Color-kinematic duality for form factors

Color-kinematic duality for form factors

Rutger H. Boels,    Bernd A. Kniehl,    Oleg V. Tarasov,    and Gang Yang
Abstract

Recently a powerful duality between color and kinematics has been proposed for integrands of scattering amplitudes in quite general gauge theories. In this paper the duality proposal is extended to the more general class of gauge theory observables formed by form factors. After a discussion of the general setup the existence of the duality is verified in two- and three-loop examples in four dimensional maximally supersymmetric Yang-Mills theory which involve the stress energy tensor multiplet. In these cases the duality reproduces known results in a particularly transparent and uniform way. As a non-trivial application we obtain a very simple form of the integrand of the four-loop two-point (Sudakov) form factor which passes a large set of unitarity cut checks.

institutetext: II. Institut für Theoretische Physik, Universität Hamburg
Luruper Chaussee 149, D- 22761 Hamburg, Germany

1 Introduction

The ability to calculate is of fundamental importance in any would-be theory of nature. In high energy physics this manifests itself in the quest to calculate to higher orders in perturbation theory, including more and more loops and legs. The basic quantities to be calculated in a gauge theory are in general a mixture of plane-wave external states and gauge invariant operators. If only on-shell states are involved these observables are scattering amplitudes, while if only gauge invariant operators are involved these are pure correlation functions. Admixtures of observables build of operators and on-shell states are called form factors.

The main motivation of this paper is the observation that in recent years many new insights have been developed for the calculation of scattering amplitudes in gauge theories (see e.g. Dixon:2011xs () and references therein), while similar developments for form factors and correlation functions have been minor in comparison. This paper is part of an effort to redress this imbalance. Recent developments have shown that maximally supersymmetric Yang-Mills theory is a good testing ground for new ideas which then should trickle down to non-supersymmetric gauge theory. This line of thought will also be followed here.

A particular insight that has recently been developed is a duality between color and kinematics that has been proposed for integrands of scattering amplitudes Bern:2008qj () Bern:2010ue (). This duality will be extended here to form factors. Color-kinematic duality in its current form states that at tree and loop level the integrand of a scattering amplitude can be arranged in such a way that whenever the color factors of an integrand obey a Jacobi relation, the kinematic parts of these diagrams can also be chosen such that they obey the same identity. In a second step which will not be of much importance for the purposes of this article, this duality can also be used to construct the integrand of a scattering amplitude in a gravity theory, whose field content is simply a tensor product of two gauge theory copies. The status of color-kinematic duality will be reviewed below.

There are two distinct ways the ideas of color-kinematic duality could apply to more general observables which include gauge invariant operators. The first is simply as a direct consequence of color-kinematic duality for scattering amplitudes. On a unitarity cut and at tree level poles of quite general form factors the duality should apply to the amplitude parts of the residues and discontinuities, see for instance Engelund:2012re (). In this paper we conjecture that the duality can be used directly for general form factors and in particular without taking any cuts. We will show in examples that the duality can be used as a powerful tool to produce consistent ansatz for high loop form factors. In particular, our results unify and greatly simplify the results in the recent literature up to three loops.

As a new non-trivial application, we compute for the first time the four-loop integrand of the Sudakov form factor in =4 super Yang-Mills theory (SYM) in a form which passes all (highly non-trivial) unitarity checks performed. The existence of such a form for the four-loop form factor is a strong evidence that the color-kinematic duality holds in general. A distinct advantage of using color-kinematic duality is that the traditionally hard to obtain non-planar integrals in the problem are fixed by the coefficients of planar integrals. The evaluation of the found integrals is in progress inprogress ().

This article is structured as follows. In section 2 the necessary background for this paper will be reviewed. This is followed by section 3 where our general calculational strategy is explained. This strategy is applied to a few examples in section 4, and in particular the four-loop form factor in section 5. The main presentation ends with a discussion and conclusion section. In the first appendix we give a representative selection of four-loop Jacobi relations. The other appendix briefly discusses color structures which appear in two-point form factors and correlation functions with two adjoint operators through eight loops.

2 Review

2.1 Color-kinematic duality for scattering amplitudes

Color-kinematic duality is the collective name for what is basically a pair of two partially proven conjectures Bern:2008qj (), Bern:2010ue (). The starting point is to write the integrand for a scattering amplitude in dimensions at loops in terms of a sum over trivalent graphs only,

 A(l)=∑Γi∫l∏j=1dDℓj1Sinicisi, (2.1)

where is the natural color factor associated to trivalent graph , is the symmetry factor associated to the trivalent graph and is the product of all internal propagators of the graph. This form of the integrand simply amounts to a rewriting of Feynman graph-based perturbation theory by parcelling out the four-vertex over three-vertices. This determines some set of numerators which is not unique. The non-trivial claim of color-kinematic duality is that there exists a set of numerators such that whenever color factors obey a Jacobi identity, the numerators do too:

 ∀{i,j,k}ci+cj+ck=0⇒ni+nj+nk=0. (2.2)

A set of numerators which obeys all Jacobi identities is called color-dual. Color-dual numerators have been constructed explicitly at tree level for all multiplicities through a variety of methods Kierm:2010xq (); Mafra:2011kj (); BjerrumBohr:2012mg (), and at loop level in specific examples in SYM for five points up to two loops Carrasco:2011mn () and for four points up to four loops Bern:2012uf (). Moreover, in pure Yang-Mills theory numerators are known at two loops for the four-point helicity equal amplitude Bern:2010ue () and at one loop for the helicity equal and one-unequal cases Boels:2013bi (). Of course, the tree level results will extend almost trivially to any generalized cut of a loop amplitude which involves tree amplitudes.

