Comments on all-loop constraints for scattering amplitudes and Feynman integrals

Recent years have witnessed enormous progress in computing and understanding analytic structures of scattering amplitudes in QFT. These developments have greatly pushed the frontier of perturbative calculations relevant for high energy experiments, and often they offer deep insights into the theory itself and exhibit surprising connections with mathematics. An outstanding example is the ${\cal N}=4$ super-Yang-Mills theory (SYM), where one can perform calculations that were unimaginable before and discover rich mathematical structures underlying them. For example, positive Grassmannian Arkani-Hamed:2016byb and the amplituhedron Arkani-Hamed:2013jha have provided a new geometric formulation for its planar integrand to all loop orders.

On the other hand, the modern bootstrap program in perturbative QFT aims at computing scattering amplitudes and other physical quantities by directly imposing analytic structure and physical constraints, without ever needing Feynman diagrams or loop integrands. Again the perfect laboratory where very impressive perturbative orders have been achieved is the planar ${\cal N}=4$ SYM amplitudes with $n=6,7$ Dixon:2011pw ; Dixon:2014xca ; Dixon:2014iba ; Drummond:2014ffa ; Dixon:2015iva ; Caron-Huot:2016owq ; Dixon:2016nkn ; Drummond:2018caf ; Caron-Huot:2019vjl ; Caron-Huot:2019bsq ; Dixon:2020cnr ; Golden:2021ggj (see Caron-Huot:2020bkp for a review). The key of the bootstrap program is the construction of a function space where the amplitude lives: for $n=6,7$ (or more generally MHV and NMHV amplitudes with any $n$) conjecturally it is the space of generalized polylogarithm functions, which depend on $3(n{-}5)$ dual conformal invariant (DCI) cross-ratios, after subtracting infrared divergences e.g. by normalizing with Bern-Dixon-Smirnov (BDS) ansatz Bern:2005iz . As another deep connection between amplitudes and mathematics, it has been realized in Golden:2013xva that cluster algebras of Grassmannian $G(4,n)$ are directly relevant for the singularities of $n$-particle amplitudes, whose kinematics can be parametrized by momentum twistor Hodges:2009hk , or the space of $G(4,n)$ mod torus action. More precisely, the so-called ${\cal A}$-coordinates of $G(4,n)$ cluster algebras are related to symbol Goncharov:2010jf ; Duhr:2011zq letters of amplitudes: the $9$ letters of six-particle amplitudes and $42$ letters of seven-particle ones are nicely explained by $A_{3}$ and $E_{6}$ cluster algebras, respectively.

Apart from the overall symbol alphabet, more refined restrictions on scattering amplitudes and Feynman integrals have proved to be crucial for the bootstrap program to higher orders for $n=6,7$. The most basic ones include the physical discontinuity Gaiotto:2011dt and the double discontinuity structure derived from the Steinmann relations Steinmann1960a ; Steinmann1960b , which have reduced possible first-two-entries to only $6$ and $28$ weight-two symbols for $n=6,7$, respectively. Remarkably, the conjectural extended Steinmann (ES) relations, which basically extend Steinmann relations to higher multiple discontinuities, have been observed to hold for amplitudes and integrals for $n=6,7$Caron-Huot:2016owq ; Dixon:2016nkn ; Caron-Huot:2019bsq . These ES relations impose similar constraints on any two consecutive entries of the symbol as Steinmann relations on the first two, which have been explicitly checked for $n=6$ (double-)pentagon ladders to arbitrary loops (and some for $n=7$) as well as $n=6$ amplitudes (up to $L=7$) and $n=7$ amplitudes (up to $L=4$), which are normalized by BDS-like subtraction Alday:2009dv and two-loop $n=8$ MHV amplitude with minimal subtraction Golden:2018gtk  ¹¹1Interesting constraints similar to extended Steinmann relations have been found for Feynman integrals Bourjaily:2020wvq , form factors Dixon:2020bbt and even the wave-function of the universe Benincasa:2020aoj .. In these cases, the ES relations can be alternatively formulated as cluster adjacency conditions Drummond:2017ssj ; Drummond:2018caf ; Drummond:2018dfd : only ${\cal A}$ coordinates that belong to the same cluster (for $A_{3}$ and $E_{6}$ for $n=6,7$ respectively) can appear adjacent in the symbol. As far as we know, ES relations/cluster adjacency has remained (mysterious) conjectures, and it is natural to ask if they extend to amplitudes and integrals for $n\geq 8$. It is in particular highly desirable to apply it to finite Feynman integrals to relatively high orders, as well as finite components of amplitudes, for higher multiplicities; among other things, such relations can provide powerful constraints on the function space for higher $n$ just as what they have achieved for $n=6,7$. This is the main question we hope to address in the paper.

