(2\pi)^{4-2\epsilon} ∫ ℓ = ∫ d 4 − 2 ϵ ℓ / ( 2 π ) 4 − 2 ϵ δ ^ = 2 π δ \hat{\delta}=2\pi\delta δ ^ = 2 π δ D i j = ( ℓ i − ℓ j ) 2 D_{ij}=(\ell_i-\ell_j)^2 D ij = ( ℓ i − ℓ j ) 2 D q 3 = ( ℓ 3 + q ) 2 D_{q3}=(\ell_3+q)^2 D q 3 = ( ℓ 3 + q ) 2 D 04 = ℓ 4 2 D_{04}=\ell_4^2 D 04 = ℓ 4 2 1
In the conservative sector of Ref. [15 ] the ϵ \epsilon ϵ ( γ − 3 ) (\gamma-3) ( γ − 3 ) [15 ] ,
I 1 ( M ) = 1 15 ( 8 π ) 4 ϵ + O ( ϵ 0 ) , I_1^{(M)} = \frac{1}{15(8\pi)^4\epsilon} + \mathcal{O}(\epsilon^0), I 1 ( M ) = 15 ( 8 π ) 4 ϵ 1 + O ( ϵ 0 ) , I 2 ( M ) = − 5 c M 6 ( 8 π ) 4 ϵ 2 + O ( ϵ − 1 ) . (12) I_2^{(M)} = -\frac{5c_M}{6(8\pi)^4\epsilon^2} + \mathcal{O}(\epsilon^{-1}). \tag*{(12)} I 2 ( M ) = − 6 ( 8 π ) 4 ϵ 2 5 c M + O ( ϵ − 1 ) . (12) Driesse et al. [15 ] fix c M c_M c M D 31 D_{31} D 31 D 24 D_{24} D 24 γ \gamma γ γ = 3 \gamma=3 γ = 3 (10) , that it cancels, and under it c M c_M c M [15 ] . The constant first enters the conservative scattering angle [15 ] at fifth post-Newtonian order, through 64 5 c M − 11456 135 \frac{64}{5}c_M-\frac{11456}{135} 5 64 c M − 135 11456 c M = 1 c_M=1 c M = 1 [40 ] .
Maximal cuts and the Landau singularity
For the planar K3 sector we show that the node of the Apéry K3 surface lies on the integration contour of the maximal cut, which makes γ = 3 \gamma=3 γ = 3 { x , 1 ± x } \{x,1\pm x\} { x , 1 ± x } [59 ] . We also expand the torus period of the cut in a series, continue it beyond its disk of convergence and prove that it satisfies the Apéry equation. New, to our knowledge, are the location of the node on the integration torus, the Landau analysis of the sector’s cut surface and subtopologies, the route from this cut surface to the lattice identity (18) , and the proof of L A ϖ = 0 L_A\varpi=0 L A ϖ = 0
For each of the four geometries in Table 1 we then compute the maximal-cut leading singularity of the top sector, Eq. (3) , and the constant relating it to the holomorphic period ϖ 0 \varpi_0 ϖ 0
The cut surface and the Apéry–Fermi K3
The K3 sector of the planar 5PM-2SF family, topology 40 of Brammer et al. [26 ] , is governed by the Apéry operator L A L_A L A (6) , which Klemm et al. established from IBP relations and differential equations on its nine master integrals [22 ] . Duhr et al. obtained L A L_A L A x x x [23 ] .
Here we study the cut surface itself, starting from the loop-by-loop Baikov representation of Brammer et al. In their Eq. (3.26) the leading singularity of the sector is proportional to the torus integral of 1 / Q 6 1/\sqrt{Q_6} 1/ Q 6
Q 6 = ( x ( 1 + t 1 ) ( t 1 + t 2 2 ) − ( 1 + x 2 ) t 1 t 2 ) 2 − 4 x 2 t 1 2 t 2 2 . Q_6=\left(x(1+t_1)(t_1+t_2^2)-(1+x^2)t_1t_2\right)^2-4x^2t_1^2t_2^2. Q 6 = ( x ( 1 + t 1 ) ( t 1 + t 2 2 ) − ( 1 + x 2 ) t 1 t 2 ) 2 − 4 x 2 t 1 2 t 2 2 . The cut surface is the double cover y 2 = Q 6 y^2=Q_6 y 2 = Q 6 ( t 1 , t 2 ) (t_1,t_2) ( t 1 , t 2 ) Q 6 = 0 Q_6=0 Q 6 = 0 w : = ( 1 + t 1 ) ( t 1 + t 2 2 ) / ( t 1 t 2 ) w:=(1+t_1)(t_1+t_2^2)/(t_1t_2) w := ( 1 + t 1 ) ( t 1 + t 2 2 ) / ( t 1 t 2 ) 2 γ = x + 1 / x 2\gamma=x+1/x 2 γ = x + 1/ x Q 6 Q_6 Q 6 ( 1 + x 2 ) ± 2 x = ( 1 ± x ) 2 (1+x^2)\pm2x=(1\pm x)^2 ( 1 + x 2 ) ± 2 x = ( 1 ± x ) 2
Q 6 = P + P − , P ± : = x ( 1 + t 1 ) ( t 1 + t 2 2 ) − ( 1 ± x ) 2 t 1 t 2 = x t 1 t 2 ( w − 2 γ ∓ 2 ) , (13) Q_6=P_+P_-,\qquad P_\pm:=x(1+t_1)(t_1+t_2^2)-(1\pm x)^2t_1t_2=x t_1t_2(w-2\gamma\mp2), \tag*{(13)} Q 6 = P + P − , P ± := x ( 1 + t 1 ) ( t 1 + t 2 2 ) − ( 1 ± x ) 2 t 1 t 2 = x t 1 t 2 ( w − 2 γ ∓ 2 ) , (13) or
Q 6 = x 2 t 1 2 t 2 2 [ ( w − 2 γ ) 2 − 4 ] , Q_6=x^2t_1^2t_2^2\left[(w-2\gamma)^2-4\right], Q 6 = x 2 t 1 2 t 2 2 [ ( w − 2 γ ) 2 − 4 ] , a spectral-curve form. Thus the two components P ± = 0 P_\pm=0 P ± = 0 w = 2 γ ± 2 w=2\gamma\pm2 w = 2 γ ± 2 w w w t 1 t_1 t 1 t 2 t_2 t 2
The loop-by-loop reduction of Duhr et al. (Section 7.2.1 of their paper) leads instead to the fully factored polynomial
P 4 , 4 = ( 1 − t 1 2 ) ( 1 − t 2 2 ) [ ( 1 − x ) 2 + 4 x t 1 t 2 ] [ ( 1 + x ) 2 + 4 x t 1 t 2 ] . P_{4,4}=(1-t_1^2)(1-t_2^2)\left[(1-x)^2+4xt_1t_2\right]\left[(1+x)^2+4xt_1t_2\right]. P 4 , 4 = ( 1 − t 1 2 ) ( 1 − t 2 2 ) [ ( 1 − x ) 2 + 4 x t 1 t 2 ] [ ( 1 + x ) 2 + 4 x t 1 t 2 ] . To map the cut surface to the Apéry–Fermi K3 of Peters and Stienstra, we lift the square root with a Poisson kernel in a third unit-circle variable s s s w + s + 1 / s = 2 γ w+s+1/s=2\gamma w + s + 1/ s = 2 γ ξ 1 = t 1 / t 2 \xi_1=t_1/t_2 ξ 1 = t 1 / t 2 ξ 2 = t 2 \xi_2=t_2 ξ 2 = t 2 w = ξ 1 + ξ 1 − 1 + ξ 2 + ξ 2 − 1 w=\xi_1+\xi_1^{-1}+\xi_2+\xi_2^{-1} w = ξ 1 + ξ 1 − 1 + ξ 2 + ξ 2 − 1 ξ 3 = s \xi_3=s ξ 3 = s ξ 1 + ξ 1 − 1 + ξ 2 + ξ 2 − 1 + ξ 3 + ξ 3 − 1 = 2 γ \xi_1+\xi_1^{-1}+\xi_2+\xi_2^{-1}+\xi_3+\xi_3^{-1}=2\gamma ξ 1 + ξ 1 − 1 + ξ 2 + ξ 2 − 1 + ξ 3 + ξ 3 − 1 = 2 γ [61 ] , and other Feynman-integral K3 surfaces include the two-parameter families of Duhr and Maggio [91 ] .
The node on the integration torus
On the unit torus ( t j = e i ϕ j ) (t_j=e^{i\phi_j}) ( t j = e i ϕ j ) w = 4 cos ϕ 1 2 cos ϕ 1 − 2 ϕ 2 2 w=4\cos\frac{\phi_1}{2}\cos\frac{\phi_1-2\phi_2}{2} w = 4 cos 2 ϕ 1 cos 2 ϕ 1 − 2 ϕ 2 ∣ w ∣ ≤ 4 |w|\le4 ∣ w ∣ ≤ 4 w = 4 w=4 w = 4 t 1 = t 2 = 1 t_1=t_2=1 t 1 = t 2 = 1 w = 2 γ ± 2 w=2\gamma\pm2 w = 2 γ ± 2 P ± = 0 P_\pm=0 P ± = 0 γ > 3 \gamma>3 γ > 3 γ = 3 \gamma=3 γ = 3 1 < γ < 3 1<\gamma<3 1 < γ < 3 x → x ± i 0 x\to x\pm i0 x → x ± i 0 γ = 3 \gamma=3 γ = 3 [59 , 61 ] , which we locate at t 1 = t 2 = 1 t_1=t_2=1 t 1 = t 2 = 1 P − = 0 P_-=0 P − = 0 (13) , a point of the integration torus. Hence the torus period, Eq. (15) below, is pinched there, as at a conifold point [92 ] .
The finite singular points of L A L_A L A z 2 − 34 z + 1 z^2-34z+1 z 2 − 34 z + 1 x x x { Q 6 = ∂ t 1 Q 6 = ∂ t 2 Q 6 = 0 } \{Q_6=\partial_{t_1}Q_6=\partial_{t_2}Q_6=0\} { Q 6 = ∂ t 1 Q 6 = ∂ t 2 Q 6 = 0 } [93 ] . By an iterated resultant we find
disc t Q 6 ∝ x ( x 2 − 1 ) ( x 4 − 34 x 2 + 1 ) = x ( x 2 − 1 ) ( x 2 − 6 x + 1 ) ( x 2 + 6 x + 1 ) , (14) \operatorname{disc}_{t} Q_6 \propto x\left(x^2-1\right)\left(x^4-34x^2+1\right)=x\left(x^2-1\right)\left(x^2-6x+1\right)\left(x^2+6x+1\right), \tag*{(14)} disc t Q 6 ∝ x ( x 2 − 1 ) ( x 4 − 34 x 2 + 1 ) = x ( x 2 − 1 ) ( x 2 − 6 x + 1 ) ( x 2 + 6 x + 1 ) , (14) whose zeros in z z z x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 ) x = 3 − 2 2 x=3-2\sqrt{2} x = 3 − 2 2 γ = ( 1 + x 2 ) / 2 x = 3 \gamma=(1+x^2)/2x=3 γ = ( 1 + x 2 ) /2 x = 3 [15 ] . Up to the sign of the radicand, the square root of the quartic x 4 − 34 x 2 + 1 x^4-34x^2+1 x 4 − 34 x 2 + 1 r ( x ) r(x) r ( x )
The torus period
In terms of a : = x / ( 1 + x ) 2 a:=x/(1+x)^2 a := x / ( 1 + x ) 2 b : = x / ( 1 − x ) 2 b:=x/(1-x)^2 b := x / ( 1 − x ) 2 z : = x 2 z:=x^2 z := x 2
ϖ ( z ) : = 1 ( 2 π i ) 2 ∮ T 2 d t 1 d t 2 Q 6 = 1 1 − z ∑ m ≥ 0 ( 2 m m ) 2 16 m ∑ j + k = 2 m ( 2 j j ) ( 2 k k ) a j b k = ∑ n ≥ 0 A n z n . (15) \varpi(z):=\frac{1}{(2\pi i)^2}\oint_{\mathbb{T}^2}\frac{\mathrm{d}t_1\,\mathrm{d}t_2}{\sqrt{Q_6}}=\frac{1}{1-z}\sum_{m\geq0}\frac{\binom{2m}{m}^2}{16^m}\sum_{j+k=2m}\binom{2j}{j}\binom{2k}{k}a^j b^k=\sum_{n\geq0}A_nz^n. \tag*{(15)} ϖ ( z ) := ( 2 π i ) 2 1 ∮ T 2 Q 6 d t 1 d t 2 = 1 − z 1 m ≥ 0 ∑ 1 6 m ( m 2 m ) 2 j + k = 2 m ∑ ( j 2 j ) ( k 2 k ) a j b k = n ≥ 0 ∑ A n z n . (15) Here A n A_n A n (9) . To derive the second equality we use, in the variables ξ 1 \xi_1 ξ 1 ξ 2 \xi_2 ξ 2
CT t 1 , t 2 [ w n ] = ( n n / 2 ) 2 [ n e v e n ] for all n ≥ 0. (16) \operatorname{CT}_{t_1,t_2}\left[w^n\right]=\binom{n}{n/2}^2\left[n\ \mathrm{even}\right]\qquad\text{for all }n\geq0. \tag*{(16)} CT t 1 , t 2 [ w n ] = ( n /2 n ) 2 [ n even ] for all n ≥ 0. (16) We apply this identity in Eq. (15) , with ( w − 2 γ ) 2 − 4 = ( w − 1 / a ) ( w − 1 / b ) (w-2\gamma)^2-4=(w-1/a)(w-1/b) ( w − 2 γ ) 2 − 4 = ( w − 1/ a ) ( w − 1/ b ) w w w ϖ \varpi ϖ L A L_A L A z = 0 z=0 z = 0 L A ϖ = 0 L_A\varpi=0 L A ϖ = 0 x x x (18) below. Therefore the Taylor coefficients c n c_n c n ϖ \varpi ϖ n 3 c n − ( 2 n − 1 ) ( 17 n 2 − 17 n + 5 ) c n − 1 + ( n − 1 ) 3 c n − 2 = 0 n^3c_n-(2n-1)(17n^2-17n+5)c_{n-1}+(n-1)^3c_{n-2}=0 n 3 c n − ( 2 n − 1 ) ( 17 n 2 − 17 n + 5 ) c n − 1 + ( n − 1 ) 3 c n − 2 = 0 c 0 = 1 c_0=1 c 0 = 1
Closed forms of this period are known: modular [94 ] , hypergeometric [95 ] , and the square of a second-order solution [96 ] . The full leading singularity of Eq. (3.26) of Brammer et al. is L S ∝ − z 1 − z ϖ ( z ) \mathrm{LS}\propto-\frac{z}{1-z}\varpi(z) LS ∝ − 1 − z z ϖ ( z )
Beyond the disk in which the series converges we evaluate the period as an integral over w w w w = 4 cos α cos β w=4\cos\alpha\cos\beta w = 4 cos α cos β ( α , β (\alpha,\beta ( α , β Q 6 Q_6 Q 6 (13) , the torus period (15) is the average
ϖ ( z ) = ⟨ 1 x ( w − 2 γ ) 2 − 4 ⟩ T 2 = 1 x ∫ − 4 4 d w ρ ( w ) ( w − 2 γ ) 2 − 4 , ρ ( w ) = 1 2 π 2 K ( 1 − w 2 16 ) , (17) \varpi(z)=\left\langle\frac{1}{x\sqrt{(w-2\gamma)^2-4}}\right\rangle_{\mathbb{T}^2}=\frac{1}{x}\int_{-4}^{4}\frac{\mathrm{d}w\,\rho(w)}{\sqrt{(w-2\gamma)^2-4}},\qquad\rho(w)=\frac{1}{2\pi^2}K\left(1-\frac{w^2}{16}\right), \tag*{(17)} ϖ ( z ) = ⟨ x ( w − 2 γ ) 2 − 4 1 ⟩ T 2 = x 1 ∫ − 4 4 ( w − 2 γ ) 2 − 4 d w ρ ( w ) , ρ ( w ) = 2 π 2 1 K ( 1 − 16 w 2 ) , (17) with the branch of the square root that tends to 2 γ − w 2\gamma-w 2 γ − w x → 0 x\to0 x → 0 K K K (17) is analytic in x x x γ ∉ [ − 3 , 3 ] \gamma\notin[-3,3] γ ∈ / [ − 3 , 3 ] 3 − 2 2 ≤ ∣ x ∣ < 1 3-2\sqrt{2}\leq|x|<1 3 − 2 2 ≤ ∣ x ∣ < 1 ∑ n A n z n \sum_n A_nz^n ∑ n A n z n z = x 2 z=x^2 z = x 2 w = 2 γ − 2 w=2\gamma-2 w = 2 γ − 2 x → x ± i 0 x\to x\pm i0 x → x ± i 0 ± i / 4 − ( 2 γ − w ) 2 \pm i/\sqrt{4-(2\gamma-w)^2} ± i / 4 − ( 2 γ − w ) 2
We compared the average (17) with the holomorphic Frobenius solution ϖ 0 \varpi_0 ϖ 0 L A L_A L A z = 1 / 200 z=1/200 z = 1/200 Table 2 ). At the four real points of the table, which lie beyond the singular point γ = 3 \gamma=3 γ = 3 Im z > 0 \operatorname{Im}z>0 Im z > 0 x + i 0 x+i0 x + i 0 Im z < 0 \operatorname{Im}z<0 Im z < 0 x − i 0 x-i0 x − i 0 ∣ z ∣ < ( 3 − 2 2 ) 2 |z|<(3-2\sqrt{2})^2 ∣ z ∣ < ( 3 − 2 2 ) 2 tables/apery_torus_period_values.txt).
Table 2. The torus period ϖ \varpi ϖ (17) , outside the disk ∣ z ∣ < ( 3 − 2 2 ) 2 |z|<(3-2\sqrt{2})^2 ∣ z ∣ < ( 3 − 2 2 ) 2 L A L_A L A z = x 2 z=x^2 z = x 2 x x x x + i 0 x+i0 x + i 0 x − i 0 x-i0 x − i 0
With the lift of § 3.1 we obtain an identity between the torus period and a lattice sum,
( 1 + z ) ϖ ( z ) = ∑ N ≥ 0 B N ( z ( 1 + z ) 2 ) N (18) (1+z)\varpi(z)=\sum_{N\geq0}B_N\left(\frac{z}{(1+z)^2}\right)^N \tag*{(18)} ( 1 + z ) ϖ ( z ) = N ≥ 0 ∑ B N ( ( 1 + z ) 2 z ) N (18) where B N B_N B N [97 ] (OEIS A002896). This identity relates the holomorphic periods of the Apéry pencil and the Fermi K3, which share a double cover. It follows from Peters and Stienstra and is given by Bertin [98 ] , and we obtain it from the cut surface, with the parameter map 1 / ( 4 γ 2 ) = z / ( 1 + z ) 2 1/(4\gamma^2)=z/(1+z)^2 1/ ( 4 γ 2 ) = z / ( 1 + z ) 2 z = x 2 z=x^2 z = x 2 (18) , ( 1 + z ) ϖ (1+z)\varpi ( 1 + z ) ϖ [97 ] .
Landau equations and first-type letters
In the Landau equations in Feynman parameters we find, on the subtopologies and polytope faces that we could resolve, no letter of topology 40 beyond { x , 1 ± x } \{x,1\pm x\} { x , 1 ± x } γ = 3 \gamma=3 γ = 3 [12 , 13 ] contain both quadratic graviton propagators D j = ℓ j 2 D_j=\ell_j^2 D j = ℓ j 2 2 u a ( e ) ⋅ ℓ e 2u_{a(e)} \cdot\ell_e 2 u a ( e ) ⋅ ℓ e a ( e ) a(e) a ( e ) k l k_l k l ℓ i = ∑ l c i l k l + r i \ell_i = \sum_l c_{il} k_l + r_i ℓ i = ∑ l c i l k l + r i α i \alpha_i α i [29 ] are
∑ j α j c j l ℓ j + ∑ e α e c e l u a ( e ) = 0 ( l = 1 , … , L ) , α j ℓ j 2 = 0 , α e ( 2 u a ( e ) ⋅ ℓ e ) = 0 , (19) \sum_j \alpha_j c_{jl} \ell_j + \sum_e \alpha_e c_{el} u_{a(e)} = 0 \quad(l = 1,\ldots,L), \qquad\alpha_j \ell_j^2 = 0, \qquad\alpha_e \left(2u_{a(e)} \cdot\ell_e\right) = 0, \tag*{(19)} j ∑ α j c j l ℓ j + e ∑ α e c e l u a ( e ) = 0 ( l = 1 , … , L ) , α j ℓ j 2 = 0 , α e ( 2 u a ( e ) ⋅ ℓ e ) = 0 , (19) where j j j e e e U U U F F F α e \alpha_e α e γ \gamma γ F F F α e α e ′ ( u a ( e ) ⋅ u a ( e ′ ) ) \alpha_e\alpha_{e'}(u_{a(e)} \cdot u_{a(e')}) α e α e ′ ( u a ( e ) ⋅ u a ( e ′ ) ) U U U U = 0 U = 0 U = 0 γ \gamma γ
At one and two loops we solve the Landau equations by saturated Gröbner-basis elimination of the Feynman parameters [62 ] . For the 2PM box and, at 3PM, for the H graph and the graph with three worldline propagators [44 , 45 ] , we find the first-type Landau locus
{ x , 1 − x , 1 + x } , (20) \{x,\ 1-x,\ 1+x\}, \tag*{(20)} { x , 1 − x , 1 + x } , (20) which is the polylogarithmic alphabet at 3PM [51 , 83 ] , while the alphabet at 2PM contains only the letter x x x γ \gamma γ ⟨ 1 ⟩ \langle1\rangle ⟨ 1 ⟩
For the three-loop K3 sector at 4PM we carry out the same elimination on its 96 subtopologies with a linearized propagator of each worldline and find only γ = ± 1 \gamma= \pm1 γ = ± 1 x = 0 , ± 1 x = 0, \pm1 x = 0 , ± 1 ϵ \epsilon ϵ [50 , 51 ] , and the full three-loop system of the second reference contains in addition the kernel x / ( 1 + x 2 ) x/(1+x^2) x / ( 1 + x 2 ) 1 + x 2 1+x^2 1 + x 2 [57 ] , is a first-type component of none of these topologies. It arises on no face, coordinate or mixed, of the Newton polytope of U + F U+F U + F
For the four-loop topology 40 the second Symanzik polynomial F F F γ \gamma γ F F F F F F γ \gamma γ γ \gamma γ Q \mathbb{Q} Q γ \gamma γ γ = ± 1 \gamma= \pm1 γ = ± 1 { x , 1 ± x } \{x,1 \pm x\} { x , 1 ± x } γ \gamma γ γ = ± 1 \gamma= \pm1 γ = ± 1
On the ten-propagator topology the elimination at symbolic γ \gamma γ α i ≠ 0 \alpha_i \ne0 α i = 0 U = 0 U = 0 U = 0 γ \gamma γ U = 0 U = 0 U = 0 γ 0 ∈ { 0 , ± 1 2 , ± 3 } \gamma_0 \in\{0, \pm\frac{1}{2}, \pm3\} γ 0 ∈ { 0 , ± 2 1 , ± 3 } γ \gamma γ 1 + x 2 1+x^2 1 + x 2 x 2 ± x + 1 x^2 \pm x+1 x 2 ± x + 1 x 2 ± 6 x + 1 x^2 \pm6x+1 x 2 ± 6 x + 1 U ≠ 0 U \ne0 U = 0 γ \gamma γ γ \gamma γ U U U { x , 1 ± x } \{x,1\pm x\} { x , 1 ± x } U = 0 U=0 U = 0
The Newton polytope of U + F U+F U + F [63 , 99 ] has 23003 faces, coordinate and mixed, of which 9468 have initial forms that depend on γ \gamma γ γ = 3 \gamma=3 γ = 3 0 0 0 1 2 \frac{1}{2} 2 1 γ = 7 / 11 \gamma=7/11 γ = 7/11 γ = 3 \gamma=3 γ = 3 γ = 3 \gamma=3 γ = 3 γ = 0 \gamma=0 γ = 0 γ = 1 2 \gamma=\frac{1}{2} γ = 2 1 { 1 + x 2 , x 2 ± x + 1 } \{1+x^2,\ x^2\pm x+1\} { 1 + x 2 , x 2 ± x + 1 } (14) (for 1 + x 2 1+x^2 1 + x 2
The second-type singularity at γ = 3 \gamma=3 γ = 3
On the maximal cut, though not in the Feynman-parameter equations above, the singularity at γ = 3 \gamma=3 γ = 3 [34 ] : the pinch lies at infinite loop momentum. We show this in the loop-by-loop Baikov representation [87 ] of Brammer et al., in which the maximal cut is the integral of [ G ( u 1 , k 2 ) G ( u 2 , k 2 ) B 3 B 2 ] − 1 / 2 [G(u_1,k_2)G(u_2,k_2)B_3B_2]^{-1/2} [ G ( u 1 , k 2 ) G ( u 2 , k 2 ) B 3 B 2 ] − 1/2 k 2 2 k_2^2 k 2 2 ( k 2 − q ) 2 (k_2-q)^2 ( k 2 − q ) 2 2 u 1 ⋅ k 2 2u_1\cdot k_2 2 u 1 ⋅ k 2 2 u 2 ⋅ k 2 2u_2\cdot k_2 2 u 2 ⋅ k 2 G ( u i , k 2 ) = k 2 2 − ( u i ⋅ k 2 ) 2 G(u_i,k_2)=k_2^2-(u_i\cdot k_2)^2 G ( u i , k 2 ) = k 2 2 − ( u i ⋅ k 2 ) 2 B 2 B_2 B 2 k 2 k_2 k 2 u 1 u_1 u 1 u 2 u_2 u 2 q q q k 3 k_3 k 3 4 B 3 = [ ( γ − 1 ) ( k 2 − q ) 2 − 2 ( u 1 ⋅ k 2 ) ( u 2 ⋅ k 2 ) ] [ ( γ + 1 ) ( k 2 − q ) 2 − 2 ( u 1 ⋅ k 2 ) ( u 2 ⋅ k 2 ) ] 4B_3=[(\gamma-1)(k_2-q)^2-2(u_1\cdot k_2)(u_2\cdot k_2)][(\gamma+1)(k_2-q)^2-2(u_1\cdot k_2)(u_2\cdot k_2)] 4 B 3 = [( γ − 1 ) ( k 2 − q ) 2 − 2 ( u 1 ⋅ k 2 ) ( u 2 ⋅ k 2 )] [( γ + 1 ) ( k 2 − q ) 2 − 2 ( u 1 ⋅ k 2 ) ( u 2 ⋅ k 2 )] ε = 1 / k 2 2 \varepsilon=1/k_2^2 ε = 1/ k 2 2 R = ( k 2 − q ) 2 / k 2 2 R=(k_2-q)^2/k_2^2 R = ( k 2 − q ) 2 / k 2 2 c i = 2 u i ⋅ n c_i=2u_i\cdot n c i = 2 u i ⋅ n n = k 2 / k 2 2 n=k_2/\sqrt{k_2^2} n = k 2 / k 2 2 k 2 k_2 k 2 q 2 = − 1 q^2=-1 q 2 = − 1
d k 2 2 d ( k 2 − q ) 2 d ( 2 u 1 ⋅ k 2 ) d ( 2 u 2 ⋅ k 2 ) G ( u 1 , k 2 ) G ( u 2 , k 2 ) B 3 B 2 = − 32 ε d ε d R d c 1 d c 2 ( 4 − c 1 2 ) ( 4 − c 2 2 ) b 3 b 2 , \frac{d k_2^2\,d(k_2-q)^2\,d(2u_1\cdot k_2)\,d(2u_2\cdot k_2)}{\sqrt{G(u_1,k_2)G(u_2,k_2)B_3B_2}}
=-\frac{32}{\varepsilon}\frac{d\varepsilon\,dR\,dc_1\,dc_2}{\sqrt{(4-c_1^2)(4-c_2^2)}\,b_3b_2}, G ( u 1 , k 2 ) G ( u 2 , k 2 ) B 3 B 2 d k 2 2 d ( k 2 − q ) 2 d ( 2 u 1 ⋅ k 2 ) d ( 2 u 2 ⋅ k 2 ) = − ε 32 ( 4 − c 1 2 ) ( 4 − c 2 2 ) b 3 b 2 d ε d R d c 1 d c 2 , with b 3 = [ 2 ( γ − 1 ) R − c 1 c 2 ] [ 2 ( γ + 1 ) R − c 1 c 2 ] b_3=[2(\gamma-1)R-c_1c_2][2(\gamma+1)R-c_1c_2] b 3 = [ 2 ( γ − 1 ) R − c 1 c 2 ] [ 2 ( γ + 1 ) R − c 1 c 2 ] b 2 = ( γ 2 − 1 ) ( 1 − R ) 2 + ε [ 2 ( γ 2 − 1 ) ( 1 + R ) + c 1 2 + c 2 2 − 2 γ c 1 c 2 ] + ε 2 ( γ 2 − 1 ) b_2=(\gamma^2-1)(1-R)^2+\varepsilon[2(\gamma^2-1)(1+R)+c_1^2+c_2^2-2\gamma c_1c_2]+\varepsilon^2(\gamma^2-1) b 2 = ( γ 2 − 1 ) ( 1 − R ) 2 + ε [ 2 ( γ 2 − 1 ) ( 1 + R ) + c 1 2 + c 2 2 − 2 γ c 1 c 2 ] + ε 2 ( γ 2 − 1 ) ε = 0 \varepsilon=0 ε = 0 k 2 2 k_2^2 k 2 2 b 2 = ( γ 2 − 1 ) ( 1 − R ) 2 b_2=(\gamma^2-1)(1-R)^2 b 2 = ( γ 2 − 1 ) ( 1 − R ) 2 R = 1 R=1 R = 1 b 3 = ( c 1 c 2 − 2 γ ) 2 − 4 b_3=(c_1c_2-2\gamma)^2-4 b 3 = ( c 1 c 2 − 2 γ ) 2 − 4 c i = τ i + 1 / τ i c_i=\tau_i+1/\tau_i c i = τ i + 1/ τ i t 1 = τ 1 2 t_1=\tau_1^2 t 1 = τ 1 2 t 2 = τ 1 τ 2 t_2=\tau_1\tau_2 t 2 = τ 1 τ 2
32 γ 2 − 1 d c 1 d c 2 ( 4 − c 1 2 ) ( 4 − c 2 2 ) [ ( c 1 c 2 − 2 γ ) 2 − 4 ] = 32 x 2 1 − x 2 d t 1 d t 2 Q 6 , \frac{32}{\sqrt{\gamma^2-1}}\frac{dc_1\,dc_2}{\sqrt{(4-c_1^2)(4-c_2^2)[(c_1c_2-2\gamma)^2-4]}}
=\frac{32x^2}{1-x^2}\frac{dt_1\,dt_2}{\sqrt{Q_6}}, γ 2 − 1 32 ( 4 − c 1 2 ) ( 4 − c 2 2 ) [( c 1 c 2 − 2 γ ) 2 − 4 ] d c 1 d c 2 = 1 − x 2 32 x 2 Q 6 d t 1 d t 2 , the integrand of Eq. (3.26) of Brammer et al. The first residue is the one that they take at t 1 = 0 t_1=0 t 1 = 0
