On gauge amplitudes first appearing at two loops

The study of scattering amplitudes has seen great advances in recent years. On the more applied side, computing higher-point and higher-loop amplitudes in the Standard Model has allowed for more precise comparisons to data collected at particle colliders (see e.g. refs. Heinrich:2020ybq ; Andersen:2024czj and references therein). On the more formal side, amplitudes are fascinating theoretical objects in their own right. They provide insight into the behavior and symmetries of a theory, as well as exhibiting previously unforeseen mathematical structures. Having explicit analytic expressions for amplitudes is paramount for finding such structures, and for better understanding aspects of quantum field theory.

Often, direct calculation of amplitudes by evaluating Feynman diagrams can be bypassed for more computationally efficient methods. In particular, a general understanding of the singular behavior of amplitudes can allow them to be “bootstrapped” to higher orders in perturbation theory, or for a greater number of scattering particles. This program has had remarkable success in $\mathcal{N}=4$ supersymmetric Yang-Mills in the planar limit (see e.g. refs. Caron-Huot:2020bkp ; Dixon:2022rse and references therein).

Amplitudes in ordinary, non-supersymmetric Yang-Mills (YM) theory remain more challenging. There have been remarkable recent advances in computing the full-color all-helicity massless QCD amplitudes for $2\to 3$ scattering at two loops Agarwal:2023suw ; DeLaurentis:2023nss ; DeLaurentis:2023izi and for $2\to 2$ scattering at three loops Caola:2021rqz ; Caola:2021izf ; Caola:2022dfa . These amplitudes have a rather intricate analytic structure, and pushing directly to one more loop or one more leg may be difficult.

Another avenue for progress, which we will pursue here, is to investigate the simplest possible helicity configuration, called “all-plus”, when all $n$ outgoing gluons have the same positive helicity. Such amplitudes vanish for any $n$ in any supersymmetric massless gauge theory Grisaru:1976vm ; Grisaru:1977px ; Parke:1985pn , and therefore they vanish at tree level in YM theory. At one loop, in any massless gauge theory, their unitarity cuts vanish in four dimensions, and they are infrared (IR) and ultraviolet (UV) finite, rational functions of the spinor products of the external momenta, which are known for an arbitrary number of gluons Mahlon:1993si ; Bern:1993qk .

Self-dual Yang-Mills theory (sdYM) Yang:1977zf involves path integrals over only self-dual gauge field configurations. Classically, sdYM is integrable Belavin:1978pa ; Tze:1982gf ; Chau:1982mn . For free plane waves, such configurations include only the positive-helicity gluons. Interactions between plane waves include a $({-}{+}{+})$ vertex Parkes:1992rz ; Chalmers:1996rq , but not the parity conjugate $({+}{-}{-})$ vertex. At tree level, one can build the one-minus amplitude $({-}{+}{+}\cdots{+})$ by sewing together $({-}{+}{+})$ vertices, but this vanishes on shell. At loop level, the same sewing leads to the one-loop all-plus amplitudes Cangemi:1996rx ; Cangemi:1996pf , which validates the suggestion that the non-vanishing of these amplitudes can be considered an anomaly in the conservation of the currents associated with integrability of sdYM Bardeen:1995gk ; Bittleston:2022nfr .

