BRST-Lagrangian Double Copy of Yang–Mills Theory

Leron Borsten, ∗ Branislav Jurčo, † Hyungrok Kim, ‡ Tommaso Macrelli, § Christian Saemann, ¶ and Martin Wolf ∗∗ Maxwell Institute for Mathematical Sciences Department of Mathematics, Heriot–Watt University Edinburgh EH14 4AS, United Kingdom Charles University Prague Faculty of Mathematics and Physics, Mathematical Institute Prague 186 75, Czech Republic Department of Mathematics, University of Surrey Guildford GU2 7XH, United Kingdom (Dated: July 29, 2020)


INTRODUCTION AND SUMMARY
Yang-Mills scattering amplitudes have been conjectured to satisfy a color-kinematic (CK) duality [1][2][3]: each amplitude can be written as a sum over purely trivalent graphs such that the kinematical numerators satisfy the same antisymmetry/Jacobi identities as the color contributions. CK duality has been shown to hold at tree level [4][5][6][7][8][9][10][11][12]. If it holds, replacing the color contributions of a Yang-Mills amplitude with another copy of the kinematical contributions yields a gravity amplitude [3]. This is known as the double copy prescription. For reviews and references see [13][14][15].
We make the crucial observation that CK duality violations due to longitudinal gluon modes can be compensated by harmless field redefinitions of the Nakanishi-Lautrup (NL) field. The Ward identities of the BRST symmetry then allow us to transfer CK duality from gluon amplitudes to those involving ghosts. Finally, onshell tree-level CK duality on the BRST-extended field space turns out to suffice to show that the BRST-Lagrangian double copied theory provides the loop integrands of a consistent perturbative quantization of N = 0 supergravity.
A longer paper explaining the origin of the double copy in terms of homotopy algebras and giving explicit expressions for many of the steps discussed only abstractly in the following is in preparation [51].

THE BRST-LAGRANGIAN DOUBLE COPY
We start with an abstract perspective on the double copy. Any Lagrangian field theory is equivalent to a field theory with exclusively cubic interaction terms, by blowing up higher order vertices using auxiliary fields, cf. also [52,53]. A generic such action is where the fields Φ I are elements of some field space F and the DeWitt index I encodes all field labels (including position x). Summation and space-time integration over repeated indices are understood. We are interested in theories invariant under a gauge symmetry described by a BRST operator Q.
Applying the strictification procedure also to the Batalin-Vilkovisky (BV) action before gauge fixing, it is not hard to see that one can always reduce the gauge transformations of all fields to be at most cubic in the fields: We further require that fields split into left and right components (with independent left and right ghost numbers), but over a common space-time point. Consequently, we write a DeWitt index I as (α,ᾱ, x) and assume that with g αβ andḡᾱβ graded (with respect to the ghost numbers) symmetric, and f δA βγ , etc., differential operators with constant coefficients. The indices A andĀ range over the summands in f IJK . To simplify notation, we define Suppressing the position dependence, the Lagrangian of the theory becomes where we used the shorthand f αβγ fᾱβγ for the evident expression in (3c). Analogously, we want the BRST operator acting on left and right indices separately. Splitting the BRST operator Q into Q L + Q R , we require wherefᾱ βγδ = 3fᾱ εδfε βγ and similarly for Q R Φ.
To double copy means to replace the left (or right) sector with a copy of the right (or left) sector of some, not necessarily the same, theory written in the form (4). If the resulting action S and BRST operator Q satisfy again the relations Q 2 = 0, QS = 0, we obtain a consistently gauge-fixed theory ready for quantization.
It is not hard to see that Q 2 L/R = 0 iff Q 2 L/R = 0; the condition Q L Q R + Q L Q R = 0 may induce further conditions. For Yang-Mills theory, one readily computes that CK duality suffices for the condition QS = 0. If CK duality fails to hold up to certain terms, then QS = 0 also fails to hold up to the same terms, possibly multiplied by other fields and their derivatives. (Mathematically, the terms describing the failure of CK duality generate an ideal in the algebra of fields and their derivatives. The expressions QS and Q 2 take values in this ideal.)