The second conjecture usually taken to be part of color-kinematic duality is that if a set of color-dual numerators exists in a gauge theory, then an amplitude in a gravitational theory may be constructed as

 M(l)=∑Γi∫l∏j=1dDℓj1Sini~nisi, (2.3)

where and are color-dual numerators for two in general distinct gauge theories. This conjecture has been proven at tree level, assuming that local numerators exist Bern:2010yg (). The field content of the gravity theory is the direct product of the field contents of the gauge theories. In this way ’squaring’ SYM gives the maximal (ungauged) super-gravity theory. Similarly, squaring pure Yang-Mills theory gives Einstein gravity coupled to a dilaton and a two-form. The latter theory is sometimes referred to as supergravity. The color-kinematic duality relation of gauge theories to gravitational theories will not play a central role in this article.

As an illustration of the general idea, consider the four-point tree amplitude in gauge theory which can be written in terms of three trivalent diagrams as

 A(0)4(1,2,3,4)=csnss+ctntt+cunuu . (2.4)

The color factors denoted as are given by the product of associated to each trivalent vertex

 cs=~fa1a2b~fba3a4,ct=~fa2a3b~fba4a1,cu=~fa1a3b~fba2a4, (2.5)

where

 ~fabc=i√2fabc=Tr([Ta,Tb]Tc). (2.6)

They satisfy the Jacobi identity

 cs=ct+cu . (2.7)

By color-kinematic duality one can also have 111Note that we have necessarily made some choice of signs for the color factors. One can choose other conventions, and the relation between should change correspondingly. The final result is given in terms of the product , and therefore is independent of this choice.

 ns=nt+nu . (2.8)

The solution to the Jacobi conditions may be given explicitly as

 ns=α ,nt=−tA(0)4(1,2,3,4)−αts ,nu=tA(0)4(1,2,3,4)−αus . (2.9)

This can be shown to reproduce the full color-dressed amplitude. The solution depends on an arbitrary parameter . This freedom in specifying numerators is referred to as a generalized gauge transformation Bern:2008qj (). In general this gauge freedom is the freedom to choose another set of kinematic numerators

 n′i=ni+Δi , (2.10)

which do not change the tree amplitudes as long as

 ∑ΓiΔicisi=0 (2.11)

holds. Moreover, the should satisfy Jacobi identities.

It should be obvious that through unitarity cuts tree-level color-kinematics duality provides constraints to the loop integrand. This occurs whenever a tree amplitude is isolated by the cuts. For example as shown in Figure 2, if we cut four propagators , we have222Sums over all states to appear in the cuts are understood but not shown explicitly in the formula.

where in the second line the Bern-Carrasco-Johansson (BCJ) Bern:2008qj () representation of four-point tree amplitudes of equation (2.4) was used and has been expressed in terms of some loop integrals. The third line is simply given by defining

 Cj,s≡cjcs,Nj,s≡njns . (2.13)

Because of , we have

 Nj,s∣∣ℓ21=ℓ22=ℓ23=ℓ24=0=(Nj,t+Nj,u)∣∣ℓ21=ℓ22=ℓ23=ℓ24=0, (2.14)

which may be thought of as the dual relation of that of color factors, . Beyond amplitudes it was noted in Engelund:2012re () that a similar observation should hold for unitarity cuts333This includes the case of tree level poles. of general form factors, whenever a scattering amplitude is isolated on the cuts.

Beyond the constraints on loop integrands from unitarity cuts it was conjectured that there exists a representation where the above duality relations are truly off-shell, i.e. without doing cuts to impose on-shell conditions Bern:2010ue (). More explicitly, given a representation that

there is an off-shell color-kinematics duality

 Ci=Cj+Ck⇒Ni=Nj+Nk. (2.16)

This is a highly non-trivial claim. This has been checked on a case-by-case basis, but no general proof exists. Some supporting evidence was recently uncovered in Oxburgh:2012zr (). It may be taken as an evidence of the existence of a Lagrangian formulation of gauge theory which has explicitly color-dual structure. This much more powerful version of color-kinematic duality at loop level will be generalized to a class of form factors in this article.

2.2 Form factors in N=4 supersymmetric Yang-Mills theory

As recalled in the introduction, form factors are the most generic class of observables in any Yang-Mills theory. They are mixtures of gauge invariant operators and on-shell states ,

 ~F(O1…Oi,s1,…sj) (2.17)

In this article only the simplest form factors with one insertion of a gauge invariant operator will be considered. This will be referred to as a -point form factor. The simplest example is the two-point form factor which is also called the Sudakov form factor. In this article these form factors are considered in the context of SYM. The operator inserted will belong to the half-BPS stress-energy multiplet: the multiplet containing the (on-shell) stress-energy tensor as well as the self-dual and anti-self-dual field strengths. Another member of the multiplet which is usually useful for practical calculation is

 O=Tr(ϕ34ϕ34), (2.18)

where the field with anti-symmetric indices parameterizes the six scalars of the super Yang-Mills theory.