Hence the whole integration torus lies at infinite k 2 2 k_2^2 k 2 2 ( k 2 − q ) 2 / k 2 2 = 1 (k_2-q)^2/k_2^2=1 ( k 2 − q ) 2 / k 2 2 = 1 2 k 1 ⋅ k 2 = k 2 2 = − 2 k 4 ⋅ k 2 2k_1\cdot k_2=k_2^2=-2k_4\cdot k_2 2 k 1 ⋅ k 2 = k 2 2 = − 2 k 4 ⋅ k 2 2 k 3 ⋅ ( k 2 − q ) = ( k 2 − q ) 2 2k_3\cdot(k_2-q)=(k_2-q)^2 2 k 3 ⋅ ( k 2 − q ) = ( k 2 − q ) 2 k 2 k_2 k 2 k 2 2 \sqrt{k_2^2} k 2 2 ( 1 − R ) k 2 2 (1-R)k_2^2 ( 1 − R ) k 2 2 k 1 k_1 k 1 k 3 k_3 k 3 k 4 k_4 k 4 ε → 0 \varepsilon\to0 ε → 0 R → 1 R \to1 R → 1 k 2 k_2 k 2 ( u 1 ⋅ n ) 2 = ( 1 + t 1 ) 2 / ( 4 t 1 ) (u_1 \cdot n)^2 = (1+t_1)^2/(4t_1) ( u 1 ⋅ n ) 2 = ( 1 + t 1 ) 2 / ( 4 t 1 ) ( u 2 ⋅ n ) 2 = ( t 1 + t 2 2 ) / ( 4 t 1 t 2 2 ) (u_2 \cdot n)^2 = (t_1+t_2^2)/(4t_1t_2^2) ( u 2 ⋅ n ) 2 = ( t 1 + t 2 2 ) / ( 4 t 1 t 2 2 ) w = 4 ( u 1 ⋅ n ) ( u 2 ⋅ n ) = c 1 c 2 w=4(u_1\cdot n)(u_2\cdot n)=c_1c_2 w = 4 ( u 1 ⋅ n ) ( u 2 ⋅ n ) = c 1 c 2
In the variables c i c_i c i ∣ c i ∣ ≤ 2 |c_i|\le2 ∣ c i ∣ ≤ 2 c i = 2 cos θ i c_i=2\cos\theta_i c i = 2 cos θ i t 1 = e 2 i θ 1 t_1=e^{2i\theta_1} t 1 = e 2 i θ 1 t 2 = e i ( θ 1 + θ 2 ) t_2=e^{i(\theta_1+\theta_2)} t 2 = e i ( θ 1 + θ 2 ) c i = ± 2 c_i=\pm2 c i = ± 2 G ( u i , k 2 ) / k 2 2 G(u_i,k_2)/k_2^2 G ( u i , k 2 ) / k 2 2 c 1 c 2 = 2 γ ± 2 c_1c_2=2\gamma\pm2 c 1 c 2 = 2 γ ± 2 B 3 / ( k 2 2 ) B_3/(k_2^2) B 3 / ( k 2 2 ) γ = 3 \gamma=3 γ = 3 c 1 c 2 = 2 γ − 2 c_1c_2=2\gamma-2 c 1 c 2 = 2 γ − 2 c 1 = c 2 = ± 2 c_1=c_2=\pm2 c 1 = c 2 = ± 2 t 1 = t 2 = 1 t_1=t_2=1 t 1 = t 2 = 1 G ( u 1 , k 2 ) G(u_1,k_2) G ( u 1 , k 2 ) G ( u 2 , k 2 ) G(u_2,k_2) G ( u 2 , k 2 ) B 3 B_3 B 3 k 2 2 k_2^2 k 2 2 k 2 2 k_2^2 k 2 2 k 2 k_2 k 2 G ( u 1 , k 2 ) = 0 G(u_1,k_2)=0 G ( u 1 , k 2 ) = 0 u 1 u_1 u 1 u 2 u_2 u 2 q q q γ = ± 1 \gamma=\pm1 γ = ± 1
The other three geometries
At 4PM we derive the cut surface of the K3 sector from the Gram determinant, whereas [25 ] obtained it loop by loop. On the maximal cut the worldline propagators δ ( 2 u i ⋅ k ) \delta(2u_i\cdot k) δ ( 2 u i ⋅ k ) [100 ] , and each integration produces an Euler Beta function and lowers the order of the determinant by one. After four such integrations we reduce the d = 4 d=4 d = 4
L S s h a p e ( x ) = ∫ − ∞ − 1 / 2 d c 2 − 2 c 2 − 1 ∮ d b 2 ( b 2 2 + c 2 2 ( γ 2 − 1 ) ) ( b 2 2 + 2 c 2 + 1 ) , γ = 1 + x 2 2 x . (21) \mathrm{LS}_{\mathrm{shape}}(x)=\int_{-\infty}^{-1/2}\frac{dc_2}{\sqrt{-2c_2-1}}\oint\frac{db_2}{\sqrt{(b_2^2+c_2^2(\gamma^2-1))(b_2^2+2c_2+1)}},\qquad\gamma=\frac{1+x^2}{2x}. \tag*{(21)} LS shape ( x ) = ∫ − ∞ − 1/2 − 2 c 2 − 1 d c 2 ∮ ( b 2 2 + c 2 2 ( γ 2 − 1 )) ( b 2 2 + 2 c 2 + 1 ) d b 2 , γ = 2 x 1 + x 2 . (21) For c 2 < − 1 2 c_2<-\frac{1}{2} c 2 < − 2 1 b 2 = ± − 2 c 2 − 1 b_2=\pm\sqrt{-2c_2-1} b 2 = ± − 2 c 2 − 1 ∮ d b 2 \oint db_2 ∮ d b 2 b 2 b_2 b 2 c 2 c_2 c 2 L S s h a p e ( x ) / ( x K ( x 2 ) 2 ) \mathrm{LS}_{\mathrm{shape}}(x)/(xK(x^2)^2) LS shape ( x ) / ( x K ( x 2 ) 2 )
L S s h a p e ( x ) = 8 x K ( x 2 ) 2 = 2 π 2 x ϖ 0 ( x 2 ) , (22) \mathrm{LS}_{\mathrm{shape}}(x)=8xK(x^2)^2=2\pi^2x\,\varpi_0(x^2), \tag*{(22)} LS shape ( x ) = 8 x K ( x 2 ) 2 = 2 π 2 x ϖ 0 ( x 2 ) , (22) where ϖ 0 ( z ) = ( 2 / π ) 2 K ( z ) 2 \varpi_0(z)=(2/\pi)^2K(z)^2 ϖ 0 ( z ) = ( 2/ π ) 2 K ( z ) 2 z ϖ 0 ( z ) \sqrt{z}\,\varpi_0(z) z ϖ 0 ( z ) L K 3 L_{\mathrm{K3}} L K3 (4) .
Dlapa et al. treated the 4PM K3 sector with differential equations [50 , 51 ] . Reducing it instead by IBP on the maximal cut, we find five master integrals, two of which are odd under the velocity reflection ( u 1 , u 2 ) ↦ ( − u 1 , − u 2 ) (u_1,u_2)\mapsto(-u_1,-u_2) ( u 1 , u 2 ) ↦ ( − u 1 , − u 2 ) ( k 1 2 ) (k_1^2) ( k 1 2 ) ( 2 k 1 ⋅ u 2 ) 2 (2k_1\cdot u_2)^2 ( 2 k 1 ⋅ u 2 ) 2 [50 ] . The first-type Landau locus (20) of the one- and two-loop topologies coincides with the points x = 0 , ± 1 x = 0, \pm1 x = 0 , ± 1 ϵ \epsilon ϵ ϵ 0 \epsilon^{0} ϵ 0
{ x K ( x 2 ) 2 , x K ( x 2 ) K ′ ( x 2 ) , x K ′ ( x 2 ) 2 } , (23) \left\{x K(x^{2})^{2},\ x K(x^{2}) K'(x^{2}),\ x K'(x^{2})^{2}\right\}, \tag*{(23)} { x K ( x 2 ) 2 , x K ( x 2 ) K ′ ( x 2 ) , x K ′ ( x 2 ) 2 } , (23) which are the solutions of its Picard–Fuchs equation given in Ref. [51 ] , Eq. (5.31), with K ′ ( z ) ≡ K ( 1 − z ) K'(z) \equiv K(1-z) K ′ ( z ) ≡ K ( 1 − z )
The two master integrals with a doubled propagator in the basis of Dlapa et al. are, at order ϵ 0 \epsilon^{0} ϵ 0 (22) is, for Ψ 1 ∝ K ( x 2 ) \Psi_{1} \propto K(x^{2}) Ψ 1 ∝ K ( x 2 ) x Ψ 1 2 x\Psi_{1}^{2} x Ψ 1 2 [50 ] , Eq. (3.23), to define the first canonical integral of the elliptic block, while Ref. [51 ] divides by x K ′ ( x 2 ) 2 / π 2 xK'(x^{2})^{2}/\pi^{2} x K ′ ( x 2 ) 2 / π 2
The hypergeometric threefold C Y 3 \mathrm{CY}_{3} CY 3 [22 , 26 ] , and its loop-by-loop Baikov polynomial is P 8 P_{8} P 8 d 3 t / P 8 \mathrm{d}^{3}t/\sqrt{P_{8}} d 3 t / P 8 T 3 T^{3} T 3 P 8 \sqrt{P_{8}} P 8 x x x x = 0 x=0 x = 0
1 ( 2 π i ) 3 ∫ T 3 d 3 t P 8 = − i 2 1 − x 2 4 F 3 ( 1 2 , 1 2 , 1 2 , 1 2 ; 1 , 1 , 1 ; x 4 ) , (24) \frac{1}{(2\pi i)^{3}}\int_{T^{3}}\frac{\mathrm{d}^{3}t}{\sqrt{P_{8}}}
=
-\frac{i}{2\sqrt{1-x^{2}}}\,{}_{4}F_{3}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1,1;x^{4}\right), \tag*{(24)} ( 2 π i ) 3 1 ∫ T 3 P 8 d 3 t = − 2 1 − x 2 i 4 F 3 ( 2 1 , 2 1 , 2 1 , 2 1 ; 1 , 1 , 1 ; x 4 ) , (24) where 4 F 3 {}_{4}F_{3} 4 F 3 ϖ 0 \varpi_{0} ϖ 0 (8) of L C Y 3 L_{\mathrm{CY}_{3}} L CY 3 Table 2 of Brammer et al.), and its maximal cut is not proportional to ϖ 0 \varpi_{0} ϖ 0
The Hadamard threefold C Y 3 ′ \mathrm{CY}'_{3} CY 3 ′ Q 8 Q_{8} Q 8 [24 ] , Eq. (3.8) of Brammer et al. We evaluate its maximal-cut integral over T 3 T^{3} T 3 x ϖ m a x c u t ( x ) = 1 8 ϖ 0 ( z ) x\varpi_{\mathrm{maxcut}}(x)=\frac{1}{8}\varpi_{0}(z) x ϖ maxcut ( x ) = 8 1 ϖ 0 ( z ) z z z ( t 1 , t 2 ) (t_{1},t_{2}) ( t 1 , t 2 ) Q 8 Q_{8} Q 8 a n = 1 , 112 , 47376 , … a_{n}=1,112,47376,\ldots a n = 1 , 112 , 47376 , … ϖ 0 \varpi_{0} ϖ 0
a n = 16 n ( 2 n n ) 2 J n , a_{n}=16^{n}\binom{2n}{n}^{2}J_{n}, a n = 1 6 n ( n 2 n ) 2 J n , J n = ∑ j , k = 0 n ( 2 n n − j ) ( 2 n n − k ) ( − 1 ) j + k ( n + 1 2 ) j + k j ! k ! = ∑ j = 0 n ( − 1 ) j ( 2 n n − j ) ( n − 1 2 − j n ) ( n + 1 2 ) j j ! , (25) \begin{aligned}
J_{n}=\sum_{j,k=0}^{n}\binom{2n}{n-j}\binom{2n}{n-k}\frac{(-1)^{j+k}\left(n+\frac{1}{2}\right)_{j+k}}{j!k!} \\
&=\sum_{j=0}^{n}(-1)^{j}\binom{2n}{n-j}\binom{n-\frac{1}{2}-j}{n}\frac{\left(n+\frac{1}{2}\right)_{j}}{j!}, \tag*{(25)}
\end{aligned} J n = j , k = 0 ∑ n ( n − j 2 n ) ( n − k 2 n ) j ! k ! ( − 1 ) j + k ( n + 2 1 ) j + k = j = 0 ∑ n ( − 1 ) j ( n − j 2 n ) ( n n − 2 1 − j ) j ! ( n + 2 1 ) j , (25) where the second form follows from the Chu–Vandermonde identity. We prove Eq. (25) for all n n n L C Y 3 ′ L_{\mathrm{CY}'_{3}} L CY 3 ′
A similar formula follows from the Hadamard representation of Klemm et al., who relate ϖ 0 \varpi_{0} ϖ 0 C ∗ i C * i C ∗ i [81 ] . The double sum that follows from this relation agrees with Eq. (25) for n ≤ 40 n \le40 n ≤ 40 i i i [96 ] relate case 2.33 by a pullback to the Hadamard square that is entry 115 of Ref. [80 ] and note its second MUM point at z ~ = 1 / 256 \tilde{z}=1/256 z ~ = 1/256 z ~ = x 2 / 256 \tilde{z}=x^2/256 z ~ = x 2 /256 x = 0 x=0 x = 0 x = 1 x=1 x = 1 [56 ] and, on its maximal cut, Duhr et al. [23 ] have constructed ϵ \epsilon ϵ
At four loops the nested Beta-function integration of the Gram determinant removes eight of the twelve irreducible scalar products for topology 3 and six of the eleven for topology 37, but does not yield the three-dimensional cut surfaces. For the 5PM rows of Table 1 we therefore take as input the loop-by-loop Baikov polynomials Q 6 Q_6 Q 6 P 8 P_8 P 8 Q 8 Q_8 Q 8 Q 6 Q_6 Q 6 P 8 P_8 P 8
For each constant of the table we evaluate the contour integral at three to five rational points inside the Frobenius disk around the MUM point and divide it by the Frobenius series of the Picard–Fuchs operator. The ratio is constant across the points to at least 34 significant figures. It is unity by construction for the Apéry row and is otherwise found by an integer-relation search. Outside the disks we made the comparison only for topology 40, since for topologies 3 and 37 it would require continuation of the period or a deformed contour, which we have not attempted.
The dissipative function space
In the four dissipative families of 5PM-2SF we keep both routings of the uncut worldline propagators, and the function space of these families has not been determined. We count their master integrals at first stage, find linear relations among them from the response matrices of the auxiliary-mass flow, and compute boundary constants of the smallest family, PX2, at one kinematic point.
Master-integral counts and parity
We reduce the four dissipative families with Kira [84 , 85 , 86 ] using its automatic sector symmetries (§ 2.1 ). Table 3 lists the first-stage master-integral counts we obtain, the parity split for the crossed family PX2 and the conservative counts of Ref. [15 ] . The four dissipative counts sum to 731, the conservative total. Since the dissipative counts are taken at a fixed seeding depth and the conservative counts after imposing relations that hold only under time symmetry, we regard this equality as a coincidence and infer no correspondence between the bases.
Table 3. First-stage master-integral counts of the four dissipative 5PM-2SF families and the conservative counts of Ref. [15 ] . The first-stage counts are taken at the fixed seeding of Appendix B.1 and are not converged basis dimensions: the PX2 count is unchanged under the deeper seeding described there, and the counts of P, PX1 and NP have not been tested. The 36 sectors in parentheses are the further top sectors of PX2 that contain the K3 and Calabi–Yau subtopologies. The parity split is determined for PX2 only.
Table 3.
For the reductions we allow derivatives of the cut delta functions and normalize the integrals of Eq. (2) , in the kinematics of § 2.1 , as
I n 1 ⋯ n 22 = ∫ ∏ i = 1 4 d d k i i π d / 2 ∏ c = 1 4 2 π δ ( n c − 1 ) ( D c ) ∏ i = 5 22 D i n i , d = 4 − 2 ϵ , (26) I_{n_1\cdots n_{22}}=\int\prod_{i=1}^{4}\frac{\mathrm{d}^{d}k_i}{i\pi^{d/2}}\frac{\prod_{c=1}^{4}2\pi\delta^{(n_c-1)}(D_c)}{\prod_{i=5}^{22}D_i^{n_i}},\qquad d=4-2\epsilon, \tag*{(26)} I n 1 ⋯ n 22 = ∫ i = 1 ∏ 4 i π d /2 d d k i ∏ i = 5 22 D i n i ∏ c = 1 4 2 π δ ( n c − 1 ) ( D c ) , d = 4 − 2 ϵ , (26) where the first four propagators are cut: n c = 1 n_c = 1 n c = 1 n c = 2 n_c = 2 n c = 2 n i < 0 n_i < 0 n i < 0 D i ∣ n i ∣ D_i^{|n_i|} D i ∣ n i ∣ u i → − u i u_i \to-u_i u i → − u i D 5 , … , D 8 D_5,\ldots,D_8 D 5 , … , D 8 − 1 -1 − 1 [23 ] .
Response matrices
In the auxiliary-mass flow [33 , 82 , 101 ] a subset of the propagators of a top sector is deformed, D i → D i − η D_i \to D_i - \eta D i → D i − η η → ∞ \eta\to\infty η → ∞ η → 0 \eta\to0 η → 0 [102 , 103 ] ). Since the evolution is linear in the boundary data, we can write
M ( ϵ ) = A ( ϵ ) b ( ϵ ) , (27) M(\epsilon) = A(\epsilon)b(\epsilon), \tag*{(27)} M ( ϵ ) = A ( ϵ ) b ( ϵ ) , (27) with M M M η \eta η η → 0 \eta\to0 η → 0 b b b η → ∞ \eta\to\infty η → ∞ A ( ϵ ) A(\epsilon) A ( ϵ ) v v v A A A v T A = 0 v^{T}A = 0 v T A = 0
Table 4 lists the propagators of the PX2 top sectors S 1 S_1 S 1 S 2 S_2 S 2 S 3 S_3 S 3 A ( ϵ ) A(\epsilon) A ( ϵ ) ϵ \epsilon ϵ ϵ \epsilon ϵ
Table 4. Response matrices of the three analyzed top sectors of PX2 and of the first NP top sector: the rank deficits give ten flow-free directions on PX2 and two on NP. n J n_J n J n b n_b n b η \eta η n v = n J − rank A n_v=n_J-\operatorname{rank} A n v = n J − rank A n s u p p n_{\mathrm{supp}} n supp response matrices/.
v ( S 1 ) = ( 1 , 16 3 , − 6 ) on ( I 0 , I 2 , I 3 ) , v 0 ( S 2 ) = ( − 1 6 , 1 , − 8 9 ) on ( I r 0 , I r 2 , I r 3 ) , (28) v^{(S_1)} = \left(1, \frac{16}{3}, -6\right)\ \text{on}\ (I_0,I_2,I_3), \qquad v_0^{(S_2)} = \left(-\frac{1}{6},1,-\frac{8}{9}\right)\ \text{on}\ (I_{r0},I_{r2},I_{r3}), \tag*{(28)} v ( S 1 ) = ( 1 , 3 16 , − 6 ) on ( I 0 , I 2 , I 3 ) , v 0 ( S 2 ) = ( − 6 1 , 1 , − 9 8 ) on ( I r 0 , I r 2 , I r 3 ) , (28) with I r k I_{rk} I r k k k k S 2 S_2 S 2 ϵ \epsilon ϵ I 0 I_0 I 0 I 2 I_2 I 2 I 3 I_3 I 3 S 1 S_1 S 1 (29) , contain none of the sector’s uncut linearized propagators D 5 , … , D 8 D_5,\ldots,D_8 D 5 , … , D 8
For the NP family we solve the η \eta η Table 4 lists the first of its nine top sectors.
The supporting master integrals
The left null vectors of § 4.2 are supported on 31 rows: I 0 I_0 I 0 I 2 I_2 I 2 I 3 I_3 I 3 S 1 S_1 S 1 I r k I_{rk} I r k S 2 S_2 S 2 I ~ k \widetilde{I}_k I k S 3 S_3 S 3 tables/PX2_supporting_31.txt. The three rows of S 1 S_1 S 1
I 0 = ∫ ∏ i = 1 4 d d k i i π d / 2 ( 2 π ) 4 δ ( 2 k 1 ⋅ u 1 ) δ ( 2 k 2 ⋅ u 1 ) δ ( 2 k 3 ⋅ u 2 ) δ ( 2 k 4 ⋅ u 2 ) ( k 1 − k 2 ) 2 ( k 2 − k 4 ) 2 ( k 3 − k 4 ) 2 ( k 1 + q ) 2 k 3 2 , (29) I_0=\int\prod_{i=1}^{4}\frac{\mathrm{d}^d k_i}{\mathrm{i}\pi^{d/2}}\frac{(2\pi)^4\delta(2k_1\cdot u_1)\delta(2k_2\cdot u_1)\delta(2k_3\cdot u_2)\delta(2k_4\cdot u_2)}{(k_1-k_2)^2(k_2-k_4)^2(k_3-k_4)^2(k_1+q)^2 k_3^2}, \tag*{(29)} I 0 = ∫ i = 1 ∏ 4 i π d /2 d d k i ( k 1 − k 2 ) 2 ( k 2 − k 4 ) 2 ( k 3 − k 4 ) 2 ( k 1 + q ) 2 k 3 2 ( 2 π ) 4 δ ( 2 k 1 ⋅ u 1 ) δ ( 2 k 2 ⋅ u 1 ) δ ( 2 k 3 ⋅ u 2 ) δ ( 2 k 4 ⋅ u 2 ) , (29) I 2 = I 0 [ ( k 1 − k 3 ) 2 ] , I 3 = I 0 [ ( k 1 − k 4 ) 2 ] , I_2=I_0\left[(k_1-k_3)^2\right],\qquad I_3=I_0\left[(k_1-k_4)^2\right], I 2 = I 0 [ ( k 1 − k 3 ) 2 ] , I 3 = I 0 [ ( k 1 − k 4 ) 2 ] , where I 0 [ N ] I_0[N] I 0 [ N ] I 0 I_0 I 0 N N N I r 0 I_{r0} I r 0 S 2 S_2 S 2 D 9 D 11 D 14 D 16 D 21 D_9D_{11}D_{14}D_{16}D_{21} D 9 D 11 D 14 D 16 D 21 I r 1 = I r 0 [ D 5 ] I_{r1}=I_{r0}[D_5] I r 1 = I r 0 [ D 5 ] I r 2 = I r 0 [ D 10 ] I_{r2}=I_{r0}[D_{10}] I r 2 = I r 0 [ D 10 ] I r 3 = I r 0 [ D 12 ] I_{r3}=I_{r0}[D_{12}] I r 3 = I r 0 [ D 12 ] I r 4 I_{r4} I r 4 D 14 D_{14} D 14 I r 5 I_{r5} I r 5 D 5 D_5 D 5 I r 6 = I r 5 [ D 12 ] I_{r6}=I_{r5}[D_{12}] I r 6 = I r 5 [ D 12 ] I r 7 = I r 5 [ D 13 ] I_{r7}=I_{r5}[D_{13}] I r 7 = I r 5 [ D 13 ] S 3 S_3 S 3 I ~ 0 , … , I ~ 7 \widetilde{I}_0,\ldots,\widetilde{I}_7 I 0 , … , I 7
We relate these integrals by three changes of variables. (i) The relabeling k 1 ↔ k 2 k_1\leftrightarrow k_2 k 1 ↔ k 2 S 1 S_1 S 1 D 9 D 13 D 14 D 15 D 21 D_9D_{13}D_{14}D_{15}D_{21} D 9 D 13 D 14 D 15 D 21 S 2 S_2 S 2 I r 0 = I 0 I_{r0}=I_0 I r 0 = I 0 I r 3 = I 2 I_{r3}=I_2 I r 3 = I 2
(ii) The reflection k i → − k i − q k_i\to-k_i-q k i → − k i − q i i i D 14 + j ↔ D 18 + j D_{14+j}\leftrightarrow D_{18+j} D 14 + j ↔ D 18 + j D 9 , … , D 14 D_9,\ldots,D_{14} D 9 , … , D 14 u i q = 0 u_iq=0 u i q = 0 S 2 S_2 S 2 S 3 S_3 S 3 ( − 1 ) m (-1)^m ( − 1 ) m m m m σ \sigma σ I ~ 0 = I r 0 \widetilde{I}_0=I_{r0} I 0 = I r 0 I ~ 2 = I r 2 \widetilde{I}_2=I_{r2} I 2 = I r 2 I ~ 3 = I r 3 \widetilde{I}_3=I_{r3} I 3 = I r 3 I ~ 4 = I r 4 \widetilde{I}_4=I_{r4} I 4 = I r 4 I ~ 1 = − I r 1 \widetilde{I}_1=-I_{r1} I 1 = − I r 1 I r 5 I_{r5} I r 5 I r 6 I_{r6} I r 6 I r 7 I_{r7} I r 7 σ \sigma σ § 4.4 ), it also gives I ~ 5 , 6 , 7 = I r 5 , 6 , 7 \widetilde{I}_{5,6,7}=I_{r5,6,7} I 5 , 6 , 7 = I r 5 , 6 , 7 σ \sigma σ S 1 S_1 S 1 k 1 → − k 3 − q k_1 \to-k_3-q k 1 → − k 3 − q k 2 → − k 4 − q k_2 \to-k_4-q k 2 → − k 4 − q k 3 → − k 1 − q k_3 \to-k_1-q k 3 → − k 1 − q k 4 → − k 2 − q k_4 \to-k_2-q k 4 → − k 2 − q u 1 ↔ u 2 u_1 \leftrightarrow u_2 u 1 ↔ u 2 D 15 ↔ D 21 D_{15} \leftrightarrow D_{21} D 15 ↔ D 21 D 9 ↔ D 14 D_9 \leftrightarrow D_{14} D 9 ↔ D 14 D 11 ↔ D 12 D_{11} \leftrightarrow D_{12} D 11 ↔ D 12 I r 2 = I 0 [ D 12 ] = I 0 [ D 11 ] = I 3 I_{r2}=I_0[D_{12}]=I_0[D_{11}]=I_3 I r 2 = I 0 [ D 12 ] = I 0 [ D 11 ] = I 3
Thus, with the identifications of the three corner integrals alone, the 31 rows involve at most 25 distinct integrals, and fewer with the reflection (ii). Reducing the corner sectors with numerator integrals as seeds, we find I r 4 I_{r4} I r 4 I 0 I_0 I 0 D 9 D_9 D 9 D 14 D_{14} D 14
Boundary constants of PX2
At one reference kinematic point,
γ ≡ u 1 ⋅ u 2 = 5 4 , q 2 = − 1 , x = γ − γ 2 − 1 = 1 2 , (30) \gamma\equiv u_1 \cdot u_2=\frac{5}{4}, \qquad q^2=-1, \qquad x=\gamma-\sqrt{\gamma^2-1}=\frac{1}{2}, \tag*{(30)} γ ≡ u 1 ⋅ u 2 = 4 5 , q 2 = − 1 , x = γ − γ 2 − 1 = 2 1 , (30) we compute twelve of the at most 25 distinct master integrals on which the ten flow-free directions are supported. With the six relabeled corner rows and I ~ 4 = I r 4 \widetilde{I}_4=I_{r4} I 4 = I r 4 S 1 S_1 S 1 S 2 S_2 S 2 I r 0 I_{r0} I r 0 I r 7 I_{r7} I r 7 S 3 S_3 S 3 I ~ 0 , … , I ~ 7 \widetilde{I}_0,\ldots,\widetilde{I}_7 I 0 , … , I 7 I r 9 I_{r9} I r 9 I r 10 I_{r10} I r 10 I r 18 I_{r18} I r 18 I r 19 I_{r19} I r 19 S 3 S_3 S 3 D 7 D_7 D 7 D 8 D_8 D 8 D 5 D_5 D 5
The measure is that of Eq. (29) , with d = 4 − 2 ϵ d=4-2\epsilon d = 4 − 2 ϵ e γ E ϵ e^{\gamma_E\epsilon} e γ E ϵ i 0 i0 i 0 (2) , are Feynman propagators. The finite parts of the corner integrals and the order-ϵ \epsilon ϵ I r 4 I_{r4} I r 4 ln 3 \ln3 ln 3 π 2 \pi^2 π 2 Li 2 ( 1 4 ) \operatorname{Li}_2(\frac{1}{4}) Li 2 ( 4 1 ) Table 5 ) are exact and lie in the ring i π 5 Q [ ln 2 , γ E ] i\pi^5\mathbb{Q}[\ln2,\gamma_E] i π 5 Q [ ln 2 , γ E ] π 4 \pi^4 π 4
Table 5. Boundary constants of PX2 at the reference point (30) , with Feynman propagators on the graviton lines. The finite parts of the corner integrals, given in closed form in Eq. (31) , are quoted to ten figures, truncated. I r k I_{rk} I r k I ~ k \widetilde{I}_k I k k k k S 2 S_2 S 2 S 3 S_3 S 3 σ = + 1 \sigma=+1 σ = + 1 (33) ).
