ℒ_W=J_MNJM'□^-1∂_JJ'J^   J'_MN M'+J_MM'□^-11/2∂^NM'J^M_ N
-J_MNM'N'□^-1(∂^J_ M'J_MNJN'-J_MNM'N')
+1/2J_MN□^-1(J_MN-∂_NM'J^ M'_M).
Let us rearrange these terms and write it in a convenient way.

+J^MNJM'□^-1∂_JN'J^   N'_MN M'-
1/2J^MM'□^-1∂_NM'J^ N_M.
This is the final form of the generating Lagrangian. The propagators can be obtained by hitting it with appropriate derivatives. For instance, to get the $\langle HH\rangle$ propagator, one needs to apply partial derivatives w.r.t the field $J^{PQP'Q'}$ twice on the generating function above. However, the form of the generating Lagrangian is quite simple such that the propagators can be directly read off from here. We have for the $HH$ propagator

⟨H^PQP'Q'(k)H_STS'T'(-k)⟩=1/ik^2ϵ^(P_ Sϵ^Q)_ Tϵ^P'_ S'ϵ^Q'_ T'.

where we use the spinor translation for the Minkowski metric as a product of two spinor metrics. There is no copy of momentum in the numerator whereas there is a factor of $k^2$ in the denominator and this makes the tetrad propagator trivial. The symmetrization of the unprimed indices of the tetrad perturbation is captured by the symmetric product of two $\epsilon$ on the right hand side. Let us now write the other propagator in this theory, which connects the tetrad to the redefined connection field.

⟨H^MN Q'_   Z'(k) Ω_JLZG'(-k)⟩= 1/k^2ϵ^M_  Jϵ^N_  Lϵ^Q'_  G' k_ZZ'.
In the $\langle H\Omega\rangle$ propagator, we find that there is both a factor of the spinor metric and a single copy of momentum sitting in the numerator. The spinor metric connects two unprimed indices, which do not have any symmetry between them while the momentum connects the third primed index of the tetrad field to the third unprimed index of the connection.  The $k^2$ factor appears as usual in the denominator. This is quite analogous to the $\langle ba\rangle$ propagator in the chiral Yang Mills. It is interesting to observe the fact that the $\langle H\Omega\rangle$ propagator can be directly related to the derivative of the $\langle HH\rangle$ propagator. This can be understood from (<ref>), where if we omit the current, we have

Ω^MNJM'=-∂^J_ N'H^MNN'M',
which then leads to the relation
⟨Ω_PQCC' (k) H^ABA'B'(-k)⟩= -ik_CD'⟨H^   D'_PQ  C'(k)H^ABA'B'(-k)⟩.

⟨h^B_ D(k) ω_AD'(-k)⟩= 1/k^2ϵ^B_  A k_DD'.
The $\langle h\omega\rangle$ propagator is quite similar to the $\langle H\Omega\rangle$ except that in the later case there are few more unprimed and primed index contractions. The propagator is non-trivial in one of the primed and the other unprimed indices because the momentum connects these two. In the rest of the work, as we will demonstrate, this propagator does not contribute to the amplitudes.

⟨h^AB(k) h_MN(-k)⟩= 1/k^2ϵ^A_ Mϵ^B_ N.
Similar to that of $\langle HH\rangle$ propagator, this propagator is trivial. There are no momentum factors in the numerator and only the unprimed spinors are contracted. This too will not contribute in the computation of amplitudes as we will see in the subsequent chapters.

§.§.§ Ghost propagators

The BRST quantization which we described in the previous sections leads to a gauge fixed Lagrangian with ghost terms. There are two kinds of ghosts: diffeomorphisms and local Lorentz. The Lagrangian consists of mixed terms between these ghosts besides the usual kinetic terms. Thus there are a total of three such terms to which we can add the relevant currents coupled with the ghosts and deduce the generating Lagrangian as before. The free ghost Lagrangian with currents coupled is given by

ℒ_ghost   =-c̅^JJ'∂^KK'∂_KK'c_JJ'+2c̅^JK'∂_KK'b^K_ J-b̅^JK∂_MM'∂^MM'b_JK

We follow the method outlined for the tetrad and connection propagators. The propagators can be obtained by hitting the generating Lagrangian with appropriate derivatives. However, the form of the generating Lagrangian is quite simple such that the propagators can be directly read off from there. We do not repeat the calculations here. Let us first write down the propagator for the diffeomorphism ghost field. We have

⟨c̅^MM'(k)c_NN'(-k)⟩=1/ik^2ϵ^M_ Nϵ^M'_ N'.

There is no copy of momentum in the numerator as should be the case, whereas there is a factor of $k^2$ in the denominator. Further, there is no symmetrization of any indices. This makes the diffeormphism ghost field propagator trivial. This is quite analogous to that of the ghost field propagator in chiral YM theory. Next, we have the propagator for the Lorentz ghost field

⟨b^MN(k) b_PQ(-k)⟩= 1/ik^2ϵ^M_ Pϵ^N_ Q.
This propagator connects the Lorentz ghost with its anti-ghost. There is no symmetrization in the indices. The $k^2$ factor appears as usual in the denominator. There is no factor of momentum in the numerator and this makes the propagator trivial as before. The remaining propagator is the one which connects the Lorentz ghost with the diffeomorphism anti-ghost. It is given by

⟨c̅^A_ B'(k) b_PQ(-k)⟩= 1/k^2ϵ^A_ Q k_PB'.
The $\langle \bar{c}b\rangle$ propagator is quite similar to the $\langle h\omega\rangle$ one except that in the latter case there are few more unprimed and primed index contractions.

§.§ Interaction

There are three kinds of interaction terms. One is the single derivative cubic term, which will play the main role in amplitudes. The other two are the non-derivative cubic interaction and the non-derivative quartic interaction. In the single derivative cubic interaction, there are again three different kinds of terms. One is of the form $HH\partial\Omega$ and the other two are $hh\partial\Omega$ and $hH\partial\Omega$ respectively.
It is the $HH\partial\Omega$ interaction which is relevant in the same helicity amplitude computations. Despite being a single derivative interaction, as we will show, it is possible to write the connection variable in terms of the tetrad perturbation, resulting in the two derivative interaction vertex, as is relevant for gravity. Let us then first write the complete set of interactions. We write this in two parts. The single derivative cubic interaction reads

L_hh∂ω=-1/2(H^AR_  M'N'H^BSM'N'∂_RR'Ω_ABS^    R'-H^A    R'_ MM'H^BMM'S'∂_RR'Ω_AB S'^   R)
-h^ARh^BS∂_RR'Ω_ABS^R'-h^ANH^B  R'S'_  N∂_RR'Ω_AB  S'^   R.
Note that the above interaction only contains one of the components of the connection field. The other component, namely $\omega^{BB'}$ does not enter the interaction. We can write this interaction in a better way. In particular, it is possible to combine the first two terms into a single effective term and rewrite the complete interaction. This then gives

L_hh∂ω=-2H^AR_  (R'N')H^BS(M'N')∂_RM'Ω_ABS^    R'-2h^ARh^BS∂_RR'Ω_ABS^R'
+2h^ANH^B  (R'S')_  N∂_RR'Ω_AB  S'^   R.
The only change in the tetrad components is that the primed spinors are now symmetrised. Let us also write the other components of the interaction. The $h\omega\omega$ and $hh\omega\omega$ parts of the interaction read
L_hωω=2h^CC'DD'(ω^A_ CDCω^B_ ABD'-ω^A_ CBD'ω^B_ ADC'),
L_hhωω=2h^CC'DD'h^A  BA'_ C'(ω^ E_C BD'ω_EADA'-ω^ E_C DA'ω_EABD').

§.§ Vertices

There are both cubic and quartic interaction terms in the linearised Lagrangian. However, as we mentioned earlier, the only relevant interaction for our purposes is the $HHd\Omega$. It is possible to use the relation of the connection variable in terms of the tetrad, as is shown in (<ref>) and write this interaction term with just the tetrad variable. This leads to an effective Feynman rule, where one of the legs of the tetrad is marked special. Two such special legs can never contract because the $\langle \Omega \Omega\rangle$ propagator vanishes in our theory. Let us first write the vertex factor in the form which is there in the interaction Lagrangian.