At two loops, the connection to sdYM becomes less clear. Two-loop all-plus gauge theory amplitudes were first computed for four gluons using generalized unitarity Bern:2000dn ; Bern:2002tk . For five external gluons, the leading-color terms were computed first numerically Badger:2013gxa , and later analytically Gehrmann:2015bfy ; Dunbar:2016aux . The nonplanar integrands in the pure-glue theory were found in ref. Badger:2015lda , and the complete nonplanar results are available in refs. Agarwal:2023suw ; DeLaurentis:2023izi . For $n>5$, the polylogarithmic part of the leading-color result was proposed for arbitrarily many gluons in ref. Dunbar:2016cxp , and the rational part was computed using an augmented recursion relation for $n=6$ Dunbar:2016gjb and $n=7$ Dunbar:2017nfy . (The planar $n=6$ integrand was presented in ref. Badger:2016ozq .) Full-color results for $n=6$ in pure gauge theory were given in ref. Dalgleish:2020mof . The all-$n$ result for a particular color structure has been conjectured in ref. Dunbar:2020wdh , and checked numerically for $n=8$ and 9 in refs. Kosower:2022bfv ; Kosower:2022iju (where the $n<8$ rational results were also checked). Many of these results rely on $D$-dimensional generalized unitarity for the construction of integrands, although the polylogarithmic results in refs. Dunbar:2016aux ; Dunbar:2016cxp ; Dunbar:2019fcq carry out the cuts four-dimensionally, and the rational parts in refs. Dunbar:2016aux ; Dunbar:2016gjb ; Dunbar:2017nfy ; Dunbar:2019fcq ; Dalgleish:2020mof ; Dunbar:2020wdh are constructed recursively.

The connection between twistors, string theory, and tree-level gluon scattering amplitudes of (mostly) positive helicity goes back to Nair Nair:1988bq , Witten Witten:2003nn , the MHV rules of Cachazo, Svrček and Witten Cachazo:2004kj , and the derivation of these rules from the YM action by Mason Mason:2005zm . They have also been derived from a twistor action Boels:2007qn . These works clarify the relations between sdYM and tree level amplitudes. The MHV rules were applied to compute tree-level form factors of operators composed of anti-self-dual field strengths, e.g. $\text{tr}(F^{2}_{\rm ASD})$ Dixon:2004za . The all-plus and one-minus form factors for this operator were computed at one loop in a non-supersymmetric $SU(N)$ theory in ref. Berger:2006sh .

Recently, a novel bootstrap method for amplitudes in special theories has been suggested in ref. Costello:2022wso . It stems from a combination of ideas from celestial holography, twisted holography, and twistor theory. In some sense, it is a loop level generalization of the earlier tree-level work Mason:2005zm ; Boels:2007qn . In this method, the cancellation of an anomaly in a theory that lives in twistor space allows for the existence of a chiral algebra, the elements of which are in bijection with the states of the theory. The correlators of the chiral algebra correspond to form factors of the theory. The operator product expansions (OPEs) between the elements of the chiral algebra are used to constrain the pole-structure of correlators, the residues of these poles being lower-loop or lower-point correlators. In this way, one can bootstrap the form factors of these theories.

In ref. Costello:2023vyy , this celestial chiral algebra (CCA) bootstrap was used to compute a two-loop $n$-gluon all-plus-helicity form factor in sdYM with Weyl fermions transforming in the representation

$$R_{0}\equiv 8F\oplus 8\bar{F}\oplus{\wedge^{2}F}\oplus{\wedge^{2}\bar{F}}$$

(1)

of the Lie algebra of $SU(N)$. Here $F$ is the fundamental representation, and ${\wedge^{2}F}$ is the antisymmetric tensor representation. In terms of Dirac fermions, the representation has 8 fundamentals (quarks) plus one antisymmetric tensor. It solves the anomaly cancellation condition from the six-dimensional twistor-space theory Costello:2022wso ,

$$\text{tr}_{R_{0}}(X^{4})=\text{tr}_{G}(X^{4}),$$

(2)

for any generator $X$ of the $SU(N)$ Lie algebra, where $G$ denotes the adjoint representation. The form factor is for an operator $\tfrac{1}{2}\text{tr}(B\wedge B)$, involving an adjoint-valued, antisymmetric, anti-self-dual tensor field $B_{\mu\nu}$, which is used to enforce self-duality of the gauge field.