 F(1,…,n;q)=∫d4xe−iqx⟨1⋯n|O(x)|0⟩ . (2.19)

The momentum of the inserted operator in general does not square to zero, . In this sense it is ’off-shell’. This does not mean it does not satisfy field equations: the conservation of the stress-energy tensor in field theory for instance is typically only guaranteed up to field equations. For the form factor these are satisfied for all inserted operators in the problem.

Compared to progress on amplitude computation, form factors are much less studied and have only attracted renewed attention very recently, see for example vanNeerven:1985ja (); Alday:2007he (); Maldacena:2010kp (); Brandhuber:2010ad (); Bork:2010wf (); Brandhuber:2011tv (); Bork:2011cj (); Henn:2011by (); Gehrmann:2011xn (); Brandhuber:2012vm (); Bork:2012tt (). The current list of achievements at weak coupling beyond one loop in SYM is

Moreover, the three-loop result was only obtained after the much harder QCD computation Baikov:2009bg () (see also Gehrmann:2010ue ()). The same holds for the two-loop three-point computation whose QCD counterpart can be found in Gehrmann:2011aa (). Form factors at strong coupling (in 2D kinematics) were considered in Alday:2007he (); Maldacena:2010kp (). In the following mainly two- and three-point form factors will be considered, unless indicated otherwise. These have both special kinematics as well as special color properties. The two-point form factor has to be proportional to the color delta function and can only be a function of the momentum squared of the operator leg as well as, in gauge theory, the number of colors and the gauge coupling . Moreover, the two-point form factor has to be proportional to the tree form factor by on-shell supersymmetry,

 F(l)(1,2;q)=δabF(0)(1,2,q)f(q2,Nc,gym). (2.20)

In particular, the function does not depend on the inserted operator. As a function of and it can be rewritten in terms of powers of the ’t Hooft coupling,

 λ=g2ymNc, (2.21)

and, in general, at a finite loop order as a finite taylor series in . Non-trivial terms in this series are called non-planar corrections. Actually, as far as the color structure is concerned, the two-point form factor is the same as the two-point correlator which is computed to extract the function at high loop order vanRitbergen:1997va (). From this it follows that the first non-planar correction to the two-point form factor occurs at four loops, regardless of gauge theory under study. This point is worked out more fully in appendix B.

For general gauge group the number of colors in an calculation is replaced by appropriate Casimir operators. The three-point form factor can have more interesting dynamics and more complicated color factors. In particular, in addition to the structure constant for the gauge group the completely symmetric structure constant might make an appearance. On-shell supersymmetry still restricts the form factor to be proportional to the tree. Since the latter is known to be absent at two loops Brandhuber:2012vm () in SYM, the first appearance of this can be at three loops. Based on the details of the computation of the beta function at three loops in Avdeev:1980bh (); Tarasov:1980au (); Tarasov:1980kx () it seems likely it indeed does appear at this loop order. For more points more general color structures and operator-dependent pre-factors can and do appear.

3 General formalism

In this section first color-kinematic duality will be generalized to form factors. This is first done at tree level for clarity and then extended to loop level. In the second half of the section we outline how this conjecture can be used in general to compute the integrand of the form factor. This technique is then applied in the following sections in specific examples.

3.1 Color-kinematic duality for form factors

The color structure of form factors is different from that of amplitudes. To get our bearings a few explicit examples will be studied first. The simplest example is the two-point case which always has the trivial color factor where and are the color indices of the gluons. In particular, there is no Jacobi identity. This also holds for the three-point form factor at tree level as it can be written as the sum of three trivalent diagrams as

 F(0)3=~fa1a2a3δ(8)(λ1η1+λ2η2+λ3η3)⟨12⟩⟨23⟩⟨31⟩=~fa1a2bδa3bn12s12+~fa2a3bδa1bn23s23+~fa3a1bδa2bn31s31, (3.1)

where

 nij=sij3⟨12⟩⟨23⟩⟨31⟩δ(8)(λ1η1+λ2η2+λ3η3). (3.2)

The fermionic delta function is included, see e.g. Brandhuber:2010ad (); Bork:2010wf (). Hence this form factor is proportional to a single structure constant.

The four-point case becomes more non-trivial. There are different trivalent diagrams: there are five ways the gauge invariant operator can be inserted in the trivalent graphs of a four-particle amplitude. Some of the graphs are illustrated in figure 3. On poles of the Feynman graphs where a tree level amplitude is isolated color kinematic duality should hold. Consider for instance the sum of the three diagrams shown in Figure 3(a),

 D(4)=δa4cs123⎛⎜⎝~fa1a2b~fba3cn(4)12s12+~fa2a3b~fbca1n(4)23s23+~fa1a3b~fba2cn(4)13s13⎞⎟⎠. (3.3)

At the kinematic pole the residue given by the expression in the bracket becomes a four-point amplitude. Color-kinematic duality for the tree amplitude then imposes

 n(4)12∣∣s123=0=(n(4)23+n(4)31)∣∣s123=0. (3.4)

This is the lesser sense in which color kinematic duality should hold for form factors. The conjecture we make in this paper is that the duality also holds before taking any poles, in a sense to be made precise below.