⟨H^PQ  S'_   N' H^BSM'Q' ∂_SS' Ω_JRDD' ⟩=-2iϵ^P_ Jϵ^Q_ Dϵ^Q'_ D'ϵ^M'_ N'ϵ^B_ Rk^S_ S'.
Now we replace the connection variable and put a copy of the tetrad with a derivative in front of it. This then becomes an effective $\langle HHH\rangle$ vertex, but with the rule that the particular leg which is replaced is marked special. For convenience, we put a cilia in this leg and correspondingly in the expression of the vertex. When this leg is external, we project it to the positive helicity state of the connection. We have for the effective $\langle HHH\rangle$ vertex

⟨H^PQ  S'_   N' H^BSM'Q' ∂_SS'∂_DE' H^∙  E'_JR  D' ⟩=2iϵ^P_ Jϵ^Q_ Dϵ^Q'_ D'ϵ^M'_ N'ϵ^B_ Rk_SS'k_DE'.

§.§ Helicity states

Let us introduce the helicity spinors which we will use subsequently in the amplitude computation. The fields $H,\Omega$ can be on-shell. The fields $h,\omega$ are not physical and thus they vanish off-shell. The metric perturbation $H$ is supported by both the positive and negative helicities, given by the usual states

ϵ^HH'MM'_+(k)=k^Hq^H'k^Mq^M'/ [qk]^2.
where $q^A$ and $q^{A'}$ are auxiliary 2-spinors.

For the field $\Omega$, we solve the gauge fixed Lagrangian from (<ref>) and (<ref>) on-shell. This gives
Ω_PQRS'=∂_RJ'H_PQ  S'^   J'.
The corresponding helicity state for the field $\Omega$ is

which is obtained by using the helicity states of the tetrad perturbation and going to the momentum space with the substitution $\partial_{MM'}=-ik_{M}k_{M'}$. The spinor $k_{M'}$ contracts with its copy in the negative helicity state of $H$ and thus it makes it vanish. The only non-vanishing helicity supported by the connection field is the positive one in (<ref>).

CHAPTER: SELF-DUAL GRAVITY

The purpose of this chapter is to describe a recently developed covariant formulation of self-dual gravity (SDGR) in flat space [33], relevant for computing scattering amplitudes. SDGR is a theory in four dimensions, whose solutions are Einstein metrics with either the self-dual or the anti self-dual part of the Weyl curvature vanishing. Such solutions of the field equations of gravity are known as gravitational instantons. When appropriately linearised around an instanton background, the theory describes two propagating degrees of the graviton. However, the two polarization states are described on a different footing as we shall see. The interesting feature of the theory is the fact that despite having a negative dimension coupling constant, it is not just renormalizable but ultraviolet finite, unlike pure gravity in four dimensions. SDGR can only describe metrics of Euclidean or split signatures because in Lorentzian signature, the vanishing of either the self-dual or the anti self-dual part of the Weyl curvature means the vanishing of all of it. There are strong similarities between SDYM and SDGR. In particular, SDGR is one loop exact and there are no divergences as can be seen by computing the one-loop effective action. This is described in [32]. The tree amplitudes in the theory vanish in an analogous way like SDYM, whereas the one-loop amplitudes are rational functions of the momenta involved. Let us then start describing the flat space version of SDGR, which we will use to compute amplitudes in the subsequent chapter.

§ SDGR IN FLAT SPACE

There are many non covariant formulations for SDGR which exist in the literature. For a generic reference, see Plebanski's description [60]. The well known covariant formulation is proposed in [34] for zero scalar curvature case and then in [32] for non-zero scalar curvature. We are going to describe two such actions relevant for the zero scalar curvature. One of them can be obtained from the chiral Einstein-Cartan in (<ref>) by removing the part of the curvature which is quadratic in the spin connection. This action is also contained in the bosonic sector of the corresponding supergravity action in [34]. The resulting covariant action in flat space background is very similar to SDYM. In particular, it contains just a single propagator and a cubic vertex. The quartic vertex which is present in the full gravity action in (<ref>) gets eliminated as a result of removing the $\omega\omega$ term from the curvature. However, the gauge fixing for this action is rather non-trivial and thus it complicates perturbative computations. The other action has a simple gauge fixing procedure and the Feynman rules are analogous to its YM cousin, in that there is just one propagator and a cubic vertex. One can also compute the BG currents in this theory and then it is easy to infer that all tree amplitudes vanish, in analogy with SDYM. The interesting part is the one-loop same helicity amplitude sector which is correctly captured by SDGR and we will expand on it in the next chapter.

§.§ Action

To write the action in self-dual gravity, let us first consider the chiral Einstein-Cartan action once again in spinor notations with coupling factors included. It reads
𝒮[e,ω]=i∫e^A_ C'∧e^BC'∧(dω_AB+κω_AC∧ω^C_ B),
where $e^{A}_{~C'}\wedge e^{BC'}$ is the self-dual 2-form constructed from the frame field and $\kappa=8\pi G$. The action in (<ref>) is equivalent to the action in (<ref>) in that the connection field is just re-scaled by a factor of $\kappa$. We now construct the SDGR action by simply taking the $\kappa\rightarrow 0$ limit. This removes the $\omega\omega$ part of the curvature. The resulting field which appears after this is to be distinguished from the connection. Thus we call it $\xi_{AB}$. The SDGR action reads
𝒮_SDGR[e,ξ]=i∫e^A_ C'∧e^BC'∧dξ_AB.
The action comprises of an exact 2-form $d\xi$ on which the self-dual 2-form acts and projects out its self dual component. This is because any arbitrary 2-form can be expanded into its self-dual and anti-self-dual parts. Then one can use the fact that the wedge product between a self-dual 2-form and an anti-self 2-form vanishes. Thus it projects out the self-dual part of $d\xi$. One can vary this action with respect to both the tetrad/frame field and the 1-form. If we vary with respect to the 1-form $\xi_{AB}$, we get an equation which says that the self-dual part of the spin connection vanishes. Thus, the curvature constructed from the self-dual part of the spin connection also vanish and this then implies that the metric obtained from the tetrad field is the one with only non-zero anti-self-dual part of the Riemann curvature. Also, varying with respect to the tetrad gives an equation which is equivalent to Einstein equation. This is explained in details in [39]. Overall, this gives the correct description of SDGR where the metric is Einstein and the self-dual part of the Riemann curvature vanishes. However, like the chiral Einstein-Cartan, the gauge fixing of this action is non-trivial and thus it is not so useful to do perturbative calculations with it.

We next explain another covariant formalism of SDGR which is recently proposed in [33]. It is relevant for metrics with zero scalar curvature and is thus suitable for perturbative computations. The action proposed can be motivated from another closely related formalism for SDGR [32] in the case of non-zero scalar curvature. In particular, the new action can be obtained by removing the quadratic in connection term from the latter. It is also illuminating because it is much closer to the SDYM action (<ref>) as we described in the previous part, in that there is only one propagator and just a cubic vertex, when expanded around an appropriate background. Another nice feature of the new formalism is that it has a simpler gauge fixing procedure and this is why it is most useful for amplitude computations. The action reads

𝒮_SDGR[ψ, a]=1/2∫ψ^mnda^m∧da^n,
where $\psi^{mn}$ is a tracefree and symmetric field. Thus $\psi^{mn}\delta_{mn}=0$. The action consists of another field, namely the 1-form $a^m$ and the way this field enters in the Lagrangian is via an exact 2-form $da^m$. The action above can be understood to arise from another covariant action for SDGR, which is described in [32] by a process where the connection-connection term in the curvature 2-form is set to zero. Then the curvature 2-form becomes an exact 2-form, which is the same as $da^m$ and thus results in the action (<ref>). Next, we describe the perturbative expansion of this action around a flat background and construct the Feynman rules.

