Berends-Giele currents for extended gravity

In this short paper, we write down the Berends-Giele (BG) currents for extended gravity explicitly and discuss the unifying relations of these BG currents. This new tool, different from the double field theory current formally, may deepen our understanding of the current Kawai-Lewellen-Tye (KLT) relation.


Introduction
Gravity is still the most mysterious thing in our nature.The exploration of a quantum theory of gravity never stops.The first trial of quantum gravity is the perturbative gravity theory [1,2,3], which regards the gravitons as the fluctuation of the flat metric.In this way, we can obtain the Lagrangian of the pure graviton theory by evaluating the Einstein-Hilbert action.The derivation of the Feynman rules and the amplitudes are straightforward.After the amazing discovery of duality between the Anti-de Sitter gravity and the Conformal field theory (AdS/CFT) [4,5], we began to learn how to deal with the non-perturbative gravity theory from the field theory perspective.
For a more general gravity theory, say extended gravity theory, we also include the antisymmetric B-fields and dilatons.The theory, as a generalization of the traditional gravity, comes from the effective action of string theory.This theory is also the result of the double copy prescription of the Yang-Mills (YM) theory [6].Our knowledge of the double copy between the YM amplitudes and the extended gravity amplitudes is very rich.On the contrary, the "double copy" property for the off-shell objects of both sides is still not clear.In addition, many relations for amplitudes do not hold exactly for off-shell objects.Hence to see how these amplitude relations are realized on off-shell objects will be interesting.In this paper, we will focus on the simplest off-shell object, Berends-Giele (BG) currents, which can be defined as amplitudes with one leg off-shell [7].BG currents also have a nice recursion property, and the recursion relation can be obtained from the classical equation of motion.This property makes BG currents easy to generalize to different theories and even different background metrics as long as we can write down the classical equation of motion of those theories.An equivalent way to obtain the gravity currents is using the double field theory [8].In [8], they discuss the current KLT relation, as a generalization of the KLT relation for amplitudes [9,10], for pure graviton currents and write down the explicit formalism for some low-point examples.Our work can be regarded as a complement to this nice work 1 .In addition, some double copy relations for effective theory currents have been studied in [11].
In this paper, we will write down the BG currents of extended gravity explicitly and explore some properties of them.Then we will consider the unifying relation for amplitudes and finally find that this relation does not hold exactly for gravity BG currents.The paper will be organized as follows.In section 2, we review how to obtain the BG currents of the pure graviton theory.In section 3, we write down the Lagrangian of the extended gravity and derive the BG currents of gravitons, B-fields, and dilatons.In section 4, we derive "off-shell" unifying relations for 2-pt, 3-pt, and 4-pt currents2 to fix the parameter of the theory and to find some hints of the off-shell double copy relation.

Review: BG currents for pure gravitons
In this section, we will review how to obtain BG currents for pure gravitons.We will follow the process in [12].More discussion about pure graviton currents can be seen in [13].Let us consider the following Einstein-Hilbert action: where g is the determinant of the metric g µν and R is the corresponding scalar curvature.In the following discussion, we will choose κ = 1.The equation of motion (EOM) can be found from the condition δS/δg µν = 0: where R µν is the Ricci tensor and g µν R µν = R.For D > 2, we will find R = 0 and the EOM can be simplified as Now we consider the following perturbiner ansatz: where we always choose I to be the order of the natural numbers.From g µν g νγ = δ γ µ , we have where I = X ∪ Y means that I ∈ X ¡ Y3 .For example, if I = 123, then I=X∪Y means that we need to sum over these terms: (X, Y ) = (1, 23), (12,3), (13,2), (23,1), (3,12), (2,13).
The Riemann curvature tensor can be obtained by the following equation: and R µν = R ρ µρν .The Christoffel symbol Γ σ µν = g σρ Γ µνρ is given by Substituting (2.4) into the EOM and choosing the gauge η µν Γ µνρ = 0 which corresponds to we will find that (2.10) Here . Finally, we need to give the single-particle states.The singleparticle states of gravitons are symmetric tensors: with η µν h iµν = η µν k µ h iνρ = 0 which are the traceless condition in the flat spacetime and the transverse condition respectively.The amplitude for pure gravitons is given by H Iab h ncd η ac η bd . (2.12)