Before writing more general formulas, let us illustrate the idea in the case of the tree level four-point form factor. In this case the form factor can be written over two classes of diagrams: those with the operator inserted on an external leg (class , see equation (3.3)) or the class with the operator inserted on the internal leg (),

 F(0)4=4∑i=1D(i)+E. (3.5)

Explicitly, follows from the diagrams such as shown in Figure 3(b)

 E=δbc~fa1a2b~fa3a4cn(12,34)s12s34+δbc~fa2a3b~fa4a1cn(23,41)s23s41+δbc~fa1a3b~fa2a4cn(13,24)s13s24, (3.6)

We conjecture that numerators can always be found such that they mirror the Jacobi identities of the color structures they multiply, i.e.

 n(i)12=n(i)23+n(i)31, (3.7)

as well as

 n(12,34)=n(23,41)+n(13,24). (3.8)

This is quite plausible as a simple counting argument shows. At tree level, there is a minimal basis for color factors modulo all Jacobi relations as explained in DelDuca:1999rs () (see also Boels:2012sy ()). For four particle color factors (“colormatics”) this minimal solution has two elements. Implementing this for both color as well as kinematic factors in equation (3.5) gives two equations in terms of unknowns, i.e. 2 for each and 2 for the type factors. This set of equations has many solutions. In fact, it is straightforward to check that the resulting equations have solutions with additional symmetries. In particular, one would expect that the different type diagrams are related by simple interchange symmetries.

One solution is to set all type numerators to zero and

 n(4)12 = 14(s12(s23+s24)F(1234)+s12s24F(1324)), (3.9) n(4)23 = 14(s24(s23+s12)F(1324)+s12s24F(1234)), (3.10)

where

 sij=(^pi+^pj)2with^pi=pi,i=1,2,3^p4=p4+q, (3.11)

and as well as are color-ordered form factors with the indicated ordering. The other are simply defined by interchanging with in the above formula. This solution was obtained by explicitly solving the system of equations.

In particular, in this solution it is seen that no constraints on the form factors are necessary: there are no BCJ-type Bern:2008qj () relations for the four-point tree level form factor. This is to be contrasted with the four-point amplitude computation where a similar computation shows that the duality imposes additional relations on the amplitudes. The color kinematic duality is expected to only hold for amplitude subgraphs separately. The sets of color-dual numerators do not in general sum to anything physical separately.

The above observation for four-point form factors can be generalized to form factors at both tree and loop level. At tree level, one should decompose the tree level form factor in terms of the place the gauge invariant operator is inserted,

 F(0)n=∑iF(i),0n. (3.12)

Then for each separately we conjecture color-dual numerators can be found. There is a considerable number of freedom to choose the numerators as a simple counting argument of equations and unknowns will yield.

Color-kinematic duality for loop form factors

At loop level a similar decomposition may be imposed. Then for every one introduces numerators and color factors as in the amplitude integrand case,

 F(k),ln=∑Γi∫l∏j=1dlDj1Sin(k)icis(k)i. (3.13)

Moreover, one can impose numerator symmetries as in the -type contributions above: if two diagrams are related by a symmetry, then their numerators are related. At tree level, by an extension of the counting argument above it is quite plausible that color-dual numerators always exist. At loop level it will be shown below that the conjecture gives strong constraints for the integrands of form factors.

This conjecture at loop level will be explored below in several examples in the next section. In the remainder of this section a complete overview is given over the calculational setup. The basic outline of this mirrors to quite some extent the setup for scattering amplitudes up to four loops as described in Bern:2012uf ().

3.2 Calculational strategy

The full integrand of a -loop form factor can be given as

 F(L)n=Kn∑σn∑i1SiCiIi , (3.14)

where we sum over a set of basis integrals,444The coupling constant will usually be suppressed in our results.

which are associated to -loop diagrams with only trivalent vertices. The classification of trivalent graphs will be discussed in the following subsection. We sum over all permutations of external on-shell legs. is some uniform kinematic factor. For two-point and three-point cases that we will consider

 K2=s212F(0)2,K3=F(0)3. (3.16)

The ’s are color factors. For a given trivalent graph, each cubic vertex can be dressed with a factor . In the form factor there is also a special vertex, the one connected to the off-shell leg that is dressed with the factor . The whole color factor is then given by taking the product of all ’s and the single . The ’s are symmetry factors. They account for the over-counting both from the sum over the permutations of external on-shell legs and the internal symmetries of the graph. In practice, for a given graph can be computed by randomly shuffling the list of edges and then counting the number of inequivalent isomorphic maps.

The main challenge is to determine the numerator for each basis integral. In principle, one can apply Feynman rules and sum over all Feynman diagrams. However, this becomes very complicated for higher loops, in particular for =4 SYM with its large number of fields. An alternative is to construct the amplitude based on traditional unitarity methods.