§.§ Linearised action

The starting point for doing perturbative calculations is to expand the above action around a flat background. The relevant background which describes flat space is
where $M^2$ is introduced for dimensional purposes and $\Sigma^m$ are the already described self-dual 2-forms. When we expand around this background, the action reads

The first term is the kinetic term for gravitons and the second term is a 2-derivative cubic interaction. It is interesting to see that the structure of the linearised Lagrangian is analogous to that in SDYM. Indeed, as we described in SDYM, the linearised Lagrangian contains a kinetic piece and a cubic term. The only difference is that in the case of SDYM, the interaction is a non-derivative one, while in the case of SDGR, the interaction contains two derivatives. To do perturbative computations, it is most useful to express the above action in spinor notations. Thus let us start describing everything in terms of spinors. The field $a^m_{\mu}$ is a 1-form and translating the index $m$ into spinor indices, it becomes $a^{MN}_{\mu}$. The unprimed indices $M,N,...$ and the primed ones $M',N',...$ are the $sl(2,\mathbb{C})$ spinor indices. We also translate the spacetime index $\mu$ and then we get the object $a^{MN}_{~~~~PP'}$. The field $\psi^{mn}$ gets translated to the spinor $\psi^{MNPQ}$, which is totally symmetric. The spinor translation of the derivatives are already described in the previous part. For completeness, let us state that the spinorial translation of the partial derivative is $\partial_{M'N'}$. The exterior derivative then becomes $\partial_{PP'}a^{MN}_{~~~~QQ'}$. Next, when the self dual 2-forms are wedged with the exterior derivative it results in the self-dual projection of the latter, which in spinor notation reads $\partial_{PP'}a^{MN~~P'}_{~~~~Q}$. Overall, the linearised kinetic term reads

ℒ_2= ψ_MNPQ∂^M_ M'a^NPQM'.
The kinetic term above is normalised, because we have absorbed the dimensionful parameter $M^2$ into it. Thus, the mass dimension of $\psi$ is two whereas that of $a$ is one. It is convenient to use a notation where all the symmetrised indices of a particular field is denoted by the same letter. Thus, we denote the completely symmetrised field $\psi^{MNPQ}$ as $\psi^{MMMM}$. The equations of motion are now immediate to write down. Variation with respect to the field $a$ gives

∂_N^  M'ψ^MMMN=0.
The field $\psi$ thus describes the $+2$ helicity state of the graviton. Its equation can be understood as the Bianchi identity for the Weyl tensor. The variation of the kinetic part with respect to $\psi$ gives

∂^M_  N'a^MMMN'=0.
This equation describes the negative helicity state of the graviton. Let us next write the interaction. There is just a single cubic vertex. The 2-form $\partial_{PP'}a^{MN}_{~~~QQ'}$ can be split into a self-dual part and an anti-self-dual part. The self dual part results in a spinor of the form which is described earlier. The anti-self-dual part in spinor notation reads $\partial_{SP'}a^{MN~~S}_{~~~~Q'}\epsilon_{PQ}$. Then, the two copies of $da$ are wedged. In this process, there is a contraction of the primed indices of two copies of the anti-self-dual parts and the unprimed indices of the corresponding self-dual parts. The wedge product of a self-dual and anti-self-dual part vanishes. Overall, the cubic interaction can be written as

ℒ_3=ψ_MMMM(∂_PS'a^MM  S'_    Q∂^P_ G'a^MMQG'+∂_SP'a^MN  S_    Q'∂^ P'_Ga^MMQ'G).

§.§ Gauge fixing and Feynman rules

Let us now describe the gauge fixing of our kinetic term. The linearized action around the flat background in (<ref>) is invariant under both diffeomorphisms and shifts of the 1-forms $a^m$ by exact 1-forms. Thus, the relevant transformations for the fields are

δψ^mn   =i_ηdψ^mn,

δa^m   =i_ηda^m+i_ηΣ^m+dχ^m.

where the Lie derivative of the 1-form $a^m$ along some vector field $\eta$ is given by
$\mathcal{L}_{\eta}a^m=d(i_{\eta}a^m)+i_{\eta}da^m$ and we have absorbed the $d(i_{\eta}a^m)$ part of it in the definition of $\chi^m$. It is convenient for our purposes to write the above in spinor notations. The spinor translation of the vector field results in a bi-spinor $\eta^{MM'}$ where the conversion is done via the background vierbien 1-form $e^{MM'}_{\mu}$. Then the symmetries take the following form

δψ^MMMM   =0,

δa^MN   =η^M_ N'e^NN'+dχ^MN.

The field $a^{MN}$, where we now suppress the world index, admits a decomposition into its irreducible components. The decomposition reads
a^MN=e_PP'Φ^MNP,P'+e^M_ N'Φ^NN'.
Now the second component can be gauged away using the diffeomorphism symmetry. This is because it can be reabsorbed into the definition of $\eta^{M}_{~N'}$ in (<ref>) and thus gets eliminated. The kinetic term in (<ref>) is independent of this component and hence the diffeomorphism gauge is fixed. The resulting fields in the theory admit the following linearised transformations
δψ^MMMM=0,    δΦ^MMM,M'=∂^MM'ξ^MM.
It is clear that the only symmetry remaining to be fixed is the $\delta\Phi^{MMM,M'}=\partial^{MM'}\xi^{MM}$. It is now fixed by imposing a variant of the Lorentz gauge. To do so, one considers a gauge fixing fermion of the form
Ψ=c̅_l(MN)∂^C_ C'Φ^MNCC'≡c̅_l(MNϵ_P)Q∂^Q_ C'Φ^MNPC' .
where $\bar{c_l}$ is the Lorentz ghost. BRST variation of this gives the bosonic contribution $\psi_{(MN}\epsilon_{P)Q}\partial^{Q}_{~C'}\Phi^{MNPC'}$ where $\psi_{(MN)}$ is the auxiliary field. The fermionic contribution is just the kinetic term for the ghosts, which is $\bar{c}_{lMN}\partial^C_{~C'}\partial^{C'}_{~C}c_l^{MN}$. One can then notice the structure of the bosonic contribution and combine the $\psi_{(MN}\epsilon_{P)Q}$ part of it to the field $\psi_{MNPQ}$ to obtain a new field, which is symmetric in its first three indices.
The new field $\tilde{\psi}$ is symmetric in its first three indices. The final gauge fixed action then depends on two fields, namely $\tilde{\psi}^{MMMN}$ and $\Phi^{MMM,N}$ and each of them contains the same number of components. It reads
∫ψ̃^MMMN∂^  N'_NΦ_MMM,N'.
The operator that maps one field to the other in the above term can now be inverted and the propagator is immediate. Let us write it.

§.§.§ Propagator

⟨ψ̃^MNPQ(k) Φ_ABC,Q'(-k)⟩= 1/k^2ϵ^(M_  Aϵ^N_  Bϵ^P)_  C k^Q_  Q'.
where the indices $M,N,P$ belong to a single symmetrized group and the symmetrization of these indices is denoted by a round bracket in the right hand side of (<ref>).

§.§ Helicity states

As we described, the physical fields which enter the gauge fixed Lagrangian are $\Phi^{MMM,N}$ and $\tilde{\psi}^{MMMN}$. These constitute the two helicities of the graviton. The convention we take is that the negative helicity is described by $\Phi$ and the positive helicity by $\tilde{\psi}$. We can then solve the momentum space version of the linearised equations in (<ref>) and (<ref>) and get the corresponding polarisation tensors. They are

ϵ^-_MMM,M'(k)=Mq_Mq_Mq_Mk_M'/⟨qk⟩^3,   ϵ^+_MMMM(k)=M^-1k_Mk_Mk_Mk_M.
where the momentum is null, i.e $k_{MM'}=k_Mk_{M'}$ and $q^M$ is the auxiliary spinor as usual. The negative polarisation tensor is dimensionless and this is why we introduced the dimensionful parameter $M$ in its definition. The positive polarisation tensor is dimensionful, with its mass dimension being one. As we can see, the two polarisation states of the graviton are treated on a different footing. The negative state consists of both primed and unprimed spinors while the positive state just has only unprimed ones. Let us then apply this to compute the BG current in this theory.

§.§ Relation to metric helicity state

Let us see how the usual negative metric helicity state arise from the first expression in (<ref>). This has been described in details in [59]. The metric perturbation is obtained by the operation of taking the anti-self-dual part of the two form $d\Phi^i$. In the spinorial notation, this then can be described by the following relation, upto numerical factors

h_MNM'N'∼∂^B_ M'Φ_BMN,N'.
Here $\partial_{MM'}=-e^{\mu}_{MM'}\partial_{\mu}$ is the Dirac operator. We can apply this to the negative helicity spinor (<ref>). Modulo numerical factors, it is easy to see that the usual negative helicity is reproduced
Let us explain how the negative helicity state arises. In the passage to momentum space, the partial derivative in (<ref>) becomes the momentum vector $k$. This then splits into a primed and an unprimed spinor due to on-shell condition. Contracting one of the $q$ spinor by the unprimed momentum spinor $k$ cancels one of the $\langle qk\rangle$ factor from the denominator. This results in two copies of primed momentum spinors in the numerator and two copies of the $q$ spinor, which is the usual expression for the negative helicity state of the metric.