PRELIMINARY OBSERVATIONS
We start with some general, field theoretic observations. We are interested in perturbative aspects and omit any non-perturbative issues. Also, we are interested in n-point amplitudes up to ℓ loops for n and ℓ finite. Thus, there is always a number N ∈ AE so that monomials of degree m > N can be neglected in the Lagrangian. We always use the term "amplitude" for onshell states and the term "correlator" for offshell states.
Observation 1. If two field theories have the same tree amplitudes, then the minimal models of their L ∞ -algebras coincide, cf. [52,53]. If they have the same field content and kinetic parts, then they are related by a local (invertible) field redefinition.
Observation 2. Two field theories are quantum equivalent, if all their correlators agree. Since correlators can be glued together from tree level correlators (up to regularization issues), it suffices if the latter agree.
Observation 3. A shift of a field by products of fields and their derivatives which do not involve the field itself does not change the path integral measure. Local field redefinitions that are trivial at linear order produce a Jacobian that is regulated to unity in dimensional regularization [54], see also [55]. Therefore, they preserve quantum equivalence.
We now turn to the BRST symmetry of Yang-Mills theory, starting from the BV form [56] of the Yang-Mills Lagrangian on Minkowski space, using canonical notation for all fields, with g the Yang-Mills coupling constant. We use the gauge fixing fermion Ψ : where ψ a is of ghost number 0 and depends at least quadratically on the fields and their derivatives. We obtain the gauge-fixed Lagrangian For ψ a = 0, we recover the R ξ -gauges. The BRST transformations are satisfying Q 2 YM = 0 offshell. The non-physical fields enlarge the one-particle field space of asymptotic onshell states by four types of states: the two unphysical polarizations of the gluon, called forward and backward and denoted by A ↑ and A ↓ , and the ghost and antighost states [36]. All amplitudes will be built from this BRST-extended onshell n-particle field space, which carries an action of the linearization of (9) denoted by Q lin YM . The physical polarizations are singlets, Q lin YM A phys = 0, and we have two more doublets: where the ellipsis indicates terms that arise from Ψ 1 . Performing shifts b a → b a + X a and Ψ 1 → Ψ 1 + Ξ 1 with Ξ 1 := d d xc a Y a induces a shift of (8) by If X a is independent of the NL field b a , this modification preserves the theory at the quantum level by Observation 3. Furthermore, if X a is at least quadratic in the fields, this transformation preserves the action of Q lin YM on the BRST-extended onshell field space. Consider now the special case ψ a = 0 and X a independent of b a and fix Y a iteratively such that the linear terms in b a of (11) vanish: This leads to the following observation: Observation 4. Terms in the Lagrangian of the form (∂ µ A µ ) a X a with X a at least quadratic in the fields and their derivatives but independent of the NL field can be removed in R ξ -gauges by shifting the NL field. This creates additional terms (11) which are at least of fourth order and preserve the amplitudes by Observation 3.
Observation 5. Terms in the action that are proportional to a NL field can be absorbed by choosing a suitable term ψ a . This leaves the physical sector invariant but it may modify the ghost sector. Because NL fields appear as trivial pairs in the BV action, it is not hard to see that this extends to general gauge theories, e.g. with several NL fields and ghosts-for-ghosts.
Observation 6. The set of connected correlation functions is BRST-invariant because the connected correlation functions can be written as linear combinations of products of correlation functions.
Crucial to our discussion are Ward identities. Consider first the supersymmetric onshell Ward identities (see e.g. [57,58]) for the supersymmetry generated by the BRST operator.
Since the free vacuum is invariant under the action of Q lin YM , we have the following onshell Ward identities: We now apply the onshell Ward identity to O 1 · · · O n = A ↑c (cc) k A n−2k−2 phys and obtain 0|(cc) k+1 A n−2k−2 phys |0 ∼ 0|A ↑ (cc) k bA n−2k−2 phys |0 . (14) Thus: Observation 7. Any amplitude with k + 1 ghostantighost pairs and all gluons transversely polarized is given by a sum of amplitudes with k ghost pairs.