Here the conjectured color-kinematics duality for the loop integrand is used to construct a consistent ansatz which is then verified through unitarity cuts. For a given trivalent graph, one can apply the Jacobi relations to relate it to two other graphs, such as those shown in Figure 2,

 Ci=Cj+Ck⇒Ni=Nj+Nk. (3.17)

Note that no cuts for loop propagators are imposed here. For each propagator (except the two connected to the off-shell leg ), there is one such equation. By considering all graphs, one can obtain a set of equations for the numerators. This gives strong constraints on the numerators. The main job is to find a solution which satisfies the full set of equations, while also consistent with the unitarity cuts. Below the various substeps in the calculation are discussed in yet more detail.

3.2.1 Generating topologies

The first step in applying color-kinematics duality is to generate all trivalent graphs at a given loop order. There are mature techniques to generate inequivalent topologies; we have opted for DiaGen () because of its simplicity and ability to handle higher loop orders. Manipulations of graphs have generically been performed with Mathematica. For the problem at hand we need to make some particular choices.

1. The diagrams are trivalent, one-particle irreducible graphs, without bubble sub-graphs. For the form factor graph the vertex which is connected to the off-shell leg (corresponding to the operator) is special.

2. Since we mainly focus on =4 SYM, we further exclude most diagrams which contain triangle subgraphs, due to the no-triangle property of =4 SYM Bern:1994zx (). For form factors, a single triangle is allowed if it is connected to the off-shell leg . The triangle subgraphs we exclude are the one-loop and two-loop non-planar triangles555 For two-loop non-planar sub-triangles this concerns graphs like . . This is enough up to four loops. For five and higher loops, there can be three-loop or higher-loop triangle subgraphs, which may be excluded and can reduce further the number of topologies. (This further subtraction is not considered in the following table.)

3. The diagrams usually have the maximal number of loop propagators. In the construction with color-kinematics duality, one also needs to consider the graphs with less loop propagators but with more complicated trivalent tree legs, for example, the “snail-like” diagrams Bern:2012uf (). Such diagrams can be easily obtained by replacing a single leg with multi-leg trivalent tree graphs in lower-point cases. One could also generate these graphs by applying color-kinematics relations to those graphs containing maximal numbers of propagators. (In the following table only graphs with maximal numbers of loop propagators are counted.)

Based on the above selection rules, we obtain the number of different topologies for two- and three-point form factors and four- and five-point amplitudes up to six loops, as shown in Table 1.

From the table it follows that for higher-loop two-point form factors the number of graphs grows about as fast as for the four-point amplitude, but is typically smaller than the four point amplitude by a factor of .

3.2.2 Solving color-kinematics dualities

After generating the trivalent graphs, the constraints imposed by color-kinematic duality can be studied.

1. Given a trivalent graph, for each internal propagator (except the two connected to the off-shell leg ), one can apply the Jacobi identity to generate two other graphs. These two graphs can both be part of the basis set. Often, one of the two contains a triangle subgraph. In this case the corresponding numerator is set to zero. By considering all propagators of all graphs, one generates a set of equations for the numerator functions of the trivalent graphs in the basis.

2. Typically the numerators of a small set of graphs can be identified from which all other numerators can be obtained by the set of equations derived from the duality. The corresponding graphs for these numerators are called master integrals since Carrasco:2011mn (). These can be identified by inspecting the full set of equations: there is usually no unique choice.

3. One can construct the most general ansatz for the master integrals, under several simple constraints derived from physical expectations:

• One would expect the numerators to be local. In particular they should be polynomial functions of momenta (both loop and external).

• The full expression should not break the excellent UV properties of SYM. In practice this implies that for any -point one-loop subgraph, the numerators contain no more than powers of the loop momentum for that loop Bern:2012uf (). The exception is that if the one-loop subgraph is a one-loop form factor, one allows the maximal power to be .

• The numerator should preserve the symmetry of the corresponding graph.

• The numerator should be consistent with the maximal cut Bern:2007ct () (where all internal legs are cut). One can apply the simple “rung rule” in such cuts Bern:1998ug ().

By applying these constraints one obtains an ansatz for the master integrals with a number of free parameters.

4. Given the ansatz for master integrals, one can obtain the numerators of all other integrals in the basis by using a small set of Jacobi relations. We then impose the same constraints considered above for all integrals. This typically fixes a large number of parameters. It is then necessary to check if the obtained numerators satisfy the full set of color-kinematics equations. If all Jacobi relations are satisfied, we have found a color-dual solution.

5. It is not guaranteed that the solution obtained is physical and in general free parameters are left at this stage. To both fix these parameters and to make sure that the solution of the integrand is indeed physical, one should check it against a complete set of (preferably D-dimensional) unitarity cuts Bern:1994zx (); Britto:2004nc (). In the next subsection, we discuss how to do these unitarity checks in a systematic way.

3.2.3 Unitarity checks

On a given unitarity cut one can compute the cut integrand both from the ansatz as well as the product of lower-loop amplitudes and form factors, summed over the complete spectrum of the theory under study. For the solution obtained from the procedure outlined above the two expressions must be consistent with each other i.e

 Kn∑σn∑i1SiCiIi∣∣cuts=∫∏id4ηℓiFtree∏aAtreea . (3.18)

In practice it is convenient to consider color-stripped cuts involving tree amplitudes. The tree inputs are color-ordered amplitudes or form factors, and the cut integrand on the LHS can be also constructed without considering color factors.