§.§ Berends-Giele current

The first computation of the currents in gravity was done by Berends and Giele using the recursion relations [9]. However, unlike in YM, the recursion in this case is quite complicated to solve and thus the MHV amplitude for gravity was not obtained using this procedure. We will give a brief outline of the computation of the 2-current from the SDGR Feynman rules which we just described and suggest the generalization to $n$-current. Like usual, we define a current as the sum of all tree level Feynman graphs with all but one leg on-shell. Thus one can have different states inserted into the on-shell legs and the off-shell leg is taken with the propagator on that leg. However, the most interesting current arises if we take all the on-shell legs to be of same helicity. By convention, we take the negative helicity state inserted into these legs. Let us write the form the general $n$-current.

The one-current is the negative helicity polarization state itself. The scalar part of the current can be written with its index structure stripped off
The two-current is obtained by taking the $\tilde{\psi}\partial \Phi\partial \Phi$ vertex in (<ref>), projecting the two $d\Phi$ legs into negative helicity states and then applying the final leg propagator. Upon inserting the negative helicity state, one of the terms in the vertex vanishes. We then read off the result for the scalar part of this current

The three current can be obtained by taking the two current and insert the third polarisation state to it, with a sum over permutations. Three such diagrams arise, one is the attachment of the third gluon to the current $J(1,2)$, the other being the attachment of the second gluon to $J(1,3)$ and the third one is the attachment of the first gluon to the current $J(2,3)$. The combinatorics is already complicated to solve. The details of this calculation is given in [32]. We outline the final result for the scalar part of the 3-current.
𝕁(1,2,3)= 1/⟨q1⟩^2⟨q3⟩^2[12][23]/⟨12⟩⟨23⟩+1/⟨q1⟩^2⟨q2⟩^2[23][31]/⟨23⟩⟨31⟩+1/⟨q2⟩^2⟨q3⟩^2[31][12]/⟨31⟩⟨12⟩.
The pattern is becoming clear. The scalar part of the general $n$th-order current can be written as

This is a compact way of writing the general formula for the current. Let us explain the notation. The first product stands for all elements of $\mathcal{M}$. The notation $\mathcal{T}(\mathcal{M})$ denotes all the tree graphs with elements of $\mathcal{M}$ as vertices. Thus the sum after the product is over all such tree graphs from the set $\mathcal{T}(\mathcal{M})$. Next, the notation $H(T)$ denotes the set of edges for a tree graph $T$. The product is then over all such spinor brackets $\langle im\rangle$ which are in $H(T)$. The $q$ in $\langle qi\rangle^2$ stands for the usual auxiliary momentum spinor. Although the formula is complicated, it can be verified for simple cases, like $n=1,2$, etc. For the case when $n=2$, we have the set $\mathcal{M}=\{1,2\}$. In this case, there is just one tree graph and thus the sum reduces to just one term, which is the result in (<ref>). For the case when n=3, we have $\mathcal{M}=\{1,2,3\}$. In this case, there are three terms contributing to the current as can be seen from the formula. This is true because there can be three different permutations for the three legs of the current and then all of this is summed over. The formula can also be proved in general, see [19]. It therefore is the closed form expression for our current.

§ GRAVITATIONAL INSTANTONS

In analogy with the Yang Mills case, one can also  construct Euclidean solutions for the Einstein field equations which minimize the action functional. Such solutions are Einstein metrics with half of the Weyl curvature vanishing. These are the so called gravitational instantons. Thus, we have the decomposition of the Riemann curvature (for zero cosmological constant)
W^+     Ricci|_tracefree

Ricci|_tracefree    W^-
where $W^+$ and $W^-$ are the self dual and anti-self-dual parts of the Weyl curvature. The instanton condition implies that the tracefree part of Ricci and half of the Weyl curvature vanish. Such solutions then automatically correspond to self-dual or anti-self dual gravity depending on which half of the Weyl curvature is non-zero. Thus, for a gravitational instanton which is described by the anti-self-dual part of the Weyl curvature, we have the following condition

The description of instantons with non-zero cosmological constant is given in [32]. In this reference, a connection description of instantons is explained. Here we briefly review the case for zero cosmological constant, which has already been described in details in [33]. Consider the flat SDGR Lagrangian once again
where we have a triple of two-forms $\xi^m=da^m$ on some manifold X. The two forms are assumed to be closed and the variation of the action with respect to the field $\psi^{mn}$ leads to the Euler Lagrange equation

Any triple of two forms which are closed and satisfy the above equation describes a gravitational instanton. Let us explain this. First, any closed two-form satisfying (<ref>) define a Riemannian signature metric $g_{\xi}$, see [33]. Further, when the two-forms are closed, it implies that the self-dual part of the spin connection vanishes. This also means that the curvature composed of just the self-dual part of the spin connection too vanish. However, we know that the curvature of the self-dual part of the spin connection can be decomposed into its self-dual and anti-self-dual parts. The self-dual part contains one half of the Weyl curvature and the scalar curvature, whereas the anti-self-dual part contains the tracefree part of the Ricci tensor. Vanishing of the curvature thus implies vanishing of both these parts. This then corresponds to the condition (<ref>) and therefore describes gravitational instantons.

CHAPTER: AMPLITUDES IN SELF-DUAL GRAVITY

We now arrive at the final chapter of the thesis. In this chapter, we describe our results on the construction and computation of same helicity one-loop gravity amplitudes, particularly the one at four points using the covariant formulation of flat space SDGR. Before expanding on the loop amplitudes, let us briefly comment on the the same helicity tree amplitudes. These amplitudes with $n>3$ points vanish, analogous to the ones in SDYM. This is because in order to construct such amplitudes, one has to remove the final leg propagator in the BG current and project it on-shell. This is done by multiplying the current with a factor of $k^2$ where $k$ is the momentum in the final leg. However, on-shell, the momentum is null and there is no pole to cancel. Thus it renders the amplitude to vanish. The only non-vanishing tree amplitude is at 3-points. We quote the result from [32]
The same helicity one-loop amplitudes in full GR is correctly captured by SDGR Feynman rules. These amplitudes are exactly analogous to their YM cousin, in the sense that they are cut-free and are rational functions of the momenta involved. Thus, inspite of being loop amplitudes, their behaviour is very similar to tree amplitudes. These amplitudes have not been computed using SDGR Feynman rules before. We provide a construction of such an amplitude at four points using the SDGR Feynman rules and show that they are finite. We then explain that it is also possible to construct it using the chiral Einstein-Cartan Feynman rules and both of these give exactly the same set of Feynman diagrams. We then provide a partial attempt to compute it using the one-loop bubbles but our computation stops because unlike in YM, it is not clear how to interpret the shift parameters in this case. Let us begin to describe all these. First we review the literature expression for these amplitudes

§ LITERATURE EXPRESSION FOR ONE-LOOP AMPLITUDES

The same helicity one-loop amplitudes were conjectured on the basis of soft and collinear limit arguments in [22]. In the same paper, it was also explained that they can be computed explicitly by using supersymmetry where a graviton in the loop is replaced by a massless scalar. The general expression of these amplitudes are written in terms of soft functions. Let us introduce and explain these functions.

§.§ Soft functions

We define a set of functions which have some simple behaviour in the soft limits. Let us write the first three of a series of such functions and subsequently write their soft limits.

h(x,{1},y)   =1/⟨x1⟩^2⟨1y⟩^2,

h(x,{1,2},y)   =[12]/⟨12⟩⟨x1⟩⟨1y⟩⟨x2⟩⟨2y⟩,

h(x,{1,2,3},y)   =[12][23]/⟨12⟩⟨23⟩⟨x1⟩⟨1y⟩⟨x3⟩⟨3y⟩+[23][31]/⟨23⟩⟨31⟩⟨x2⟩⟨2y⟩⟨x1⟩⟨1y⟩

where $x,y$ are arbitrary momenta and $\langle xi\rangle, \langle yj\rangle$ for $i,j =1,2,3...$ are the usual unprimed spinor contractions. These functions satisfy the soft limits
h(x,M,y) -𝒮_m(x,M,y)×h(x,M-m,y),      for $m\in M$,
and $M$ is a set containing the remaining legs other than $x,y$.