From the construction of amplitudes via Feynman diagrams, it follows that we also have the following onshell Ward identity for an approximate BRST symmetry.
Observation 8. Suppose that QS = 0 and Q 2 = 0 only onshell. Then, we still have (13) together with a corresponding identification of amplitudes with k + 1 ghostantighost pairs and all gluons transversely polarized and a sum of amplitudes with k ghost pairs.
We shall also need the offshell Ward identities for the BRST symmetry, where j µ is the BRST current. The left-hand side vanishes after integration over x, and using Observation 6, we can restrict to connected correlators at a particular order in the coupling constant g and then further to lowest order in , i.e. to tree level. Consider now operators O i (x i ) for those restricted Ward identities which are linear in the fields.
Observation 9. The onshell relations between tree amplitudes from Observation 7 induced by (13) extend to (offshell) tree-level connected correlators. For example, We also make the following three observations regarding the double copy.
Observation 11. For amplitudes in CK-dual form, there is a corresponding Lagrangian whose partial amplitudes produce the kinematical numerators [46]. Observation 12. Double copying the Yang-Mills tree amplitudes in CK-dual form yields the tree amplitudes of N = 0 supergravity [1][2][3].

CK-DUAL YANG-MILLS THEORY
In order to BRST-Lagrangian double copy Yang-Mills theory, we first must bring its action into the normalized form (4). Our goal will be to construct abstractly a Lagrangian which guarantees tree-level CK duality for the BRST-extended onshell field space. CK duality of the Feynman diagrams for the field space of physical gluons can be guaranteed by adding terms to the Lagrangian [3,46] and subsequently strictifying these, i.e. introducing a set of auxiliary fields such that all interaction vertices are cubic. This strictification is mostly determined by the color and momentum structure of the additional terms in the Lagrangian.
It remains to ensure CK duality for tree amplitudes involving ghosts or backward polarized gluon states, which we do by introducing compensating terms, preserving quantum equivalence. (Forward polarized gluons can be absorbed by residual gauge transformations and therefore do not appear in the Lagrangian. Thus, they cannot contribute to CK duality violations.) We implement the necessary changes iteratively for npoint amplitudes, starting with n = 4, and within each n iteratively for the number k of ghost-antighost pairs.
We start at n = 4, k = 0. First, we compensate for CK duality violations due to backward polarized gluons, which can be done by introducing terms of the form (∂ µ A µ ) a X a . By Observation 4, we can produce such terms, preserving quantum equivalence. This shift also produces terms − ξ 2 (X a ) 2 , which does not affect CK duality of higher n-point amplitudes since it preserves the gluon amplitudes (and thus their strictification).
We then increase k by 1 and consider connected treelevel correlators of the form ccA 2 phys . Each of these correlators is determined by 4-gluon correlators with a forward-backward gluon pair and a pair of physical gluons by Observation 9. We use the strictification of the 4-gluon amplitudes to derive a CK-dual description of the amplitude with one ghost-antighost pair. Using Observation 11, we then construct new ghost terms in the Lagrangian, manifestly preserving tree-level correlators and thus quantum equivalence, cf. Observation 2. Finally, we again compensate for CK duality violations in amplitudes due to backward polarized gluons in the same manner as for the 4-gluon amplitudes.
The iteration should then be evident: for each n, iterate over the possible numbers k of ghost-antighost pairs, and create new ghost terms with subsequent compensation for contributions of backward polarized gluons. Once completed, set k = 0 and increase n by one. Perform the compensation for contributions of backward polarized gluons to (n + 1)-point gluon amplitudes; then start increasing the ghost number again. We iterate this prescription until we reach the highest point tree-level correlator that can contribute to the loop order in which we are interested.
The resulting Lagrangian L CK YM is of the form (4) and quantum equivalent to the Lagrangian L YM given in (8).