Although the idea is straightforward, in practice there are two technical problems. One is to compute the product of tree amplitudes, where the main challenge is to do the summation over the on-shell states appearing on cut legs. The other is to extract the cut integrand from the ansatz, in particular when there is a large number of basis integrals.

Tree input

We can start from supersymmetric expressions for tree amplitudes and form factors, and take the product of them by integrating out the fermionic variable for each cut leg . For example, in the four-loop computation, we need to consider a quintuple cut which needs the following tree input

 ∫5∏i=1d4ηli[FMHV5(−l1,−l2,−l3,−l4,−l5)A¯¯¯¯¯¯¯¯¯¯¯MHV7(p1,p2,l5,l4,l3,l2,l1)+ (3.19) FNMHV5(−l1,−l2,−l3,−l4,−l5)AN2MHV7(p1,p2,l5,l4,l3,l2,l1)+ FN2MHV5(−l1,−l2,−l3,−l4,−l5)ANMHV7(p1,p2,l5,l4,l3,l2,l1)+ Fmax-non-MHV5(−l1,−l2,−l3,−l4,−l5)AMHV7(p1,p2,l5,l4,l3,l2,l1)] .

The main complexity is the fermionic integration as it involves high maximally helicity violating (MHV) degrees. This has been discussed for example in Elvang:2008vz (); Bern:2009xq (). In practice we find it convenient to apply the MHV rules as in Elvang:2008vz (). The supersymmetric form factor was developed in Brandhuber:2011tv (); Bork:2011cj ().

Cut integrand from the ansatz

For the higher-loop integrand given by a large number of integrals, there would be hundreds of cut diagrams contributing to some unitarity cuts. For example, for the above quintuple cut of the four-loop form factor, there are 192 cut diagrams to sum over. Obviously, one has to develop a systematic method to automatize the computation. In principle, one could start from the basis graphs, consider various possible cuts of them, and then collect the contributions. In practice, we find the following way more convenient. A similar idea has been used in Bern:2010tq ().

1. Rather than starting from the basis and then doing cuts, we “inverse” the procedure. We first construct the trivalent tree diagrams of amplitudes and form factors. We take them as the cut tree diagrams on the r.h.s. of (3.18).

2. Given the trivalent tree graphs, the next step is to construct loop diagrams by sewing trivalent tree diagrams in all possible ways. Since the tree diagrams are trivalent, the loop diagrams obtained are also trivalent. Not all such loop diagrams correspond to basis integrals. If a loop diagram is isomorphic to one of the basis, then it corresponds to a cut contribution of that basis. If it is not isomorphic to any basis, we can simply neglect it. In this way, by sewing all possible trivalent diagrams, we are guaranteed to obtain all possible cut contributions from all basis integrals, without double counting.

3. The cut integrand for each cut diagram is given by the product of the corresponding numerator and tree propagators. In particular, the numerator for a given cut diagram should be re-expressed in terms of external and cut momenta. As we mentioned that it is convenient to consider color-stripped cuts, it is not necessary to consider color factors here666We add a comment on the non-planar cut, which can be obtained by permuting the external legs of tree amplitudes. For these cases it is crucial to obtain the correct sign factor for each cut diagram that is related to the order of tree legs. One way to obtain these sign factors is to assign a color factor for each trivalent vertex of tree graphs, and compare the “glued” color factor with the color factor of the basis.. The full cut integrand for a given cut is then given by summing over all cut diagrams

 Kn∑cut graphsiNi∏(tree propagators) , (3.20)

which should be compared with the integrand obtained from the product of tree amplitudes.

The above procedure has been applied to a few known results up to three loops and always consistency has been found, which shows that the method is robust. We then apply it to the four-loop form factor, such as the above quintuple cut of four-loop form factors, which also provides consistent results.

4 Examples

In this section the general technology outlined in the previous section is applied to various form factors in SYM above one loop777It is easier to work out the one-loop two- and three-point form factors following the higher-loop examples considered here. which have appeared in the literature. Although the results are known, our techniques illustrate the existence and the power of the duality.

4.1 Two-point two-loop form factor

As a warm-up exercise, we consider first the two-loop two-point form factor. This result has been computed by Feynman graph methods in vanNeerven:1985ja ().

First, by equation (2.20) the two-point form factor in SYM is trivially dependent of the inserted operator through a tree factor, as long as it’s in the stress-energy tensor multiplet. This tree factor is factored out, as will be done in every two-point calculation in this article.

By the rules introduced in the previous section, there are only two trivalent graphs to consider as shown in Figure 4: a planar ladder and a non-planar ladder diagram. The Jacobi relations simply tell us that the numerators of both integrals are the same.

By the power counting constraint explained in the previous section, we find that the numerator should be independent of loop momenta. Hence by the kinematics of the problem the numerator should be proportional to a power of , which will be absorbed into the whole kinematic factor . The numerator is then a purely numerical constant. This numerical constant can be easily fixed by considering any (color-stripped) unitarity cut, which turns out to be one. At the same time, this unitarity cut verifies that nothing has been missed in the construction. In this simple example it is not hard to explicitly compute and verify all possible unitarity cuts, verifying that the result is physical.