The sdYM form factor computed in ref. Costello:2023vyy should reproduce scattering amplitudes in YM for arbitrary $n$. Due to the anomaly cancellation condition, the one-loop amplitude should vanish in this theory. As we will see, this condition implies identities among the QCD all-plus partial amplitudes. The identities include the “three-photon vanishing” relations first noticed in ref. Bern:1993qk . A more general set of linear relations was found in ref. Bjerrum-Bohr:2011jrh ; we will show that these relations are all explained by the vanishing of the one-loop all-plus amplitude for representation $R_{0}$.

The relevant two-loop sdYM form factor was computed for all $n$ in ref. Costello:2023vyy . The four-point result is

$\displaystyle\begin{split}\mathcal{A}_{4,\text{sdYM}}^{\text{2-loop}}\ =\ \frac{g^{6}}{(4\pi)^{4}}\rho\bigg{[}&\bigg{(}12N-4\frac{s^{2}+4st+t^{2}}{st}-\frac{24}{N}\bigg{)}\big{(}\text{tr}(1234)+\text{tr}(1432)\big{)}\\ &+\bigg{(}24+\frac{24}{N}\bigg{)}\text{tr}(12)\text{tr}(34)\bigg{]}+\mathcal{C}(234),\end{split}$

(3)

where

$$\rho=i\frac{[12][34]}{\langle 12\rangle\langle 34\rangle},$$

(4)

and $s=(k_{1}+k_{2})^{2}$ and $t=(k_{2}+k_{3})^{2}$ are the four-point Mandelstam variables. We use the shorthand notation

$$\text{tr}_{R}(ij\cdots k)=\text{tr}_{R}(t^{a_{i}}t^{a_{j}}\cdots t^{a_{k}}),$$

(5)

which is the trace over the generators $t^{a}$ of the Lie algebra of $SU(N)$ in an arbitrary representation $R$. Throughout this paper, traces without a subscript, as in eq. (3), will mean the trace over fundamental-representation generators. The “$+\mathcal{C}(234)$” instructs one to add the two non-trivial cyclic permutations of $(2,3,4)$ acting on the previous expression.

In this paper, we wish to investigate the relation between the sdYM form factor given in eq. (3) and all-plus amplitudes in ordinary YM. The two-loop all-plus four-point amplitude in QCD was computed in dimensional regularization in refs. Bern:2000dn ; Bern:2002tk . Here we will replace the fermion loops for QCD (i.e. for fermions in the fundamental (+ antifundamental) representation only) with fermion loops in the representation $R_{0}$ in eq. (1). Then we can directly compare the form factor in sdYM to the two-loop amplitude in YM. The double-trace term is not provided in ref. Costello:2023vyy , so we compute it in Appendix B. Our results agree only after UV renormalization and after subtracting off the universal two-loop IR divergences given by Catani Catani:1998bh . This statement does not disprove eq. (3); rather, the discrepancy most likely arises from the fact that the CCA bootstrap technique keeps all momenta four-dimensional, in contrast to dimensional regularization. We resolve the discrepancy by using a different IR regularization scheme, namely a mass regularization of the loop integrands. With this scheme, the two-loop four-point sdYM form factor equals the YM amplitude, and we suppose that the same will be true for $n>4$. We also argue that the $n$-gluon sdYM result gives the finite remainder of the YM amplitude in dimensional regularization. This result could provide a check of higher-point two-loop all-plus helicity amplitudes, once all the fermionic and subleading-color terms become available.

In this section, we provide a non-rigorous overview of a method used to bootstrap certain two-loop amplitudes Costello:2023vyy . We will refer to this method as the celestial chiral algebra (CCA) bootstrap. Positive- and negative-helicity states of sdYM on twistor space are in one-to-one correspondence with local operators in an (extended) chiral algebra. The conformal blocks of this algebra are the local operators in the self-dual theory. Therefore, correlation functions of the chiral algebra in a given conformal block correspond to form factors of the gauge theory. Moreover, the OPEs in the algebra are collinear limits of states in the field theory. This suggests that one can use the chiral algebra to “bootstrap” form factors of sdYM by using the analytic properties of the OPEs.