In the two worldline frequencies a a a b b b ( k 2 − k 4 ) 2 (k_2-k_4)^2 ( k 2 − k 4 ) 2 a 2 + b 2 − 2 a b γ a^2+b^2-2ab\gamma a 2 + b 2 − 2 abγ γ > 1 \gamma>1 γ > 1 K K K [15 ] that we evaluate in § 6 , with an ultraviolet pole from a factor Γ ( 4 ϵ − 1 ) \Gamma(4\epsilon-1) Γ ( 4 ϵ − 1 ) § 11 , together with the dotted and static-slice integrals below, and find for their pole coefficients and finite parts
I 0 ∣ ϵ − 1 = π 4 80 ln 2 − 48 27 + i π 5 27 , I_0\big|_{\epsilon^{-1}}=\pi^4\frac{80\ln2-48}{27}+\frac{i\pi^5}{27}, I 0 ϵ − 1 = π 4 27 80 ln 2 − 48 + 27 i π 5 , I 2 ∣ ϵ − 1 = π 4 150 − 220 ln 2 27 − 7 i π 5 864 , I_2\big|_{\epsilon^{-1}}=\pi^4\frac{150-220\ln2}{27}-\frac{7i\pi^5}{864}, I 2 ϵ − 1 = π 4 27 150 − 220 ln 2 − 864 7 i π 5 , I 3 ∣ ϵ − 1 = π 4 1128 − 1640 ln 2 243 − i π 5 972 , I_3\big|_{\epsilon^{-1}}=\pi^4\frac{1128-1640\ln2}{243}-\frac{i\pi^5}{972}, I 3 ϵ − 1 = π 4 243 1128 − 1640 ln 2 − 972 i π 5 , I 0 ∣ ϵ 0 + 4 γ E I 0 ∣ ϵ − 1 = π 4 ( − 80 3 + 22 9 π 2 + 400 27 ln 2 − 796 27 ln 2 2 − 238 27 Λ ) + i π 5 ( 41 27 + 976 27 ln 2 − 638 27 ln 3 ) , I_0\big|_{\epsilon^0}+4\gamma_E I_0\big|_{\epsilon^{-1}}
=\pi^4\left(-\frac{80}{3}+\frac{22}{9}\pi^2+\frac{400}{27}\ln2-\frac{796}{27}\ln^2 2-\frac{238}{27}\Lambda\right)
+i\pi^5\left(\frac{41}{27}+\frac{976}{27}\ln2-\frac{638}{27}\ln3\right), I 0 ϵ 0 + 4 γ E I 0 ϵ − 1 = π 4 ( − 3 80 + 9 22 π 2 + 27 400 ln 2 − 27 796 ln 2 2 − 27 238 Λ ) + i π 5 ( 27 41 + 27 976 ln 2 − 27 638 ln 3 ) , I 2 ∣ ϵ 0 + 4 γ E I 2 ∣ ϵ − 1 = π 4 ( 3995 54 − 977 144 π 2 − 3985 162 ln 2 + 17593 216 ln 2 2 + 10553 432 Λ ) I_2\big|_{\epsilon^0}+4\gamma_E I_2\big|_{\epsilon^{-1}}
=\pi^4\left(\frac{3995}{54}-\frac{977}{144}\pi^2-\frac{3985}{162}\ln2+\frac{17593}{216}\ln^2 2+\frac{10553}{432}\Lambda\right) I 2 ϵ 0 + 4 γ E I 2 ϵ − 1 = π 4 ( 54 3995 − 144 977 π 2 − 162 3985 ln 2 + 216 17593 ln 2 2 + 432 10553 Λ ) + i π 5 ( − 20035 5184 − 5287 54 ln 2 + 28153 432 ln 3 ) , I 3 ∣ ϵ 0 + 4 γ E I 3 ∣ ϵ − 1 = π 4 ( 14900 243 − 911 162 π 2 − 14140 729 ln 2 + 16399 243 ln 2 2 + 9839 486 Λ ) + i π 5 ( − 18559 5832 − 19684 243 ln 2 + 26239 486 ln 3 ) , (31) \begin{aligned}
&+i\pi^5\left(-\frac{20035}{5184}-\frac{5287}{54}\ln2+\frac{28153}{432}\ln3\right),\\
\left.I_3\right|_{\epsilon^0}+4\gamma_E\left.I_3\right|_{\epsilon^{-1}}={}&\pi^4\left(\frac{14900}{243}-\frac{911}{162}\pi^2-\frac{14140}{729}\ln2+\frac{16399}{243}\ln^2 2+\frac{9839}{486}\Lambda\right)\\
&+i\pi^5\left(-\frac{18559}{5832}-\frac{19684}{243}\ln2+\frac{26239}{486}\ln3\right), \tag*{(31)}
\end{aligned} I 3 ∣ ϵ 0 + 4 γ E I 3 ∣ ϵ − 1 = + i π 5 ( − 5184 20035 − 54 5287 ln 2 + 432 28153 ln 3 ) , π 4 ( 243 14900 − 162 911 π 2 − 729 14140 ln 2 + 243 16399 ln 2 2 + 486 9839 Λ ) + i π 5 ( − 5832 18559 − 243 19684 ln 2 + 486 26239 ln 3 ) , (31) with Λ = Li 2 ( 1 4 ) − 2 ln 2 ln 3 \Lambda=\operatorname{Li}_2\left(\frac{1}{4}\right)-2\ln2\ln3 Λ = Li 2 ( 4 1 ) − 2 ln 2 ln 3
The dotted integral I r 4 = I ~ 4 I_{r4}=\widetilde{I}_4 I r 4 = I 4 S 2 S_2 S 2 ( k 3 − k 4 ) 2 (k_3-k_4)^2 ( k 3 − k 4 ) 2
I r 4 = − π 4 ϵ − π 4 ( 4 + 20 3 ln 2 − 4 γ E ) + 2 3 i π 5 − ϵ π 4 ( 28 + 7 3 π 2 + 80 3 ln 2 − 232 3 ln 2 2 − 64 3 Λ − 16 γ E − 80 3 γ E ln 2 + 8 γ E 2 ) − ϵ i π 5 ( − 8 3 + 232 3 ln 2 − 164 3 ln 3 + 8 3 γ E ) + O ( ϵ 2 ) . (32) \begin{aligned}
I_{r4}={}&-\frac{\pi^4}{\epsilon}-\pi^4\left(4+\frac{20}{3}\ln2-4\gamma_E\right)+\frac{2}{3}i\pi^5\\
&-\epsilon\pi^4\left(28+\frac{7}{3}\pi^2+\frac{80}{3}\ln2-\frac{232}{3}\ln^2 2\right.\\
&\left.\qquad-\frac{64}{3}\Lambda-16\gamma_E-\frac{80}{3}\gamma_E\ln2+8\gamma_E^2\right)\\
&-\epsilon i\pi^5\left(-\frac{8}{3}+\frac{232}{3}\ln2-\frac{164}{3}\ln3+\frac{8}{3}\gamma_E\right)+O(\epsilon^2). \tag*{(32)}
\end{aligned} I r 4 = − ϵ π 4 − π 4 ( 4 + 3 20 ln 2 − 4 γ E ) + 3 2 i π 5 − ϵ π 4 ( 28 + 3 7 π 2 + 3 80 ln 2 − 3 232 ln 2 2 − 3 64 Λ − 16 γ E − 3 80 γ E ln 2 + 8 γ E 2 ) − ϵ i π 5 ( − 3 8 + 3 232 ln 2 − 3 164 ln 3 + 3 8 γ E ) + O ( ϵ 2 ) . (32) Its pole arises at a static endpoint of the radial integration, off the radiation cone, which first contributes at the finite order through the + 2 3 i π 5 +\frac{2}{3}i\pi^5 + 3 2 i π 5 I 0 I_0 I 0 I 2 I_2 I 2 I 3 I_3 I 3 ϵ \epsilon ϵ I r 4 I_{r4} I r 4 [15 , 42 ] , on evaluations at 330 and 220 digits. The relations hold for more than 100 digits beyond those used in the fit. The overall sign of I r 4 I_{r4} I r 4 I 0 I_0 I 0 γ \gamma γ
The odd-parity master integrals I r 5 , I r 6 , I r 7 I_{r5},I_{r6},I_{r7} I r 5 , I r 6 , I r 7 S 2 S_2 S 2 I ~ 5 , I ~ 6 , I ~ 7 \widetilde{I}_5,\widetilde{I}_6,\widetilde{I}_7 I 5 , I 6 , I 7 S 3 S_3 S 3 1 / ( 2 k ⋅ u + i σ 0 ) 1/(2k\cdot u+i\sigma0) 1/ ( 2 k ⋅ u + iσ 0 ) [67 , 72 ] . We split it into a principal value and a delta function,
1 b + i σ 0 = PV 1 b − i π σ δ ( b ) . (33) \frac{1}{b+i\sigma0}=\operatorname{PV}\frac{1}{b}-i\pi\sigma\delta(b). \tag*{(33)} b + iσ 0 1 = PV b 1 − iπ σ δ ( b ) . (33) A reflection of both frequency axes reverses b b b k 1 − k 4 k_1-k_4 k 1 − k 4 δ ( b ) \delta(b) δ ( b ) i 0 i0 i 0 σ \sigma σ Γ \Gamma Γ ϵ \epsilon ϵ Γ ( 4 ϵ − 1 ) \Gamma(4\epsilon-1) Γ ( 4 ϵ − 1 )
The master integrals I r 1 I_{r1} I r 1 S 2 S_2 S 2 I ~ 1 \widetilde{I}_1 I 1 S 3 S_3 S 3 ϵ \epsilon ϵ
I r 1 = I ~ 1 = 0. (34) I_{r1}=\widetilde{I}_1=0. \tag*{(34)} I r 1 = I 1 = 0. (34) We collect the boundary constants in Table 5 , having derived each of them and the pole and finite part in Eq. (32) at least twice. We know the values for both signs of σ \sigma σ [67 ] , and we leave this choice open. Retarded graviton propagators are Feynman propagators plus cut terms [67 , 70 ] . A cut term is not even in the graviton momentum, so Eq. (34) and the vanishing of the principal-value parts do not extend to retarded propagators. We have not evaluated the cut integrals and give no in-in boundary constants.
Relations among boundary constants
The three corner integrals satisfy
I 0 + 16 3 I 2 − 6 I 3 = 0 (35) I_0+\frac{16}{3}I_2-6I_3=0 \tag*{(35)} I 0 + 3 16 I 2 − 6 I 3 = 0 (35) order by order in ϵ \epsilon ϵ (31) cancels separately. The weights are the components of v ( S 1 ) v^{(S_1)} v ( S 1 ) (28) , fixed by the differential system independently of the values. Equation (35) is one instance of a general relation. Under three assumptions stated in Appendix § 9 , of which the physical one is the completeness of the enumerated boundary integrals (assumption (ii), not proved here), every flow-free direction v v v
v T A = 0 ⟹ v T M = 0 (36) v^{T}A=0\quad\Longrightarrow\quad v^{T}M=0 \tag*{(36)} v T A = 0 ⟹ v T M = 0 (36) identically in ϵ \epsilon ϵ A ( ϵ ) A(\epsilon) A ( ϵ )
The ten relations v T M = 0 v^{T}M=0 v T M = 0 S 1 S_{1} S 1 S 2 S_{2} S 2 S 3 S_{3} S 3 [104 ] , including for these families [15 ] . With these integrals as targets we obtain all ten relations with coefficients rational in d d d γ \gamma γ (35) for all d d d γ \gamma γ
Seven of the ten flow-free directions are supported on computed master integrals: the direction of Eq. (35) in each of the three sectors (on S 3 S_{3} S 3 ϵ \epsilon ϵ S 2 S_{2} S 2 S 3 S_{3} S 3 (36) holds exactly: in rational arithmetic at the pole on S 1 S_{1} S 1 S 2 S_{2} S 2 (31) at the finite order on all three sectors. At ϵ = 0 \epsilon=0 ϵ = 0 ϵ \epsilon ϵ ϵ \epsilon ϵ d = 4 d=4 d = 4 γ = 5 / 4 \gamma=5/4 γ = 5/4 boundary_constants_PX2.json.
The relation (36) applies separately to each assignment of graviton propagators that is even in the graviton momenta (Feynman, anti-Feynman or their average), since these share the differential system and differ only in boundary data.
On S 2 S_{2} S 2 I r 1 I_{r1} I r 1 { I r 5 , I r 6 } \{I_{r5},I_{r6}\} { I r 5 , I r 6 } { I r 5 , I r 7 } \{I_{r5},I_{r7}\} { I r 5 , I r 7 } I r 5 I_{r5} I r 5 (36) fixes the other two,
I r 6 ∣ ϵ = 0 = − 8 27 I r 5 ∣ ϵ = 0 = − 128 81 i π 5 , I r 7 ∣ ϵ = 0 = − 2 27 I r 5 ∣ ϵ = 0 = 32 81 i π 5 , (37) \begin{aligned}
I_{r6}\big|_{\epsilon=0} &= -\frac{8}{27} I_{r5}\big|_{\epsilon=0} = -\frac{128}{81} i\pi^{5},\\
I_{r7}\big|_{\epsilon=0} &= -\frac{2}{27} I_{r5}\big|_{\epsilon=0} = \frac{32}{81} i\pi^{5}, \tag*{(37)}
\end{aligned} I r 6 ϵ = 0 I r 7 ϵ = 0 = − 27 8 I r 5 ϵ = 0 = − 81 128 i π 5 , = − 27 2 I r 5 ϵ = 0 = 81 32 i π 5 , (37) and, applied at fixed ϵ \epsilon ϵ ϵ \epsilon ϵ Table 5 . The two directions occur on S 3 S_{3} S 3 I ~ 5 \widetilde{I}_{5} I 5 I ~ 6 \widetilde{I}_{6} I 6 I ~ 7 \widetilde{I}_{7} I 7 ϵ \epsilon ϵ S 2 S_{2} S 2 M M M σ \sigma σ σ \sigma σ
In the remaining ϵ \epsilon ϵ S 2 S_{2} S 2 (36) fixes, at the pole, a combination of the four S 2 S_{2} S 2
1 2 I r 9 ( − 1 ) + 5 12 I r 10 ( − 1 ) + 1 8 I r 18 ( − 1 ) + I r 19 ( − 1 ) = 5 12 I r 0 ( − 1 ) = π 4 100 ln 2 − 60 81 + 5 324 i π 5 . (38) \begin{aligned}
\frac{1}{2}I_{r9}^{(-1)}+\frac{5}{12}I_{r10}^{(-1)}+\frac{1}{8}I_{r18}^{(-1)}+I_{r19}^{(-1)}
&=\frac{5}{12}I_{r0}^{(-1)}\\
&=\pi^{4}\frac{100\ln2-60}{81}+\frac{5}{324}i\pi^{5}. \tag*{(38)}
\end{aligned} 2 1 I r 9 ( − 1 ) + 12 5 I r 10 ( − 1 ) + 8 1 I r 18 ( − 1 ) + I r 19 ( − 1 ) = 12 5 I r 0 ( − 1 ) = π 4 81 100 ln 2 − 60 + 324 5 i π 5 . (38) Similarly, the two remaining directions of S 3 S_{3} S 3
K3 and Calabi–Yau sectors
The K3 and Calabi–Yau sectors at 5PM-2SF [22 , 23 , 24 , 26 , 56 ] are subtopologies with six or seven graviton propagators, whereas the leading top sectors of dissipative PX2 have five. (Table 4 ). Since a period can enter a master integral only through a sector contained in its top sector, no K3 or Calabi–Yau period appears in any of the 37 master integrals of the leading top sectors. Hence any such period of PX2 must enter through one of its 36 further top sectors (Table 3 ), and we have not determined which of them couple to the dissipative master integrals. Brammer et al. predict that the 4PM K3 reappears in the dissipative sector [26 ] .
The planar family P contains the Apéry K3 sector, topology 40 of Brammer et al. On its maximal cut we reduce the even-parity block with integration-by-parts and Lorentz-invariance identities [105 , 106 , 107 ] to eleven master integrals. When we also impose the sector’s worldline-exchange symmetry we obtain nine, the number found in Ref. [23 ] . At one kinematic point and two numerical values of d d d
Continuation to bound kinematics
As solutions of the Picard–Fuchs equations (4) –(7) ), the K3 and Calabi–Yau periods of the 4PM and 5PM integrals [22 , 26 , 57 ] extend analytically from γ > 1 \gamma> 1 γ > 1 γ < 1 \gamma< 1 γ < 1 [15 , 57 ] , which converge on part of the bound domain, and leave the transition to the bound case pending. Here we continue the full Frobenius bases of the four operators at z = 0 z = 0 z = 0 γ < 1 \gamma< 1 γ < 1 4 F 3 {}_{4}F_{3} 4 F 3
Bound kinematics and Frobenius frame
For bound motion we write
γ = cos θ ∈ ( 0 , 1 ) , x = e ± i θ , (39) \gamma= \cos\theta\in(0,1), \qquad x = e^{\pm i\theta}, \tag*{(39)} γ = cos θ ∈ ( 0 , 1 ) , x = e ± i θ , (39) and bound orbits map to the unit circle ∣ x ∣ = 1 |x| = 1 ∣ x ∣ = 1 s + i 0 s + i0 s + i 0 s s s i 0 + i0^{+} i 0 + s + i 0 s + i0 s + i 0 γ + i 0 \gamma+ i0 γ + i 0 d γ / d x < 0 \mathrm{d}\gamma/\mathrm{d}x < 0 d γ / d x < 0 0 < x < 1 0 < x < 1 0 < x < 1 x = e − i θ x = e^{-i\theta} x = e − i θ θ ∈ ( 0 , π / 2 ] \theta\in(0,\pi/2] θ ∈ ( 0 , π /2 ]
On the arc we use a separate variable for each geometry,
z K 3 = z K 3 ′ = x 2 , z C Y 3 = x 4 2 8 , z C Y 3 ′ = 1 − γ 2 2 10 , (40) z_{\mathrm{K3}} = z_{\mathrm{K3}'} = x^2,\qquad z_{\mathrm{CY}_3} = \frac{x^4}{2^8},\qquad z_{\mathrm{CY}_3'} = \frac{1-\gamma^2}{2^{10}}, \tag*{(40)} z K3 = z K3 ′ = x 2 , z CY 3 = 2 8 x 4 , z CY 3 ′ = 2 10 1 − γ 2 , (40) the last from γ = ( x + 1 / x ) / 2 \gamma= (x + 1/x)/2 γ = ( x + 1/ x ) /2 L K 3 L_{\mathrm{K3}} L K3 (4) has indicial exponent 1 2 \frac{1}{2} 2 1 z = 0 z = 0 z = 0 3 θ 2 + 2 θ + 1 2 ) + z 2 ( θ + 1 ) 3 3\theta^2+2\theta+\frac{1}{2})+z^2(\theta+1)^3 3 θ 2 + 2 θ + 2 1 ) + z 2 ( θ + 1 ) 3 z = 0 z=0 z = 0 ϖ 0 = 2 F 1 ( 1 2 , 1 2 ; 1 ; z ) 2 = ( 2 π ) 2 K ( z ) 2 \varpi_0={}_2F_1\left(\frac{1}{2},\frac{1}{2};1;z\right)^2=\left(\frac{2}{\pi}\right)^2K(z)^2 ϖ 0 = 2 F 1 ( 2 1 , 2 1 ; 1 ; z ) 2 = ( π 2 ) 2 K ( z ) 2 L ~ K 3 \widetilde{L}_{K3} L K 3 L K 3 L_{K3} L K 3 z = 0 z=0 z = 0 θ r \theta^r θ r r r r
ϖ k ( z ) = ∑ j = 0 k ln j z j ! h k − j ( z ) , k = 0 , … , r − 1 , h 0 ( 0 ) = 1 , h j ( 0 ) = 0 ( j ≥ 1 ) , (41) \varpi_k(z)=\sum_{j=0}^{k}\frac{\ln^j z}{j!}h_{k-j}(z),\qquad k=0,\ldots,r-1,\qquad h_0(0)=1,\qquad h_j(0)=0\ (j\geq1), \tag*{(41)} ϖ k ( z ) = j = 0 ∑ k j ! ln j z h k − j ( z ) , k = 0 , … , r − 1 , h 0 ( 0 ) = 1 , h j ( 0 ) = 0 ( j ≥ 1 ) , (41) with power series h j h_j h j z = 0 z=0 z = 0 ln z → ln z + 2 π i \ln z\to\ln z+2\pi i ln z → ln z + 2 π i M 0 − 1 M_0^{-1} M 0 − 1 (48) .
Singular points on the bound domains
The finite singular points of each operator are the zeros of its leading symbol s r ( z ) s_r(z) s r ( z ) p r ( z ) = z r s r ( z ) p_r(z)=z^r s_r(z) p r ( z ) = z r s r ( z ) r r r
s 2 K 3 ( z ) = ( 1 − z ) 2 , s 3 K 3 ′ ( z ) = 1 − 34 z + z 2 , s 4 C Y ( z ) = 1 − 256 z , s 4 C Y ′ ( z ) = ( 1 − 1024 z ) 3 , (42) s^{\mathrm{K3}}_2(z)=(1-z)^2,\qquad s^{\mathrm{K3'}}_3(z)=1-34z+z^2,\qquad s^{\mathrm{CY}}_4(z)=1-256z,\qquad s^{\mathrm{CY'}}_4(z)=(1-1024z)^3, \tag*{(42)} s 2 K3 ( z ) = ( 1 − z ) 2 , s 3 K 3 ′ ( z ) = 1 − 34 z + z 2 , s 4 CY ( z ) = 1 − 256 z , s 4 C Y ′ ( z ) = ( 1 − 1024 z ) 3 , (42) and we write z + z_+ z + z = 0 z=0 z = 0 Figure 1 ). The Apéry symbol has roots z ± = 17 ∓ 12 2 z_\pm=17\mp12\sqrt{2} z ± = 17 ∓ 12 2 z + ≈ 0.0294 z_+\approx0.0294 z + ≈ 0.0294 γ = 3 \gamma=3 γ = 3 z − z_- z − γ = 3 \gamma=3 γ = 3 γ < 1 \gamma<1 γ < 1 [42 ] and enters the continuation only through the monodromy around z + z_+ z +
Figure 1. Of the four bound domains, only the Apéry K3′ arc has no singular point on it or at its endpoints. Each panel shows one geometry of Table 1 in its variable z z z (40) : singular points (crosses), the MUM point z = 0 z=0 z = 0 γ > 1 \gamma>1 γ > 1 (44) (open circles), and the two continuation paths from the MUM point (A, solid, and B, dashed). The short counterclockwise path in the CY3 _3 3 (48) , and the dotted loop in the K3′ panel has monodromy M ( z + ) M(z_+) M ( z + ) (47) .
For the 4PM K3 sector [22 , 50 , 51 ] the arc z = e − 2 i θ z=e^{-2i\theta} z = e − 2 i θ ∣ z ∣ = 1 |z|=1 ∣ z ∣ = 1 z = 1 z=1 z = 1 γ → 1 \gamma\to1 γ → 1
ϖ 0 = ( 2 π ) 2 K ( z ) 2 ∼ 1 π 2 ln 2 16 1 − z ( γ → 1 − ) . (43) \varpi_0=\left(\frac{2}{\pi}\right)^2K(z)^2\sim\frac{1}{\pi^2}\ln^2\frac{16}{1-z}\qquad(\gamma\to1^-). \tag*{(43)} ϖ 0 = ( π 2 ) 2 K ( z ) 2 ∼ π 2 1 ln 2 1 − z 16 ( γ → 1 − ) . (43) The 4PM results [48 ] , applied at bound-orbit energies [9 , 108 , 109 , 110 ] , contain K ( γ − 1 γ − 1 + 1 ) = K ( 1 − z 2 ) K\left(\frac{\gamma^{-1}}{\gamma^{-1}+1}\right)=K\left(\frac{1-z}{2}\right) K ( γ − 1 + 1 γ − 1 ) = K ( 2 1 − z ) γ < 1 \gamma<1 γ < 1 γ = 0 \gamma=0 γ = 0 z = − 1 z=-1 z = − 1
The 4 F 3 4F_3 4 F 3 z = 2 − 8 e − 4 i θ z=2^{-8}e^{-4i\theta} z = 2 − 8 e − 4 i θ ∣ z ∣ = 2 − 8 |z|=2^{-8} ∣ z ∣ = 2 − 8 z + z_+ z + θ \theta θ ( 0 , π / 2 ] (0,\pi/2] ( 0 , π /2 ] z + = 2 − 8 z_+=2^{-8} z + = 2 − 8 γ = 1 \gamma=1 γ = 1 γ = 0 \gamma=0 γ = 0 M 0 − 1 M_0^{-1} M 0 − 1
The bound domain z ∈ ( 0 , 2 − 10 ) z\in(0,2^{-10}) z ∈ ( 0 , 2 − 10 ) C Y ′ 3 \mathrm{CY'}_3 C Y ′ 3 γ = 1 \gamma=1 γ = 1 z = 2 − 10 z=2^{-10} z = 2 − 10 γ = 0 \gamma=0 γ = 0 { 0 , 1 2 , 1 2 , 2 } \{0,\frac{1}{2},\frac{1}{2},2\} { 0 , 2 1 , 2 1 , 2 } { 0 , 1 , 1 , 2 } \{0,1,1,2\} { 0 , 1 , 1 , 2 } [22 ] .