§.§.§ One loop amplitudes

Using these soft functions, we now explicitly write the literature expressions for the one loop amplitudes at 4, 5 and 6 points.

ℳ_4(1^-,2^-,3^-,4^-)   =h(1,{2},3)h(3,{4},1)tr^3[1234]

ℳ_5(1^-,2^-,3^-,4^-,5^-)=h(1,{2},3)h(3,{4,5},1)tr^3[123(4+5)] + perms,
+h(1,{2},3)h(3,{4,5,6},1)tr^3[123(4+5+6)]+ perms.
The general $n$-point gravity amplitude can be written as ℳ_n(1^-,2^-,...,n^-)=∑_1≤x<y≤n M,Nh(x,M,y)h(y,N,x)tr^3[xMyN],
where $M$ and $N$ are two sets forming a distinct partition of the remaining $(n-2)$ legs. $M$ and $N$ are both considered to be non-empty and $(M,N)=(N,M)$.
Interestingly, it can be seen that these soft functions coincide with the scalar part of the BG currents in (<ref>) when the two arguments $x,y$ are taken to be equal to the auxiliary spinor $q$. In this case,

h(q,{1},q)   =𝕁(1),

h(q,{1,2},q)   =𝕁(1,2),

h(q,{1,2,3},q)   =𝕁(1,2,3),

h(q,{1,2,3...n},q)   =𝕁(1,2....n).

There is yet another representation of these one-loop amplitudes in terms of the double off-shell scalar currents. This is described in [35] and we quote the results here.

§.§.§ Double off-shell scalar current

The double off-shell scalar current $S_q(l;C)$ contains $n$ on-shell positive helicity gravitons for $C={1,2...n}$ and a massless scalar line which have outgoing momenta $l$ and $-l-K_C$ at the two ends. One can then sew the off-shell ends of this scalar line and thereby reconstruct the same helicity amplitudes in gravity. The recurrence relation which is obeyed by the double off-shell currents is given by

S_q(l;C)   =-1/(l+K_C)^2∑_A⊂C B≡C-AS_q(l;A)(l+K_A)^μ(l+K_A)^αJ^μα_+q(B)

=-1/(l+K_C)^2∑_A⊂C B≡C-AS_q(l;A)h(q,B,q)⟨q^-|(l+K_A)K_B|q^+⟩^2,

where we have used the relation between the half-soft functions $h(q,C,q)$ when both the arguments in $h(a,C,b)$ are taken to be equal to the auxiliary spinor $q$ and the current $\mathbb{J}^{\mu\alpha}_{+q}(C)$
In the following representation, we equate the momenta of the two off-shell end of the scalar line for the current $S_q$ and integrate over $L$, to obtain the scalar loop contribution to $M_n(+,+,...,+)$. One can then use the supersymmetric Ward identities to see that the scalar contribution is the same as that of a graviton in the loop, up to overall numerical factors.

One also needs to insert the current $\mathbb{J}$ into the sewing. One focuses on a particular leg, say $n$, and defines a tree by which it is attached to the loop to be the current $\mathbb{J}^{\mu\alpha}(B)$ for some subset $B$. One then obtains for the same helicity $n$-point amplitude in gravity, the following representation
M_n(1^-,2^-,...,n^-)=∫d^DL/(2π)^D∑_B⊂C; n∈BS_q(L;A)/L^2⟨q^-|lK_B|q^+⟩^2h(q,B,q).

§ ONE-LOOP SAME HELICITY FOUR POINT GRAVITON AMPLITUDE

§.§ Setup

Let us explain the construction of the amplitude. The amplitude constitutes of a sum of three box diagrams, six triangle diagrams and fourteen bubbles, but for computation purposes, one diagram for each of these topologies is sufficient since we later permute the external legs in the result to obtain the other ones. We first construct the diagrams from the SDGR Feynman rules and then outline a construction from the chiral Einstein-Cartan action. Both results in the same loop integrals. The relevant diagrams to consider are the box, triangle and bubble. In SDGR at one-loop, one cannot have $\tilde{\psi}$ on external lines. This is because the propagator takes $\tilde{\psi}$ to $\Phi$ and the vertex is linear in $\tilde{\psi}$. Thus, it is not possible to construct one loop diagrams with external $\tilde{\psi}$ legs. Then we insert $d\Phi$ on external legs. In our convention, this is the all negative helicity graviton amplitude. With this, we begin with the box.

§.§.§ Box

In the above diagram, the negative helicity states from (<ref>) are inserted on the external legs. Consider any single vertex in the above diagram. From the interaction in (<ref>) it is clear that only the $\Phi^{MN}$ part of the $a^{MN}$ contributes. We now want to insert negative helicity states (<ref>) to the external legs. Upon insertion, the second term vanishes due to $\langle qq\rangle$ contraction. The non-zero contribution thus comes from the first term, which is the anti-self dual part of the wedge product of two copies of $d\Phi$. Another way to see this is that on negative helicity states, the self-dual part of $d\Phi$ vanishes because of its linearised field equation. The ASD part is non-vanishing and results in the polarisation spinor $Mq_{M}q_Nk_{M'}k_{N'}/\langle qk\rangle^2$. This is the usual negative helicity state of a graviton. The resulting loop integral, upto coupling factors is

It is worth mentioning the structure of the loop integrand which arises from our construction. Comparing with the YM box diagram in (<ref>), we can see that the numerator of the gravity box integrand is the square of the numerator of the corresponding YM one. This is the double copy property between gravity and gauge theory amplitudes and we see that it arises directly from the Feynman rules of SDYM and SDGR. The expression of the above integrand is remarkably compact. In particular, in the absence of quartic vertices, there is just one box diagram to consider and it is in parallel with the construction in SDYM. This is the main outcome of the new covariant formalism for flat space SDGR.

Let us also discuss the setup of the box diagram from the Feynman rules in chiral Einstein-Cartan theory. In this theory, the on-shell fields are $H$ and $\Omega$. We want to construct the all negative helicity diagram and thus insert it on the external $H$ leg. The field $\omega^{AA'}$ does not enter the interaction. This eliminates the possibility of $\langle h\omega\rangle$ type propagators. Therefore, neither the $h$ field nor the $\omega$ field contributes to the amplitude. This fixes the structure of the diagram, composed of $\langle H\Omega\rangle$ internal lines and $\langle HH\partial\Omega\rangle$ vertex. The vertex factor consists of two interaction terms. Consider first the second interaction term in the bracketed piece in (<ref>). Let us apply it to the vertex where leg $1$ is inserted. From the $\langle H\Omega\rangle$ propagator in (<ref>), it is clear that the loop momentum $l$ will carry the third unprimed index of the connection and the third primed index of the metric. The derivative in the second interaction term is on the connection line and consists of the third unprimed index of the connection. This then hits the derivative in the propagator. Each polarization consists of two auxiliary unprimed spinors as is clear from (<ref>), which we take identical. Thus there are a total of 8 auxiliary unprimed spinors. These can either get contracted to the unprimed indices of the loop momenta or to themselves. However, in the above case where two loop momenta have the same unprimed index and gets contracted according to $l^{AA'}l_{AB'}=\frac{1}{2}\epsilon^{A'}_{~B'}l^2$, there are fewer loop momenta left to make them contract with all the  8 $qs$. Therefore, at least two auxiliary spinors in this case will contract amongst themselves, thus giving a vanishing contribution $\langle qq\rangle=0$.

Therefore, the second interaction term in (<ref>) cannot contribute, because in all such terms one or many derivatives in the interaction hit the derivatives in the propagator, leading to $\langle qq\rangle=0$ contraction. The diagram thus consists of entirely the first interaction term. With this, it takes the particular form (<ref>)

§.§.§ Triangle

There is another non-zero contribution at one-loop, given by the triangle diagram

Using the SDGR Feynman rules, it is constructed in a similar way. The external legs are projected to negative helicity states and it is the ASD part of one of the $d\Phi$ which gives rise to the usual graviton polarization spinor. Thus we get four such spinors and these are then contracted with the other factors in the loop diagram. The result is

For completeness, let us also mention that we can equivalently use the full GR Feynman rules, and this results in the same integral as in (<ref>). The process of construction is exactly similar to the box. In particular, the second term in the interaction Lagrangian in (<ref>) does not contribute and thus the complete one loop integral is constructed using the first term.