Beyond $n=6,7$, Grassmannian cluster algebras for $G(4,n)$ for $n\geq 8$ become infinite, and a certain truncation is needed to obtain a finite symbol alphabet. As already seen for one-loop N²MHV, amplitudes with $n\geq 8$ generally involve letters that cannot be expressed as rational functions of Plücker coordinates of the kinematics $G(4,n)/T$; more non-trivial algebraic letters appear in computations based on ${\bar{Q}}$ equations CaronHuot:2011kk for two-loop NMHV amplitudes for $n=8$ and $n=9$ Zhang:2019vnm ; He:2020vob , requiring extension of Grassmannian cluster algebras to include algebraic letters. Solutions to both problems has been proposed using tropical positive Grassmannian speyer2005tropical and related tools for $n=8$ Drummond:2019qjk ; Drummond:2019cxm ; Henke:2019hve ; Arkani-Hamed:2019rds ; Arkani-Hamed:2020cig ; Herderschee:2021dez and $n=9$ Henke:2021avn ; Ren:2021ztg , as well as using Yangian invariants or the associated collections of plabic graphs Mago:2020kmp ; He:2020uhb ; Mago:2020nuv ; Mago:2021luw .

On the other hand, ${\cal N}=4$ SYM has been an extremely fruitful laboratory for the study of Feynman integrals (c.f. ArkaniHamed:2010gh ; Drummond:2010cz ; Caron-Huot:2018dsv ; Henn:2018cdp ; Bourjaily:2018aeq ; Herrmann:2019upk and references therein).The connections to cluster algebras extend to individual Feynman integrals as well, e.g. the same $A_{3}$ and $E_{6}$ control $n=6,7$ multi-loop integrals in ${\cal N}=4$ SYM Caron-Huot:2018dsv ; Drummond:2017ssj . Cluster algebra structures have also been discovered for Feynman integrals beyond those in planar ${\cal N}=4$ SYM Chicherin:2020umh including a five-particle alphabet which has played an important role in recent two-loop computations Abreu:2018aqd ; Chicherin:2018old ; Chicherin:2018yne . The knowledge of alphabet and more refined information can be used for bootstrapping Feynman integrals Chicherin:2017dob ; Henn:2018cdp (see also Dixon:2020bbt ). In He:2021esx ; He:2021non , we have identified (truncated) cluster algebras for the alphabets of a class of finite, dual conformal invariant (DCI) Drummond:2006rz ; Drummond:2007aua Feynman integrals to high loops, based on recently-proposed Wilson-loop $d\log$ representation He:2020uxy ; He:2020lcu . For ladder integrals with possible “chiral pentagons” on one or both ends (without any square roots), we find a sequence of cluster algebras $D_{2},D_{3},\cdots,D_{6}$ for their alphabets, depending on $n$ and the kinematic configurations; for cases with square root, such as the $n=8$ double-penta ladder integrals, we find a truncated affine $D_{4}$ cluster algebra which (minimally) contain $25$ rational letters and $5$ algebraic ones.

In this paper, we would like to use this rich collection of data for a systematic check of extended Steinmann relations beyond $n=6,7$. Before proceeding, let us briefly list all finite integrals we use as data in this paper. As the main example, we consider $n=8$ double-penta ladder integral, $I^{(L)}_{\text{dp}}(1,4,5,8)$ with four massless corners labelled by $1,4,5,8$, up to $L=4$ (whose alphabet contains $25+5$ letters mentioned above). Its collinear limit $8\to 7$ gives $n=7$ $I^{(L)}_{\text{dp}}(1,4,5,7)$ with an alphabet of $D_{4}$ cluster algebra ($16$ rational letters). Simpler examples include penta-box ladder $I^{(L)}_{\text{pb}}$ up to $L=4$ (alphabet is $D_{3}$ cluster algebra) and the well-known box ladder $I^{(L)}_{\text{b}}$ to all loops (alphabet is $D_{2}\sim A_{1}^{2}$ cluster algebra).