Periods on the bound arc
We evaluate the Frobenius basis of each operator at the bound-orbit energies
γ ∈ { 199 200 , 99 100 , 49 50 , 97 100 , 8 9 , 23 25 , 9 10 , 3 2 , 4 5 , 2 2 , 3 5 , 1 2 } . (44) \gamma\in\left\{\frac{199}{200},\frac{99}{100},\frac{49}{50},\frac{97}{100},\sqrt{\frac{8}{9}},\frac{23}{25},\frac{9}{10},\frac{\sqrt{3}}{2},\frac{4}{5},\frac{\sqrt{2}}{2},\frac{3}{5},\frac{1}{2}\right\}. \tag*{(44)} γ ∈ { 200 199 , 100 99 , 50 49 , 100 97 , 9 8 , 25 23 , 10 9 , 2 3 , 5 4 , 2 2 , 5 3 , 2 1 } . (44) The grid is densest at the weak-binding end and includes γ = 8 / 9 ≈ 0.9428 \gamma=\sqrt{8/9}\approx0.9428 γ = 8/9 ≈ 0.9428 [111 ] along two homotopic polygonal paths, A and B in Figure 1 . Each integration step has a rigorous remainder bound, and every value has an enclosure radius below 10 − 167 10^{-167} 1 0 − 167
The ancillary files bound_frobenius/values_*.txt contain, for each geometry and each energy (48 points in all), the full Frobenius matrix
W i k ( z ) = d i − 1 d z i − 1 ϖ k − 1 ( z ) , i , k = 1 , … , r , (45) W_{ik}(z)=\frac{d^{i-1}}{dz^{i-1}}\varpi_{k-1}(z),\qquad i,k=1,\ldots,r, \tag*{(45)} W ik ( z ) = d z i − 1 d i − 1 ϖ k − 1 ( z ) , i , k = 1 , … , r , (45) on the physical branch, in the frame (41) (for K3 the matrix of L ~ K 3 \widetilde{L}_{\mathrm{K3}} L K3 W 11 = ( 2 / π ) 2 K ( z ) 2 W_{11}=(2/\pi)^2K(z)^2 W 11 = ( 2/ π ) 2 K ( z ) 2 θ \theta θ d / d z d/dz d / d z operators.txt and in bound_frobenius/. For example, the holomorphic period of the Apéry K3′ at γ = 8 / 9 \gamma=\sqrt{8/9} γ = 8/9 ϖ 0 = 0.3805554073 − 0.3468311375 i \varpi_0=0.3805554073-0.3468311375i ϖ 0 = 0.3805554073 − 0.3468311375 i
The values of ϖ 0 \varpi_0 ϖ 0 3 _3 3 3 _3 3 γ = 2 / 2 \gamma=\sqrt{2}/2 γ = 2 /2 z = − 2 − 8 z=-2^{-8} z = − 2 − 8 3 _3 3 γ \gamma γ 1 − γ 2 \sqrt{1-\gamma^2} 1 − γ 2 z z z z ˉ \bar z z ˉ ∣ z ∣ = 2 − 8 |z|=2^{-8} ∣ z ∣ = 2 − 8 ϖ 0 \varpi_0 ϖ 0 M 0 M_0 M 0
The K3′ period ϖ ^ \hat{\varpi} ϖ ^ [15 ] is normalized at threshold by the velocity expansion
( 1 + x 2 ) ϖ ^ = 1 + 15 32 v 2 + 351 1024 v 4 + O ( v 6 ) (1+x^2)\hat{\varpi}=1+\frac{15}{32}v^2+\frac{351}{1024}v^4+O(v^6) ( 1 + x 2 ) ϖ ^ = 1 + 32 15 v 2 + 1024 351 v 4 + O ( v 6 ) with v = γ 2 − 1 / γ v=\sqrt{\gamma^2-1}/\gamma v = γ 2 − 1 / γ [57 ] are given at x = 1 x=1 x = 1 1 − x 1-x 1 − x v 2 = 8 9 v^2=\frac{8}{9} v 2 = 9 8 x = 0 x=0 x = 0 ∣ v 2 ∣ < 8 9 |v^2|<\frac{8}{9} ∣ v 2 ∣ < 9 8 ∣ 1 − x ∣ < 1 |1-x|<1 ∣1 − x ∣ < 1 γ > 3 / 17 ≈ 0.73 \gamma>3/\sqrt{17}\approx0.73 γ > 3/ 17 ≈ 0.73 γ > 1 2 \gamma>\frac{1}{2} γ > 2 1 ϖ ^ \hat{\varpi} ϖ ^ ( 1 + x 2 ) ϖ ^ (1+x^2)\hat{\varpi} ( 1 + x 2 ) ϖ ^ γ < 1 \gamma<1 γ < 1 (41) of L A L_A L A
ϖ ^ = c ^ 0 ϖ 0 + c ^ 1 ϖ 1 + c ^ 2 ϖ 2 , c ^ 0 = 0.3888737022 − 0.5499504638 i , \hat{\varpi}=\hat{c}_0\varpi_0+\hat{c}_1\varpi_1+\hat{c}_2\varpi_2,\qquad
\hat{c}_0=0.3888737022-0.5499504638\,i, ϖ ^ = c ^ 0 ϖ 0 + c ^ 1 ϖ 1 + c ^ 2 ϖ 2 , c ^ 0 = 0.3888737022 − 0.5499504638 i , c ^ 1 = − 0.5251640086 , c ^ 2 = 0.1182034314 + 0.1671648958 i . (46) \hat{c}_1=-0.5251640086,\qquad
\hat{c}_2=0.1182034314+0.1671648958\,i. \tag*{(46)} c ^ 1 = − 0.5251640086 , c ^ 2 = 0.1182034314 + 0.1671648958 i . (46) The ancillary file tables/threshold_frame_K3prime.txt gives the three constants to thirty decimal places.
The physical K3′ path runs from the scattering segment z ∈ ( z + , 1 ) z\in(z_+,1) z ∈ ( z + , 1 ) ( z − i 0 ) (z-i0) ( z − i 0 ) ∣ z ∣ < z − |z|<z_- ∣ z ∣ < z − M 0 M_0 M 0 z + z_+ z +
M ( z + ) = ( 0 0 π 2 / 3 0 1 0 3 / π 2 0 0 ) , M ( z + ) 2 = 1. (47) M(z_+)=
\begin{pmatrix}
0 & 0 & \pi^2/3 \\
0 & 1 & 0 \\
3/\pi^2 & 0 & 0
\end{pmatrix},
\qquad M(z_+)^2=\mathbb{1}. \tag*{(47)} M ( z + ) = 0 0 3/ π 2 0 1 0 π 2 /3 0 0 , M ( z + ) 2 = 1. (47) We continue the K3′ basis to z = e − i π / 3 z=e^{-i\pi/3} z = e − iπ /3 γ = 3 / 2 \gamma=\sqrt{3}/2 γ = 3 /2 z + z_+ z + z − z_- z − z + z_+ z + (47) entrywise (local exponents 0 , 1 2 , 1 0,\frac{1}{2},1 0 , 2 1 , 1 z + z_+ z +
On the 4 F 3 _4F_3 4 F 3 γ = 1 / 2 \gamma=1/2 γ = 1/2 Figure 1 , which reaches the same z z z z = 0 z=0 z = 0
( M 0 − 1 ) j k = ( − 2 π i ) k − j ( k − j ) ! , k ≥ j , (48) \left(M_0^{-1}\right)_{jk}=\frac{(-2\pi i)^{k-j}}{(k-j)!},\qquad k\ge j, \tag*{(48)} ( M 0 − 1 ) j k = ( k − j )! ( − 2 π i ) k − j , k ≥ j , (48) and zero below the diagonal.
Local exponents at threshold
At the threshold x = 1 x=1 x = 1 ϵ ≠ 0 \epsilon\ne0 ϵ = 0 [23 ] for the nine-master Apéry sector [22 ] on its maximal cut, with the subsector rows not included, is singular there, and we determine its local exponents. The nine-element basis I 1 , … , I 9 I_1,\ldots,I_9 I 1 , … , I 9 ∂ x I = Ω ( x , ϵ ) I \partial_x I=\Omega(x,\epsilon)I ∂ x I = Ω ( x , ϵ ) I Ω \Omega Ω I 2 I_2 I 2 I 3 I_3 I 3 § 4.4 , are the first and second derivatives of I 1 I_1 I 1
For ϵ ≠ 0 \epsilon\ne0 ϵ = 0 Ω \Omega Ω x = 1 x=1 x = 1 t = x − 1 t=x-1 t = x − 1 J = diag ( 1 , t , t 2 , 1 , t , t , 1 , 1 , 1 ) I J=\operatorname{diag}(1,t,t^2,1,t,t,1,1,1)I J = diag ( 1 , t , t 2 , 1 , t , t , 1 , 1 , 1 ) I t ∂ t J = B ( t ) J t\partial_tJ=B(t)J t ∂ t J = B ( t ) J B B B t = 0 t = 0 t = 0 x = 1 x = 1 x = 1 B ( 0 ) B(0) B ( 0 )
1 , − 6 ϵ , 2 + 10 ϵ , 1 − 4 ϵ (twice) , − 4 ϵ (twice) , 10 ϵ (twice) (49) 1,\quad-6\epsilon,\quad2 + 10\epsilon,\quad1 - 4\epsilon\ \text{(twice)},\quad-4\epsilon\ \text{(twice)},\quad10\epsilon\ \text{(twice)} \tag*{(49)} 1 , − 6 ϵ , 2 + 10 ϵ , 1 − 4 ϵ (twice) , − 4 ϵ (twice) , 10 ϵ (twice) (49) of which one − 4 ϵ -4\epsilon − 4 ϵ 10 ϵ 10\epsilon 10 ϵ I 8 I_8 I 8 I 9 I_9 I 9
Since a rational change of basis normalization shifts the exponents by integers, we give them modulo integers: they are 0 0 0 − 6 ϵ -6\epsilon − 6 ϵ 10 ϵ 10\epsilon 10 ϵ − 4 ϵ -4\epsilon − 4 ϵ 1 1 1 1 1 1 3 3 3 4 4 4 ϵ \epsilon ϵ t ρ t^\rho t ρ t t t ϵ \epsilon ϵ x = − 1 x = -1 x = − 1
Bound-orbit observables
A bound-orbit observable requires, beyond the continued periods, a map from scattering data to bound dynamics. For the local-in-time conservative sector such a map is the boundary-to-bound (B2B) dictionary of Kälin and Porto [6 , 7 ] . It holds for all local-in-time contributions and generic orbits [112 ] and has been applied at 4PM [48 , 49 , 109 ] and 5PM-1SF [113 ] . Of the periods that we continue, only the Apéry K3′ remains in the conservative 5PM-2SF result [15 , 58 ] .
Nonlocal-in-time (tail) terms can be mapped only at large eccentricity and, logarithms aside, do not continue smoothly between hyperbolic and elliptic motion [49 , 112 ] . Moreover, Kankani and McWilliams find with numerical relativity that the B2B relations do not hold non-perturbatively [114 ] .
At 4PM the dissipative observables contain no tail logarithms and, as Jakobsen et al. note, it may then be possible to map the fluxes directly to bound orbits [74 ] . The flux, reconstructed from scattering data, is the natural object for generic bound orbits, since the per-period losses of the adiabatic map vanish at even PM orders [51 , 112 , 115 ] . The γ \gamma γ γ < 1 \gamma< 1 γ < 1 s + i 0 s + i0 s + i 0
Evaluation of the memory-region integrals
We evaluate the memory-region boundary integrals I 1 ( M ) I_1^{(M)} I 1 ( M ) I 2 ( M ) I_2^{(M)} I 2 ( M ) (11) , at the static point and extract the constant c M c_M c M (12) . The two memory gravitons are routed as in the γ \gamma γ [15 ] , which we take as input, together with the frequency representations and the integration-by-parts identity of its supplement. We derive the Laurent coefficients of I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M ) (51) ) in closed form through order ϵ 1 \epsilon^1 ϵ 1 (63) .
Frequency representations
At the static point x = 1 x = 1 x = 1 I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M ) [15 ] . Since c M c_M c M x = 1 x = 1 x = 1 x x x c M c_M c M (52) , depend on γ \gamma γ γ 2 − 1 \sqrt{\gamma^2-1} γ 2 − 1 γ = 2 \gamma=\sqrt{2} γ = 2
We write ω 3 = ω 1 + ω 2 \omega_3=\omega_1+\omega_2 ω 3 = ω 1 + ω 2 ∫ ω = ∫ − ∞ ∞ d ω / ( 2 π ) \int_\omega=\int_{-\infty}^{\infty}d\omega/(2\pi) ∫ ω = ∫ − ∞ ∞ d ω / ( 2 π ) K ϵ K_\epsilon K ϵ 2
I 1 ( M ) = N 1 ( ϵ ) ∫ ω 1 ω 2 ∏ j = 1 , 2 ( 0 + − i ω j ) 1 − 2 ϵ ∏ i = 1 3 ∣ ω i ∣ − ϵ K ϵ ( ∣ ω i ∣ ) , I_1^{(M)}=N_1(\epsilon)\int_{\omega_1\omega_2}\prod_{j=1,2}(0^+-i\omega_j)^{1-2\epsilon}\prod_{i=1}^{3}|\omega_i|^{-\epsilon}K_\epsilon(|\omega_i|), I 1 ( M ) = N 1 ( ϵ ) ∫ ω 1 ω 2 j = 1 , 2 ∏ ( 0 + − i ω j ) 1 − 2 ϵ i = 1 ∏ 3 ∣ ω i ∣ − ϵ K ϵ ( ∣ ω i ∣ ) , N 1 ( ϵ ) = − 2 7 ϵ − 1 Γ 2 ( 2 ϵ − 1 2 ) Γ ( 4 ϵ − 1 ) Γ ( 2 − 5 ϵ ) ( 4 π ) 5 − 4 ϵ . (50) N_1(\epsilon)=-2^{7\epsilon-1}\frac{\Gamma^2\left(\frac{2\epsilon-1}{2}\right)\Gamma(4\epsilon-1)}{\Gamma(2-5\epsilon)(4\pi)^{5-4\epsilon}}. \tag*{(50)} N 1 ( ϵ ) = − 2 7 ϵ − 1 Γ ( 2 − 5 ϵ ) ( 4 π ) 5 − 4 ϵ Γ 2 ( 2 2 ϵ − 1 ) Γ ( 4 ϵ − 1 ) . (50) and I 3 ( M ) I_3^{(M)} I 3 ( M ) I 1 ( M ) I_1^{(M)} I 1 ( M ) − ( ℓ 3 ⋅ v 1 ) 3 ( ℓ 4 ⋅ v 1 ) 3 -(\ell_3\cdot v_1)^3(\ell_4\cdot v_1)^3 − ( ℓ 3 ⋅ v 1 ) 3 ( ℓ 4 ⋅ v 1 ) 3
I 3 ( M ) = N 3 ( ϵ ) ∫ ω 1 ω 2 ω 1 3 ω 2 3 ∏ j = 1 , 2 ( 0 + − i ω j ) − 1 − 2 ϵ ∏ i = 1 3 ∣ ω i ∣ − ϵ K ϵ ( ∣ ω i ∣ ) , I_3^{(M)}=N_3(\epsilon)\int_{\omega_1\omega_2}\omega_1^3\omega_2^3\prod_{j=1,2}(0^+-i\omega_j)^{-1-2\epsilon}\prod_{i=1}^{3}|\omega_i|^{-\epsilon}K_\epsilon(|\omega_i|), I 3 ( M ) = N 3 ( ϵ ) ∫ ω 1 ω 2 ω 1 3 ω 2 3 j = 1 , 2 ∏ ( 0 + − i ω j ) − 1 − 2 ϵ i = 1 ∏ 3 ∣ ω i ∣ − ϵ K ϵ ( ∣ ω i ∣ ) , N 3 ( ϵ ) = 2 7 ϵ − 3 Γ 2 ( 2 ϵ + 1 2 ) Γ ( 4 ϵ − 2 ) Γ ( 3 − 5 ϵ ) ( 4 π ) 5 − 4 ϵ . (51) N_3(\epsilon)=2^{7\epsilon-3}\frac{\Gamma^2\left(\frac{2\epsilon+1}{2}\right)\Gamma(4\epsilon-2)}{\Gamma(3-5\epsilon)(4\pi)^{5-4\epsilon}}. \tag*{(51)} N 3 ( ϵ ) = 2 7 ϵ − 3 Γ ( 3 − 5 ϵ ) ( 4 π ) 5 − 4 ϵ Γ 2 ( 2 2 ϵ + 1 ) Γ ( 4 ϵ − 2 ) . (51) The integration-by-parts identity of the supplement, whose derivation uses propagator symmetries respected by both the γ \gamma γ I 2 ( M ) I_2^{(M)} I 2 ( M ) I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M )
I 2 ( M ) = r 1 ( ϵ ) I 3 ( M ) + r 2 ( ϵ ) I 1 ( M ) , (52) I_2^{(M)}=r_1(\epsilon)I_3^{(M)}+r_2(\epsilon)I_1^{(M)}, \tag*{(52)} I 2 ( M ) = r 1 ( ϵ ) I 3 ( M ) + r 2 ( ϵ ) I 1 ( M ) , (52) with
r 1 ( ϵ ) = 28 ( 5 ϵ − 2 ) ( 14 ϵ − 9 ) ( 1088 ϵ 4 − 898 ϵ 3 + 275 ϵ 2 − 38 ϵ + 2 ) 3 ϵ 3 ( 96 ϵ 3 − 116 ϵ 2 + 43 ϵ − 5 ) , r_1(\epsilon)=\frac{28(5\epsilon-2)(14\epsilon-9)(1088\epsilon^4-898\epsilon^3+275\epsilon^2-38\epsilon+2)}{3\epsilon^3(96\epsilon^3-116\epsilon^2+43\epsilon-5)}, r 1 ( ϵ ) = 3 ϵ 3 ( 96 ϵ 3 − 116 ϵ 2 + 43 ϵ − 5 ) 28 ( 5 ϵ − 2 ) ( 14 ϵ − 9 ) ( 1088 ϵ 4 − 898 ϵ 3 + 275 ϵ 2 − 38 ϵ + 2 ) , r 2 ( ϵ ) = ( 1 − 2 ϵ ) 2 ( 121024 ϵ 6 − 237340 ϵ 5 + 181552 ϵ 4 − 70363 ϵ 3 + 14767 ϵ 2 − 1612 ϵ + 72 ) 6 ϵ 3 ( 96 ϵ 3 − 116 ϵ 2 + 43 ϵ − 5 ) . (53) r_2(\epsilon)=\frac{(1-2\epsilon)^2(121024\epsilon^6-237340\epsilon^5+181552\epsilon^4-70363\epsilon^3+14767\epsilon^2-1612\epsilon+72)}{6\epsilon^3(96\epsilon^3-116\epsilon^2+43\epsilon-5)}. \tag*{(53)} r 2 ( ϵ ) = 6 ϵ 3 ( 96 ϵ 3 − 116 ϵ 2 + 43 ϵ − 5 ) ( 1 − 2 ϵ ) 2 ( 121024 ϵ 6 − 237340 ϵ 5 + 181552 ϵ 4 − 70363 ϵ 3 + 14767 ϵ 2 − 1612 ϵ + 72 ) . (53) Both coefficients scale as ϵ − 3 \epsilon^{-3} ϵ − 3 I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M ) ϵ − 1 \epsilon^{-1} ϵ − 1 ϵ − 4 \epsilon^{-4} ϵ − 4 ϵ − 3 \epsilon^{-3} ϵ − 3 r 1 I 3 ( M ) + r 2 I 1 ( M ) r_1I_3^{(M)}+r_2I_1^{(M)} r 1 I 3 ( M ) + r 2 I 1 ( M ) ϵ − 2 \epsilon^{-2} ϵ − 2 (12) , and to extract c M c_M c M I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M ) ϵ − 1 \epsilon^{-1} ϵ − 1
The i 0 + i0^+ i 0 + ( 0 + − i s ω ) a − q ϵ (0^+-is\omega)^{a-q\epsilon} ( 0 + − i s ω ) a − q ϵ a a a q q q s = ± 1 s=\pm1 s = ± 1 a a a − q ϵ -q\epsilon − q ϵ n n n 3 − 2 ϵ 3-2\epsilon 3 − 2 ϵ [ − ( ω + i 0 ) 2 ] 3 / 2 − n − ϵ = ( 0 + − i ω ) 3 − 2 n − 2 ϵ [-(\omega+i0)^2]^{3/2-n-\epsilon}=(0^+-i\omega)^{3-2n-2\epsilon} [ − ( ω + i 0 ) 2 ] 3/2 − n − ϵ = ( 0 + − iω ) 3 − 2 n − 2 ϵ [15 ] . For a retarded (s = + 1 s=+1 s = + 1 s = − 1 s=-1 s = − 1
ln ( 0 + − i s ω ) = ln ∣ ω ∣ − i π 2 s sign ( ω ) , (54) \ln(0^{+}-is\omega)=\ln|\omega|-\frac{i\pi}{2}s\operatorname{sign}(\omega), \tag*{(54)} ln ( 0 + − i s ω ) = ln ∣ ω ∣ − 2 iπ s sign ( ω ) , (54) whereas for a Feynman propagator 1 / ( ω 2 − k 2 + i 0 ) n 1/(\omega^2-k^2+i0)^n 1/ ( ω 2 − k 2 + i 0 ) n − ω 2 − i 0 = ( 0 + − i ∣ ω ∣ ) 2 -\omega^2-i0=(0^{+}-i|\omega|)^2 − ω 2 − i 0 = ( 0 + − i ∣ ω ∣ ) 2 − ( ω + i 0 ) 2 -(\omega+i0)^2 − ( ω + i 0 ) 2 ( 0 + − i ∣ ω ∣ ) a − q ϵ (0^{+}-i|\omega|)^{a-q\epsilon} ( 0 + − i ∣ ω ∣ ) a − q ϵ ln ∣ ω ∣ − i π / 2 \ln|\omega|-i\pi/2 ln ∣ ω ∣ − iπ /2 ω \omega ω (50) and (51) both memory gravitons have retarded propagators, s 1 = s 2 = + 1 s_1=s_2=+1 s 1 = s 2 = + 1 γ \gamma γ ( ω 1 , ω 2 ) → ( − ω 1 , − ω 2 ) (\omega_1,\omega_2)\to(-\omega_1,-\omega_2) ( ω 1 , ω 2 ) → ( − ω 1 , − ω 2 )
Closed forms of I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M )
The three Bessel factors of Eqs. (50) and (51) are coupled through ω 3 = ω 1 + ω 2 \omega_3=\omega_1+\omega_2 ω 3 = ω 1 + ω 2 [15 , 57 ] , we Fourier transform to the time domain, where the frequency integrals reduce to one-dimensional moments. For two functions A A A C C C a ( k ) = ∫ ω e i k ω A ( ω ) a(k)=\int_\omega e^{ik\omega}A(\omega) a ( k ) = ∫ ω e ik ω A ( ω ) c ( k ) = ∫ ω e i k ω C ( ω ) c(k)=\int_\omega e^{ik\omega}C(\omega) c ( k ) = ∫ ω e ik ω C ( ω )
∫ ω 1 ω 2 A ( ω 1 ) A ( ω 2 ) C ( ω 1 + ω 2 ) = ∫ − ∞ ∞ d k c ( k ) a ( − k ) 2 , (55) \int_{\omega_1\omega_2}A(\omega_1)A(\omega_2)C(\omega_1+\omega_2)=\int_{-\infty}^{\infty}dk\,c(k)a(-k)^2, \tag*{(55)} ∫ ω 1 ω 2 A ( ω 1 ) A ( ω 2 ) C ( ω 1 + ω 2 ) = ∫ − ∞ ∞ d k c ( k ) a ( − k ) 2 , (55) which replaces the two coupled frequency integrals by one integral over the conjugate variable k k k ω 3 \omega_3 ω 3 C = ∣ ω ∣ − ϵ K ϵ ( ∣ ω ∣ ) C=|\omega|^{-\epsilon}K_\epsilon(|\omega|) C = ∣ ω ∣ − ϵ K ϵ ( ∣ ω ∣ ) g ( k ) = G ( 1 + k 2 ) ϵ − 1 / 2 g(k)=G(1+k^2)^{\epsilon-1/2} g ( k ) = G ( 1 + k 2 ) ϵ − 1/2 G = 2 − ϵ − 1 Γ ( 1 2 − ϵ ) / π G=2^{-\epsilon-1}\Gamma(\frac{1}{2}-\epsilon)/\sqrt{\pi} G = 2 − ϵ − 1 Γ ( 2 1 − ϵ ) / π [116 Eq. 10.32.11] .
