Remainder

For scattering amplitudes, the BDS ansatz [59,60] states that the finite part of the logarithm of the loop correction (3.3) is entirely determined by the one-loop result. While this ansatz gives correct predictions for four and five points, it misses certain contributions starting at six points [64–67]. The contributions missed by the BDS ansatz are known as remainder functions. In addition to scattering amplitudes, remainder functions were also studied for form factors of BPS operators [134, 139]. In both cases, they were found to be of maximal uniform transcendentality and could be vastly simplified using the so-called symbol techniques [68, 69].

For form factors of non-protected operators that renormalise non-diagonally, the defi-nition of the remainder function has to be generalised to

R⁽²⁾=I⁽²⁾(ε)−1 2

I⁽¹⁾(ε)2

−f⁽²⁾(ε)I⁽¹⁾(2ε) +O(ε), (5.9) where

f⁽²⁾(ε) =−2ζ₂−2ζ₃ε−2ζ₄ε² (5.10) as in the amplitude case [59] and the renormalised interaction operators I^(ℓ) have been defined in (3.12). In particular,R⁽²⁾is an operator itself. The termf⁽²⁾(ε)I⁽¹⁾(2ε) subtracts the result expected due to the universality and exponentiation of the IR divergences. In general,f^(ℓ)is connected to the cusp and collinear anomalous dimensions asf^(ℓ)= ¹₈γ_cusp^(ℓ) + ε₄^ℓG0^(ℓ)+O(ε²) [59].

As all disconnected contributions in I⁽²⁾ are cancelled by the square of I⁽¹⁾ in (5.9), the remainder can be written in terms of a density of range three:

R⁽²⁾= XL

i=1

r_{i i+1}⁽²⁾ _i+2. (5.11)

5While there is in general a freedom to define the dilatation operator density (5.8), the requirement of having a finite density effectively eliminates this freedom for the remainder function; see the discussion below. Here, we give the dilatation operator density that corresponds to the finite remainder function density.

This density is explicitly given by Note that there is a freedom of defining the densityr⁽²⁾_{i i+1i+2}, which is related to distributing the contributions with effective range two between the first and second pair of neighbouring legs. In order to obtain a finite density, however, these contributions have to be distributed equally; hence the prefactors of ¹₂ in front of the respective terms in (5.12).⁶ The occurring renormalisation constants are depicted in analogy to the interaction operators. They can be obtained from the dilatation operator via (3.8).

The different matrix elements of the remainder density satisfy analogous identities to those ofI_{i i+1}⁽²⁾ _i+2:

(r⁽²⁾_i )^{Y XX}_XXY + (r_i⁽²⁾)^{XY X}_XXY + (r⁽²⁾_i )^XXY_XXY = (r⁽²⁾_i )^XXX_XXX, (r⁽²⁾_i )^{XY X}_{XY X}+ (r_i⁽²⁾)^{Y XX}_{XY X}+ (r⁽²⁾_i )^XXY_{XY X} = (r⁽²⁾_i )^XXX_XXX, (r_i⁽²⁾)^{XY X}_XXY + (r⁽²⁾_i )^{Y XX}_XXY = (r_i⁽²⁾)^XXY_{XY X}+ (r⁽²⁾_i )^XXY_{Y XX}.

(5.13)

These identities are a consequence of SU(2) symmetry and follow from

[J^A, R⁽²⁾] = 0, (5.14)

which can be derived from (3.35) and (5.5). Accordingly, we can write r⁽²⁾_{i i+1i+2} = (r_i⁽²⁾)^XXX_XXX−(r⁽²⁾_i )^{XY X}_{Y XX}−(r_i⁽²⁾)^{XY X}_XXY

1i i+1i+2

+ (r⁽²⁾_i )^{XY X}_{Y XX}−(r_i⁽²⁾)^{Y XX}_XXYP_{i i+1}+ (r⁽²⁾_i )^{XY X}_XXY −(r_i⁽²⁾)^XXY_{Y XX}P_i+1i+2 +(r⁽²⁾_i )^{Y XX}_XXYP_{i i+1}P_i+1i+2+ (r⁽²⁾_i )^XXY_{Y XX}P_i+1i+2P_{i i+1}.

(5.15)

Taking into account the symmetry under inverting the order of the fields, it is hence sufficient to calculate (r⁽²⁾_i )^XXX_XXX, (r⁽²⁾_i )^{XY X}_XXY and (r⁽²⁾_i )^{Y XX}_XXY.

Each of these matrix elements depends on the Mandelstam variabless_{i i+1},s_i+1i+2 and s_i+2i mainly through the ratios⁷

u_i = s_{i i+1}

s_{i i+1i+2} , v_i = s_i+1i+2

s_{i i+1i+2} , w_i = s_i+2i

s_{i i+1i+2}, (5.16)

6For other choices that lead to divergent densities, the divergent contributions cancel in the sum (5.11).

7The BPS remainder studied in [139] depends only on the rations ui, vi and wi. The fact that also

s_{i i+1}

µ² , ^si+1_µ2ⁱ⁺² and ^si+2_µ2ⁱ occur in (5.21), (5.22) and (5.23) reflects the fact that we are considering form factors of composite operators that are in general non-protected. As required, the latter terms cancel for the BPS case.