Besides these ladder integrals up to $n=8$, we will also use the most general two-loop double-pentagon integrals computed in He:2020lcu , which we denote as $\mathcal{I}_{\text{dp}}(i,j,k,l)$. These integrals become generic at $n=12$ e.g. $(i,j,k,l)=(1,4,7,10)$ (depending on $13$ variables) whose alphabet has $164$ rational letters and $5$ multiplicative independent algebraic letters for each of the $16$ distinct square roots. For $n=8$, we have e.g. $\mathcal{I}_{\text{dp}}(1,3,5,7)$ (depending on $9$ variables) whose alphabet has $100$ rational letters and $2\times 5$ algebraic ones (with two different square roots).

Remarkably, these integrals directly give IR finite components of two-loop NMHV amplitudes to all multiplicities! More precisely, for non-adjacent $i,j,k,l$ the component $\chi_{i}\chi_{j}\chi_{k}\chi_{l}$ of NMHV amplitude (after stripping off MHV tree)/Wilson loop is simply given by ${\cal I}_{\rm dp}(i,j,k,l)-{\cal I}_{\rm dp}(j,k,l,i)$; they represent the simplest class of NMHV components (while most generic for large $n$), which are also free of square roots Bourjaily:2019igt (as first computed for $n=8$ and for higher $n$ using $\bar{Q}$ method Zhang:2019vnm ; He:2020vob ).

In section 3, we will explicitly check ES relations and find positive answer for all these finite Feynman integrals (up to $L=4$) and two-loop NMHV (finite) amplitudes with $n\geq 8$; note that the latter provides all-multiplicity evidence for ES relations for finite component amplitudes without the need of any subtraction/normalization. We will also confirm ES relations using the three-loop $n=8$ MHV amplitudes obtained very recently using ${\bar{Q}}$ equations by one of the authors and Chi Zhang Li2021 . Similar to the BDS-like subtraction for $n=6,7$ case, we need such normalization for higher-point MHV amplitudes and in particular for the case when $n$ is a multiple of $4$, such as $n=8$, the BDS-like subtraction is not defined and we need e.g. minimal subtraction. Note that for two-loop MHV amplitudes, cluster adjacency, which is closely related to ES relations, has been checked in Golden:2019kks using Sklyanin bracket, which also requires minimal subtraction for $n=8$. It is not at all obvious to us that with the same subtraction, three-loop $n=8$ MHV amplitude also satisfies ES relations!

In all these cases except for $I_{\rm pb}$ and $I_{\rm dp}(1,4,5,7)$, we need to take into account algebraic letters, and while most ES relations directly hold for the rational part, naively there is a violation of ES relations in this part for remaining pairs which appear in cross-ratio of four-mass kinematics; it is remarkable that by including the algebraic part which also depends on such four-mass kinematics, the violation is cancelled and we have ES relations for the full answer. Note that while ES relations can be imposed without explicitly referring to the alphabet, it is more subtle to extend cluster adjacency to higher $n$ whose alphabet also contains algebraic letters; for rational ones, the so-called Sklyanin bracket Sklyanin:1983ig can be used, and we will see that for all those rational letters which do not violate ES relations, we can test that cluster adjacency is also satisfied.

In addition to ES relations which apply to all consecutive entries of the symbol, it is well known that the first two entries in general and last two entries for MHV amplitudes are not the same as before. As we have mentioned, given the physical discontinuity or the first entry condition, Steinmann-satisfying first two entries are much more constrained than subsequent pairs. In section 2, we will list all such (integrable) first-two-entries, which amounts to having the weight-$2$ space of Steinmann-satisfying dilogarithm functions. We list all such weight-$2$ functions for any $n$, in terms of ${\rm Li}_{2}$ and $\log\log$. As extracted from the box functions, entries of the $\operatorname{Li}_{2}$ part will automatically DCI, while entries of $\log\log$ functions are simply $x_{ij}^{2}$. We will give the general counting of these first-two-entries to all $n$, and check this ansatz by all the data we have, including amplitudes and finite integrals.

In section 4 which has a different flavor than the rest of the paper, we will also provide a list of all possible last two entries for MHV amplitudes. The reason we study them is that last entries of MHV amplitudes have been known since Caron-Huot:2011dec which can be derived from ${\bar{Q}}$ equations and have played an important role for the bootstrap program. It is known again for $n=6,7$ that similar constraints can be derived for last two entries, again by using ${\bar{Q}}$ equations. We will explicitly go through the situations for $n\leq 9$ and get all possible $\mathcal{A}$-coordinates pairs on the last-two-entries. Furthermore, for $n=7$ we reorganize the pairs into manifestly DCI integrable symbols, showing that there are $139$ allowed combinations. We will see that numbers of the last-two-entries from $\bar{Q}$ equations remain very limited as $n$ grows, leaving stronger constraints on the function space of MHV amplitudes.

