Uniform transcendentality decomposition

−2s13s23

2Gb2E2+π∇₀E2

τ₂²

+2s³₁₂

_π∇₀E2

τ²₂ −2 3

π∇₀E3

τ₂² − (1+2E2)_Gb₂

^(6.70)

+2s12s₁₃s₂₃

(2E2+5E3+ζ3)Gb2+ π∇₀E2

τ₂² − 2 3

π∇₀E3

τ²₂

+O(α⁰⁴)

I^(4,0)

12|34 Gb²₂−2s12Gb²₂+s²₁₂

G4+Gb²₂(3+2E2) − 2_Gb_2π^∇_0E2 τ₂²

−2s13s₂₃

G4+Gb²₂E2+Gb2π∇₀E2

τ²₂

+s12s13s23

(4E2+5E3+ζ3)bG²₂−2G4 (6.71) +(π∇₀)²E3

τ₂⁴ + _Gb_2π^∇_0E2 τ²₂ − 4

3

Gb2π∇₀E3

τ₂²

+s³₁₂

−2G4−4bG²₂(1+E2) +(π∇₀)²E3

3τ₂⁴ + 4Gb2π∇₀E2

τ²₂ − 4 3

Gb2π∇₀E3

τ²₂

+O(α⁰⁴)

after applying the basis decompositions from Section5.7. Similar to the representations (6.58) and (6.59) of the planar integrals, we have used the substitutions (6.57). Together with the planar results and the expression (3.138) forI⁽0,0)

, (6.70) and (6.71) complete the ingredients for theτintegrand (6.24) of the heterotic-string amplitude.

6.2.3 Uniform transcendentality decomposition

As mentioned in the introduction to this chapter, one of the remarkable features of type-II amplitudes is that they exhibit so-called uniform transcendentality at each order inα⁰. In this section, we will study the transcendentality properties of the heterotic string by restricting to the salient points that require a rewriting of the basis integralsI^(a,0)

... ; additional details can be found in AppendixC.

In analogy with the superstring, we associate transcendental weights to the various objects appearing in the low-energy expansion of the heterotic integrals over the punctures as follows. The Eisenstein series Gk and Ek as well as ζk are assigned transcendental weight k, i.e. π has transcendental weight one, whereasτand∇₀have transcendental

weight zero. Accordingly, one finds transcendental weight one for both π∇₀ and y πτ2, i.e. weight k+p for (π∇₀)^pEk, and weight 4+p for (π∇₀)^pE_2,2. A more general definition of transcendental weight in terms of iterated integrals is given in AppendixC, but the assignment above suffices for the discussion of this section.

Inspecting the α⁰ expansions of the planar single-trace integrals I⁽0,0)

and I^(2,0)

1234 of (3.138) and (6.58), one sees that their k^th order consistently involves MGFsof weight k and k+2, respectively. Thus, these two integrals are referred to as uniformly transcendental.

By contrast,I^(4,0)

1234 in (6.59) violates uniform transcendentality since the same type of transcendental object appears at different orders in the α⁰expansion. For instance, G4of transcendentality four appears with 1 +6s13+. . . and thus at different orders in α⁰. Similarly, the integralsI^(a,0)

12|34in (6.70) and (6.71) from the double-trace sector violate uniform transcendentality. This can for instance be seen from the terms

∼ (1+2E2)along withs²₁₂inI^(2,0)

12|34and∼ (3+2E2)along withs₁₂² inI^(4,0)

12|34, respectively.

This violation of uniform transcendentality can be traced back to the following phenomenon. For the planar integralI^(4,0)

1234 we see from (6.38) that there is a leading contribution with a closed cycle of the form f₁₂⁽¹⁾f₂₃⁽¹⁾f₃₄⁽¹⁾f₄₁⁽¹⁾. This cycle exhibits purely holomorphic modular weight (4,0)and is thus amenable toHSRas discussed in Section5.4. However, the formula (5.73) for dihedralHSRgenerically produces explicit factors of_Gb_2G2⁻_τ^π

2 which are clearly not of uniform transcendental weight.

We therefore expect thatallclosed cycles f₁₂⁽¹⁾f₂₃⁽¹⁾. . . f_k1⁽¹⁾in then-point integrand break uniform transcendentality, including those withkn.

This is in marked contrast with the genus-zero situation where only closedsubcycles(z₁₂z₂₃. . .z_k1)⁻¹in the integrand withk ≤ n−2 violate uniform transcendentality [89,186,199,200].⁸The subcycles f₁₂⁽¹⁾f₂₁⁽¹⁾in the integrands of the non-uniformly transcendental integralsI^(2,0)

12|34and I^(4,0)

12|34in the non-planar sector confirm the general expectation.

At genus zero, any non-uniformly transcendental disk or sphere integral over n punctures can be expanded in a basis of uniformly transcendental integrals, see [113,201,202] for a general argument and [88, 186, 200,203] for examples and methods. This basis, known as Parke–Taylor basis, consists of(n−3)! elements [88,201,204] and spans the twisted cohomology defined by the Koba–Nielsen factor made out of|zi j|^{−si j} [113].