Under the retarded routing both lines ω 1 , ω 2 \omega_1,\omega_2 ω 1 , ω 2 ∣ ω ∣ − 3 ϵ K ϵ ( ∣ ω ∣ ) e i π ϵ sign ω |\omega|^{-3\epsilon}K_\epsilon(|\omega|)e^{i\pi\epsilon\operatorname{sign}\omega} ∣ ω ∣ − 3 ϵ K ϵ ( ∣ ω ∣ ) e iπ ϵ sign ω − i ω -i\omega − iω I 1 ( M ) I_1^{(M)} I 1 ( M ) i ω 2 i\omega^2 i ω 2 I 3 ( M ) I_3^{(M)} I 3 ( M ) f R = cos ( π ϵ ) C ϵ − sin ( π ϵ ) S ϵ f_R=\cos(\pi\epsilon)C_\epsilon-\sin(\pi\epsilon)S_\epsilon f R = cos ( π ϵ ) C ϵ − sin ( π ϵ ) S ϵ
C ϵ ( k ) = 1 π ∫ 0 ∞ ω − 3 ϵ K ϵ ( ω ) cos ( k ω ) d ω , S ϵ ( k ) = 1 π ∫ 0 ∞ ω − 3 ϵ K ϵ ( ω ) sin ( k ω ) d ω , (56) C_\epsilon(k)=\frac{1}{\pi}\int_0^\infty\omega^{-3\epsilon}K_\epsilon(\omega)\cos(k\omega)\,d\omega,\qquad S_\epsilon(k)=\frac{1}{\pi}\int_0^\infty\omega^{-3\epsilon}K_\epsilon(\omega)\sin(k\omega)\,d\omega, \tag*{(56)} C ϵ ( k ) = π 1 ∫ 0 ∞ ω − 3 ϵ K ϵ ( ω ) cos ( k ω ) d ω , S ϵ ( k ) = π 1 ∫ 0 ∞ ω − 3 ϵ K ϵ ( ω ) sin ( k ω ) d ω , (56) and the powers of ω \omega ω k k k C ϵ C_\epsilon C ϵ S ϵ S_\epsilon S ϵ k k k
I 1 ( M ) = N 1 ∫ g ( f R ′ ) 2 d k = N 1 [ cos 2 ( π ϵ ) E 1 + sin 2 ( π ϵ ) O 1 ] , (57) I_1^{(M)}=N_1\int g(f_R')^2\,dk=N_1\left[\cos^2(\pi\epsilon)E_1+\sin^2(\pi\epsilon)O_1\right], \tag*{(57)} I 1 ( M ) = N 1 ∫ g ( f R ′ ) 2 d k = N 1 [ cos 2 ( π ϵ ) E 1 + sin 2 ( π ϵ ) O 1 ] , (57) I 3 ( M ) = − N 3 ∫ g ( f R ′ ′ ) 2 d k = − N 3 [ cos 2 ( π ϵ ) E 3 + sin 2 ( π ϵ ) O 3 ] . I_3^{(M)}=-N_3\int g(f_R'')^2\,dk=-N_3\left[\cos^2(\pi\epsilon)E_3+\sin^2(\pi\epsilon)O_3\right]. I 3 ( M ) = − N 3 ∫ g ( f R ′′ ) 2 d k = − N 3 [ cos 2 ( π ϵ ) E 3 + sin 2 ( π ϵ ) O 3 ] . where E 1 = ∫ g ( C ϵ ′ ) 2 d k E_1=\int g(C_\epsilon')^2dk E 1 = ∫ g ( C ϵ ′ ) 2 d k O 1 = ∫ g ( S ϵ ′ ) 2 d k O_1=\int g(S_\epsilon')^2dk O 1 = ∫ g ( S ϵ ′ ) 2 d k E 3 E_3 E 3 O 3 O_3 O 3 ( 0 + − i ∣ ω ∣ ) a − 2 ϵ = ( − i ) a e i π ϵ ∣ ω ∣ a − 2 ϵ (0^+-i|\omega|)^{a-2\epsilon}=(-i)^ae^{i\pi\epsilon}|\omega|^{a-2\epsilon} ( 0 + − i ∣ ω ∣ ) a − 2 ϵ = ( − i ) a e iπ ϵ ∣ ω ∣ a − 2 ϵ S ϵ S_\epsilon S ϵ I 1 , F ( M ) = − e 2 i π ϵ N 1 O 1 I_{1,F}^{(M)}=-e^{2i\pi\epsilon}N_1O_1 I 1 , F ( M ) = − e 2 iπ ϵ N 1 O 1 I 3 , F ( M ) = e 2 i π ϵ N 3 O 3 I_{3,F}^{(M)}=e^{2i\pi\epsilon}N_3O_3 I 3 , F ( M ) = e 2 iπ ϵ N 3 O 3
Bessel’s equation for ω − 3 ϵ K ϵ ( ω ) \omega^{-3\epsilon}K_\epsilon(\omega) ω − 3 ϵ K ϵ ( ω ) C ϵ C_\epsilon C ϵ S ϵ S_\epsilon S ϵ
( 1 + k 2 ) f ′ ′ + ( 3 − 6 ϵ ) k f ′ + ( 1 − 2 ϵ ) ( 1 − 4 ϵ ) f = 0 , (58) (1+k^2)f''+(3-6\epsilon)kf'+(1-2\epsilon)(1-4\epsilon)f=0, \tag*{(58)} ( 1 + k 2 ) f ′′ + ( 3 − 6 ϵ ) k f ′ + ( 1 − 2 ϵ ) ( 1 − 4 ϵ ) f = 0 , (58) with C ϵ ′ ( 0 ) = 0 C_\epsilon'(0)=0 C ϵ ′ ( 0 ) = 0 S ϵ ( 0 ) = 0 S_\epsilon(0)=0 S ϵ ( 0 ) = 0 K ϵ K_\epsilon K ϵ [116 Eq. 10.43.19] , C ϵ ( 0 ) = 2 − 3 ϵ − 1 Γ ( 1 2 − ϵ ) Γ ( 1 2 − 2 ϵ ) / π C_\epsilon(0)=2^{-3\epsilon-1}\Gamma\left(\frac{1}{2}-\epsilon\right)\Gamma\left(\frac{1}{2}-2\epsilon\right)/\pi C ϵ ( 0 ) = 2 − 3 ϵ − 1 Γ ( 2 1 − ϵ ) Γ ( 2 1 − 2 ϵ ) / π S ϵ ′ ( 0 ) = 2 − 3 ϵ Γ ( 1 − ϵ ) Γ ( 1 − 2 ϵ ) / π S_\epsilon'(0)=2^{-3\epsilon}\Gamma(1-\epsilon)\Gamma(1-2\epsilon)/\pi S ϵ ′ ( 0 ) = 2 − 3 ϵ Γ ( 1 − ϵ ) Γ ( 1 − 2 ϵ ) / π k = sinh t k=\sinh t k = sinh t f = f 0 cosh − 1 + 3 ϵ t y ( t ) f=f_0\cosh^{-1+3\epsilon}t\,y(t) f = f 0 cosh − 1 + 3 ϵ t y ( t ) f 0 = C ϵ ( 0 ) f_0=C_\epsilon(0) f 0 = C ϵ ( 0 ) y E y_E y E f 0 = S ϵ ′ ( 0 ) f_0=S_\epsilon'(0) f 0 = S ϵ ′ ( 0 ) y O y_O y O (58) becomes y ′ ′ = [ ϵ 2 + ( 3 ϵ − 9 ϵ 2 ) sech 2 t ] y y''=\left[\epsilon^2+(3\epsilon-9\epsilon^2)\operatorname{sech}^2t\right]y y ′′ = [ ϵ 2 + ( 3 ϵ − 9 ϵ 2 ) sech 2 t ] y y E ( 0 ) = 1 y_E(0)=1 y E ( 0 ) = 1 y E ′ ( 0 ) = 0 y_E'(0)=0 y E ′ ( 0 ) = 0 y O ( 0 ) = 0 y_O(0)=0 y O ( 0 ) = 0 y O ′ ( 0 ) = 1 y_O'(0)=1 y O ′ ( 0 ) = 1 ϵ \epsilon ϵ L = ln cosh t L=\ln\cosh t L = ln cosh t
y E = 1 + 3 ϵ L + ϵ 2 ( 9 2 L 2 − 4 t 2 ) + O ( ϵ 3 ) , y O = t + ϵ y 1 + ϵ 2 y 2 + O ( ϵ 3 ) , (59) y_E=1+3\epsilon L+\epsilon^2\left(\frac{9}{2}L^2-4t^2\right)+O(\epsilon^3),\qquad y_O=t+\epsilon y_1+\epsilon^2y_2+O(\epsilon^3), \tag*{(59)} y E = 1 + 3 ϵ L + ϵ 2 ( 2 9 L 2 − 4 t 2 ) + O ( ϵ 3 ) , y O = t + ϵ y 1 + ϵ 2 y 2 + O ( ϵ 3 ) , (59) where y 1 ′ = 3 ( t tanh t − L ) y_1'=3(t\tanh t-L) y 1 ′ = 3 ( t tanh t − L ) y 2 ′ ′ = 3 sech 2 t y 1 + ( 1 − 9 sech 2 t ) t y_2''=3\operatorname{sech}^2t\,y_1+(1-9\operatorname{sech}^2t)t y 2 ′′ = 3 sech 2 t y 1 + ( 1 − 9 sech 2 t ) t y 1 ( 0 ) = y 2 ( 0 ) = y 2 ′ ( 0 ) = 0 y_1(0)=y_2(0)=y_2'(0)=0 y 1 ( 0 ) = y 2 ( 0 ) = y 2 ′ ( 0 ) = 0 (58) , we remove the derivatives of f f f z = sech 2 t z=\operatorname{sech}^2t z = sech 2 t
∫ g ( f ′ ) 2 d k = G f 0 2 ∫ − ∞ ∞ e 8 ϵ L B 1 y 2 d t , ∫ g ( f ′ ′ ) 2 d k = G f 0 2 ∫ − ∞ ∞ e 8 ϵ L B 2 y 2 d t , (60) \int g(f')^2\,dk=Gf_0^2\int_{-\infty}^{\infty}e^{8\epsilon L}B_1y^2\,dt,\qquad\int g(f'')^2\,dk=Gf_0^2\int_{-\infty}^{\infty}e^{8\epsilon L}B_2y^2\,dt, \tag*{(60)} ∫ g ( f ′ ) 2 d k = G f 0 2 ∫ − ∞ ∞ e 8 ϵ L B 1 y 2 d t , ∫ g ( f ′′ ) 2 d k = G f 0 2 ∫ − ∞ ∞ e 8 ϵ L B 2 y 2 d t , (60) where B 1 = ( 5 − 18 ϵ + 16 ϵ 2 ) z 2 − ( 6 − 16 ϵ + 8 ϵ 2 ) z 3 B_1=(5-18\epsilon+16\epsilon^2)z^2-(6-16\epsilon+8\epsilon^2)z^3 B 1 = ( 5 − 18 ϵ + 16 ϵ 2 ) z 2 − ( 6 − 16 ϵ + 8 ϵ 2 ) z 3 B 2 = ( 130 − 834 ϵ + 1960 ϵ 2 − 1992 ϵ 3 + 736 ϵ 4 ) z 3 − ( 375 − 2142 ϵ + 4308 ϵ 2 − 3528 ϵ 3 + 960 ϵ 4 ) z 4 + ( 252 − 1332 ϵ + 2376 ϵ 2 − 1584 ϵ 3 + 288 ϵ 4 ) z 5 B_2=(130-834\epsilon+1960\epsilon^2-1992\epsilon^3+736\epsilon^4)z^3-(375-2142\epsilon+4308\epsilon^2-3528\epsilon^3+960\epsilon^4)z^4+(252-1332\epsilon+2376\epsilon^2-1584\epsilon^3+288\epsilon^4)z^5 B 2 = ( 130 − 834 ϵ + 1960 ϵ 2 − 1992 ϵ 3 + 736 ϵ 4 ) z 3 − ( 375 − 2142 ϵ + 4308 ϵ 2 − 3528 ϵ 3 + 960 ϵ 4 ) z 4 + ( 252 − 1332 ϵ + 2376 ϵ 2 − 1584 ϵ 3 + 288 ϵ 4 ) z 5 t t t n n n s s s
∫ − ∞ ∞ sech n t e s t d t = 2 n − 1 Γ ( n + s 2 ) Γ ( n − s 2 ) Γ ( n ) . (61) \int_{-\infty}^{\infty}\operatorname{sech}^n t\,e^{st}\,dt=\frac{2^{n-1}\Gamma\left(\frac{n+s}{2}\right)\Gamma\left(\frac{n-s}{2}\right)}{\Gamma(n)}. \tag*{(61)} ∫ − ∞ ∞ sech n t e s t d t = Γ ( n ) 2 n − 1 Γ ( 2 n + s ) Γ ( 2 n − s ) . (61) A factor t 2 t^2 t 2 s s s s = 0 s=0 s = 0 L L L − ∂ n -\partial_n − ∂ n S n = ∫ sech n t d t = π Γ ( n 2 ) / Γ ( n + 1 2 ) S_n=\int\operatorname{sech}^n t\,dt=\sqrt{\pi}\Gamma\left(\frac{n}{2}\right)/\Gamma\left(\frac{n+1}{2}\right) S n = ∫ sech n t d t = π Γ ( 2 n ) /Γ ( 2 n + 1 ) L n = ∫ sech n t L d t = − ∂ n S n L_n=\int\operatorname{sech}^n t\,L\,dt=-\partial_nS_n L n = ∫ sech n t L d t = − ∂ n S n T n = ∫ sech n t t 2 d t = 1 2 S n ψ 1 ( n 2 ) T_n=\int\operatorname{sech}^n t\,t^2dt=\frac{1}{2}S_n\psi_1\left(\frac{n}{2}\right) T n = ∫ sech n t t 2 d t = 2 1 S n ψ 1 ( 2 n ) ϵ \epsilon ϵ e 8 ϵ L y E 2 = 1 + 14 ϵ L + ϵ 2 ( 98 L 2 − 8 t 2 ) + O ( ϵ 3 ) e^{8\epsilon L}y_E^2=1+14\epsilon L+\epsilon^2(98L^2-8t^2)+O(\epsilon^3) e 8 ϵ L y E 2 = 1 + 14 ϵ L + ϵ 2 ( 98 L 2 − 8 t 2 ) + O ( ϵ 3 ) sin 2 ( π ϵ ) \sin^2(\pi\epsilon) sin 2 ( π ϵ ) ϵ 2 \epsilon^2 ϵ 2
At ϵ = 0 \epsilon=0 ϵ = 0 E 1 = 1 30 E_1=\frac{1}{30} E 1 = 30 1 E 3 = 8 105 E_3=\frac{8}{105} E 3 = 105 8 O 1 = 1 90 + 1 3 π 2 O_1=\frac{1}{90}+\frac{1}{3\pi^2} O 1 = 90 1 + 3 π 2 1 O 3 = 8 315 + 4 9 π 2 O_3=\frac{8}{315}+\frac{4}{9\pi^2} O 3 = 315 8 + 9 π 2 4 ∫ ω \int_\omega ∫ ω E 1 E_1 E 1 E 3 E_3 E 3
∫ R 2 ω 1 ω 2 ∏ i = 1 3 K 0 ( ∣ ω i ∣ ) d ω 1 d ω 2 = − 2 π 2 15 , \int_{\mathbb{R}^2}\omega_1\omega_2\prod_{i=1}^{3}K_0(|\omega_i|)\,d\omega_1d\omega_2=-\frac{2\pi^2}{15}, ∫ R 2 ω 1 ω 2 i = 1 ∏ 3 K 0 ( ∣ ω i ∣ ) d ω 1 d ω 2 = − 15 2 π 2 , ∫ R 2 ω 1 2 ω 2 2 ∏ i = 1 3 K 0 ( ∣ ω i ∣ ) d ω 1 d ω 2 = 32 π 2 105 . (62) \int_{\mathbb{R}^2}\omega_1^2\omega_2^2\prod_{i=1}^{3}K_0(|\omega_i|)\,d\omega_1d\omega_2=\frac{32\pi^2}{105}. \tag*{(62)} ∫ R 2 ω 1 2 ω 2 2 i = 1 ∏ 3 K 0 ( ∣ ω i ∣ ) d ω 1 d ω 2 = 105 32 π 2 . (62) The leading Laurent coefficients of the prefactors and of the coefficients (53) are N 1 ∣ ϵ − 1 = 1 / ( 2048 π 4 ) N_1|_{\epsilon^{-1}}=1/(2048\pi^4) N 1 ∣ ϵ − 1 = 1/ ( 2048 π 4 ) N 3 ∣ ϵ − 1 = 1 / ( 131072 π 4 ) N_3|_{\epsilon^{-1}}=1/(131072\pi^4) N 3 ∣ ϵ − 1 = 1/ ( 131072 π 4 ) r 2 ∣ ϵ − 3 = − 12 5 r_2|_{\epsilon^{-3}}=-\frac{12}{5} r 2 ∣ ϵ − 3 = − 5 12 r 1 ∣ ϵ − 3 = − 336 5 r_1|_{\epsilon^{-3}}=-\frac{336}{5} r 1 ∣ ϵ − 3 = − 5 336 ϵ − 1 \epsilon^{-1} ϵ − 1 I 1 ( M ) I_1^{(M)} I 1 ( M ) 1 / ( 30 ⋅ 2048 π 4 ) = 1 / ( 15 ( 8 π ) 4 ) 1/(30\cdot2048\pi^4)=1/(15(8\pi)^4) 1/ ( 30 ⋅ 2048 π 4 ) = 1/ ( 15 ( 8 π ) 4 ) I 3 ( M ) I_3^{(M)} I 3 ( M ) − 8 / ( 105 ⋅ 131072 π 4 ) = − 1 / ( 1720320 π 4 ) -8/(105\cdot131072\pi^4)=-1/(1720320\pi^4) − 8/ ( 105 ⋅ 131072 π 4 ) = − 1/ ( 1720320 π 4 )
Expanding Eq. (57) with the series of N 1 N_1 N 1 N 3 N_3 N 3 G G G C ϵ ( 0 ) C_\epsilon(0) C ϵ ( 0 ) S ϵ ′ ( 0 ) S_\epsilon'(0) S ϵ ′ ( 0 ) I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M ) I 2 ( M ) I_2^{(M)} I 2 ( M ) [15 ] . We write λ = ln 2 π − γ E \lambda=\ln2\pi-\gamma_E λ = ln 2 π − γ E ( 4 π ) − 4 ϵ (4\pi)^{-4\epsilon} ( 4 π ) − 4 ϵ Γ \Gamma Γ N 1 , 3 N_{1,3} N 1 , 3 e γ E ϵ e^{\gamma_E\epsilon} e γ E ϵ
I 1 ( M ) = 1 15 ( 8 π ) 4 [ 1 ϵ + 4 λ + 232 15 + ϵ ( 8 λ 2 + 928 15 λ + 50944 225 − 7 π 2 ) ] + O ( ϵ 2 ) , I^{(M)}_1=\frac{1}{15(8\pi)^4}\left[\frac{1}{\epsilon}+4\lambda+\frac{232}{15}+\epsilon\left(8\lambda^2+\frac{928}{15}\lambda+\frac{50944}{225}-7\pi^2\right)\right]+\mathcal{O}(\epsilon^2), I 1 ( M ) = 15 ( 8 π ) 4 1 [ ϵ 1 + 4 λ + 15 232 + ϵ ( 8 λ 2 + 15 928 λ + 225 50944 − 7 π 2 ) ] + O ( ϵ 2 ) , I 3 ( M ) = − 1 420 ( 8 π ) 4 [ 1 ϵ + 4 λ + 182 15 + ϵ ( 8 λ 2 + 728 15 λ + 116782 675 − 7 π 2 ) ] + O ( ϵ 2 ) . (63) I^{(M)}_3=-\frac{1}{420(8\pi)^4}\left[\frac{1}{\epsilon}+4\lambda+\frac{182}{15}+\epsilon\left(8\lambda^2+\frac{728}{15}\lambda+\frac{116782}{675}-7\pi^2\right)\right]+\mathcal{O}(\epsilon^2). \tag*{(63)} I 3 ( M ) = − 420 ( 8 π ) 4 1 [ ϵ 1 + 4 λ + 15 182 + ϵ ( 8 λ 2 + 15 728 λ + 675 116782 − 7 π 2 ) ] + O ( ϵ 2 ) . (63) Here λ \lambda λ e 4 λ ϵ e^{4\lambda\epsilon} e 4 λ ϵ
The value of c M c_M c M
For I 2 ( M ) = r 1 I 3 ( M ) + r 2 I 1 ( M ) I^{(M)}_2=r_1I^{(M)}_3+r_2I^{(M)}_1 I 2 ( M ) = r 1 I 3 ( M ) + r 2 I 1 ( M ) (60) into
Q ( z , ϵ ) = ϵ 3 [ r 1 N 3 N 1 B 2 − r 2 B 1 ] = q 0 + ϵ q 1 + … , N 3 N 1 = 1 − 2 ϵ 32 ( 2 − 5 ϵ ) . (64) Q(z,\epsilon)=\epsilon^3\left[r_1\frac{N_3}{N_1}B_2-r_2B_1\right]=q_0+\epsilon q_1+\ldots,\qquad\frac{N_3}{N_1}=\frac{1-2\epsilon}{32(2-5\epsilon)}. \tag*{(64)} Q ( z , ϵ ) = ϵ 3 [ r 1 N 1 N 3 B 2 − r 2 B 1 ] = q 0 + ϵ q 1 + … , N 1 N 3 = 32 ( 2 − 5 ϵ ) 1 − 2 ϵ . (64) whose leading term is a total derivative, q 0 = d 2 R / d t 2 q_0=d^2R/dt^2 q 0 = d 2 R / d t 2 R = 3 4 z 2 − 151 40 z 3 + 147 40 z 4 R=\frac{3}{4}z^2-\frac{151}{40}z^3+\frac{147}{40}z^4 R = 4 3 z 2 − 40 151 z 3 + 40 147 z 4 a k a_k a k c k c_k c k ϵ k \epsilon^k ϵ k Re ∫ e 8 ϵ L Q y 2 ′ 2 d t \operatorname{Re}\int e^{8\epsilon L}Q\,y_2^{\prime\,2}\,dt Re ∫ e 8 ϵ L Q y 2 ′ 2 d t Re ∫ e 8 ϵ L Q y 2 ′ 2 d t \operatorname{Re}\int e^{8\epsilon L}Q\,y_2^{\prime\,2}\,dt Re ∫ e 8 ϵ L Q y 2 ′ 2 d t
I 2 ( M ) = − N 1 G ϵ 3 [ cos 2 ( π ϵ ) C ϵ ( 0 ) 2 ∑ k a k ϵ k + sin 2 ( π ϵ ) S ϵ ′ ( 0 ) 2 ∑ k c k ϵ k ] , I^{(M)}_2=-\frac{N_1G}{\epsilon^3}\left[\cos^2(\pi\epsilon)C_\epsilon(0)^2\sum_k a_k\epsilon^k+\sin^2(\pi\epsilon)S'_\epsilon(0)^2\sum_k c_k\epsilon^k\right], I 2 ( M ) = − ϵ 3 N 1 G [ cos 2 ( π ϵ ) C ϵ ( 0 ) 2 k ∑ a k ϵ k + sin 2 ( π ϵ ) S ϵ ′ ( 0 ) 2 k ∑ c k ϵ k ] , I 2 , F ( M ) = e 2 i π ϵ N 1 G S ϵ ′ ( 0 ) 2 ϵ 3 ∑ k c k ϵ k . (65) I^{(M)}_{2,F}=\frac{e^{2i\pi\epsilon}N_1G\,S'_\epsilon(0)^2}{\epsilon^3}\sum_k c_k\epsilon^k. \tag*{(65)} I 2 , F ( M ) = ϵ 3 e 2 iπ ϵ N 1 G S ϵ ′ ( 0 ) 2 k ∑ c k ϵ k . (65) with
a 0 = a 1 = 0 , a 2 = 2 3 , a 3 = − 26 3 − 28 3 ln 2 , a_0=a_1=0,\qquad a_2=\frac{2}{3},\qquad a_3=-\frac{26}{3}-\frac{28}{3}\ln2, a 0 = a 1 = 0 , a 2 = 3 2 , a 3 = − 3 26 − 3 28 ln 2 , c 0 = 2 3 , c 1 = − 26 3 − 4 3 ln 2 , c 2 = 40 + 52 3 ln 2 + 4 3 ln 2 2 − 1 2 π 2 . (66) c_0=\frac{2}{3},\qquad c_1=-\frac{26}{3}-\frac{4}{3}\ln2,\qquad c_2=40+\frac{52}{3}\ln2+\frac{4}{3}\ln^2 2-\frac{1}{2}\pi^2. \tag*{(66)} c 0 = 3 2 , c 1 = − 3 26 − 3 4 ln 2 , c 2 = 40 + 3 52 ln 2 + 3 4 ln 2 2 − 2 1 π 2 . (66) Here c 2 c_2 c 2 y 2 y_2 y 2 ∫ sech n t y 2 ′ d t \int\operatorname{sech}^n t\,y'_2\,dt ∫ sech n t y 2 ′ d t S n S_n S n L n L_n L n T n T_n T n 2 ∫ 0 ∞ sech 2 t y 1 d t = 6 ln 2 + 3 ln 2 2 − 1 2 π 2 2\int_0^\infty\operatorname{sech}^2t\,y_1\,dt=6\ln2+3\ln^2 2-\frac{1}{2}\pi^2 2 ∫ 0 ∞ sech 2 t y 1 d t = 6 ln 2 + 3 ln 2 2 − 2 1 π 2 N 1 N_1 N 1 ϵ − 1 \epsilon^{-1} ϵ − 1 a 0 = a 1 = 0 a_0=a_1=0 a 0 = a 1 = 0 ϵ − 4 \epsilon^{-4} ϵ − 4 ϵ − 3 \epsilon^{-3} ϵ − 3 I 2 ( M ) I^{(M)}_2 I 2 ( M ) ϵ − 1 \epsilon^{-1} ϵ − 1 I 1 ( M ) I^{(M)}_1 I 1 ( M ) (12) , and for I 2 ( M ) I^{(M)}_2 I 2 ( M )
I 2 ( M ) = − 5 6 ( 8 π ) 4 ϵ 2 + O ( ϵ − 1 ) , c M = 1. (67) I^{(M)}_2=-\frac{5}{6(8\pi)^4\epsilon^2}+\mathcal{O}(\epsilon^{-1}),\qquad c_M=1. \tag*{(67)} I 2 ( M ) = − 6 ( 8 π ) 4 ϵ 2 5 + O ( ϵ − 1 ) , c M = 1. (67) exactly. Of this value, the cos 2 \cos^2 cos 2 (65) contributes 1 5 \frac{1}{5} 5 1 sin 2 \sin^2 sin 2 4 5 \frac{4}{5} 5 4 [15 ] generally evaluate the frequency integrals numerically and reconstruct analytic results from numerical data. With the closed forms above we confirm their value of c M c_M c M
With Feynman propagators on D 31 D_{31} D 31 D 24 D_{24} D 24 (50) and (51) become ( 0 + − i ∣ ω j ∣ ) 1 − 2 ϵ (0^+-i|\omega_j|)^{1-2\epsilon} ( 0 + − i ∣ ω j ∣ ) 1 − 2 ϵ ( 0 + − i ∣ ω j ∣ ) − 1 − 2 ϵ (0^+-i|\omega_j|)^{-1-2\epsilon} ( 0 + − i ∣ ω j ∣ ) − 1 − 2 ϵ ϵ 0 \epsilon^0 ϵ 0 − ω 1 ω 2 -\omega_1\omega_2 − ω 1 ω 2 I 1 ( M ) I^{(M)}_1 I 1 ( M ) − ∣ ω 1 ω 2 ∣ -|\omega_1\omega_2| − ∣ ω 1 ω 2 ∣ ϵ − 1 \epsilon^{-1} ϵ − 1 I 1 ( M ) I^{(M)}_1 I 1 ( M ) − O 1 / E 1 = − 1 3 − 10 π 2 = − 1.3465 … -O_1/E_1=-\frac{1}{3}-\frac{10}{\pi^2}=-1.3465\ldots − O 1 / E 1 = − 3 1 − π 2 10 = − 1.3465 … (12) . Because the ϵ − 1 \epsilon^{-1} ϵ − 1 I 3 ( M ) I^{(M)}_3 I 3 ( M ) − O 3 / E 3 = − 1 3 − 35 6 π 2 = -O_3/E_3=-\frac{1}{3}-\frac{35}{6\pi^2}= − O 3 / E 3 = − 3 1 − 6 π 2 35 = − 0.9243 … -0.9243\ldots − 0.9243 … ϵ − 4 \epsilon^{-4} ϵ − 4 I 2 , F ( M ) I_{2,F}^{(M)} I 2 , F ( M ) (52) survives, with coefficient 1 / ( 6144 π 6 ) 1/(6144\pi^6) 1/ ( 6144 π 6 ) § 11 ). The second line of Eq. (65) gives the three leading poles,
I 2 , F ( M ) = 1 6144 π 6 [ 1 ϵ 4 + 1 ϵ 3 + 4 λ + 2 i π ϵ 3 + 8 λ 2 + 8 i π λ − 23 12 π 2 ϵ 2 ] + O ( ϵ − 1 ) , (68) I_{2,F}^{(M)}=\frac{1}{6144\pi^6}\left[\frac{1}{\epsilon^4}+\frac{1}{\epsilon^3}+\frac{4\lambda+2i\pi}{\epsilon^3}+\frac{8\lambda^2+8i\pi\lambda-\frac{23}{12}\pi^2}{\epsilon^2}\right]+O(\epsilon^{-1}), \tag*{(68)} I 2 , F ( M ) = 6144 π 6 1 [ ϵ 4 1 + ϵ 3 1 + ϵ 3 4 λ + 2 iπ + ϵ 2 8 λ 2 + 8 iπ λ − 12 23 π 2 ] + O ( ϵ − 1 ) , (68) and no constant c M c_M c M (12) is defined. Reference [15 ] reports an uncanceled ϵ − 4 \epsilon^{-4} ϵ − 4 i 0 + i0^+ i 0 + ( γ − 3 ) − 15 / 2 (\gamma-3)^{-15/2} ( γ − 3 ) − 15/2
Porto and Riva [40 ] instead define a local-in-time conservative sector through a Poincaré–Bertrand decomposition of the principal-value integrals that Feynman’s i 0 + i0^+ i 0 + [75 ] . In their fifth-post-Newtonian computation the γ \gamma γ [15 ] at c M = 1 c_M=1 c M = 1 O ( G 5 ν 2 ) O(G^5\nu^2) O ( G 5 ν 2 ) I 1 ( M ) I_1^{(M)} I 1 ( M ) I 2 ( M ) I_2^{(M)} I 2 ( M ) (12) at c M = 1 c_M=1 c M = 1 γ = 3 \gamma=3 γ = 3 ϵ → 0 \epsilon\to0 ϵ → 0 [40 ] . These values are not evaluations of Eqs. (50) –(51) with Feynman propagators on D 31 D_{31} D 31 D 24 D_{24} D 24 [15 ] does not bear on them.