As a warm-up exercise, let us first present our conjecture for all possible first-two-entries of integrals/amplitudes. The starting point is that physical discontinuities, or the first entries of the symbol, for any amplitudes/integrals in planar ${\cal N}=4$ SYM, must correspond to planar variables $x_{a,b}^{2}=0$; this is even true for any planar theory, and what is special about SYM is the dual conformal symmetry which forces the first entries to be DCI combinations. Note that out of all the $n(n{-}3)/2$ planar variables, which we write as $\langle a{-}1ab{-}1b\rangle$, $n$ of them are frozen variables ($\langle ii{+}1i{+}2i{+}3\rangle$ for $i=1,\cdots,n$), thus we have $n(n{-}5)/2$ unfrozen Plücker variables which correspond to diagonals of $n$-gon except for the $n$ “shortest” ones. Since we are interested in variables that are unfrozen, these are all we can have for the first entry, and equivalently one can construct the same number of DCI combinations e.g. $v_{i,j}:=x_{i,j}^{2}x_{i{+}2,j{+}2}^{2}/(x_{i,i+2}^{2}x_{j,j{+}2}^{2})$. For example, for $n=6,7$, these correspond to the well known $3$ and $7$ first entries, and for $n=8$ it is easy to see that we have $12$ first entries, which are given by $x^{2}_{i,i{+}3}$ for $i=1,\cdots,8$ and $x^{2}_{i,i{+}4}$ for $i=1,\cdots,4$ (or corresponding $v_{i,i{+}3}$ and $v_{i,i{+}4}$).

What about the first-two-entries? The most important constraint on the first-two-entries is the Steinmann relations, which says that the double discontinuities taken in overlapping channels of any amplitude should vanish. A convenient way to proceed is to start at one loop, where all finite amplitudes/integrals can be expanded in terms of (finite part of) box integrals. First we note that the only finite box integrals are the four-mass ones; given four dual points $x_{a},x_{b},x_{c},x_{d}$ with $a,b,c,d$ cyclically ordered and each adjacent pair at least separated by $2$, we have the (normalized) four-mass box integral

where $u_{a,b,c,d}:=x_{a,b}^{2}x_{c,d}^{2}/(x_{a,c}^{2}x_{b,d}^{2})$ and similarly for $u_{b,c,d,a}$. It is straightforward to check that $I_{a,b,c,d}$ satisfies Steinmann relations as we will see in the next section. We can also go to lower-mass cases but they suffer from infra-red (IR) divergences, and certain regularization is needed. One way to preserve DCI is to introduce the DCI-regulator as in Bourjaily:2013mma , but we find that the resulting finite part of the lower-mass box integrals violate Steinmann relations. More precisely, it is possible to write the ${\rm Li}_{2}$ part of lower-mass boxes in a way that respects Steinmann relations, but the $\log\log$ part does not, thus we will discuss these two parts separately for them. Our main conjecture is that for finite integrals and amplitudes to all loops, the first two entries that satisfy Steinmann relations can only be extracted from such box functions where the $\log\log$ part need to be treated separately. In addition to the four-mass box cases, we will encounter the following ${\rm Li}_{2}$ functions for three-mass and two-mass-easy boxes ²²2For two-mass-hard or one-mass boxes, only a constant ${\rm Li}_{2}(1)$ can appear in this part; as we are listing first-two-entries at the symbol level, we ignore such constant for now.:

where one can check that indeed Steinmann relations are satisfied: the second-entry that can enter are just $1-1/u_{c,a{-}1,a,b}$ and $1-1/u_{c,a{-}1,a,c{-}1}$. It is easy to count the total number of four-mass boxes and such ${\rm Li}_{2}$ functions for lower-mass cases: there are $n\choose 4$ box functions and $n(n{-}4)$ of them are two-mass-hard/one-mass boxes which only give ${\rm Li}_{2}(1)$, thus we have $n(n{-}5)(n^{2}-n-18)/24$ such functions. For example, for $n=6,7$, there are $3$ and $14$ ${\rm Li_{2}}$ functions, which can be chosen to be ${\rm Li}_{2}(1-a_{i})$ for $i=1,2,3$ and ${\rm Li}_{2}(1-a_{1,i})$ for $i=1,\cdots,7$, respectively Dixon:2016nkn . Now for $n=8$, we have $2$ four-mass boxes, and $36$ ${\rm Li}_{2}$ functions from lower-mass boxes, which we can choose to be $\operatorname{Li}_{2}(1-1/u)$ with $u$ in