§.§.§ Bubble

Finally we consider the bubble contribution to this amplitude. There are two types, bubbles on internal lines and bubbles on external legs. Let us describe the internal bubble diagram.

The loop integral for the diagram is

This is a nominally divergent integral. The denominator of the integrand is exactly like the one in YM, while the numerator is the square of the corresponding one in YM. Thus, in the form it is written above, the same arguments using Lorentz invariant regularization can be applied here. Indeed, the non-vanishing contribution in the numerator of the integrand comes from the quadratic, cubic and quartic in $l$ parts. However, using momentum conservation, the factor $\langle q|(1+2)\circ(3+4)|q\rangle$ vanishes by $\langle qq\rangle$ contraction and thus the quadratic and cubic parts do not contribute. The only non-vanishing contribution comes from the quartic part of the integral. This is proportional to
With any Lorentz invariant regularization, this can only be proportional to $(1+2)_{\mu}(1+2)_{\nu}\eta_{\alpha\beta}$+perm or to $(1+2)^4(\eta_{\mu\nu}\eta_{\alpha\beta}+\textrm{perm})$. Whatever the combination, the presence of the metric causes causes $\langle qq\rangle$ contraction and the diagram therefore vanishes. A similar thing holds for the bubbles on external lines and they too vanish. As we can see, this kind of argument is dependent on the form of the integrand, in conjunction with the SDYM case.
However, the integral in (<ref>) is quartic divergent and thus one again needs to be careful before concluding it to be vanishing. Indeed, like the SDYM bubble, one can try to shift the loop momenta by some arbitrary momenta and then check if the argument applies. We will come to this in a subsequent section, where we discuss the self energy diagram in gravity.

§.§ One-loop finiteness

The box and triangle diagrams in (<ref>) and (<ref>) are nominally divergent to start with. However, we now show that they are actually convergent. Let us consider the box integral in (<ref>) and the same argument will hold for the triangle. We use the Feynman parametrization technique to rewrite the integrand. Thus, we assemble the quadratic factors of the denominator and complete the square by shifting the loop momentum parameter $l$ by an appropriate amount. We then rewrite the integral in terms of the shifted momentum and Feynman parameters

iℳ_box=∫dX ∫d^4l/(2π)^4(⟨q|l-k|4]⟨q|l-k|1]⟨q|l+1-k|2]⟨q|l-4-k|3])^2/(l^2+D)^4(∏_j=1^4⟨qj⟩)^2,

∫dX     = ∫_0^1dx_1∫_0^1-x_1dx_2∫_0^1-x_1-x_2dx_3,

D     = -(x_31-x_14+x_2(1+2))^2+

k     = x_31-x_14+x_2(1+2).

Let us consider the numerator of (<ref>). Note that the terms which have an odd number of loop momenta vanish when integrated over all momenta, since the denominator is symmetric under the exchange $l\rightarrow -l$, while the numerator is anti-symmetric. Next we collect the terms with an even number of loop momenta. In such terms, the Lorentz structure is carried only by the loop momenta and therefore can only be proportional to the metric tensor or sum of products of metric tensors. Let us consider the simplest cases of two and four loop momenta in the numerator. Such terms, by virtue of Lorentz invariance, can be written as

∫d^4l/(2π)^4l_μ_1l_μ_2/(l^2+D)^4   =η_μ_1μ_2∫d^4l/(2π)^4l^2/(l^2+D)^4,

∫d^4l/(2π)^4l_μ_1...l_μ_4/(l^2+D)^4   =(η_μ_1μ_2η_μ_3μ_4+η_μ_1μ_3η_μ_2μ_4+η_μ_1μ_4η_μ_2μ_3)∫d^4l/(2π)^4l^4/(l^2+D)^4.

The generalization to higher powers of loop momenta is now obvious.
In 4-dimensions, the metric tensor can be written as a product of spinor metrics, $\eta_{\mu\nu}=\epsilon_{MN}\epsilon_{M'N'}$. Therefore, the combination of metric tensors produce combinations of a pair of spinor metrics, primed and unprimed. The unprimed spinor metrics contract the $q$s coming from the polarizations leading to $\langle qq\rangle=0$. Thus, we conclude that the terms with an even number of loop momenta also vanish when integrated over all momentum space. The remaining terms in the numerator do not contain any loop momentum factor and are therefore power counting finite. We have thus established that the integral in (<ref>) is convergent. A similar argument renders the triangle integral to be power counting finite.

Having established that the diagrams we consider are finite, it is reasonable to proceed towards computing these and extract the result for the amplitude. However, we encounter a problem at this point. The box and triangle integrals as in (<ref>) and (<ref>) are quartic divergent to start with. The propagator factors in the denominator of these integrals are exactly similar to the ones in SDYM. Thus, it may seem reasonable to cancel propagators and decompose these into bubble like integrals with two propagator factors in the denominator. However, in this process, the degree of divergence of the individual integrals will increase as a result of the decrement of loop momentum factors in the denominator. In particular, the degree of divergence of the resulting bubble-like integrals will be more than or equal to five. Moreover, it is not clear if these bubble like integrals will vanish after shifts, since the numerator in this case will be a complicated function of the loop momentum and external momenta and it is not guaranteed that one can use Lorentz invariant arguments in this case. Even if the integrals vanish after shifting the loop momentum parameters, the computation of shifts will be a daunting task. It is then, if not impossible, almost impenetrable to do a shift computation and extract the result of the amplitude in this route.

We thus follow a different route, namely a possible generalization of the bubble computation in the case of gravity. The motivation behind this follows from the close relations between SDYM and SDGR. As we have seen, the structure of one-loop diagrams in both these theories are on similar footing. The numerator of the integrand of the self-energy in this case is again the square of the one in SDYM, as we shall see below. Also, analogous to SDYM, the notion of self-dual BG currents is present in SDGR. Recall, it is the gluing of these currents to the self energy which leads to the construction of the amplitudes in the former case. We thus expect a similar thing will follow in the latter. Indeed, as we shall see below, it is possible to perform a shift computation of the self energy, except that one needs to calculate shifts upto quartic order in this case. However, it is possible to extract the result by arguing on general grounds. Once the self-energy is computed, one can glue BG currents to it and recover all possible bubble diagrams which can contribute to the amplitude. Let us describe all this

§ SELF ENERGY BUBBLE

Let us now consider the self-energy diagram in the theory of SDGR. As we shall see, the property of this diagram is in parallel to the one in SDYM. In particular, the diagram is sensitive to shifts and is in general non-vanishing.

In the diagram above, the external lines are projected to two negative helicity states and the convention being all external momenta incoming. The amplitude can be written as

\begin{equation}
\label{segr}
i\Pi^{--}_{SDGR}=\int \frac{d^4l}{(2\pi)^4}\frac{(\langle q|l|p]\langle q|l+p|p])^2}{l^2(l+p)^2\langle qp\rangle^4}.
\end{equation}

In this form, the amplitude can be argued to vanish. Indeed, this amplitude is similar to the bubble diagrams, in that the form of the integrand as it is written is the square of the corresponding SDYM self-energy. Also, the diagram is quartic divergent. The only difference in this case is, the propagator which originates from the two vertices of the bubble is truncated and the external lines are projected to helicity states. Thus, the momenta $p$ flowing inside the loop is null. A similar argument like in the case of the bubble diagrams still applies and it is easy to see that with any Lorentz invariant regularisation, the diagram vanishes. However, as we explained earlier, the diagram is quartic divergent and thus one should be careful in analyzing it. Let us shift the loop momentum $l= x+ s'$, where $s'$ is some momentum and $x$ is the new integration variable. The shifted integral is

The argument above depends on the specific form of the integrand in (<ref>) and is no longer applicable to the shifted integrand. Let us then compute the difference between the shifted and the unshifted integral, analogous to the SDYM case. The difference with SDYM however is that, the integral in SDGR is quartic divergent and thus we have to compute shifts upto quartic order. The computation is complicated and we refer the reader to Appendix E for details. Here, we instead give an argument and give the general form of the result.