The singular point and graviton routing
Driesse et al. [15 ] find that the potential-region contribution diverges at γ = 3 \gamma=3 γ = 3 (12) , for any c M c_M c M § 3.2 and § 3.5 we showed that γ = 3 \gamma=3 γ = 3 [22 ] (topology 40 [26 ] ). The ten on-shell conditions of topology 40 have no simultaneous solution at real loop momenta, because a real null momentum orthogonal to a timelike velocity vanishes. By the Coleman–Norton theorem [35 ] , this singularity corresponds to no real classical process (cf. [40 ] ).
The divergence arises in the potential-region integrals, which are by construction insensitive to the graviton i 0 + i0^+ i 0 + [15 ] , whereas c M c_M c M x = 1 x=1 x = 1 D 31 D_{31} D 31 D 24 D_{24} D 24 I 1 , 2 ( M ) I_{1,2}^{(M)} I 1 , 2 ( M ) (14) , but the value of c M c_M c M
Under the in-in worldline rules [67 , 70 ] , every propagator is retarded toward the outgoing vertex of its diagram. In topology 40, when that vertex is the middle one of three on its worldline, both radiative gravitons are retarded into their common vertex, and otherwise one is advanced relative to it (cf. [40 ] ). We do not identify these gravitons with D 31 D_{31} D 31 D 24 D_{24} D 24 I 1 , 2 ( M ) I_{1,2}^{(M)} I 1 , 2 ( M ) γ = 3 \gamma=3 γ = 3 § 12 .§ 2 ). However, the complete observable requires this integral in d = 4 − 2 ϵ d=4-2\epsilon d = 4 − 2 ϵ
Conclusions
Next-generation gravitational-wave detectors need more accurate waveform models, and the post-Minkowskian expansion of the two-body problem that enters them has reached fifth order, four loops. At this order its integrals involve K3 surfaces and Calabi–Yau threefolds, as do Feynman integrals in particle physics, and one can ask, as for scattering amplitudes, which singularities they have and which functions they evaluate to. Many Feynman-integral methods carry over, such as the Landau equations and maximal cuts, and this paper has explored some applications of those methods.
For example, in 5th order in the PM expansion at second order in the mass ratio, the potential-region contribution diverges at relative Lorentz factor γ = 3 \gamma= 3 γ = 3 [15 ] . Nothing is special at that speed. We traced the divergence to the maximal cut of the planar K3 sector, where it comes from a pinch at infinite loop momentum that corresponds to no real classical process [35 ] . It must therefore cancel, and the only region that can cancel it is the memory region, where two gravitons are radiative [15 ] .
How the memory region is treated also affects the conservative part of the result, which is the part that carries over to bound orbits [6 , 7 ] and effective-one-body models [8 , 9 ] . Separating the conservative from the dissipative part requires an i 0 i0 i 0 [15 , 40 ] . The choice should affect only how the full observable is divided, but no rule for it has yet been derived from first principles. The γ = 3 \gamma= 3 γ = 3 [15 ] we derived the first memory-region integral in closed form through order ϵ \epsilon ϵ c M = 1 c_M = 1 c M = 1 ϵ − 4 \epsilon^{-4} ϵ − 4 [15 ] , and we derived its coefficient and the next two poles.
A full 5PM-2SF calculation, with radiation reaction, needs the remaining sectors of the four-loop families, and these families contain no geometries beyond two K3 surfaces and two Calabi–Yau threefolds [26 ] . We fixed the maximal-cut normalizations of the periods of all four geometries, proved a closed formula for the Frobenius coefficients of the period of the threefold at second order in the mass ratio, and continued the four Frobenius bases to bound-orbit energies. For the dissipative families we made a first count of the master integrals, and in the PX2 family we found ten exact relations with the auxiliary-mass flow [33 , 82 ] and evaluated twelve master integrals at one velocity with Feynman graviton propagators, working from maximal cuts and boundary limits without the integration-by-parts reduction of the four-loop integrand. With that reduction and the remaining sectors, the differential equations would continue these boundary values to general kinematics and show whether K3 and Calabi–Yau periods enter the radiated energy, as a threefold period does at first order in the mass ratio [57 ] .
Acknowledgments
We thank Rafael A. Porto for comments on a draft of this manuscript. MDS was supported in part by the U.S. Department of Energy, Office of Science, under grant DE-SC0013607. Support and funding for computations in this project were provided by Anthropic. Calculations in this paper were largely performed by AI (Claude Fable 5.1) and AI drafted the manuscript, which the author then edited. The author takes full responsibility for the validity of all the calculations in the paper.
Auxiliary-mass flow on worldline families
As in phase-space integrals [102 ] , on worldline families the cut propagators 2 π δ ( 2 k ⋅ u ) 2\pi\delta(2 k\cdot u) 2 π δ ( 2 k ⋅ u ) η \eta η [33 , 82 ] . We do not evaluate these boundary integrals for the 2SF sectors: the boundary constants of § 4 are master integrals that we compute directly at the reference point (30) , and the response matrices are independent of the boundary-integral values.
Assumptions for the boundary relation
The relation (36) holds under three assumptions. (i) Closure: the η \eta η J ( η , ϵ ) J(\eta,\epsilon) J ( η , ϵ ) M = lim η → 0 + J M=\lim_{\eta\to0+}J M = lim η → 0 + J ∂ η J = C ( η , ϵ ) J \partial_\eta J=C(\eta,\epsilon)J ∂ η J = C ( η , ϵ ) J η → ∞ \eta\to\infty η → ∞ J ( η ) = Φ ( η , ϵ ) b ( ϵ ) + o ( 1 ) J(\eta)=\Phi(\eta,\epsilon)b(\epsilon)+o(1) J ( η ) = Φ ( η , ϵ ) b ( ϵ ) + o ( 1 ) Φ \Phi Φ η \eta η A ( ϵ ) A(\epsilon) A ( ϵ ) ϵ \epsilon ϵ (36) then follows from linearity in the boundary data. We make no claim where the rank jumps, and the relation does not extend to the fixed-point family of Appendix § 13 .§ 3 , where the boundary vector coincides with M M M
The solver
We carry out the η \eta η [31 , 32 ] , our generalization of AMFlow [33 ] to the boundary integrals of cut worldline sectors. The auxiliary-mass flow with linear and cut propagators [102 , 103 ] is due to Liu, Ma and collaborators. AMFlow has been applied to one-loop post-Minkowskian integrals and their cuts [117 ] and to one-worldline integrals in gravitational Compton scattering at three and four loops [118 , 119 , 120 ] , including retarded and cut configurations [119 , 120 ] . The sectors that we treat here have two worldlines. PMFlow consists of a driver, in the BootLoops repository, and a fork of AMFlow.cpp [121 ] . We do not distribute the reduction tables.
In the fork we evaluate several classes of boundary integrals of cut sectors in closed form, using the static-source and cusp integrals of heavy-quark effective theory [122 , 123 ] . Along a linearized worldline propagator, where the standard enumeration ℓ ∼ η 0 \ell\sim\eta^0 ℓ ∼ η 0 ℓ ∼ η 1 / 2 \ell\sim\eta^{1/2} ℓ ∼ η 1/2 [103 ] , we include the region ℓ ∼ η \ell\sim\eta ℓ ∼ η η \eta η
{ − 1 2 , − 3 2 , − 1 , + 1 2 } + 4 ϵ (69) \left\{ -\frac{1}{2},-\frac{3}{2},-1,+\frac{1}{2} \right\}+4\epsilon \tag*{(69)} { − 2 1 , − 2 3 , − 1 , + 2 1 } + 4 ϵ (69) at η → ∞ \eta\to\infty η → ∞ t t t t 2 t^2 t 2
We have derived three further classes of two-velocity boundary integrals, which arise in the coupled sectors of the O ( G 3 ) O(G^3) O ( G 3 )
As a test of the solver, we derive graviton-cut boundary data from the closed forms of Ref. [66 ] for two master integrals of its two-loop soft family. With these data the flow reproduces order by order in ϵ \epsilon ϵ [33 ] . The O ( G 3 ) O(G^3) O ( G 3 ) γ = 5 4 \gamma= \frac{5}{4} γ = 4 5 [124 ] from the cut integrand of Ref. [65 ] and the master values of Ref. [66 ] , matches the closed form of Ref. [68 ] . On the 4PM cut-eikonal K3 family [50 , 51 ] one set of values of the 43 master integrals satisfies the constraints of two flows with different η \eta η ({xref}Appendix E.3`).
Master integrals and identities
This appendix gives the seeding of the first-stage reductions of § 4 and our master integrals for the Apéry sector, which we relate to the nine-element basis of Duhr et al. [23 ] .
First-stage reductions of the four families
The counts of Table 3 come from one Laporta reduction per family. We seeded the integrals of the top sectors with at most r r r s s s n d n_d n d ( r , s , n d ) = ( 14 , 0 , 1 ) (r,s,n_d)=(14,0,1) ( r , s , n d ) = ( 14 , 0 , 1 ) ( 14 , 1 , − ) (14,1,-) ( 14 , 1 , − ) ( r = 13 , s = n d = 0 ) (r=13,s=n_d=0) ( r = 13 , s = n d = 0 ) tables/first-stage_seeding.txt; the four master lists are in masters/). Repeating the PX2 reduction with the seeds ( r , s , n d ) = ( 15 , 2 , 2 ) (r,s,n_d)=(15,2,2) ( r , s , n d ) = ( 15 , 2 , 2 )
The PX2 count holds for the family as defined, in which the uncut linearized propagators have a common routing sign σ \sigma σ S 1 S_1 S 1 S 2 S_2 S 2 k i ↦ k i + q k_i \mapsto k_i+q k i ↦ k i + q q ↦ − q q \mapsto-q q ↦ − q S 3 S_3 S 3 2 × ( 30 + 28 ) = 116 2 \times(30+28)=116 2 × ( 30 + 28 ) = 116
The master integrals of the response-matrix rows are not in general among the 37, since the auxiliary-mass flow uses the master integrals of the η \eta η
The Apéry-sector master integrals
The Apéry sector (K 3 ′ \mathrm{K3}' K3 ′ [26 ] . Our family for it (ancillary file masters/t40_even_44_eleven.txt) has the ten denominators P 1 , … , P 10 P_1,\ldots,P_{10} P 1 , … , P 10 P 7 , … , P 10 P_7,\ldots,P_{10} P 7 , … , P 10 P 11 , … , P 22 P_{11},\ldots,P_{22} P 11 , … , P 22 P 13 , … , P 16 P_{13},\ldots,P_{16} P 13 , … , P 16 γ = 23 / 17 \gamma=23/17 γ = 23/17 γ \gamma γ M 1 , … , M 11 M_1,\ldots,M_{11} M 1 , … , M 11
The worldline exchange u 1 ↔ u 2 u_1 \leftrightarrow u_2 u 1 ↔ u 2 k 1 → k 2 + k 4 k_1 \to k_2+k_4 k 1 → k 2 + k 4 k 3 → k 2 − k 3 − q k_3 \to k_2-k_3-q k 3 → k 2 − k 3 − q k 4 → k 1 − k 2 k_4 \to k_1-k_2 k 4 → k 1 − k 2 ( P 1 P 4 ) ( P 2 P 6 ) ( P 3 P 5 ) ( P 7 P 10 ) ( P 8 P 9 ) (P_1P_4)(P_2P_6)(P_3P_5)(P_7P_{10})(P_8P_9) ( P 1 P 4 ) ( P 2 P 6 ) ( P 3 P 5 ) ( P 7 P 10 ) ( P 8 P 9 ) M 2 M_2 M 2 M 5 M_5 M 5 M 1 M_1 M 1 M 2 M_2 M 2 M 3 M_3 M 3 M 4 M_4 M 4 M 6 M_6 M 6 M 7 M_7 M 7 M 8 M_8 M 8 M 9 M_9 M 9 M 11 M_{11} M 11
Relation to the nine-element basis
Klemm et al. [22 ] report nine master integrals in the Apéry sector. Duhr et al. give in their ancillary files an explicit nine-element basis on the maximal cut, obtained with Feynman propagators and all symmetry relations, together with its differential equation and the rotation to ϵ \epsilon ϵ M ^ 1 , … , M ^ 8 \hat{M}_1,\ldots,\hat{M}_8 M ^ 1 , … , M ^ 8 tables/t40_duhr_propagators_integrals.txt) and the first and second x x x M ^ 1 \hat{M}_1 M ^ 1 x x x ϵ \epsilon ϵ M ^ 7 \hat{M}_7 M ^ 7 M ^ 8 \hat{M}_8 M ^ 8
The loop momenta of Duhr et al. are related to ours by K 1 = k 2 − k 3 − q K_1=k_2-k_3-q K 1 = k 2 − k 3 − q K 2 = k 1 − k 3 − q K_2=k_1-k_3-q K 2 = k 1 − k 3 − q K 3 = − k 3 − k 4 − q K_3=-k_3-k_4-q K 3 = − k 3 − k 4 − q K 4 = − k 3 − q K_4=-k_3-q K 4 = − k 3 − q M ^ 1 \hat{M}_1 M ^ 1 M ^ 3 \hat{M}_3 M ^ 3 2 4 2^4 2 4 M 1 M_1 M 1 M 3 M_3 M 3 M ^ 2 \hat{M}_2 M ^ 2 2 4 2^4 2 4 P 1 P_1 P 1 M 4 M_4 M 4 2 4 2^4 2 4
At γ = 23 / 17 \gamma=23/17 γ = 23/17 d = 97 / 23 d=97/23 d = 97/23 d = 113 / 31 d=113/31 d = 113/31 masters/t40_duhr_map_point.json). The last two elements contain the numerators P 11 = k 2 2 P_{11}=k_2^2 P 11 = k 2 2 P 18 = ( k 3 + q ) 2 P_{18}=(k_3+q)^2 P 18 = ( k 3 + q ) 2 P 11 P_{11} P 11 P 18 P_{18} P 18
M ^ 7 = − 32 ( d − 4 ) M 6 , (70) \hat{M}_7=-32\left(d-4\right)M_6, \tag*{(70)} M ^ 7 = − 32 ( d − 4 ) M 6 , (70) M ^ 8 = ( 3358 15 − 1540 27 d ) M 1 + 578 135 d − 5 d − 4 M 3 − 16 3 M 4 + 1564 135 M 6 + 578 45 M 7 + 578 45 d − 3 d − 4 M 8 − 782 135 M 9 . (71) \begin{aligned}
\hat{M}_8={}&\left(\frac{3358}{15}-\frac{1540}{27}d\right)M_1+\frac{578}{135}\frac{d-5}{d-4}M_3-\frac{16}{3}M_4\\
&+\frac{1564}{135}M_6+\frac{578}{45}M_7+\frac{578}{45}\frac{d-3}{d-4}M_8-\frac{782}{135}M_9. \tag*{(71)}
\end{aligned} M ^ 8 = ( 15 3358 − 27 1540 d ) M 1 + 135 578 d − 4 d − 5 M 3 − 3 16 M 4 + 135 1564 M 6 + 45 578 M 7 + 45 578 d − 4 d − 3 M 8 − 135 782 M 9 . (71) Equation (B.1) is a combination of two integration-by-parts identities and one exchange relation, none of which contains γ \gamma γ γ \gamma γ d d d γ = 23 / 17 \gamma=23/17 γ = 23/17 d d d masters/t40_duhr_map_u8_u9.json), and we have not determined its dependence on γ \gamma γ d d d
Derivations for the boundary integrals
We reduce the boundary constants of PX2 quoted in § 4.4 to frequency integrals, in the conventions of that section. We then extend the closed forms of the memory-region integrals of § 6 by one order in ϵ \epsilon ϵ
The corner integrals I 0 I_0 I 0 I 2 I_2 I 2 I 3 I_3 I 3
In the corner integral I 0 I_0 I 0 (29) and its numerator partners I 2 I_2 I 2 I 3 I_3 I 3 k 1 , k 2 ∈ V 1 = u 1 ⊥ k_1,k_2 \in V_1 = u_1^\perp k 1 , k 2 ∈ V 1 = u 1 ⊥ k 3 , k 4 ∈ V 2 = u 2 ⊥ k_3,k_4 \in V_2 = u_2^\perp k 3 , k 4 ∈ V 2 = u 2 ⊥ d ~ = 3 − 2 ϵ \tilde d = 3 - 2\epsilon d ~ = 3 − 2 ϵ k 2 = b e ^ 1 + k 2 T k_2 = b\hat e_1 + k_{2T} k 2 = b e ^ 1 + k 2 T k 4 = a e ^ 2 + k 4 T k_4 = a\hat e_2 + k_{4T} k 4 = a e ^ 2 + k 4 T e ^ 1 ∈ V 1 \hat e_1 \in V_1 e ^ 1 ∈ V 1 e ^ 2 ∈ V 2 \hat e_2 \in V_2 e ^ 2 ∈ V 2 u 1 u_1 u 1 u 2 u_2 u 2 k i T k_{iT} k i T T = 2 − 2 ϵ T = 2 - 2\epsilon T = 2 − 2 ϵ a a a b b b k 1 k_1 k 1 k 3 k_3 k 3 V 1 V_1 V 1 V 2 V_2 V 2
∫ d d ~ k π d ~ / 2 1 k 2 ( k + P ) 2 = G 11 ( P 2 ) − 1 2 − ϵ , G 11 = Γ ( 1 2 + ϵ ) Γ ( 1 2 − ϵ ) 2 Γ ( 1 − 2 ϵ ) , (72) \int\frac{\mathrm{d}^{\tilde d} k}{\pi^{\tilde d/2}} \frac{1}{k^2 (k+P)^2} = G_{11} (P^2)^{-\frac{1}{2}-\epsilon}, \qquad G_{11} = \frac{\Gamma\left(\frac{1}{2}+\epsilon\right)\Gamma\left(\frac{1}{2}-\epsilon\right)^2}{\Gamma(1-2\epsilon)}, \tag*{(72)} ∫ π d ~ /2 d d ~ k k 2 ( k + P ) 2 1 = G 11 ( P 2 ) − 2 1 − ϵ , G 11 = Γ ( 1 − 2 ϵ ) Γ ( 2 1 + ϵ ) Γ ( 2 1 − ϵ ) 2 , (72) with P 2 = b 2 + ∣ k 2 T + q ∣ 2 P^2 = b^2 + \lvert k_{2T} + q\rvert^2 P 2 = b 2 + ∣ k 2 T + q ∣ 2 k 1 k_1 k 1 P 2 = a 2 + ∣ k 4 T ∣ 2 P^2 = a^2 + \lvert k_{4T}\rvert^2 P 2 = a 2 + ∣ k 4 T ∣ 2 k 3 k_3 k 3 1 2 P μ \frac{1}{2}P^\mu 2 1 P μ I 2 I_2 I 2 I 3 I_3 I 3
( k 1 − k 3 ) 2 → N 2 = p 1 ⋅ q − 1 + γ 2 s w 1 w 2 r 2 + 1 2 p 1 ⋅ p 3 − q ⋅ p 3 , (73) (k_1-k_3)^2 \to\mathcal{N}_2 = p_1\cdot q - 1 + \frac{\gamma}{2}s\frac{w_1w_2}{r^2} + \frac{1}{2}p_1\cdot p_3 - q\cdot p_3, \tag*{(73)} ( k 1 − k 3 ) 2 → N 2 = p 1 ⋅ q − 1 + 2 γ s r 2 w 1 w 2 + 2 1 p 1 ⋅ p 3 − q ⋅ p 3 , (73) ( k 1 − k 4 ) 2 → N 3 = p 1 ⋅ q − 1 + γ s w 1 w 2 r 2 + p 1 ⋅ p 3 − 2 q ⋅ p 3 − ( a 2 + ∣ k 4 T ∣ 2 ) , (k_1-k_4)^2 \to\mathcal{N}_3 = p_1\cdot q - 1 + \gamma s\frac{w_1w_2}{r^2} + p_1\cdot p_3 - 2q\cdot p_3 - \left(a^2+\lvert k_{4T}\rvert^2\right), ( k 1 − k 4 ) 2 → N 3 = p 1 ⋅ q − 1 + γ s r 2 w 1 w 2 + p 1 ⋅ p 3 − 2 q ⋅ p 3 − ( a 2 + ∣ k 4 T ∣ 2 ) , where p 1 = k 2 T + q p_1 = k_{2T} + q p 1 = k 2 T + q p 3 = k 4 T p_3 = k_{4T} p 3 = k 4 T a b = s w 1 w 2 / r 2 ab = s w_1w_2/r^2 ab = s w 1 w 2 / r 2 s = ± 1 s = \pm1 s = ± 1
The graviton propagator ( k 2 − k 4 ) 2 (k_2-k_4)^2 ( k 2 − k 4 ) 2
− ( k 2 − k 4 ) 2 = F + ∣ k 2 T − k 4 T ∣ 2 , F = a 2 + b 2 − 2 a b γ , (74) -(k_2-k_4)^2 = F + \lvert k_{2T}-k_{4T}\rvert^2, \qquad F = a^2+b^2-2ab\gamma, \tag*{(74)} − ( k 2 − k 4 ) 2 = F + ∣ k 2 T − k 4 T ∣ 2 , F = a 2 + b 2 − 2 abγ , (74) which vanishes for γ > 1 \gamma> 1 γ > 1 a / b = x ± 1 a/b = x^{\pm1} a / b = x ± 1 F → F − i 0 F \to F-i0 F → F − i 0
After the two bubble integrations, a two-loop chain in the transverse space remains: a line of mass b b b 1 2 + ϵ \frac{1}{2}+\epsilon 2 1 + ϵ F \sqrt{F} F a a a 1 2 + ϵ \frac{1}{2}+\epsilon 2 1 + ϵ r r r ( b , a ) = ( w 1 , w 2 ) / r (b,a) = (w_1,w_2)/r ( b , a ) = ( w 1 , w 2 ) / r [116 Eq. 10.22.43] . With ν = 1 2 − 2 ϵ \nu= \frac{1}{2}-2\epsilon ν = 2 1 − 2 ϵ w ν K ν ( w ) w^\nu K_\nu(w) w ν K ν ( w ) − w ν + 1 K ν + 1 ( w ) -w^{\nu+1}K_{\nu+1}(w) − w ν + 1 K ν + 1 ( w ) F ^ − ϵ / 2 K ϵ ( F ^ − i 0 ) \hat F^{-\epsilon/2}K_\epsilon\left(\sqrt{\hat F-i0}\right) F ^ − ϵ /2 K ϵ ( F ^ − i 0 ) F ^ = w 1 2 + w 2 2 − 2 s γ w 1 w 2 \hat F = w_1^2+w_2^2-2s\gamma w_1w_2 F ^ = w 1 2 + w 2 2 − 2 s γ w 1 w 2 I ( w 1 , w 2 , s ) \mathcal{I}(w_1,w_2,s) I ( w 1 , w 2 , s )
W ( ϵ ) = 2 ∑ s = ± 1 ∫ 0 ∞ ∫ 0 ∞ d w 1 d w 2 sh 1 ( w 1 ) sh 3 ( w 2 ) F ^ − ϵ / 2 K ϵ ( F ^ − i 0 ) I ( w 1 , w 2 , s ) , (75) W(\epsilon)=2\sum_{s=\pm1}\int_0^\infty\!\int_0^\infty\mathrm{d}w_1\,\mathrm{d}w_2\,\operatorname{sh}_1(w_1)\operatorname{sh}_3(w_2)\hat F^{-\epsilon/2}K_\epsilon\left(\sqrt{\hat F-i0}\right)\mathcal{I}(w_1,w_2,s), \tag*{(75)} W ( ϵ ) = 2 s = ± 1 ∑ ∫ 0 ∞ ∫ 0 ∞ d w 1 d w 2 sh 1 ( w 1 ) sh 3 ( w 2 ) F ^ − ϵ /2 K ϵ ( F ^ − i 0 ) I ( w 1 , w 2 , s ) , (75) where s h 1 , 3 \mathrm{sh}_{1,3} sh 1 , 3 s = + 1 s=+1 s = + 1 w 1 / w 2 ∈ ( x , 1 / x ) w_1/w_2\in(x,1/x) w 1 / w 2 ∈ ( x , 1/ x ) F ^ \hat{F} F ^ K ν ( − i y ) = i π 2 e i ν π / 2 H ν ( 1 ) ( y ) K_\nu(-iy)=\frac{i\pi}{2}e^{i\nu\pi/2}H_\nu^{(1)}(y) K ν ( − i y ) = 2 iπ e i ν π /2 H ν ( 1 ) ( y ) i 0 i0 i 0
Collecting the constants, we obtain
I 0 ( ϵ ) = − π 2 5 ϵ Γ ( 1 2 − ϵ ) 4 Γ ( 4 ϵ − 1 ) Γ ( 1 − 2 ϵ ) 2 Γ ( 2 − 5 ϵ ) W A ( ϵ ) , (76) I_0(\epsilon)=-\pi\,2^{5\epsilon}\frac{\Gamma\left(\frac{1}{2}-\epsilon\right)^4\Gamma(4\epsilon-1)}{\Gamma(1-2\epsilon)^2\Gamma(2-5\epsilon)}W_A(\epsilon), \tag*{(76)} I 0 ( ϵ ) = − π 2 5 ϵ Γ ( 1 − 2 ϵ ) 2 Γ ( 2 − 5 ϵ ) Γ ( 2 1 − ϵ ) 4 Γ ( 4 ϵ − 1 ) W A ( ϵ ) , (76) with W A W_A W A I 2 I_2 I 2 I 3 I_3 I 3 Γ ( 4 ϵ − 2 ) \Gamma(4\epsilon-2) Γ ( 4 ϵ − 2 ) A A A B B B s w 1 w 2 s w_1w_2 s w 1 w 2 C C C E E E ϵ = 0 \epsilon=0 ϵ = 0
W A ( 0 ) = π ( 320 ln 2 − 192 ) 27 + 4 π 2 27 i , W_A(0)=\frac{\pi(320\ln2-192)}{27}+\frac{4\pi^2}{27}i, W A ( 0 ) = 27 π ( 320 ln 2 − 192 ) + 27 4 π 2 i , W B ( 0 ) = π ( 3056640 − 4409344 ln 2 ) 2187 + 352 π 2 2187 i , W_B(0)=\frac{\pi(3056640-4409344\ln2)}{2187}+\frac{352\pi^2}{2187}i, W B ( 0 ) = 2187 π ( 3056640 − 4409344 ln 2 ) + 2187 352 π 2 i , W C ( 0 ) = π ( 20480 ln 2 − 14592 ) 243 − 68 π 2 243 i , W_C(0)=\frac{\pi(20480\ln2-14592)}{243}-\frac{68\pi^2}{243}i, W C ( 0 ) = 243 π ( 20480 ln 2 − 14592 ) − 243 68 π 2 i , W E ( 0 ) = π ( 4014080 ln 2 − 2777088 ) 2187 + 1252 π 2 2187 i . (77) W_E(0)=\frac{\pi(4014080\ln2-2777088)}{2187}+\frac{1252\pi^2}{2187}i. \tag*{(77)} W E ( 0 ) = 2187 π ( 4014080 ln 2 − 2777088 ) + 2187 1252 π 2 i . (77) With the residues of the Γ \Gamma Γ (31) , for example π 3 4 W A ( 0 ) \frac{\pi^3}{4}W_A(0) 4 π 3 W A ( 0 ) I 0 I_0 I 0
Feynman-parameter representation and the finite parts
For the finite parts we need the cores at first order in ϵ \epsilon ϵ ( w 1 , w 2 ) ∈ R 2 (w_1,w_2)\in\mathbb{R}^2 ( w 1 , w 2 ) ∈ R 2 s s s w 1 w 2 w_1w_2 w 1 w 2 F = w 1 2 + w 2 2 − 2 γ w 1 w 2 F=w_1^2+w_2^2-2\gamma w_1w_2 F = w 1 2 + w 2 2 − 2 γ w 1 w 2
∫ − ∞ ∞ d w e − i k w ∣ w ∣ ν K ν ( ∣ w ∣ ) = π 2 ν Γ ( ν + 1 2 ) ( 1 + k 2 ) − ν − 1 2 , ∫ d 2 w e − i ϑ ⋅ w F − ϵ / 2 K ϵ ( F − i 0 ) = i 2 1 − ϵ π Γ ( 1 − ϵ ) γ 2 − 1 ( 1 + Π + i 0 ) − 1 + ϵ , (78) \begin{aligned}
\int_{-\infty}^{\infty}dw\,e^{-ikw}|w|^\nu K_\nu(|w|)&=\sqrt{\pi}\,2^\nu\Gamma\left(\nu+\frac{1}{2}\right)(1+k^2)^{-\nu-\frac{1}{2}},\\
\int d^2w\,e^{-i\vartheta\cdot w}F^{-\epsilon/2}K_\epsilon(\sqrt{F-i0})&=\frac{i\,2^{1-\epsilon}\pi\Gamma(1-\epsilon)}{\sqrt{\gamma^2-1}}(1+\Pi+i0)^{-1+\epsilon}, \tag*{(78)}
\end{aligned} ∫ − ∞ ∞ d w e − ik w ∣ w ∣ ν K ν ( ∣ w ∣ ) ∫ d 2 w e − i ϑ ⋅ w F − ϵ /2 K ϵ ( F − i 0 ) = π 2 ν Γ ( ν + 2 1 ) ( 1 + k 2 ) − ν − 2 1 , = γ 2 − 1 i 2 1 − ϵ π Γ ( 1 − ϵ ) ( 1 + Π + i 0 ) − 1 + ϵ , (78) with Π = − κ ( ϑ 1 2 + ϑ 2 2 + 2 γ ϑ 1 ϑ 2 ) \Pi=-\kappa(\vartheta_1^2+\vartheta_2^2+2\gamma\vartheta_1\vartheta_2) Π = − κ ( ϑ 1 2 + ϑ 2 2 + 2 γ ϑ 1 ϑ 2 ) κ = 1 / ( γ 2 − 1 ) \kappa=1/(\gamma^2-1) κ = 1/ ( γ 2 − 1 ) s w 1 w 2 s w_1w_2 s w 1 w 2 ϑ 1 ϑ 2 \vartheta_1\vartheta_2 ϑ 1 ϑ 2 ϑ \vartheta ϑ
W X ( ϵ ) = i π c X 2 − 5 ϵ Γ ( n X − 5 ϵ ) γ 2 − 1 Φ X ( ϵ ) , W_X(\epsilon)=\frac{i\pi c_X2^{-5\epsilon}\Gamma(n_X-5\epsilon)}{\sqrt{\gamma^2-1}}\Phi_X(\epsilon), W X ( ϵ ) = γ 2 − 1 iπ c X 2 − 5 ϵ Γ ( n X − 5 ϵ ) Φ X ( ϵ ) , Φ X ( ϵ ) = ∫ x 1 + x 2 + x 3 = 1 d 2 x x 1 a 1 − 2 ϵ x 2 a 2 − 2 ϵ x 3 a 3 − ϵ ( x 1 x 2 − κ x 3 + i 0 ) − m X / 2 . (79) \Phi_X(\epsilon)=\int_{x_1+x_2+x_3=1}d^2x\,x_1^{a_1-2\epsilon}x_2^{a_2-2\epsilon}x_3^{a_3-\epsilon}(x_1x_2-\kappa x_3+i0)^{-m_X/2}. \tag*{(79)} Φ X ( ϵ ) = ∫ x 1 + x 2 + x 3 = 1 d 2 x x 1 a 1 − 2 ϵ x 2 a 2 − 2 ϵ x 3 a 3 − ϵ ( x 1 x 2 − κ x 3 + i 0 ) − m X /2 . (79) where the constants ( c X , n X ) (c_X,n_X) ( c X , n X ) ( a 1 , a 2 , a 3 ) (a_1,a_2,a_3) ( a 1 , a 2 , a 3 ) m X m_X m X ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 0 , 0 , 0 ) (0,0,0) ( 0 , 0 , 0 ) 1 1 1 A A A ( − 2 , 3 ) (-2,3) ( − 2 , 3 ) ( 1 , 0 , 0 ) (1,0,0) ( 1 , 0 , 0 ) 1 1 1 C C C ( 4 , 4 ) (4,4) ( 4 , 4 ) ( 1 , 1 , 0 ) (1,1,0) ( 1 , 1 , 0 ) 1 1 1 E E E ( − 2 κ γ , 3 ) (-2\kappa\gamma,3) ( − 2 κγ , 3 ) ( 1 , 1 , 1 ) (1,1,1) ( 1 , 1 , 1 ) 3 3 3 B B B B B B − 3 2 -\frac{3}{2} − 2 3 x 2 x_2 x 2 x 1 x 2 < κ x 3 x_1x_2 < \kappa x_3 x 1 x 2 < κ x 3 ( x 1 x 2 − κ x 3 + i 0 ) − 1 / 2 = − i ∣ x 1 x 2 − κ x 3 ∣ − 1 / 2 (x_1x_2-\kappa x_3+i0)^{-1/2}=-i\lvert x_1x_2-\kappa x_3\rvert^{-1/2} ( x 1 x 2 − κ x 3 + i 0 ) − 1/2 = − i ∣ x 1 x 2 − κ x 3 ∣ − 1/2 Φ X = Φ X + − i Φ X − \Phi_X=\Phi_X^+-i\Phi_X^- Φ X = Φ X + − i Φ X − Φ X ± \Phi_X^\pm Φ X ±
In the variables x 1 x_1 x 1 x 2 x_2 x 2 x 1 x 2 − κ x 3 x_1x_2-\kappa x_3 x 1 x 2 − κ x 3 x 2 x_2 x 2 ϵ \epsilon ϵ x 2 x_2 x 2 γ = 5 4 \gamma=\frac{5}{4} γ = 4 5 ϵ \epsilon ϵ x 2 x_2 x 2 ln x 1 \ln x_1 ln x 1 ln ( 1 − x 1 ) \ln(1-x_1) ln ( 1 − x 1 ) ln ( x 1 + κ ) \ln(x_1+\kappa) ln ( x 1 + κ ) arctan x 1 / κ \arctan\sqrt{x_1/\kappa} arctan x 1 / κ x 1 x_1 x 1 π \pi π π ln 2 \pi\ln2 π ln 2 π ln 3 \pi\ln3 π ln 3 Φ + \Phi^+ Φ + 1 1 1 ln 2 \ln2 ln 2 ln 3 \ln3 ln 3 ln 2 2 \ln^2 2 ln 2 2 ln 2 ln 3 \ln2\ln3 ln 2 ln 3 ln 2 3 \ln^2 3 ln 2 3 π 2 \pi^2 π 2 Li 2 ( 1 4 ) \operatorname{Li}_2(\frac{1}{4}) Li 2 ( 4 1 ) Φ − \Phi^- Φ − tables/px2_feynman_parameter_integrals.txt. Combining these relations with the Laurent expansions of the prefactors, we obtain the finite parts of I 0 I_0 I 0 I 2 I_2 I 2 I 3 I_3 I 3 (31) .