A requirement for the existence of a chiral algebra is the associativity of its OPEs. Associativity fails at the first loop correction for pure gauge theory, due to a gauge anomaly arising from the all-plus helicity amplitude on twistor space. In order to remedy this, a fourth-order scalar field that couples to the Yang-Mills topological term was introduced in refs. Costello:2021bah ; Costello:2022upu ; Costello:2022wso . However, the mechanism can only cancel double-trace contributions, and so it is necessary for the gauge group to not have an independent quartic Casimir structure. Alternatively, the anomaly can be cured by introducing fermions in special representations of the gauge group Costello:2023vyy . In particular, the requirement is that the quartic Casimir in the adjoint representation is exactly that in the (real) representation $R$

$$\text{tr}_{G}(X^{4})=\text{tr}_{R}(X^{4}).$$

(6)

For $SU(N)$ guage theory, one such example of this type of representation is $R_{0}$ given in eq. (1).

With this choice of matter representation, the one-loop OPEs are associative. Therefore, the chiral algebra exists for this theory and can be used to compute form factors. In fact, associativity constrains all form factors of self-dual Yang-Mills (plus matter) to be rational functions, with poles only in the spinor products $\langle ij\rangle$. The chiral algebra OPEs determine all possible poles in the form factor, and the residues of these poles are chiral algebra correlators that have fewer external states or are at lower loop order. In this way, one can determine the $n$-point form factors inductively.

The form factor of most interest is the one with the operator

$$\frac{1}{2}\text{tr}(B\wedge B),$$

(7)

inserted at the origin,¹¹1 We mean the origin in position space $x$. The $x$-dependence of the correlator is $\propto\exp(i\sum_{j=1}^{n}k_{j}\cdot x)$ where $k_{j}$ are the gluon momenta. where $B$ is the adjoint-valued anti-self-dual two-form appearing in the sdYM Lagrangian Chalmers:1996rq ,

$${\cal L}_{\rm sdYM}=\text{tr}(B\wedge F).$$

(8)

Deforming the self-dual Lagrangian by $\tfrac{1}{2}g^{2}\text{tr}(B\wedge B)$ and integrating out $B$ yields the regular Yang-Mills Lagrangian, up to a topological term which does not affect the perturbation theory. So form factors of self-dual Yang-Mills with the operator $\tfrac{1}{2}\text{tr}(B\wedge B)$ inserted at the origin are amplitudes of ordinary Yang-Mills theory.

Using the CCA bootstrap, massless QCD amplitudes with matter in the representation (1) were computed at tree level Costello:2022wso , one loop Costello:2022upu , and two loops Costello:2023vyy for the two-minus, one-minus, and all-plus helicity configurations, respectively. The two-loop all-plus four-point sdYM form factor is¹¹1 The double-trace term was not provided in ref. Costello:2023vyy . However, Appendix B of ref. Costello:2023vyy outlines the computation of the color factors, so that one can keep track of the double-trace terms if desired relatively easily; see Appendix B.

$$\mathcal{A}_{4,\text{sdYM}}^{\text{2-loop}}=g^{6}\Bigl{[}A^{\text{2-loop}}_{4;1,\text{sdYM }}\big{(}\text{tr}(1234)+\text{tr}(1432)\big{)}+A^{\text{2-loop}}_{4;3,\text{sdYM}}\text{tr}(12)\text{tr}(34)\Bigr{]}+\mathcal{C}(234),$$