The results including the color and symmetry factors are summarized in Table 2. The full form factor result can be obtained as

 F(2)2 = K2∑σ2b∑i=a1SiCiIi (4.1) = N2cδa1a2s212F(0)(4Ia+Ib), (4.2)

which reproduces exactly the known result vanNeerven:1985ja (). Note that the color and symmetry factors are responsible for the numerical integer factors, which are and for planar and non-planar graphs respectively.

4.2 Two-point three-loop form factor

As a more non-trivial example, the two-point form factor at three loops is calculated next by the procedure outlined above. This result has been computed by unitarity methods in Gehrmann:2011xn ().

First, by generating topologies we can find there are six trivalent diagrams, as shown in Figure 5.888There is one bubble-like graph containing a two-point tree leg which turns out not contribute. For simplicity we do not include it here. In the four-loop construction, such graphs are as shown in Figure 10..

By applying the color-kinematic relation to this set of trivalent diagrams, a set of equations can be obtained for the numerators. It turns out that one can choose the single integral (d) as the master integral. One can then make an ansatz for the numerator of this master integral by applying the following three constraints.

1. From the power counting property, the numerator should depend only linearly on the loop momentum and there should be no dependence on other loop momenta. A general ansatz is therefore given as (note that we have factorized a whole factor )

 Nansatzd(p1,p2,ℓ)=α1ℓ⋅p1+α2ℓ⋅p2+α3p1⋅p2, (4.3)

which contains three parameters .

2. The numerator should preserve the symmetry of the graph, which implies that it should be invariant under

 {p1,p2,ℓ}⟺{p2,p1,q−ℓ}, (4.4)

or more explicitly

 Nd(p1,p2,ℓ)=Nd(p2,p1,q−ℓ). (4.5)

Plugging in the ansatz (4.3), we obtain the relation

 α2=−α1. (4.6)
3. Finally, we consider the constraint of maximal cut. From the “rung rule” we read off the numerator . On the maximal cut we have

 [Nd(p1,p2,ℓ)−(ℓ−p1)2]∣∣maximal cut=0. (4.7)

This fixes the remaining two parameters

 α1=−1,α3=−1. (4.8)

Therefore, by applying the above constraints we arrive at a unique solution for the master integral

 Nansatzd=(p2−p1)⋅ℓ−p1⋅p2 . (4.9)

Given this solution, one can check that all Jacobi equations are satisfied. Other numerators can be obtained from the master integral by using the relations

 Na=Nb=Nc,Nd=−Ne=Nf, (4.10) Nb(p1,p2)=−Ne(p1,p2,ℓ)−Ne(p2,p1,ℓ) , (4.11)

where if not specified.

Now it is essential to check that the solution is indeed physical i.e. satisfies the unitarity cuts. A non-trivial quadruple cut is given as in Figure 6. The product of trees is

 ∫4∏i=1d4ηli[FMHV4(−l1,−l2,−l3,−l4)A¯¯¯¯¯¯¯¯¯¯¯MHV6(p1,p2,l4,l3,l2,l1)+ (4.12) FNMHV4(−l1,−l2,−l3,−l4)ANMHV6(p1,p2,l4,l3,l2,l1)+ Fmax-non-MHV4(−l1,−l2,−l3,−l4)AMHV6(p1,p2,l4,l3,l2,l1)] .

From the basis integrals, we obtain the cut integrand as a sum of cut diagrams. We have compared the two expressions numerically, and have found perfect agreement.

The results have been summarized including the color and symmetry factors in Table 3. The full form factor result can be obtained as

 F(3)2=s212F(0)2∑σ2e∑i=a1SiCiIi . (4.13)

Note that the graph (f) has zero color factor and therefore does not contribute to the final result of the form factor. However, it is necessarily involved in solving the Jacobi relations.

The result we obtain by applying color-kinematic duality seems quite different from that in Gehrmann:2011xn (). However, it is a simple check that the results are equivalent, by using the identities given in section 3 of Gehrmann:2011xn () between different integrals. Our result is presented in a much simpler form which involves only trivalent graphs. The numerical integer factors which have no easy interpretation in Gehrmann:2011xn () are also naturally explained here by the color and symmetry factors.

4.3 Three-point two-loop form factor

As an example containing more external legs, we present the three-point two-loop form factor in a form satisfying all Jacobi equations. The three-point two-loop form factor was computed in Brandhuber:2012vm ().

There are six trivalent diagrams, as shown in Figure. 7.

By inspecting the set of Jacobi relations, we can find that one needs at least two master integrals, which may be chosen as graph (a) and graph (d). The numerators of the other graphs can be obtained by the following relations

 Nb(p1,p2,p3)=Na(p2,p3,p1)+Na(p3,p2,p1), (4.14) Ne(p1,p2,p3,ℓ)=Na(p1,p2,p3)−Nd(p1,p2,p3,ℓ), (4.15) Nc=Nb,Nf=Nd, (4.16)

where if not specified.