Having fixed all possible ${\rm Li}_{2}$ part, we move to the $\log\log$ part, and it is clear which of them satisfy Steinmann relations:

where $(i,j)\sim(k,l)$ means that the two diagonals (neither are the shortest ones) do not cross each other. Naively these do not respect DCI, but as we have mentioned above such $x_{i,j}^{2}$ can be converted to cross-ratios (with the help of frozen ones), which need to be done carefully with respect to Steinmann relations. For example, for $n=6,7$, there are exactly $3$ and $14$ such $\log\log$ functions respectively: $\log^{2}x_{i,i{+}3}^{2}$ with $i=1,2,3$ for $n=6$, and $\log^{2}x_{i,i{+}3}^{2}$ and $\log x^{2}_{i,i{+}3}\log x^{2}_{i{+}3,i{+}6}$ with $i=1,\cdots,7$ for $n=7$. For $n=8$, we find $40$ $\log(x)\log(y)$ functions for $(x,y)$ in

For general $n$, it is straightforward to count that there are $n(n{-}3)(n{-}4)(n{-}5)/12$ such $\log\log$ functions. In total, there are $n(n{-}5)(n^{2}-n+5)/8$ first-two-entries which can be derived from box functions and satisfy Steinmann relations, and we conjecture that they are all we need for finite integrals and Steinmann-respecting amplitudes. For example, we have checked that in all two-loop $n=8$ finite Feynman integrals in Figure 1 and ${\cal I}_{\rm dp}(1,3,5,7)$, exactly these $78$ first-two-entries for $n=8$ can appear. Moreover, we have checked first-two entries of minimally normalized of two-loop and three-loop MHV amplitudes for $n=8$, exactly all $78$ first-two-entries appear.

As discussed in the last section, the Steinmann relations that restrict the for amplitudes or Feynman integrals give constraints on the first-two entries. The extended Steinmann relations are just generalizing these constraints to any two consecutive entries, thus restricting iterated discontinuity at any depth. Let us first write these relations as follows: define a linear “discontinuity” map in the space of symbols

where we normalize the discontinuity map that computes the monodromy around $x=c$ such that $\operatorname{Disc}_{x=c}\log(x-c)=1$. Then the extended Steinmann relations for a amplitude or Feynman integral $F$ of weight $w$ are

for any $1\leq s\leq w-1$ and any two overlapping channels $x_{ij}^{2}$, $x_{kl}^{2}$, i.e. $(ij)\not\sim(kl)$. For $s=1$ this reduces to the Steinmann relations (the second $\operatorname{Disc}^{1}_{x=0}$ acting on the second-entry of the original symbol). Note that for $s\neq 1$, $\operatorname{Disc}^{s}_{x=c}$ of a integrable symbol may not be integrable, so it’s not well-defined in the space of functions.

If any two letters of $F$ do not share the same branch point, $\operatorname{Disc}_{x=0}(\log(y))=0$ for any two distinct letters $x$ and $y$, then $\operatorname{Disc}^{k}_{x=0}(F)$ is given by clipping the $k$-th entry $x$ off in the symbol of $F$ (after deleting all terms whose $k$-th entry is not $x$), and ES relations simply become constraints of adjacent letters in the symbol. This is indeed how ES relations are checked for BDS-like normalized amplitudes for $n=6,7$ (up to $L=7$ and $L=4$ respectively) as well as certain Feynman integrals (such as $n=6$ double-pentagon ladders). Note that this is always the case when the alphabet contains only rational letters, which cannot share any branch point since by definition they are multiplicative independent. For instance, we have also checked the double-penta ladder integrals $I_{\text{dp}}^{(L)}(1,4,5,7)$ with $n=7$ and the penta-box ladders $I_{\text{pb}}^{(L)}$ with $n=8$ up to $L=4$, and find that they indeed satisfy ES relations in exactly the same way as finite integrals for $n=6,7$.