The dotted integral I r 4 I_{r4} I r 4
In the dotted integral I r 4 I_{r4} I r 4 k 3 k_3 k 3 ( 2 , 1 ) (2,1) ( 2 , 1 ) ϵ → 0 \epsilon\to0 ϵ → 0 Γ ( 4 ϵ ) \Gamma(4\epsilon) Γ ( 4 ϵ )
I r 4 ( ϵ ) = π 2 1 + 5 ϵ Γ ( 1 2 − ϵ ) 3 Γ ( − 1 2 − ϵ ) Γ ( 4 ϵ ) Γ ( 1 − 2 ϵ ) Γ ( − 2 ϵ ) Γ ( 1 − 5 ϵ ) W ′ ( ϵ ) , (80) I_{r4}(\epsilon)=\pi\,2^{1+5\epsilon}\frac{\Gamma\left(\frac{1}{2}-\epsilon\right)^3\Gamma\left(-\frac{1}{2}-\epsilon\right)\Gamma(4\epsilon)}{\Gamma(1-2\epsilon)\Gamma(-2\epsilon)\Gamma(1-5\epsilon)}W'(\epsilon), \tag*{(80)} I r 4 ( ϵ ) = π 2 1 + 5 ϵ Γ ( 1 − 2 ϵ ) Γ ( − 2 ϵ ) Γ ( 1 − 5 ϵ ) Γ ( 2 1 − ϵ ) 3 Γ ( − 2 1 − ϵ ) Γ ( 4 ϵ ) W ′ ( ϵ ) , (80) with the overall sign opposite to that of Eq. (C.5), because I r 4 I_{r4} I r 4 I 0 I_0 I 0
The core W ′ W' W ′ ( c X , n X ) = ( 1 2 , 1 ) (c_X,n_X)=(\frac{1}{2},1) ( c X , n X ) = ( 2 1 , 1 ) ( 0 , − 1 , 0 ) (0,-1,0) ( 0 , − 1 , 0 ) m X = 1 m_X=1 m X = 1 x 2 → 0 x_2\to0 x 2 → 0 Φ r 4 = 3 i 4 ϵ + Φ r 4 + − i Φ r 4 − \Phi_{r4}=\frac{3i}{4\epsilon}+\Phi_{r4}^+-i\Phi_{r4}^- Φ r 4 = 4 ϵ 3 i + Φ r 4 + − i Φ r 4 − Φ X ± \Phi_X^\pm Φ X ± 2 π 3 2\pi^3 2 π 3 ϵ = 0 \epsilon=0 ϵ = 0 W ′ W' W ′ − π 2 -\frac{\pi}{2} − 2 π (32) . The finite part and the order-ϵ \epsilon ϵ Φ r 4 ± \Phi_{r4}^\pm Φ r 4 ±
The static-slice integrals I r 5 I_{r5} I r 5 I r 6 I_{r6} I r 6 I r 7 I_{r7} I r 7
The corner sector of S 2 S_2 S 2 S 1 S_1 S 1 k 1 k_1 k 1 k 2 k_2 k 2 k 2 k_2 k 2 k 3 k_3 k 3 k 1 ∈ V 1 k_1\in V_1 k 1 ∈ V 1 b b b k 4 ∈ V 2 k_4\in V_2 k 4 ∈ V 2 a a a ( k 1 − k 4 ) 2 (k_1-k_4)^2 ( k 1 − k 4 ) 2 I r 5 I_{r5} I r 5 I r 6 I_{r6} I r 6 I r 7 I_{r7} I r 7 1 / ( D 5 + i σ 0 ) 1/(D_5+i\sigma0) 1/ ( D 5 + iσ 0 ) D 5 = 2 k 1 ⋅ u 2 = − 2 γ 2 − 1 b D_5=2k_1\cdot u_2=-2\sqrt{\gamma^2-1}\,b D 5 = 2 k 1 ⋅ u 2 = − 2 γ 2 − 1 b
1 D 5 + i σ 0 = PV 1 D 5 − i π σ δ ( D 5 ) , δ ( D 5 ) = δ ( b ) 2 γ 2 − 1 = 2 3 δ ( b ) ( γ = 5 4 ) . (81) \frac{1}{D_5+i\sigma0}=\operatorname{PV}\frac{1}{D_5}-i\pi\sigma\delta(D_5),\qquad\delta(D_5)=\frac{\delta(b)}{2\sqrt{\gamma^2-1}}=\frac{2}{3}\delta(b)\quad\left(\gamma=\frac{5}{4}\right). \tag*{(81)} D 5 + iσ 0 1 = PV D 5 1 − iπ σ δ ( D 5 ) , δ ( D 5 ) = 2 γ 2 − 1 δ ( b ) = 3 2 δ ( b ) ( γ = 4 5 ) . (81) For Feynman graviton propagators the principal-value part integrates to zero (§ 4.4 ). Only the slice b = 0 b=0 b = 0 F = a 2 F=a^2 F = a 2 k 1 k_1 k 1
I r 5 = σ 2 π i 3 π 2 ϵ − 1 G 11 2 J , J = ∫ − ∞ ∞ d a ∫ d T k 1 d T k 4 ( ∣ k 1 + q ∣ 2 ) − 1 2 − ϵ ( a 2 + ∣ k 4 ∣ 2 ) − 1 2 − ϵ a 2 + ∣ k 1 − k 4 ∣ 2 . (82) I_{r5}=\sigma\frac{2\pi i}{3}\pi^{2\epsilon-1}G_{11}^2J,\qquad J=\int_{-\infty}^{\infty} da\int d^Tk_1\,d^Tk_4\,\frac{\left(\lvert k_1+q\rvert^2\right)^{-\frac{1}{2}-\epsilon}\left(a^2+\lvert k_4\rvert^2\right)^{-\frac{1}{2}-\epsilon}}{a^2+\lvert k_1-k_4\rvert^2}. \tag*{(82)} I r 5 = σ 3 2 π i π 2 ϵ − 1 G 11 2 J , J = ∫ − ∞ ∞ d a ∫ d T k 1 d T k 4 a 2 + ∣ k 1 − k 4 ∣ 2 ( ∣ k 1 + q ∣ 2 ) − 2 1 − ϵ ( a 2 + ∣ k 4 ∣ 2 ) − 2 1 − ϵ . (82) with k 1 k_1 k 1 k 4 k_4 k 4
In position space J J J ∣ a ∣ |a| ∣ a ∣ ∣ a ∣ |a| ∣ a ∣ a = w / r a = w/r a = w / r r r r w w w
∫ 0 ∞ t − ϱ K μ ( t ) K ν ( t ) d t = 2 − 2 − ϱ Γ ( 1 − ϱ ) ∏ ± , ± ′ Γ ( 1 − ϱ ± μ ± ′ ν 2 ) (83) \int_0^\infty t^{-\varrho}K_\mu(t)K_\nu(t)\,\mathrm{d}t = \frac{2^{-2-\varrho}}{\Gamma(1-\varrho)}\prod_{\pm,\pm'}\Gamma\left(\frac{1-\varrho\pm\mu\pm'\nu}{2}\right) \tag*{(83)} ∫ 0 ∞ t − ϱ K μ ( t ) K ν ( t ) d t = Γ ( 1 − ϱ ) 2 − 2 − ϱ ± , ± ′ ∏ Γ ( 2 1 − ϱ ± μ ± ′ ν ) (83) where the product runs over the four sign choices. Hence I r 5 I_{r5} I r 5 Γ \Gamma Γ ϵ \epsilon ϵ 1 8 < ϵ < 1 4 \frac{1}{8}<\epsilon<\frac{1}{4} 8 1 < ϵ < 4 1 I r 6 = I r 5 [ ( k 2 − k 3 ) 2 ] I_{r6}=I_{r5}[(k_2-k_3)^2] I r 6 = I r 5 [( k 2 − k 3 ) 2 ] I r 7 = I r 5 [ ( k 2 − k 4 ) 2 ] I_{r7}=I_{r5}[(k_2-k_4)^2] I r 7 = I r 5 [( k 2 − k 4 ) 2 ] Table 5 .
Higher orders of the memory-region integrals
Carrying the expansion of Section 6.2 6.2 6.2 (63) ,
I 1 ( M ) = e 4 λ ϵ 61440 π 4 [ 1 ϵ + 232 15 + ϵ ( 50944 225 − 7 π 2 ) + ϵ 2 ( 10774648 3375 − 1624 15 π 2 − 2284 3 ζ ( 3 ) ) ] + O ( ϵ 3 ) , I_1^{(M)}=\frac{e^{4\lambda\epsilon}}{61440\pi^4}\left[\frac{1}{\epsilon}+\frac{232}{15}+\epsilon\left(\frac{50944}{225}-7\pi^2\right)+\epsilon^2\left(\frac{10774648}{3375}-\frac{1624}{15}\pi^2-\frac{2284}{3}\zeta(3)\right)\right]+O(\epsilon^3), I 1 ( M ) = 61440 π 4 e 4 λ ϵ [ ϵ 1 + 15 232 + ϵ ( 225 50944 − 7 π 2 ) + ϵ 2 ( 3375 10774648 − 15 1624 π 2 − 3 2284 ζ ( 3 ) ) ] + O ( ϵ 3 ) , I 3 ( M ) = − e 4 λ ϵ 1720320 π 4 [ 1 ϵ + 182 15 + ϵ ( 116782 675 − 7 π 2 ) + ϵ 2 ( 73302157 30375 − 1274 15 π 2 − 2284 3 ζ ( 3 ) ) ] + O ( ϵ 3 ) . (84) I_3^{(M)}=-\frac{e^{4\lambda\epsilon}}{1720320\pi^4}\left[\frac{1}{\epsilon}+\frac{182}{15}+\epsilon\left(\frac{116782}{675}-7\pi^2\right)+\epsilon^2\left(\frac{73302157}{30375}-\frac{1274}{15}\pi^2-\frac{2284}{3}\zeta(3)\right)\right]+O(\epsilon^3). \tag*{(84)} I 3 ( M ) = − 1720320 π 4 e 4 λ ϵ [ ϵ 1 + 15 182 + ϵ ( 675 116782 − 7 π 2 ) + ϵ 2 ( 30375 73302157 − 15 1274 π 2 − 3 2284 ζ ( 3 ) ) ] + O ( ϵ 3 ) . (84) From Eq. (65) we obtain the expansion of I 2 ( M ) I_2^{(M)} I 2 ( M )
I 2 ( M ) = − 5 24576 π 4 [ 1 ϵ 2 + 4 λ ϵ + 8 λ 2 − 3 5 π 2 ] + O ( ϵ ) . (85) I_2^{(M)}=-\frac{5}{24576\pi^4}\left[\frac{1}{\epsilon^2}+\frac{4\lambda}{\epsilon}+8\lambda^2-\frac{3}{5}\pi^2\right]+O(\epsilon). \tag*{(85)} I 2 ( M ) = − 24576 π 4 5 [ ϵ 2 1 + ϵ 4 λ + 8 λ 2 − 5 3 π 2 ] + O ( ϵ ) . (85) With Feynman propagators on the two memory gravitons we find from Eq. (65) that the right-hand side of Eq. (68) equals e ( 4 λ + 2 i π ) ϵ [ 1 + 1 12 π 2 ϵ 2 + O ( ϵ 3 ) ] / ( 6144 π 6 ϵ 4 ) e^{(4\lambda+2i\pi)\epsilon}[1+\frac{1}{12}\pi^2\epsilon^2+O(\epsilon^3)]/(6144\pi^6\epsilon^4) e ( 4 λ + 2 iπ ) ϵ [ 1 + 12 1 π 2 ϵ 2 + O ( ϵ 3 )] / ( 6144 π 6 ϵ 4 ) 1 / ( 6144 π 6 ) 1/(6144\pi^6) 1/ ( 6144 π 6 )
I 1 , F ( M ) = − 1 184320 π 6 [ π 2 + 30 ϵ + ( 4 λ + 2 i π ) ( π 2 + 30 ) + 232 15 π 2 + 518 − 24 ζ ( 3 ) ] + O ( ϵ ) , I_{1,F}^{(M)}=-\frac{1}{184320\pi^6}\left[\frac{\pi^2+30}{\epsilon}+(4\lambda+2i\pi)(\pi^2+30)+\frac{232}{15}\pi^2+518-24\zeta(3)\right]+O(\epsilon), I 1 , F ( M ) = − 184320 π 6 1 [ ϵ π 2 + 30 + ( 4 λ + 2 iπ ) ( π 2 + 30 ) + 15 232 π 2 + 518 − 24 ζ ( 3 ) ] + O ( ϵ ) , I 3 , F ( M ) = 1 10321920 π 6 [ 2 π 2 + 35 ϵ + ( 4 λ + 2 i π ) ( 2 π 2 + 35 ) + 364 15 π 2 + 8543 18 − 48 ζ ( 3 ) ] + O ( ϵ ) . (86) I_{3,F}^{(M)}=\frac{1}{10321920\pi^6}\left[\frac{2\pi^2+35}{\epsilon}+(4\lambda+2i\pi)(2\pi^2+35)+\frac{364}{15}\pi^2+\frac{8543}{18}-48\zeta(3)\right]+O(\epsilon). \tag*{(86)} I 3 , F ( M ) = 10321920 π 6 1 [ ϵ 2 π 2 + 35 + ( 4 λ + 2 iπ ) ( 2 π 2 + 35 ) + 15 364 π 2 + 18 8543 − 48 ζ ( 3 ) ] + O ( ϵ ) . (86) The ancillary file cM_laurent.json contains the polynomial B 2 B_2 B 2 a k a_k a k c k c_k c k ϵ 3 \epsilon^3 ϵ 3 I 1 ( M ) I_1^{(M)} I 1 ( M ) I 3 ( M ) I_3^{(M)} I 3 ( M ) ϵ 1 \epsilon^1 ϵ 1 ϵ 2 \epsilon^2 ϵ 2 ϵ − 1 \epsilon^{-1} ϵ − 1 ϵ 0 \epsilon^0 ϵ 0 I 2 , F ( M ) I_{2,F}^{(M)} I 2 , F ( M ) ϵ \epsilon ϵ tables/memory-cores.txt.
Proofs and further analysis for the Apéry sector
We prove here that the torus period of topology 40 satisfies the Apéry equation and evaluate the scalar integral of the topology in position space. We also obtain the alphabet of topology 41 from its leading singularities.
Two proofs of the torus-period identity
The first proof that the torus period (15) satisfies L A ϖ = 0 L_A\varpi=0 L A ϖ = 0 L A L_A L A (6) , is a Griffiths–Dwork reduction of the integrand at symbolic x x x ξ 1 = t 1 / t 2 \xi_1=t_1/t_2 ξ 1 = t 1 / t 2 ξ 2 = t 2 \xi_2=t_2 ξ 2 = t 2 § 3.1 we have Q 6 = t 1 2 t 2 2 R Q_6=t_1^2t_2^2R Q 6 = t 1 2 t 2 2 R R = ( x w − 1 − x 2 ) 2 − 4 x 2 R=(xw-1-x^2)^2-4x^2 R = ( x w − 1 − x 2 ) 2 − 4 x 2 ϖ \varpi ϖ ξ 1 , ξ 2 \xi_1,\xi_2 ξ 1 , ξ 2 R − 1 / 2 R^{-1/2} R − 1/2 x x x δ i = ξ i ∂ ξ i \delta_i=\xi_i\partial_{\xi_i} δ i = ξ i ∂ ξ i θ = ( x / 2 ) d / d x \theta=(x/2)d/dx θ = ( x /2 ) d / d x L A L_A L A
L A R − 1 / 2 = δ 1 ( U 1 R − 5 / 2 ) + δ 2 ( U 2 R − 5 / 2 ) , U 1 = 1 − x 2 8 ∑ i ≥ 1 , j ≥ 0 i + j ≤ 4 x i + j v i j ( ξ 1 i − ξ 1 − i ) e j ( ξ 2 ) , (87) L_A R^{-1/2}=\delta_1(U_1R^{-5/2})+\delta_2(U_2R^{-5/2}),\qquad U_1=\frac{1-x^2}{8}\sum_{\substack{i\geq1,\ j\geq0\\i+j\leq4}}x^{i+j}v_{ij}(\xi_1^i-\xi_1^{-i})e_j(\xi_2), \tag*{(87)} L A R − 1/2 = δ 1 ( U 1 R − 5/2 ) + δ 2 ( U 2 R − 5/2 ) , U 1 = 8 1 − x 2 i ≥ 1 , j ≥ 0 i + j ≤ 4 ∑ x i + j v ij ( ξ 1 i − ξ 1 − i ) e j ( ξ 2 ) , (87) where U 2 ( ξ 1 , ξ 2 ) = U 1 ( ξ 2 , ξ 1 ) U_2(\xi_1,\xi_2)=U_1(\xi_2,\xi_1) U 2 ( ξ 1 , ξ 2 ) = U 1 ( ξ 2 , ξ 1 ) e 0 = 1 e_0=1 e 0 = 1 e j ( ξ ) = ξ j + ξ − j e_j(\xi)=\xi^j+\xi^{-j} e j ( ξ ) = ξ j + ξ − j j ≥ 1 j\geq1 j ≥ 1
v 10 = x 8 − 7 x 6 − 280 x 4 − 7 x 2 + 1 , v 11 = ( x 2 + 1 ) ( 3 x 4 + 128 x 2 + 3 ) , v 12 = − 3 ( 3 x 4 + 38 x 2 + 3 ) , v 13 = 5 ( x 2 + 1 ) , v 20 = − ( x 2 + 1 ) ( x 4 − 42 x 2 + 1 ) , v 21 = − 10 ( x 4 + 14 x 2 + 1 ) , v 22 = 11 ( x 2 + 1 ) , v 30 = − x 4 − 26 x 2 − 1 , v 31 = 7 ( x 2 + 1 ) , v 40 = x 2 + 1. (88) \begin{aligned}
v_{10}&=x^8-7x^6-280x^4-7x^2+1, & v_{11}&=(x^2+1)(3x^4+128x^2+3),\\
v_{12}&=-3(3x^4+38x^2+3), & v_{13}&=5(x^2+1),\\
v_{20}&=-(x^2+1)(x^4-42x^2+1), & v_{21}&=-10(x^4+14x^2+1),\\
v_{22}&=11(x^2+1), & v_{30}&=-x^4-26x^2-1,\\
v_{31}&=7(x^2+1), & v_{40}&=x^2+1. \tag*{(88)}
\end{aligned} v 10 v 12 v 20 v 22 v 31 = x 8 − 7 x 6 − 280 x 4 − 7 x 2 + 1 , = − 3 ( 3 x 4 + 38 x 2 + 3 ) , = − ( x 2 + 1 ) ( x 4 − 42 x 2 + 1 ) , = 11 ( x 2 + 1 ) , = 7 ( x 2 + 1 ) , v 11 v 13 v 21 v 30 v 40 = ( x 2 + 1 ) ( 3 x 4 + 128 x 2 + 3 ) , = 5 ( x 2 + 1 ) , = − 10 ( x 4 + 14 x 2 + 1 ) , = − x 4 − 26 x 2 − 1 , = x 2 + 1. (88) Multiplied by R 7 / 2 R^{7/2} R 7/2 ξ 1 , ξ 2 \xi_1,\xi_2 ξ 1 , ξ 2 v i j v_{ij} v ij ( 1 − x 2 ) x i + j / 8 (1-x^2)x^{i+j}/8 ( 1 − x 2 ) x i + j /8 apery-proof/gd-certificate.json. Since the constant term of δ i G \delta_iG δ i G G G G ξ 1 , ξ 2 \xi_1,\xi_2 ξ 1 , ξ 2 L A ϖ = 0 L_A\varpi=0 L A ϖ = 0
For the second proof we use Eq. (18) and the recurrence of the return numbers B N B_N B N [59 ] , which we rederived with Zeilberger’s algorithm (ancillary file apery-proof/fermi-certificate.json). We write the return numbers as B N = ∑ j F ( N , j ) B_N=\sum_jF(N,j) B N = ∑ j F ( N , j ) F ( N , j ) = ( 2 N N ) ( 2 j j ) ( 2 j j ) F(N,j)=\binom{2N}{N}\binom{2j}{j}\binom{2j}{j} F ( N , j ) = ( N 2 N ) ( j 2 j ) ( j 2 j )
( N + 2 ) 3 F ( N + 2 , j ) − 2 ( 2 N + 3 ) ( 10 N 2 + 30 N + 23 ) F ( N + 1 , j ) + 36 ( N + 1 ) ( 2 N + 1 ) ( 2 N + 3 ) F ( N , j ) = G ( N , j + 1 ) − G ( N , j ) , G ( N , j ) = 4 j 3 ( N + 1 ) ( 2 N + 1 ) ( 2 N + 3 ) ( − 4 N + 3 j − 8 ) H ( N , j ) , H ( N , j ) = ( 2 N N ) ( 2 j j ) N ! 2 j ! 2 ( N + 2 − j ) ! 2 . (89) \begin{aligned}
(N+2)^3F(N+2,j)&-2(2N+3)(10N^2+30N+23)F(N+1,j)\\
&+36(N+1)(2N+1)(2N+3)F(N,j)=G(N,j+1)-G(N,j),\\
G(N,j)&=4j^3(N+1)(2N+1)(2N+3)(-4N+3j-8)H(N,j),\\
H(N,j)&=\binom{2N}{N}\binom{2j}{j}\frac{N!^2}{j!^2(N+2-j)!^2}. \tag*{(89)}
\end{aligned} ( N + 2 ) 3 F ( N + 2 , j ) G ( N , j ) H ( N , j ) − 2 ( 2 N + 3 ) ( 10 N 2 + 30 N + 23 ) F ( N + 1 , j ) + 36 ( N + 1 ) ( 2 N + 1 ) ( 2 N + 3 ) F ( N , j ) = G ( N , j + 1 ) − G ( N , j ) , = 4 j 3 ( N + 1 ) ( 2 N + 1 ) ( 2 N + 3 ) ( − 4 N + 3 j − 8 ) H ( N , j ) , = ( N 2 N ) ( j 2 j ) j ! 2 ( N + 2 − j ) ! 2 N ! 2 . (89) where 1 / ( N + 2 − j ) ! = 0 1/(N+2-j)!=0 1/ ( N + 2 − j )! = 0 j > N + 2 j>N+2 j > N + 2 G ( N , j ) = 0 G(N,j)=0 G ( N , j ) = 0 j ≥ N + 3 j\geq N+3 j ≥ N + 3 G ( N , 0 ) = 0 G(N,0)=0 G ( N , 0 ) = 0 0 ≤ j ≤ N + 2 0\leq j\leq N+2 0 ≤ j ≤ N + 2 § 3 of Peters and Stienstra,
N 3 B N − 2 ( 2 N − 1 ) ( 10 N 2 − 10 N + 3 ) B N − 1 + 36 ( N − 1 ) ( 2 N − 3 ) ( 2 N − 1 ) B N − 2 = 0. (90) N^3B_N-2(2N-1)(10N^2-10N+3)B_{N-1}+36(N-1)(2N-3)(2N-1)B_{N-2}=0. \tag*{(90)} N 3 B N − 2 ( 2 N − 1 ) ( 10 N 2 − 10 N + 3 ) B N − 1 + 36 ( N − 1 ) ( 2 N − 3 ) ( 2 N − 1 ) B N − 2 = 0. (90) Hence the series f ( ζ ) = ∑ N B N ζ N f(\zeta)=\sum_NB_N\zeta^N f ( ζ ) = ∑ N B N ζ N
L F = θ ζ 3 − 2 ζ ( 2 θ ζ + 1 ) ( 10 θ ζ 2 + 10 θ ζ + 3 ) + 36 ζ 2 ( θ ζ + 1 ) ( 2 θ ζ + 1 ) ( 2 θ ζ + 3 ) , θ ζ = ζ d d ζ . (91) L_F=\theta_\zeta^3-2\zeta(2\theta_\zeta+1)(10\theta_\zeta^2+10\theta_\zeta+3)+36\zeta^2(\theta_\zeta+1)(2\theta_\zeta+1)(2\theta_\zeta+3),\qquad\theta_\zeta=\zeta\frac{d}{d\zeta}. \tag*{(91)} L F = θ ζ 3 − 2 ζ ( 2 θ ζ + 1 ) ( 10 θ ζ 2 + 10 θ ζ + 3 ) + 36 ζ 2 ( θ ζ + 1 ) ( 2 θ ζ + 1 ) ( 2 θ ζ + 3 ) , θ ζ = ζ d ζ d . (91) For ζ = z / ( 1 + z ) 2 \zeta= z/(1+z)^2 ζ = z / ( 1 + z ) 2 L A L_A L A L F L_F L F L A [ ( 1 + z ) − 1 f ( ζ ) ] = ( 1 − z ) ( L F f ) ( ζ ) L_A\left[(1+z)^{-1}f(\zeta)\right] = (1-z)(L_Ff)(\zeta) L A [ ( 1 + z ) − 1 f ( ζ ) ] = ( 1 − z ) ( L F f ) ( ζ ) f f f L A ϖ = 0 L_A\varpi= 0 L A ϖ = 0 (18) .