§.§ Result

Let us understand the form of the result which can be obtained by a shift computation. The
denominator in the result can only have a factor of $\langle qp\rangle^4$. After the shift $x\rightarrow x-\tilde{s}$, the numerator can be a possible combination of two factors $\langle q|\tilde{s}|p]^m\langle q|p+\tilde{s}|p]^n$ with $m+n=4$. Let us give a name to the combination $p+\tilde{s}=s$. Then, it is easy to see that the combination of factors $\langle q|\tilde{s}|p]^m\langle q|p+\tilde{s}|p]^n$ can always be written in the form $(\langle q|s|p]\langle q|\tilde{s}|p])^2$, owing to $[pp]=0$ contraction. Overall, the amplitude takes the form

Thus, the self energy bubble projected to negative helicity states in SDGR is the square of the one in SDYM. This is again reminiscent of the double copy relation in the self-dual sector of these theories. Here we have analysed the bubble using general arguments and obtained the particular form (<ref>). Alternatively, the same result can be obtained from a direct shift calculation (<ref>). The shift in the gravity case is calculated upto quartic order, in contrast with the YM case which was upto second order. One can now interpret the result in (<ref>) as an insertion of negative helicity states to an effective propagator.

§.§ Effective gravitational propagator

The effective propagator in SDGR is the one-loop correction to the $\langle\tilde{\psi}\Phi\rangle$ propagator. Thus, it is the bubble graph with two external $\Phi$ legs which are truncated. The 2-form $d\Phi$ on the external leg can be split into a self-dual part and the anti-self-dual part. We would like to insert negative helicity states to these legs. Such an insertion removes the self-dual part of $d\Phi$ in each of these legs. The anti-self-dual part which is non-vanishing reproduces the usual negative metric helicity. One can then insert negative helicity states to these legs and this results in (<ref>). In terms of the shifts, the effective propagator reads

:=     s^ M_P' s̃_P^ M's^ N_Q's̃^ Q_N'.
Comparing with the SDYM case, the effective propagator in SDGR can be seen to be the double copy of the former. We can now insert two copies of the negative helicity polarization spinor of the graviton (<ref>) and this results in (<ref>). Overall, the shift in gravitational effective propagator is the most natural generalisation of the effective propagator in YM from spin 1 to spin 2 case.

§.§ BG current insertion into effective propagator

Similar to YM, we can construct bubble diagrams in gravity as a result of insertion of BG currents to the effective propagator. Let us explain this.

§.§.§ Internal Bubbles

There are two in-equivalent permutations of the bubbles on internal lines. In one diagram, the legs 1 and 2 join on one side of the internal line while 3 and 4 on the other side. In the other diagram, the legs $1, 4$ join on one side while the legs $2,3$ join on the other end. When written in terms of insertion of 2-currents to the effective propagator in (<ref>), the result of these diagrams read
ℳ_11=𝕁(1,2)𝕁(3,4)(⟨q|s_1∘(1+2)|q⟩⟨q|s̃_1∘(1+2)| q⟩)^2,
ℳ_12= 𝕁(1,4)𝕁(2,3)(⟨q|s_2∘(1+4)|q⟩⟨q|s̃_2∘(1+4)| q⟩)^2,
where $s_i,\tilde{s}_i$ for $i=1, 2$ are arbitrary shifts of the loop momentum variable in the bubble integrals. Let us also write down the results for the shift in external bubbles.

§.§.§ Bubbles on External Lines

There are four distinct contributions here, each of which can be written as an insertion of the 3-current and 1-current to the legs of the bubble. Let us first denote a generic 3-current by a diagram

𝕁_MNPQ'(2,3,4)    =

On the left hand side of the diagram, the dashed line indicates the field $\tilde{\psi}$ with the convention that the propagator is included as a part of the definition of the current and the blob indicates the sum of three tree diagrams.
Next we insert each of these currents into the bubbles such that for a particular graph, the internal line adjacent to a bubble is the off-shell leg of the current. The pole resulting from the denominator in the propagator of the internal line, i.e $1/p^2$, where $p=1,2,..4$ gets cancelled yielding a finite result in the on-shell limit. The external bubble diagrams can then be expressed in terms of the gluing of BG currents to the effective propagator as

ℳ_21=𝕁(1)𝕁(2,3,4)(⟨q|s_3|1]⟨q|s̃_3|1| q⟩)^2,
ℳ_22= 𝕁(2)𝕁(1,3,4)(⟨q|s_4|2]⟨q|s̃_4|1| q⟩)^2,
ℳ_23=𝕁(3)𝕁(1,2,4) (⟨q|s_5|3]⟨q|s̃_5|3| q⟩)^2,
ℳ_24=𝕁(4)𝕁(1,2,3) (⟨q|s_6|4]⟨q|s̃_6|4| q⟩)^2.
Here, we summarised the list of all possible bubble graphs for the one loop same helicity four point amplitude. The structure of the bubbles mimicks the YM ones, in that they arise by gluing BG currents to the effective propagator. However, in this case, one considers all possible permutations of the external legs in contrast to YM, where only cyclic permutations are allowed. The number of in-equivalent bubble diagrams are thus higher in the case of gravity. This is correctly captured by the structure of the currents in SDGR. In particular, the 3-currents in SDGR contains a sum of three distinct terms, unlike the ones in SDYM where there is only a single term. Thus, it is this higher number of terms in the gravity currents which help to capture all possible permutations of the bubbles and assemble them in a condensed form which we wrote above. We want to compute them explicitly and extract the result for the amplitude. However, here we face a problem. It is not clear how to interpret the shift parameters in the gravity case. In the case of YM we could associate these with the region momenta. There is no cyclic ordering in the case of gravity, and so the use of region momenta does not seem justified any longer. Thus, while there is strong evidence that there must exist a gravitational analog of the bubble interpretation of the four point amplitude and its generalization to arbitrary multiplicity as in the formula (<ref>), it is not clear how to give meaning to the momentum variables appearing in the gravitational effective propagator, and so it is not clear what form the GR analog of (<ref>) should take.

Conclusion and Outlook
In this thesis, we have studied the series of all minus amplitudes in massless QED, self-dual YM and self-dual GR. These amplitudes are rational functions of the momenta involved and are devoid of any branch cut singularities. Firstly, we have analyzed such an amplitude at four points in SDYM and massless QED, establishing that the result comes out from a computation of shifts. The important feature of such a computation is that it is done in 4 dimensions and avoids any regularization scheme. Indeed, in the absence of any singularities, except for two particle poles, the result should not depend upon any regularization and this is what we were able to demonstrate. An important insight gained from this computation is that the technique of reducing one-loop amplitudes to shifts can be an alternative variant to extract rational parts of a wider class of amplitudes.

However, the most important result of this thesis is the understanding that the all minus amplitudes in Yang-Mills can be described by a very simple formula consisting of the BG currents and an effective propagator in SDYM.

We have also managed to give an explicit proof of our formula by establishing that it is independent of the region momentum variables that are needed to make sense of the effective propagator. To establish our formula, we used the technology of shifts to show that the self-energy bubble in SDYM is ambiguous and shift dependent. Thus, it is rather a special choice of shift variables, namely the region momentum variables which produces a finite result for the bubble. The amplitudes are then built by gluing BG currents to this bubble and adding all possible cyclic permutations. The non-vanishing of such amplitudes reduces to the non-vanishing of the bubble. The formula we proposed is important for several reasons. Let us explain these.

It is well known that the CSW [23] tree-level formalism gives a prescription for how to compute tree-level YM
amplitudes with effective Feynman rules that use MHV amplitudes as vertices. An attempt to extend
this to loop amplitudes was made in [24], where it was in particular suggested that one-loop same
helicity amplitudes that are the subject of this thesis can be added as new interaction vertices, in
addition to the tree level MHV vertices. The interpretation of these one-loop amplitudes that emerges
from our work is that they are built from more elementary tree-level blocks. It thus seems unnatural
to try to build more involved amplitudes from the amplitudes that are already composed. Our results
show that these amplitudes are made from simpler tree-level amplitudes connected by an effective
propagator. Thus, it appears that it is the region momenta-dependent effective propagator that should
be added into the set of CSW rules if one is to reproduce loop amplitudes.