(9)

where

$\displaystyle\begin{split}A^{\text{2-loop}}_{4;1,\text{sdYM}}=&~{}\frac{i}{(4\pi)^{4}}\Biggl{[}(6N-4-8N^{-1})\bigg{(}\frac{[12][34]}{\langle 12\rangle\langle 34\rangle}+\frac{[14][23]}{\langle 14\rangle\langle 23\rangle}\bigg{)}-(4+8N^{-1})\frac{[13][24]}{\langle 13\rangle\langle 24\rangle}\\ &-2\frac{[12][34]}{\langle 12\rangle\langle 34\rangle}\frac{\langle 13\rangle\langle 24\rangle+\langle 14\rangle\langle 23\rangle}{\langle 12\rangle\langle 34\rangle}-2\frac{[14][23]}{\langle 14\rangle\langle 23\rangle}\frac{\langle 13\rangle\langle 24\rangle+\langle 12\rangle\langle 34\rangle}{\langle 14\rangle\langle 23\rangle}\Biggr{]}\end{split}$

(10)

and

$\displaystyle A_{4;3,\text{sdYM}}^{\text{2-loop}}=\frac{8i}{(4\pi)^{4}}(1+N^{-1})\bigg{(}\frac{[12][34]}{\langle 12\rangle\langle 34\rangle}+\frac{[13][24]}{\langle 13\rangle\langle 24\rangle}+\frac{[14][23]}{\langle 14\rangle\langle 23\rangle}\bigg{)}\,.$

(11)

This expression can be simplified using the Schouten spinor identity and four-point momentum conservation, which includes the result that $\rho$ is totally symmetric,²²2Our overall normalization of form factors and amplitudes differs from ref. Costello:2023vyy by a factor of $i$.

$$\frac{\rho}{i}=\frac{[12][34]}{\langle 12\rangle\langle 34\rangle}=\frac{[13][24]}{\langle 13\rangle\langle 24\rangle}=\frac{[14][23]}{\langle 14\rangle\langle 23\rangle}\,.$$

(12)

Then eqs. (10) and (11) collapse to eq. (3). However, when computing $n$-point form factors based on lower-point ones, one must remember not to use lower-point momentum conservation to simplify the lower-point form factors, as it is the sum of the $n$ gluon momenta that is conserved, not a subset of them.

With this in mind, the $n$-point color-ordered amplitude is constructed recursively, based on eqs. (10) and (11), and is given by

$\displaystyle\begin{split}\mathcal{A}^{\text{2-loop}}_{n,\text{sdYM}}&=g^{n+2}\Biggl{[}\sum_{\sigma\in S_{n}/\mathbb{Z}_{n}}\text{tr}(\sigma_{1}\cdots\sigma_{n})\\ &\hskip 42.67912pt\times\sum_{1\leq i<j<k<l\leq n}A^{\text{2-loop}}_{4;1,\text{sdYM}}(\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{l})\frac{\langle\sigma_{i}\sigma_{j}\rangle\langle\sigma_{j}\sigma_{k}\rangle\langle\sigma_{k}\sigma_{l}\rangle\langle\sigma_{l}\sigma_{i}\rangle}{\langle\sigma_{1}\sigma_{2}\rangle\langle\sigma_{2}\sigma_{3}\rangle\cdots\langle\sigma_{n}\sigma_{1}\rangle}\\ &+\sum_{c=3}^{\lfloor n/2\rfloor+1}\sum_{\sigma\in S_{n}/S_{n;c}}\text{tr}(\sigma_{1}\cdots\sigma_{c-1})\text{tr}(\sigma_{c}\cdots\sigma_{n})\sum_{1\leq i<j<k<l\leq n}A_{n;c,\text{sdYM}}^{\text{2-loop}}(\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{l})\Biggr{]}\,,\end{split}$