The scalar integral in position space
We consider the scalar integral of topology 40 with unit numerator and cut worldline propagators, in four dimensions and in impact-parameter space, in units ∣ b ∣ = 1 |b| = 1 ∣ b ∣ = 1 τ \tau τ τ ′ \tau' τ ′ Δ 2 \Delta_2 Δ 2
f s a m e ( γ ) = 1 2 ∫ Δ 2 < 0 d τ d τ ′ G , f m i x ( γ ) = 1 2 ∫ Δ 2 > 0 d τ d τ ′ G (92) f_{\mathrm{same}}(\gamma) = \frac{1}{2}\int_{\Delta_2<0} \mathrm{d}\tau\,\mathrm{d}\tau'\,\mathcal{G}, \qquad f_{\mathrm{mix}}(\gamma) = \frac{1}{2}\int_{\Delta_2>0} \mathrm{d}\tau\,\mathrm{d}\tau'\,\mathcal{G} \tag*{(92)} f same ( γ ) = 2 1 ∫ Δ 2 < 0 d τ d τ ′ G , f mix ( γ ) = 2 1 ∫ Δ 2 > 0 d τ d τ ′ G (92) with
G ( τ , τ ′ ) = 2 π 1 + ( γ 2 − 1 ) τ 2 1 + ( γ 2 − 1 ) τ ′ 2 1 P 2 − Δ 2 2 ln P + P 2 − Δ 2 2 ∣ Δ 2 ∣ , \mathcal{G}(\tau,\tau') = \frac{2\pi}{\sqrt{1+(\gamma^2-1)\tau^2}\sqrt{1+(\gamma^2-1)\tau'^2}}\frac{1}{\sqrt{P^2-\Delta_2^2}}\ln\frac{P+\sqrt{P^2-\Delta_2^2}}{|\Delta_2|}, G ( τ , τ ′ ) = 1 + ( γ 2 − 1 ) τ 2 1 + ( γ 2 − 1 ) τ ′2 2 π P 2 − Δ 2 2 1 ln ∣ Δ 2 ∣ P + P 2 − Δ 2 2 , Δ 2 = τ 2 + τ ′ 2 − 2 γ τ τ ′ − 1 , P = 2 ( τ − γ τ ′ ) ( γ τ − τ ′ ) − γ Δ 2 , \Delta_2 = \tau^2+\tau'^2-2\gamma\tau\tau'-1,\qquad P=2(\tau-\gamma\tau')(\gamma\tau-\tau')-\gamma\Delta_2, Δ 2 = τ 2 + τ ′2 − 2 γ τ τ ′ − 1 , P = 2 ( τ − γ τ ′ ) ( γ τ − τ ′ ) − γ Δ 2 , P 2 − Δ 2 2 = [ ( γ − 1 ) ( τ + τ ′ ) 2 + γ + 1 ] [ ( γ + 1 ) ( τ − τ ′ ) 2 + γ − 1 ] . (93) P^2-\Delta_2^2=\left[(\gamma-1)(\tau+\tau')^2+\gamma+1\right]\left[(\gamma+1)(\tau-\tau')^2+\gamma-1\right]. \tag*{(93)} P 2 − Δ 2 2 = [ ( γ − 1 ) ( τ + τ ′ ) 2 + γ + 1 ] [ ( γ + 1 ) ( τ − τ ′ ) 2 + γ − 1 ] . (93) The scalar integral is proportional to f s a m e f_{\mathrm{same}} f same f m i x f_{\mathrm{mix}} f mix § 6.4 ). Both f s a m e f_{\mathrm{same}} f same f m i x f_{\mathrm{mix}} f mix γ > 1 \gamma> 1 γ > 1
To compare with the result of Driesse et al. [15 ] , let ϖ ^ ( x ) \hat{\varpi}(x) ϖ ^ ( x ) x = γ − γ 2 − 1 x=\gamma-\sqrt{\gamma^2-1} x = γ − γ 2 − 1 ( 1 + x 2 ) ϖ ^ = 1 + 15 32 v 2 + O ( v 4 ) (1+x^2)\hat{\varpi}=1+\frac{15}{32}v^2+O(v^4) ( 1 + x 2 ) ϖ ^ = 1 + 32 15 v 2 + O ( v 4 ) v = γ 2 − 1 / γ v=\sqrt{\gamma^2-1}/\gamma v = γ 2 − 1 / γ I [ … ; x ] \mathcal{I}[\ldots;x] I [ … ; x ] x = 1 x=1 x = 1 ϕ = x ϖ ^ \phi=x\hat{\varpi} ϕ = x ϖ ^ K 1 = ( 1 + x 2 ) ϖ ^ / x \mathcal{K}_1=(1+x^2)\hat{\varpi}/x K 1 = ( 1 + x 2 ) ϖ ^ / x K 2 = 1 / ( x r ϖ ^ ) \mathcal{K}_2=1/(xr\hat{\varpi}) K 2 = 1/ ( x r ϖ ^ ) r = − 1 + 34 x 2 − x 4 r=\sqrt{-1+34x^2-x^4} r = − 1 + 34 x 2 − x 4
F 0 p o t = 2 π 2 ϕ I [ K 2 ; x ] , F 0 m e m = − 16 ϕ I [ K 2 , K 2 , K 1 ; x ] . (94) F^{\mathrm{pot}}_0=\sqrt{2}\,\pi^2\phi\,\mathcal{I}[\mathcal{K}_2;x],\qquad F^{\mathrm{mem}}_0=-16\phi\,\mathcal{I}[\mathcal{K}_2,\mathcal{K}_2,\mathcal{K}_1;x]. \tag*{(94)} F 0 pot = 2 π 2 ϕ I [ K 2 ; x ] , F 0 mem = − 16 ϕ I [ K 2 , K 2 , K 1 ; x ] . (94) The sum of these two functions is f 20 ( O ) / ( 2 γ ) f^{(O)}_{20}/(2\gamma) f 20 ( O ) / ( 2 γ ) F 0 p o t F^{\mathrm{pot}}_0 F 0 pot F 0 m e m F^{\mathrm{mem}}_0 F 0 mem ( 3 − γ ) 1 / 2 (3-\gamma)^{1/2} ( 3 − γ ) 1/2 F 0 p o t F^{\mathrm{pot}}_0 F 0 pot F 0 m e m F^{\mathrm{mem}}_0 F 0 mem + 2 π 2 / 4 +\sqrt{2}\,\pi^2/4 + 2 π 2 /4 − 2 π 2 / 4 -\sqrt{2}\,\pi^2/4 − 2 π 2 /4 ( 3 − γ ) 1 / 2 (3-\gamma)^{1/2} ( 3 − γ ) 1/2 ( 3 − γ ) 199 / 2 (3-\gamma)^{199/2} ( 3 − γ ) 199/2
( γ 2 − 1 ) f s a m e = − 2 π ( F 0 p o t + F 0 m e m ) . (95) (\gamma^2-1)f_{\mathrm{same}}=-2\pi\left(F^{\mathrm{pot}}_0+F^{\mathrm{mem}}_0\right). \tag*{(95)} ( γ 2 − 1 ) f same = − 2 π ( F 0 pot + F 0 mem ) . (95) We also apply to f s a m e f_{\mathrm{same}} f same f m i x f_{\mathrm{mix}} f mix M = ( γ 2 − 9 ) ( γ 2 − 1 ) ∂ γ 3 + 6 γ ( γ 2 − 5 ) ∂ γ 2 + ( 7 γ 2 − 12 ) ∂ γ + γ \mathcal{M}=(\gamma^2-9)(\gamma^2-1)\partial_\gamma^3+6\gamma(\gamma^2-5)\partial_\gamma^2+(7\gamma^2-12)\partial_\gamma+\gamma M = ( γ 2 − 9 ) ( γ 2 − 1 ) ∂ γ 3 + 6 γ ( γ 2 − 5 ) ∂ γ 2 + ( 7 γ 2 − 12 ) ∂ γ + γ ϕ \phi ϕ
M [ ( γ 2 − 1 ) f s a m e ] = 16 π γ γ 2 − 1 , M [ ( γ 2 − 1 ) f m i x ] = − 16 π γ γ 2 − 1 + 20 π + 32 π γ 2 − 1 + 32 π γ ln x ( γ 2 − 1 ) 3 / 2 . (96) \begin{aligned}
\mathcal{M}\left[(\gamma^2-1)f_{\mathrm{same}}\right]&=\frac{16\pi\gamma}{\sqrt{\gamma^2-1}},\\
\mathcal{M}\left[(\gamma^2-1)f_{\mathrm{mix}}\right]&=-\frac{16\pi\gamma}{\sqrt{\gamma^2-1}}+20\pi+\frac{32\pi}{\gamma^2-1}+\frac{32\pi\gamma\ln x}{(\gamma^2-1)^{3/2}}. \tag*{(96)}
\end{aligned} M [ ( γ 2 − 1 ) f same ] M [ ( γ 2 − 1 ) f mix ] = γ 2 − 1 16 π γ , = − γ 2 − 1 16 π γ + 20 π + γ 2 − 1 32 π + ( γ 2 − 1 ) 3/2 32 π γ ln x . (96) to at least 27 digits on 1.6 ≤ γ ≤ 4.2 1.6\leq\gamma\leq4.2 1.6 ≤ γ ≤ 4.2
At γ = 3 \gamma= 3 γ = 3 f s a m e = 2.75532463456 f_{\mathrm{same}} = 2.75532463456 f same = 2.75532463456 f m i x = 1.52462878840 f_{\mathrm{mix}} = 1.52462878840 f mix = 1.52462878840 γ = 3 \gamma= 3 γ = 3 ( 3 − γ ) 1 / 2 (3-\gamma)^{1/2} ( 3 − γ ) 1/2 γ = 3 \gamma= 3 γ = 3
To show analyticity we write γ = cosh η \gamma= \cosh\eta γ = cosh η τ = sinh a / sinh η \tau= \sinh a/\sinh\eta τ = sinh a / sinh η τ ′ = sinh a ′ / sinh η \tau' = \sinh a'/\sinh\eta τ ′ = sinh a ′ / sinh η u = a − a ′ u = a-a' u = a − a ′ s = ( a + a ′ ) / 2 s = (a+a')/2 s = ( a + a ′ ) /2 C = cosh 2 s C = \cosh2s C = cosh 2 s c = cosh u c = \cosh u c = cosh u
Δ 2 sinh 2 η = ( C + γ ) ( c − γ ) , \Delta_2 \sinh^2\eta= (C+\gamma)(c-\gamma), Δ 2 sinh 2 η = ( C + γ ) ( c − γ ) , so that Δ 2 < 0 \Delta_2 < 0 Δ 2 < 0 ∣ u ∣ < η |u| < \eta ∣ u ∣ < η P = γ ( τ 2 + τ ′ 2 ) − 2 τ τ ′ + γ > 0 P = \gamma(\tau^2+\tau'^2)-2\tau\tau'+\gamma> 0 P = γ ( τ 2 + τ ′2 ) − 2 τ τ ′ + γ > 0 arcsinh ( P 2 − Δ 2 2 / ∣ Δ 2 ∣ ) \operatorname{arcsinh}(\sqrt{P^2-\Delta_2^2}/|\Delta_2|) arcsinh ( P 2 − Δ 2 2 /∣ Δ 2 ∣ )
f s a m e = 4 π sinh η ∫ 0 ∞ d s ∫ 0 η d u arcsinh X A B , f m i x = 4 π sinh η ∫ 0 ∞ d s ∫ η ∞ d u arcsinh X A B , (97) f_{\mathrm{same}} = \frac{4\pi}{\sinh\eta}\int_0^\infty ds\int_0^\eta du\,\frac{\operatorname{arcsinh}X}{\sqrt{AB}},\qquad
f_{\mathrm{mix}} = \frac{4\pi}{\sinh\eta}\int_0^\infty ds\int_\eta^\infty du\,\frac{\operatorname{arcsinh}X}{\sqrt{AB}}, \tag*{(97)} f same = sinh η 4 π ∫ 0 ∞ d s ∫ 0 η d u A B arcsinh X , f mix = sinh η 4 π ∫ 0 ∞ d s ∫ η ∞ d u A B arcsinh X , (97) with
A = ( C − 1 ) ( c + 1 ) + ( γ + 1 ) 2 , B = ( C + 1 ) ( c − 1 ) + ( γ − 1 ) 2 , X = sinh η A B ( C + γ ) ∣ c − γ ∣ . (98) A = (C-1)(c+1)+(\gamma+1)^2,\qquad B = (C+1)(c-1)+(\gamma-1)^2,\qquad X = \frac{\sinh\eta\sqrt{AB}}{(C+\gamma)|c-\gamma|}. \tag*{(98)} A = ( C − 1 ) ( c + 1 ) + ( γ + 1 ) 2 , B = ( C + 1 ) ( c − 1 ) + ( γ − 1 ) 2 , X = ( C + γ ) ∣ c − γ ∣ sinh η A B . (98) We set u = η t u = \eta t u = η t 0 < t < 1 0 < t < 1 0 < t < 1 f s a m e f_{\mathrm{same}} f same u = η + t u = \eta+t u = η + t t > 0 t > 0 t > 0 f m i x f_{\mathrm{mix}} f mix γ \gamma γ ∣ c − γ ∣ |c-\gamma| ∣ c − γ ∣ η \eta η γ − c \gamma-c γ − c c − γ c-\gamma c − γ η \eta η arccosh 3 \operatorname{arccosh}3 arccosh 3 A A A B B B X X X t t t s s s C ± 1 ≥ 0 C\pm1\geq0 C ± 1 ≥ 0 γ ± 1 \gamma\pm1 γ ± 1 C + γ C+\gamma C + γ c + 1 = 2 cosh 2 ( u / 2 ) c+1=2\cosh^2(u/2) c + 1 = 2 cosh 2 ( u /2 ) c − 1 = 2 sinh 2 ( u / 2 ) c-1=2\sinh^2(u/2) c − 1 = 2 sinh 2 ( u /2 ) ± ( c − γ ) = ± 2 sinh u + η 2 sinh u − η 2 \pm(c-\gamma)=\pm2\sinh\frac{u+\eta}{2}\sinh\frac{u-\eta}{2} ± ( c − γ ) = ± 2 sinh 2 u + η sinh 2 u − η O ( ∣ Im η ∣ ) O(|\operatorname{Im}\eta|) O ( ∣ Im η ∣ ) t t t η \eta η η \eta η e − s ( 1 − t ) − 1 / 2 e^{-s}(1-t)^{-1/2} e − s ( 1 − t ) − 1/2 e − s − t / 4 t − 1 / 2 e^{-s-t/4}t^{-1/2} e − s − t /4 t − 1/2 f s a m e f_{\mathrm{same}} f same f m i x f_{\mathrm{mix}} f mix γ = 3 \gamma= 3 γ = 3 ∣ γ − 3 ∣ = 1 2 |\gamma-3|=\frac{1}{2} ∣ γ − 3∣ = 2 1 f s a m e ( 3 ) f_{\mathrm{same}}(3) f same ( 3 ) f m i x ( 3 ) f_{\mathrm{mix}}(3) f mix ( 3 )
For a second check we use Eq. (D.10). The operator M \mathcal{M} M 0 0 0 1 2 \frac{1}{2} 2 1 1 1 1 γ = 3 \gamma= 3 γ = 3 γ < 3 \gamma< 3 γ < 3 f f f f s a m e f_{\mathrm{same}} f same f m i x f_{\mathrm{mix}} f mix ( γ 2 − 1 ) f = α ( γ ) + 8 c 1 ( 3 − γ ) 1 / 2 h ( γ ) (\gamma^2-1)f=\alpha(\gamma)+8c_1(3-\gamma)^{1/2}h(\gamma) ( γ 2 − 1 ) f = α ( γ ) + 8 c 1 ( 3 − γ ) 1/2 h ( γ ) α \alpha α γ = 3 \gamma=3 γ = 3 c 1 c_1 c 1 ( 3 − γ ) 1 / 2 (3-\gamma)^{1/2} ( 3 − γ ) 1/2 f f f h = 1 + 1 4 ( 3 − γ ) + 11 160 ( 3 − γ ) 2 + … h=1+\frac{1}{4}(3-\gamma)+\frac{11}{160}(3-\gamma)^2+\ldots h = 1 + 4 1 ( 3 − γ ) + 160 11 ( 3 − γ ) 2 + … M \mathcal{M} M f s a m e f_{\mathrm{same}} f same f m i x f_{\mathrm{mix}} f mix 2.4 < γ < 3 2.4<\gamma<3 2.4 < γ < 3 ∣ c 1 ∣ / f ( 3 ) < 10 − 74 |c_1|/f(3)<10^{-74} ∣ c 1 ∣/ f ( 3 ) < 1 0 − 74 f s a m e f_{\mathrm{same}} f same ( 3 − γ ) 1 / 2 (3-\gamma)^{1/2} ( 3 − γ ) 1/2 F 0 p o t F^{\mathrm{pot}}_0 F 0 pot F 0 m e m F^{\mathrm{mem}}_0 F 0 mem
The topology-41 alphabet
Topology 41 of Brammer et al. [26 ] is the thirteen-propagator top topology of the non-planar family. From the d log d\log d log [25 ] , and from the factorization over Q [ x ] \mathbb{Q}[x] Q [ x ]
A t o p .41 = { x , x − 1 , x + 1 , x 2 + 1 } . (99) \mathcal{A}_{\mathrm{top.41}}=\{x,\ x-1,\ x+1,\ x^2+1\}. \tag*{(99)} A top.41 = { x , x − 1 , x + 1 , x 2 + 1 } . (99) This alphabet coincides with the multiple-polylogarithm alphabet of the 5PM-1SF result [57 ] and is a subset of the kernels of the 5PM-2SF observables [15 ] . The latter x 2 + 1 x^2+1 x 2 + 1 u 1 ⋅ u 2 = 0 u_1 \cdot u_2=0 u 1 ⋅ u 2 = 0 L S e v e n ( 6 D ) = q 2 ( x 2 + 1 ) / ( x 2 − 1 ) \mathrm{LS}_{\mathrm{even}}^{(6D)}=q^2(x^2+1)/(x^2-1) LS even ( 6 D ) = q 2 ( x 2 + 1 ) / ( x 2 − 1 ) x 2 ± x + 1 x^2 \pm x+1 x 2 ± x + 1
Beyond the maximal cut at 4PM
For the three-loop K3 sector of the conservative 4PM dynamics [46 , 48 , 50 , 51 ] , we expand the solutions of the maximal-cut block of the ϵ \epsilon ϵ ϵ 2 \epsilon^2 ϵ 2 § 3 with two auxiliary-mass flows [33 , 82 ] . However, with these flows we do not determine the individual master integrals as functions of ϵ \epsilon ϵ
Sector labels and maximal-cut values
In the notation of Ref. [51 ] the sector whose maximal cut we reduce to Eq. (21) is M 000 ; 010101110 ( 112 ; + + + + ) M^{(112;++++)}_{000;010101110} M 000 ; 010101110 ( 112 ; ++++ ) G 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 0 , 0 , 1 , 1 G_{1,1,1,0,0,0,0,0,1,1,1,0,0,1,1} G 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 0 , 0 , 1 , 1 [50 ] under k 2 ↔ k 3 k_2 \leftrightarrow k_3 k 2 ↔ k 3 u 1 ↔ u 2 u_1 \leftrightarrow u_2 u 1 ↔ u 2 L S s h a p e ( x ) \mathrm{LS}_{\mathrm{shape}}(x) LS shape ( x ) x K ( x 2 ) 2 x K(x^2)^2 x K ( x 2 ) 2 (22) are in the ancillary file tables/fourpm_maxcut_values.txt.
Function space on the maximal cut
The Landau locus (20) [29 ] gives the singular points of the sector, but not the functions in which its master integrals are expressed. The ϵ \epsilon ϵ π 2 / [ x ( 1 − x 2 ) K ′ 2 ] \pi^2/[x(1-x^2)K'^2] π 2 / [ x ( 1 − x 2 ) K ′2 ] K ′ 2 K'^2 K ′2 K ′ 4 K'^4 K ′4 K ′ ≡ K ( 1 − x 2 ) K' \equiv K(1-x^2) K ′ ≡ K ( 1 − x 2 ) ϵ \epsilon ϵ (23) with coefficients linear in log [ x / ( 1 − x 2 ) ] \log[x/(1-x^2)] log [ x / ( 1 − x 2 )] π 2 / [ x ( 1 − x 2 ) K ′ 2 ] \pi^2/[x(1-x^2)K'^2] π 2 / [ x ( 1 − x 2 ) K ′2 ] ( 1 + 110 x 2 + x 4 ) K ′ 2 / [ x ( 1 − x 2 ) ] (1+110x^2+x^4)K'^2/[x(1-x^2)] ( 1 + 110 x 2 + x 4 ) K ′2 / [ x ( 1 − x 2 )]
Comparing exact series expansions at x = 1 x=1 x = 1 (20) . Which combination of the three functions appears is fixed by boundary data. Determining it requires values of the master integrals, which we do not have, and to our knowledge none are published for this sector.
A fixed point of the flow
The 4PM cut-eikonal K3 family, with five graviton propagators and three cut linearized worldline propagators − 2 k ⋅ u -2k \cdot u − 2 k ⋅ u [12 , 13 , 51 ] , is a fixed point of the boundary recursion of the auxiliary-mass flow. Because cut propagators cannot be deformed, we insert the auxiliary mass in the same five graviton propagators at every level, and the η → ∞ \eta\to\infty η → ∞ b \boldsymbol{b} b (27) coincides with the vector M \mathbf{M} M
M = A ( ϵ ) M . (100) \mathbf{M}=A(\epsilon)\mathbf{M}. \tag*{(100)} M = A ( ϵ ) M . (100) Of the rows of A A A
rank ( A − I ) = 15 , dim ker ( A − I ) = 28. (101) \operatorname{rank}(A-I)=15,\qquad\dim\ker(A-I)=28. \tag*{(101)} rank ( A − I ) = 15 , dim ker ( A − I ) = 28. (101) The fixed-point condition determines 15 independent linear combinations of the 43 master integrals, and the 28 kernel directions are the single-scale cut-eikonal vacuum integrals in 1 + η 1+\eta 1 + η
We compute the rank at two points, γ = 5 / 3 \gamma=5/3 γ = 5/3 x = 1 / 3 x=1/3 x = 1/3 γ = 13 / 5 \gamma=13/5 γ = 13/5 x = 1 / 5 x=1/5 x = 1/5 A A A ϵ \epsilon ϵ ϵ \epsilon ϵ A 2 = A A^2=A A 2 = A tr A = 28 \operatorname{tr}A=28 tr A = 28 rank ( A − I ) = 43 − tr A = 15 \operatorname{rank}(A-I)=43-\operatorname{tr}A=15 rank ( A − I ) = 43 − tr A = 15 ϵ \epsilon ϵ
In the PM literature the master integrals of such cut-eikonal families are obtained from differential equations in x x x [14 , 49 , 51 ] , which we have not implemented for this family. Instead we constrain the kernel directions with a second flow, in which we insert η \eta η M = C ( ϵ ) v \mathbf{M}=C(\epsilon)\mathbf{v} M = C ( ϵ ) v v \mathbf{v} v M \mathbf{M} M
J 63 = − i 8 π − 3 / 2 Γ ( ϵ − 1 2 ) 3 , (102) J_{63}=-\frac{i}{8}\pi^{-3/2}\Gamma\left(\epsilon-\frac{1}{2}\right)^3, \tag*{(102)} J 63 = − 8 i π − 3/2 Γ ( ϵ − 2 1 ) 3 , (102) and the equality J 111 = J 63 J_{111}=J_{63} J 111 = J 63 v \mathbf{v} v ϵ \epsilon ϵ m 0 m_0 m 0 m 1 m_1 m 1
m 1 m 0 = − 2 5 , (103) \frac{m_1}{m_0}=-\frac{2}{5}, \tag*{(103)} m 0 m 1 = − 5 2 , (103) an integration-by-parts identity recovered numerically.
Master ratios at leading order in epsilon
At the sampled values of ϵ \epsilon ϵ m 0 , … , m 7 m_0,\ldots,m_7 m 0 , … , m 7 m r / m 0 m_r/m_0 m r / m 0 r = 1 , … , 7 r=1,\ldots,7 r = 1 , … , 7 ϵ = 0 \epsilon=0 ϵ = 0 p / q p/q p / q p / q p/q p / q 10 68 10^{68} 1 0 68 tables/fourpm_master_ratios.txt).
At γ = 13 / 5 \gamma=13/5 γ = 13/5 ϵ \epsilon ϵ γ = 5 / 3 \gamma=5/3 γ = 5/3 ϵ \epsilon ϵ 3 × 10 − 11 3\times10^{-11} 3 × 1 0 − 11 K 3 K3 K 3 (23) can first enter these ratios, nor derived the ratios from the differential equation at ϵ = 0 \epsilon=0 ϵ = 0