BG currents of extended gravity
In this section, we will consider the extended gravity theory.Firstly we write down the action of extended gravity, which is the effective theory of string theory [6,14]: where Here we still use the convention κ = 1.Note that the scale of the B-field here is different from [6].The reason for this convention will be given in section 4 where we find that only when the coefficient is 1/24, the unifying relation [15], as a corollary of the KLT relation, is correct.
As usual, we can obtain the EOM from the action: 3) It is worth mentioning that we cannot cancel the factor e −4φ/(D−2) in the EOM of the B-field.Although this operation will not change the solution of the EOM, it will change BG currents which correspond to the coefficients of certain orders of the solution.To match the Feynman rules, we must keep this factor and this factor will give contact terms with any number of dilatons.For example, if we only focus on the 4-vertex φφBB, then we can truncate the power series of the factor e −4φ/(D−2) to the term (−4) 2  2!(D−2) 2 φ 2 and then calculate the BG currents.If we cancel the factor e −4φ/(D−2) on both sides, we will not obtain this 4-vertex.
To obtain BG currents, we need to expand the factor e −4κφ/(D−2) as a power series of φ.Also note that the scalar curvature can be written as After some calculations, we obtain the currents for extended gravity.The graviton currents can be found in appendix A.
The B-field current and the dilaton current can also be derived similarly.It is worth mentioning that, as a 2-form field, the gauge for B-fields is chosen to be The single-particle state for gravitons is the same as the pure graviton case, while the singleparticle state for B-fields should be chosen to be antisymmetric and the single-particle state should be chosen to be transverse diagonal [14,6].Then the BG current for fat gravitons (i.e.all external legs should be the sum of single-particle states of the graviton, the B-field, and the dilaton) can be obtained by summing over the graviton current, the B-field current, and the dilaton current (with a projective operator).Then the amplitude of fat gravitons can be obtained from the BG current as before.
As we have shown, the vertices involving dilatons make BG currents very complicated, which is also the greatest obstacle to the applications of these currents.In the next section, we will explore the applications of these currents after truncating some dilaton vertices.

Unifying relation for extended gravity currents
The ultimate goal of constructing extended gravity currents is constructing the KLT relation for BG currents, i.e. finding the extra off-shell terms after imposing the traditional KLT relation to BG currents naively, which will be zero when we take the on-shell limit and obtain amplitudes.However, finding the explicit form of the current KLT relation is extremely difficult.In [8], they calculate some low-point cases and we will not repeat this result in this paper.Instead, we discuss the unifying relation, as a corollary of the KLT relation, for extended gravity currents as a verification of this new tool.In this case, as we will see, the extra off-shell terms appear and the number of such terms explodes when the number of points of BG current increases.We will only construct the unifying relation and the off-shell terms in detail for 2-pt, 3-pt, and 4-pt currents in this paper.

Unifying relation for extended gravity
It is interesting to explore the off-shell version of the unifying relations.In [15], the unifying relation for extended gravity amplitudes has been raised, which is also a corollary of the KLT relation: where the operator T [α] is defined as The polarization vector for extended gravity has a double copy formalism ǫ µ ǫν and in this paper we choose the differential operator T [α] to act on the ǫ part.This relation converts the extended gravity amplitudes into the color-ordered Yang-Mills amplitudes with the color order α.However, for BG currents, which have 1 leg being off-shell, there may be some extra terms after acting T [α] on the extended gravity currents which will vanish when we take the on-shell limit.For the unifying relation between the Yang-Mills theory and the Yang-Mills scalar theory, there is no such term [16,17].However, things are different for extended gravity, as we will see.
From the definition of T [α], we will find that the terms that survive must not contain dilatons since the dilaton propagators do not contain η µν .Thus we can truncate all dilaton terms: For single-particle states, we will use the following double-copy prescription with ǫ i • ǫi = 0, which means that we will not consider dilaton external legs.To make this paper self-contained, we write down the gluon BG current with Lorenz gauge here: The reason for choosing the Lorenz gauge here is that if we act the operator T [α] on the gauge condition for graviton currents we will expect to obtain gluon currents with the Lorenz gauge.We will expect the following equation:

2-pt current
Let us start with the simplest case, 2-pt currents (i.e.I = 12), which only involves the 3-vertex terms.We can find out such terms: and the definition of the operator T [α], we can find each term in the gluon currents after acting T [123] on the currents H 12ab ǫ 3c ǫ3d η ac η bd and B 12ab ǫ 3c ǫ3d η ac η bd .Also note that for all X and Y such that I = X ∪ Y , only the cases I = XY and I = Y X are possible to survive from T [In].
We first investigate the pure graviton vertices in the graviton current.After acting T [123] on H 12ab ǫ 3c ǫ3d η ac η bd , the first term while the term proportional to k Iν will not contribute since such terms will not give ǫ1 • ǫ3 .The second term Note that there is a very subtle term in it: This term will not contribute to the unifying relation in the 2-pt current case.However, when we move to the higher point case, we will find that η αβ H Xαν H Y µβ has enough k • ǫ and it will survive from the acting of T [α].
Other terms come from the HBB-vertex terms.The term will give and the last term, which proportional to η µν will not contribute.
Now we can sum over all terms and obtain the following equation: For the B-field currents, we can also do the same analysis.

3-pt current
Let us turn to 3-pt currents.Now we need to consider the contribution from the 4-vertex terms and terms like We still only discuss the graviton currents as an example.From the results of the 2-pt currents, the 3-vertex terms of the gluon currents can be easily obtained.The 4-vertex terms of the graviton current −s 123 H 123ab ǫ 4c ǫ4d η ac η bd will give the following terms after acting T [1234] on it: Note that when we talk about the 4-vertex terms of the graviton currents, we also include the 2-deshuffle terms (terms with I=X∪Y ) with an inverse graviton subcurrent say F µν X , since from (2.5), such currents are also possible to contribute to 4-vertex terms.In fact, these terms contribute 1  8 From the gluon currents, we know we also need a term This term comes from and

.21)
A simple calculation shows that these two terms will also give an extra off-shell term which will vanish when will take k 4µ to be on-shell.Another extra off-shell term comes from For this term, if k • ǫ comes from factors like k ρ Y H Xρa , it will give the term which coincides with the term in the gluon currents.If k • ǫ comes from terms like H Xρa H Y σν η aσ , it will give an extra off-shell term After a similar analysis for the B-field currents, we finally find 25) The greatest difficulty in this case is that there will also be some contribution from the 3-vertex terms.In other words, unlike the unifying relation between YM and YMS, where 4-vertex only corresponds to 4-vertex, 4-vertex in YM corresponds to both 4-vertex and two 3-vertex in the extended gravity.Note that in the 3-pt case, the extra off-shell terms appear for the first time.In the 4-pt currents case, we will find that these extra off-shell terms are necessary but hard to deal with.

4-pt current
The analysis is similar to the former cases.In this case, however, we need to deal with some extra terms in addition to the terms that the gluon currents consist of.Such terms will come from the off-shell terms in the 3-pt current case, terms like (4.20) and 4-vertex.Note that in this case the 5-vertex will not contribute.
After acting T [12345] on −s 1234 H 1234ab ǫ 5c ǫ5d η ac η bd , the extra term from 3-vertex terms are (including the extra off-shell terms of the 3-pt currents and terms like (4.20)) (4.26)One will find that some extra off-shell terms appear.The last two lines of the equation have the same origin as the 3-pt off-shell terms.In the following calculation, we will see they are the only terms left.
The extra terms from the 4-vertex terms can also be obtained.We write down the answer directly: We can sum over these two parts, and finally obtain a complicated result: 28) This equation is the expression of the extra off-shell terms of the unifying relation of the 4-pt currents OFF 4 .For the 4-pt B-field current, the analysis is simpler than the graviton current, and we find that the result is the same as the 4-pt graviton case: In fact, one can obtain the extra off-shell terms for any point currents using Mathematica and finally find that the unifying relation for amplitudes can be reproduced from BG currents.However, we have not found a closed form for those extra off-shell terms yet.From the results above, we propose the following equation as an enhancement of (4.6): which we have verified up to 4-pt currents (or rank-4) in this paper.

Outlooks
In this paper, we write down the BG currents for the extended gravity explicitly, which is equivalent to the double field theory currents.Gravity BG currents are very helpful for computing any point gravity amplitudes since this is a programming-friendly method.We also discuss the unifying relations for some low-point examples.We want to see in what sense the unifying relation is realized for BG currents and finally find that there will be many off-shell terms appearing.One can always do this analysis for any point currents if patient enough.However, the general proof is still unknown since the structure of extended gravity BG currents is extremely complicated.We hope we can solve it in the future as a corollary of the current KLT relation.
There are many problems left and we can write some of them here.
The graviton current of the extended gravity theory is then given by (A.3)