(13)

where $S_{n;c}$ is the subgroup of $S_{n}$ consisting of permutations that keep the double-trace structure $\text{tr}(1,\dotsc,c-1)\text{tr}(c,\dotsc,n)$ invariant. $A_{n,c,\text{sdYM}}^{\text{2-loop}}(i,j,k,l)$ is the kinematic factor that multiplies this double-trace structure for the form factor with energy-level-1 insertions at $i,j,k,l$ (as explained in Appendix B). It is defined as

$$A_{n;c,\text{sdYM}}^{\text{2-loop}}(i,j,k,l)=\frac{A_{4;3,\text{sdYM}}^{\text{2-loop}}(i,j,k,l)\langle ij\rangle^{2}\langle kl\rangle^{2}}{\langle 12\rangle\langle 23\rangle\cdots\langle c-1,1\rangle\langle c,c+1\rangle\langle c+1,c+2\rangle\cdots\langle n,c\rangle}$$

(14)

for $1\leq i<j\leq c-1$ and $c\leq k<l\leq n$, and it is zero otherwise. In Appendix B, we prove eq. (14) using the CCA bootstrap.

Note that for fermionic matter in $R_{0}$ there is no triple trace contribution, which would be present generically. The triple-trace cancellation is a consequence of the recursive construction, and its absence for $n=4$ since $\text{tr}(t^{a})=0$ in $SU(N)$.

We wish to check eqs. (10)–(14) in the simplest case, $n=4$, via an alternative method. We will use the fact that the two-loop four-gluon amplitudes were computed in QCD in dimensional regularization Bern:2000dn ; Bern:2002tk in a color-decomposed form which makes it straightforward to modify the fermion representation to $R_{0}$.

Before doing the two-loop color algebra, we first warm up by computing the one-loop all-plus $n$-point amplitude, which vanishes (non-trivially) in this theory due to the anomaly cancellation (2).

Here, we compute the one-loop all-plus $n$-point amplitude for massless QCD with matter in the representation (1). Color-decomposition plays a crucial role in this computation. We begin by reviewing the color-decomposition of one-loop $n$-gluon amplitudes in QCD for gauge group $SU(N)$ with matter in the representation $N_{F}(F\oplus\bar{F})$, where $N_{F}$ is the number of quark flavors.

We now turn to the computation of the two-loop all-plus four-gluon amplitude for fermions in the representation $R_{0}$.

Our starting point is the two-loop four-gluon amplitude in QCD, which is given in ref. Bern:2002zk as

$$\mathcal{A}^{\text{2-loop}}_{\text{4,QCD}}=\mathcal{A}_{G}^{\text{adj}}+\mathcal{A}^{\text{fund}}_{F},$$

(33)

where $\mathcal{A}_{G}^{\text{adj}}$ is the adjoint gluon contribution and $\mathcal{A}_{F}^{\text{fund}}$ is the fundamental matter contribution. Each particle contribution above can be decomposed into a sum of “parent” diagrams,

$$\mathcal{A}^{\text{rep}}_{X}=g^{6}\sum_{D_{i}}\Big{[}(C_{\text{rep}})_{1234}^{D_{i}}A_{X1234}^{D_{i}}+(C_{\text{rep}})_{3421}^{D_{i}}A_{X3421}^{D_{i}}\Big{]}+\mathcal{C}(234),$$

(34)

where each $D_{i}$ corresponds to a parent diagram. The subscript $X\in\{G,F\}$ denotes either the pure-gluon contribution $G$, or a fermion $F$ propagating in at least one of the two loops in the diagrams. The quantities $C_{\text{rep}}$ denote the color factors associated to the kinematic factors $A_{X}$, with “rep” signifying the gauge group representation in which particle $X$ resides. The $F$ parent diagrams span the space of all independent four-gluon color-factors with a fermion-loop contribution and a non-vanishing kinematic factor. This result can be shown by applying the Jacobi identity suitably to color diagrams containing triangle subdiagrams.

We want to compute the two-loop four-gluon amplitude with matter in the representation $R_{0}$,

$$\mathcal{A}_{4}^{\text{2-loop}}=\mathcal{A}^{\text{adj}}_{G}+\mathcal{A}^{R_{0}}_{F}\,.$$

(35)