THE BRST-LAGRANGIAN DOUBLE COPY OF YANG-MILLS THEORY
We now turn to the N = 0 supergravity side. The gauge-fixed BRST Lagrangian L N =0 of this theory is readily constructed. The following two diagrams concisely summarize the theory's field content, describing the symmetrized and antisymmetrized tensor products of Yang-Mills fields: Here, the physical fields of ghost number 0 are h µν (the metric perturbation about the Minkowski vacuum), B µν (the Kalb-Ramond two-form), and depending on frame, π or δ (the dilaton). Ghost number increases by column from left to right, and all vector/form indices are made explicit. Many fields come with a triad of ghost, antighost, and NL fields as indicated counterclockwise around the field by arrows. In addition to the expected BRST field content, we have two trivial BV pairs (δ, β) and (β, π) due to the presence of the dilaton. See [51] for more details, as well as [37][38][39][40]43]. The double copy of Q YM and L CK YM yields a BRST operator Q which satisfies Q 2 = 0 onshell and a Lagrangian L for the field content (17). The latter is quantum equivalent to N = 0 supergravity, as we now argue.
(i) Kinematic equivalence: The two kinematic Lagrangians are equivalent and linked by evident suitable field redefinitions [51]. The existence of such a field redefinition is ensured by the linear double copy BRST operator Q lin [39,43], which is equivalent to the linear BRST operator Q lin N =0 of N = 0 supergravity and annihilates the quadratic double copy Lagrangian [51]. We implement the field redefinition on L N =0 , obtaining L ′ N =0 . (ii) Equivalence of physical Lagrangian: Since the classical Yang-Mills action was written in a form with purely cubic, local interactions with manifest CK duality to all points, the tree amplitudes of L for physical fields match those of L N =0 , cf. Observation 12.
The difference between L and L N =0 after integrating out all auxiliary fields and putting all unphysical fields to zero therefore consists of interaction terms proportional to Φ, for Φ a physical field. This difference can be absorbed in a local field redefinition (which can be shown not to involve derivatives), preserving quantum equivalence by Observation 3. Thus, the two theories have the same tree-level correlators for physical fields. We implement the field redefinition on L ′ N =0 , obtaining L ′′ N =0 .
(iii) Gauge fixing sector: The difference between L, after integrating out all auxiliary fields, and L ′′ N =0 proportional to any of the NL fields (β,β, ̟ µ , π, γ, α µγ ) can be absorbed in a choice of gauge fixing which will only create new terms in the ghost sector, cf. Observation 5. We implement this new gauge fixing, and take over the strictification from L, obtaining L ′′′ N =0 together with Q ′′′ N =0 . (iv) Ghost sector: We now proceed in the same way as for Yang-Mills theory: reconstruct a ghost sector via the offshell Ward identities of Observation 9, leading to a Lagrangian L CK N =0 . Since the tree-level correlators are preserved by definition, the strictified Lagrangian L CK N =0 is quantum equivalent to L ′′′ N =0 , and BRST symmetry is preserved (with induced BRST action on auxiliary fields arising from strictification).
Both L and L CK N =0 are local and have the same field content. The tree-level correlators involving physical and NL fields agree. Using the approximate Ward identities, cf. Observation 8, and the fact that Q lin and Q CK,lin N =0 agree, we deduce that all tree amplitudes involving ghost-antighost pairs agree, too. By construction, this agreement extends to individual onshell Feynman diagrams, between the strictifications L and L CK N =0 , even for auxiliary fields: we can iteratively split off external vertices with two external legs, exposing Feynman diagrams with onshell external but offshell auxiliary fields. Up to a field redefinition of the auxiliaries, these also must agree.
The only potential remaining difference between L and L CK N =0 is then interaction terms containing Γ and Γ terms for Γ a ghost field. Going through the construction, one can argue that such terms, if they are there, have to appear in the same way in L and L CK N =0 . Alternatively, one can show that both theories satisfy the same Ward identities for tree-level correlators, rendering them quantum equivalent by Observation 2. The simplest argument, however, is to use Observation 1 to note that both theories are related by a local field redefinition. Observation 3 then implies that both theories are quantum equivalent.
Data Management. No additional research data beyond the data presented and cited in this work are needed to validate the research findings in this work.