Using the numerators in Brandhuber:2012vm () for two master integrals, one can check that all the Jacobi relations are automatically satisfied. The numerator solution is given in Table 4. The color and symmetry factors are also given in Table 4. The full result can be constructed from the table as

 F(2)3=F(0)3∑σ3e∑i=a1SiCiIi . (4.17)

It is a simple check that the result is equivalent to the one presented in Brandhuber:2012vm (). We have also checked that the solution passes various triple cuts following the procedure of section 3.2.3.

5 Four-loop two-point form factor

In this section the full four-loop integrand of the two-point form factor is constructed. To our knowledge this has never been computed before.

5.1 Outline of the construction

The construction is similar to the three-loop example. Here we outline the main steps.

1. There are in total trivalent diagrams to consider, which are as given in Figures 810. For completeness, we consider also three bubble-like graphs as shown in Figure 10.

2. By applying the Jacobi relations for all possible propagators of all 37 integrals, one obtains a set of equations for the numerator functions. Part of the equations are given in appendix A. From the equations, one can identify two integrals as the master integrals: (13) and (21), which both have planar topology.

3. We then construct an ansatz for the numerators of the master integrals. By power-counting, one can write down a most general ansatz as a polynomial of loop and external momenta for the numerator. For the master integral (21), the numerator depends at most quadraticly on , so a general ansatz is

 N21 = α1(ℓ3⋅p1)2+α2(ℓ3⋅p2)2+α3(ℓ3⋅p1)(ℓ3⋅p2) (5.1)

For the master integral (13), the numerator depends linearly on and , and a general ansatz is

 N13 = β1(ℓ3⋅p1)(ℓ4⋅p1)+β2(ℓ3⋅p2)(ℓ4⋅p2)+β3(ℓ3⋅p1)(ℓ4⋅p2)+β4(ℓ3⋅p2)(ℓ4⋅p1) +[β5(ℓ3⋅ℓ4)+β6(ℓ3⋅p1)+β7(ℓ3⋅p2)+β8(ℓ4⋅p1)+β9(ℓ4⋅p2)+β10(p1⋅p2)](p1⋅p2).

There are in total parameters. The numerators of all other basis elements can be determined from these two via Jacobi relations.

4. Then we can impose the constraints from the maximal cut999We apply the maximal cut and symmetry constraints for master integrals and other integrals together.. There are five planar integrals for which one can easily read off the numerators directly from the rung rule:

 Nrr1 = s212, (5.2) Nrr6 = s12(ℓ4−p1)2, (5.3) Nrr9 = s12(ℓ3−ℓ5)2, (5.4) Nrr13 = (ℓ3−p1)2(ℓ4−p1−p2)2, (5.5) Nrr21 = (ℓ3−p1)4. (5.6)

In the maximal cut the ansatz must be consistent with these coefficients i.e.

 (Ni−Nrri)∣∣maximal cut=0. (5.7)

These equations fix ten parameters.

5. The numerators should satisfy the symmetry of the diagrams. Since the numerators of many graphs are related to each other by Jacobi relations, it is enough to consider the symmetries of the following graphs

 {(1),(6),(9),(12),(20),(21),(25),(26)} . (5.8)

For example graph (26) has the symmetry:

 {p1,p2,ℓ3,ℓ4,ℓ5,ℓ6} ↔ {p2,p1,p12−ℓ3,ℓ5+ℓ6−ℓ4,ℓ6,ℓ5}. (5.9)

These symmetry constraints can fix parameters. Together with the above constraints from the maximal cut, one can fix in total 13 parameters, which can be explicitly given as

 α2=α1,α3=−4+2α1,α5=4−2α1−α4,α6=−2α1−α4, α7=α1+α4,β3=4+β2,β4=−4+β1, β5=β6=−β7=β10=−2,β8=2−β1,β9=−β2. (5.10)

Substituting these solutions into the numerators, we find that the full set of Jacobi equations are satisfied. As a consistency check, all the numerators have the required power counting behavior. It also turns out that the numerators of the bubble-like integrals (35)(37) are zero.

6. There is another simple cut one can apply for the basis integrals (31)(34), as shown in graph (31) in Figure 9. The cut integrand involves a three-point loop amplitude, which implies that the numerator should vanish under this cut. This simple constraint fixes two more parameters:

 β1=−α1,β2=2−α42. (5.11)

Therefore, we arrive at a solution of numerators which depends on only two parameters {}.

5.2 Unitarity checks

To make sure that the solution is indeed the form factor result, we perform a large number of unitarity cuts. In particular a non-trivial quadruple cut is given as in Figure 11. From the basis integrals, we obtain the cut integrand in terms of 192 cut diagrams from the following 18 integrals:

 {(1),(2),(3),(4),(5),(6),(7),(9),(10),(12),(13),(14),(17),(19),(21),(25),(30),(31)}. (5.12)

This should be compared with the product of trees, which involves non-trivial next-to-next-to-MHV tree amplitudes

 ∫5∏i=1d4ηli[FMHV5(−l1,−l2,−l3,−l4,−l5)A¯¯¯¯¯¯¯¯¯¯¯MHV7(p1,p2,l5,l4,l3,l2,l1)+ (5.13) FN<