8 This point can be illustrated by considering a four-point integral at genus zero over closed subcycles with an integrand of the form(z₁₂z₂₁)⁻¹(z₃₄z₄₃)⁻¹. Integrating by parts inzleads to the cyclic factor(z₁₂z₂₃z₃₄z₄₁)⁻¹subtending all four punctures that is called a Parke–Taylor factor. The integration by parts also generates the rational factor_1+s^s23

12 in Mandelstam invariants that mixes different orders inα⁰. Genus-zero integrals over Parke–Taylor factors subtending all the punctures are known to be uniformly transcendental (which is for instance evident from their representation in terms of the Drinfeld associator [23]). Hence, the original subcycle expression with (z₁₂z₂₁)⁻1(z₃₄z₄₃)⁻1must violate uniform transcendentality.

At genus one, a classification of integration-by-parts inequivalent half-integrands — i.e. chiral halves for torus integrands is conjectured in [36,37]. While genus-one correlators of the open superstring exclude a variety of worldsheet functions by maximal supersymmetry [1,62], Kac–Moody correlators such as (6.22) give a more accurate picture of the problem. In (7.1), we define a generating series of Koba–Nielsen integrals whose coefficients span all the aboveI^(a,0)

... (and all other Koba–Nielsen integrals appearing in string amplitudes) via integration-by-parts and Fay identities. These equivalence classes are again referred to as twisted cohomologies, where the twist is defined by the Koba–Nielsen factor KNn.

Therefore we shall now re-express the planar and non-planar inte-grands in a basis of uniformly transcendental integrals, hoping that this will also shed light on the question of a basis for twisted coho-mologies at genus one. We present below candidate basis elements Ib_...^(a,0)of conjectured uniform transcendentality that appear suitable for the four-current correlator (6.22). Our explicit expressions at leading orders inα⁰and their different modular weights can be used to exclude relations among the^Ib

(a,0)

... . However, it is beyond the scope of this work to arrive at a reliable prediction for the basis dimension of uniform-transcendentality integrals at four points. At the level of the generating series (7.1), we will conjecture the basis dimension to be(n−1)! atn points for each chiral half, but this needs to be adjusted for the counting of the component integrals (7.5).

In the relation between the new quantities^Ib

(a,0)

... and the genus-one integralsI^(4,0)

1234,I^(2,0)

12|34andI^(4,0)

12|34all terms that break uniform transcen-dentality are contained in simple explicit coefficients like_Gb_2or⁽_1+s12⁾

−1. The manipulations necessary to arrive at the^Ib

(a,0)

... are given in detail in AppendicesC.1andC.2and driven by integration by parts, resulting again in series inMGFswhich bypass the need forHSRand avoid the conditionally convergent or divergent lattice sums caused by integration overV₂(i,j). Aspects of the computational complexity when using the I^(a,0)

... versus the^Ib

(a,0)

... can be found in AppendixC.3.

planar uniformly transcendental integrals

As derived in AppendixC.1, a decomposition of the single-trace part of the four-point gauge amplitude that exhibits uniform transcendentality is

M₄(τ)

Tr(t^a1t^a2t^a3t^a4)G²₄^Ib

(4,0) 1234 +G4

G4Gb2− 7 2G6

I^(2,0)

1234

+₄₉

6 G²₆− 10 3 G³₄

I^(0,0). (6.72)

In this expression we introduced the following combination of modular that manifestly respects uniform transcendentality to the order given.

In (C.12), we provide a closed integral form of ^Ib

(4,0)

1234(s_{i j}, τ) that we conjecture to be uniformly transcendental at every order inα⁰, with weightk+3 at the order ofα^0k. As argued above,I^(2,0)

1234 andI⁽⁰,0)

are uniformly transcendental and all non-uniformly transcendental terms in the above way of writing the planar amplitude are in the coefficients of the basis integrals. The coefficient ofs₁₃in (6.73) may also be written as 6G4(π∇₀)²E2/τ₂⁴to highlight the parallel with theMGF(π∇₀)²E3/(6τ₂⁴) at the subleading orderα⁰².

Note that the coefficient ofI⁽⁰,0)

in (6.72) can be recognized as 49

6 G²₆−10

3 G³₄−128π¹²

2025 η²⁴. (6.74)

At the level of the integrated amplitude (6.23), this cancels the factor of η⁻²⁴due to the partition function. Hence, one can import the techniques of the type-II amplitude [38, 150] to perform the modular integrals

∫

F d²τ

τ²₂ I⁽⁰,0)

in (6.72) as we shall see in Section6.2.4.

Similar to (6.74) the coefficient of I^(2,0)

1234 in (6.72) exhibits a special relative factor in the combination G4G2− ⁷

2G6 that can therefore be written as aτ-derivative

G4Gb2−7

2G6−π

τ2G4+π²qdG4

dq , (6.75)

using the Ramanujan identities (5.63). Hence, the only contribution

∼q⁰to (6.75) stems from the non-holomorphic term−^π

τ2G4. non-planar uniformly transcendental integrals

Similarly, we can also rewrite the non-planar part of the amplitude in terms of combinations that exhibit uniform transcendentality as follows

M₄(τ)

The details of the derivation of this result are given in AppendixC.2. On top of a contribution from the planar integral given in (6.73), (6.76) con-tains the non-planar integrals of conjectured uniform transcendentality Ib^(a,0) For definitions of the^Ib

(a,0)

12|34 to all orders inα⁰viaz-integrals, see (C.20) and (C.26).

Im Dokument Modular Graph Forms and Scattering Amplitudes in String Theory (Seite 182-186)