5.3 Remainder 79 wheres_{i i+1i+2} =s_{i i+1}+s_i+1i+2+s_i+2i and u_i+v_i+w_i = 1.

The first matrix element was already calculated in [139] and is of uniform transcenden-tality four:

(r_i⁽²⁾)^XXXXXX = (r_i⁽²⁾)^XXXXXX

4, (5.17)

where we denote the restriction to functions with a specific degree of transcendentality by a vertical bar. Inserting the respective expressions for the Feynman integrals into (5.12), one finds a result filling several pages. One can, however, simplify it using the symbol [68,69].⁸ The symbol of (r_i⁽²⁾)^XXX_XXX is given by [139] In [139], it was integrated to the relatively compact expression

(r⁽²⁾_i )^XXX_XXX|4 term that cannot be expressed in terms of classical polylogarithms.

The matrix element (r⁽²⁾_i )^{XY X}_XXY has mixed transcendentality of degree three to zero. Its maximally transcendental part has the symbol

S

8The symbol is implemented e.g. in theMathematicacode [238].

which can be integrated to Together with the less transcendental parts, the complete second matrix element reads

(r⁽²⁾_i )^{XY X}_XXY = (r⁽²⁾_i )^{XY X}_XXY The matrix element (r_i⁽²⁾)^{Y XX}_XXY has mixed transcendentality with degree two to zero. It is given by From the above results and the form (5.15), we find that the remainder of any operator in the SU(2) sector is given by a linear combination of one function of transcendentality four, one function of transcendentality three and two functions of transcendentality two and less. In particular, the transcendentality-four contribution to the remainder is the same for any operator in the SU(2) sector and agrees with the one of the BPS operator tr(φ^L₁₄) studied in [139]. This generalises the principle of maximal transcendentality to non-protected operators in N = 4 SYM theory. The difference with respect to the BPS remainder is of transcendentality three or less and can be written entirely in terms of classical polylogarithms.

Given the above results, it is tempting to conjecture that the universality of the leading transcendental part extends to the remainder functions of the minimal form factors of all operators, also beyond the SU(2) sector. Moreover, this suggests that the leading transcendental part of the three-point form factor remainder of any length-two operator matches the corresponding BPS remainder calculated in [134], which also matches the leading transcendental part of the Higgs-to-three-gluons amplitude calculated in [142]. It would be desirable to check this conjecture in further examples or even to prove it.⁹

Furthermore, note that the maximal degree of transcendentality t in a given matrix element is related to the shuffle number sas t= 4−s, where sis the number of sites by which a singleY among two X’s or vice versa is displaced. Interestingly, we find that the

9One approach to such a proof might be to show the universality of the leading singularities. The latter are closely related to the maximally transcendental functions in the dlog form, see e.g. [74]. Moreover, leading singularities are related to on-shell diagrams, which will be one of the subjects of the next chapter.

5.3 Remainder 81 degree-zero contributions to the remainder function are related to the two-loop dilatation operator as

D⁽²⁾_{i i+1i+2} =−4

7r⁽²⁾_{i i+1i+2}

0. (5.24)

An important property of scattering amplitudes and form factors is their behaviour under soft and collinear limits. In these limits, they in general reduce to lower-point scattering amplitudes and form factors multiplied by some universal function. Similarly, the corresponding remainders reduce to lower-point remainders. This poses important constraints on the remainders, which, together with other constraints, even made it possible to bootstrap them in several cases; see for instance [71, 72] and references therein.

As already observed for the BPS remainder in [139], also the soft and collinear limits of the remainders in the SU(2) sector considered here yield non-vanishing results. This is interesting as these remainders correspond to the minimal form factors, i.e. no non-vanishing form factors with less legs exist to which they could be proportional. In fact, a similar behaviour occurs also for scattering amplitudes. For example, several of the six-point NMHV amplitudes listed in appendix B.2 have non-vanishing soft and collinear limits that do not correspond to physical amplitudes. Via the three-particle unitarity cut, this behaviour of the amplitudes can be related to the one of the minimal form factor remainders. A better understanding of theses limits is clearly desirable.

This chapter concludes the treatment of minimal loop-level form factors in this thesis

— although many interesting questions remain unanswered. Most notably, it would be very interesting to extend the methods presented here to obtain the minimal two-loop form factor of a generic operator and from it the complete two-loop dilatation operator.

These questions are currently under active investigation. In the next chapter, we will turn to non-minimal (n-point) form factors at tree-level.

Chapter 6

Tree-level form factors

In the previous chapters, we have studied loop-level form factors with the minimal number of external legs. In this chapter, we turn to the opposite configuration — n-point form factors at tree level. We will mostly focus on the form factors of the chiral part of the stress-tensor supermultiplet, which we will briefly introduce in section 6.1. In the subse-quent sections, we extend several powerful techniques that were developed in the context of scattering amplitudes to form factors, in particular on-shell diagrams, central-charge deformations, an integrability-based construction via R operators and a formulation in terms of Graßmannian integrals. For each of these techniques, we first give a very short description in the case of scattering amplitudes and then show how to generalise them to form factors. We introduce on-shell diagrams, corresponding permutations, their construc-tion and the extension to top-cell diagrams in secconstruc-tion 6.2. In secconstruc-tion 6.3, we introduce central-charge deformations for form factors, show how form factors can be constructed via the integrability-based method of R operators and demonstrate how they can thus be found as solutions to an eigenvalue equation of the transfer matrix, which also (partially) generalises to generic operators. In the final section 6.4, we find a Graßmannian integral representation of form factors in spinor-helicity variables, which we moreover translate to twistor and momentum-twistor variables. As we are only discussing tree-level expressions, we will be dropping the superscript that indicates the loop order throughout this chapter.

The results presented in this chapter were first published in [7].