Our new interpretation is also relevant for the problem of UV divergence of quantum gravity. As we have already discussed, this divergence was linked to the non-vanishing of the same helicity one-loop amplitude
in GR. Given that the story with GR is likely mirroring that in YM, the
interpretation of this work shows that the non-vanishing of the same helicity one-loop amplitudes can
in turn be linked to the non-vanishing of the one-loop bubble. It would thus be very interesting to
better understand the significance and interpretation of the non-vanishing of the bubbles.
This leads to the question of interpretation of such a simple construction as our formula. The fact that
region momentum variables play such a prominent role points to both the worldsheet interpretation as
in [14], as well as the (momentum) twistor space interpretation as in [31]. In any case, it would be very
interesting to understand a deeper origin of our proposed formula.

In this work we have analysed only the same helicity amplitudes. These amplitudes vanish at tree level, and are non-zero at one loop. There is, however, another
amplitude at one-loop with exactly the same status, this is the all but one same helicity
amplitude. It vanishes at tree level, but is given by a purely rational expression at one
loop, see e.g. the formula (3.10) in [10]. This amplitude, however, cannot be computed in
SDYM theory. This is because in SDYM only diagrams with external gauge field lines can
be constructed at one loop. In the case of the same helicity one loop amplitudes this is sufficient. Indeed, in the conventions used in SDYM, auxiliary field can only describe the
positive helicity. When one computes the all minus amplitude, the Feynman diagrams with
the gauge field on the external lines are the only ones to consider, and these are precisely
the diagrams that get generated in SDYM. However, in the case of a $(---+)$ amplitude,
in addition to diagrams with the gauge field on external lines, there is a new diagram where
the minus helicity state gets inserted into an auxiliary field external line. This diagram
is not present in SDYM, and one needs the full YM Feynman rules to construct it. It is
possible that the methods outlined in this thesis can still be applied to this situation and the sum
of all arising diagrams gets reduced to shifts of the effective propagator, but this is far from certain and we will leave
this computation to future research.

Another important aspect of this thesis is to give the first steps of the understanding of all minus amplitudes in gravity. The main issue in gravity is that the existing Feynman rules in the usual Einstein-Hilbert formulation is a mess to deal with. Thus, we considered a recently developed chiral formulation for GR [40] which has very simple Feynman rules. The only missing parts in this formalism was the ghost Feynman rules. We thus developed them in this work by using the BRST formalism. Before discussing the all plus amplitudes in gravity, let us describe as an aside, our study of the one-loop effective action around a general Einstein background in the new formulation. We devised a generalization of the flat space gauge fixing Lagrangian which is described in [40], to a general Einstein background. We found that this introduces a novel $hh$ term in the kinetic part of the action. The novelty in the gauge fixing is that the kinetic part of the Lagrangian gets decoupled into two distinct sectors in an analogous fashion as in flat space. Another important aspect in the computation is that the BRST transform of the Lorentz and diffeomorphism ghosts are coupled and this results in a ghost Lagrangian where there are coupling terms between the two ghosts. However, using field re-definitions, we could get rid of the mixed terms and found the relevant Laplace type operator for the kinetic part. We then studied the ghost contribution to the effective action using the heat kernel methods.

It is possible to construct the all plus (or minus) amplitudes in gravity both from the full GR Feynman rules or the SDGR Feynman rules. The SDGR Feynman rules are obtained from the recently proposed action for flat SDGR in [33], where a detailed gauge fixing is outlined. The diagrams contributing to the four point all minus amplitude is constructed using these Feynman rules. The diagrams are all quartic divergent to start with. However, using Lorentz invariance arguments, we showed that the divergent pieces vanish and the diagrams are actually power counting finite. This is reminiscent of the one-loop finiteness of pure gravity. The problem arises when we attempt to compute these diagrams. The shift technology if at all applicable, is not feasible to extract the answer. This is because it is not clear if the process to cancel denominators and reducing the resulting integrand to shifts will hold in this case. Thus, we take the alternative route to compute the amplitude using a bubble computation, in analogy with SDYM. Indeed, we computed the self-energy bubble, with the result being the square of the corresponding YM one. We then glue BG currents to this bubble and the sum over all possible permutations is expected to give the amplitude. The only issue here is that is that the interpretation of the shift parameters is not clear, unlike in YM where the shifts were interpreted as region momenta. The main difference in gravity is that, unlike in YM (or SDYM), everything is permutation symmetric and thus the whole group of permutation symmetry is at play, rather than just the smaller subset of cyclic symmetry. Thus, the arguments in SDYM which depend on cyclic ordering do not apply in gravity. Thus, while there is strong evidence of an analogous formula as in (<ref>) for gravity, it is not clear what form the formula will take. We leave this for future research. Let us now summarise the main points and results of the different chapters in this thesis.

•  Chapter 4: The $\beta$-function in the chiral formulation of Yang-Mills is worked out. The Lagrangian and Feynman rules for SDYM has been sketched and a description of Yang-Mills instantons has been outlined. There are no new results in this chapter but some warm-up calculations to fix notations.

•  Chapter 5: The results in this chapter appeared in [62] and [63]. The one-loop four point amplitude is reduced to a shift computation in SDYM and massless QED (with scalar QED). The sum of integrands for this amplitude is computed in the covariant formalism. It is observed that the sum vanishes with a particular assignment of region momentum variables in the graphs. The self-energy bubble is observed to be shift dependent and the shift is computed using standard techniques. The result of the shift is interpreted as an effective propagator in SDYM. The four point amplitude is then obtained by gluing BG currents to this effective propagator and summing over cyclic permutations. A new formula for all YM amplitudes (same helicity) at one-loop is proposed and proved. The collinear and soft limits are analyzed. As an illustration of our formula, the explicit computation of the amplitude at 3, 4 and 5 points are outlined.

•  Chapter 7: Some of the results of this chapter are new. The quantization of the chiral Einstein-Cartan gravity is studied in a general Einstein background. We use the BRST formalism to develop the linearised gauge fixed Lagrangian. The non-triviality in the BRST transformations is that the transformation for the Lorentz and diffeomorphism ghosts are mixed. This results in a ghost Lagrangian where there is a coupling term between the two fields. Nevertheless, we diagonalised the arising operator in the Lagrangian and then used the heat kernel to compute the ghost contribution to the one loop effective action. This part has not appeared elsewhere.

We also deduced the ghost Feynman rules in flat space. For the sake of completeness, we mentioned the remaining Feynman rules for chiral Einstein-Cartan and this part is mostly taken from [40].

•  Chapter 8: This chapter is mostly based on [33] and [32]. The action for flat space self-dual gravity is described. There are two relevant actions and we described one of them in brief and the other in details. The Feynman rules and BG currents are outlined for the SDGR theory. A short description of gravitational instantons is also sketched.

•  Chapter 9: In this chapter, we give partial results for the one-loop same helicity amplitudes in self-dual gravity. This part is new and has not appeared elsewhere. The construction of the one-loop same helicity amplitude at four point is sketched.  The setup of the four point amplitude is outlined in terms of the box, triangle and bubble diagrams. An explanation for the finiteness at one-loop is given. The result for the self-energy bubble in SDGR is obtained, both by direct computation and general arguments. The truncated bubble is then interpreted as an effective gravitational propagator in the theory. BG currents are glued to this propagator in all possible ways and it is expected that the sum of all these bubble diagrams should result in the four point amplitude. However, this last statement remains to be seen explicitly. The only missing element is the interpretation of the shift parameters for the bubbles, which is not clear in the case of gravity. Thus, the bubble interpretation of the gravity amplitudes is incomplete.

A possible direction to approach the computation in the gravity case is as follows. Using the analogy of SDYM, we can first try to compute the sum of integrands for the one loop amplitude. This would require contribution of the box, triangles and bubbles. The technique of decomposing the box and triangles to "bubble"-like integrals is the necessary next step. This part is non-trivial because unlike the case of SDYM, in gravity we deal with quartically divergent integrals from the beginning. Once this reduction is achieved, it may be possible to consider some good choice of loop momentum variables such that the integrand vanishes. It remains to be seen what kind of variables these are. The formula for the amplitude at all multiplicity in gravity can then possibly be sketched. We however leave this for future research.

CHAPTER: FEYNMAN PARAMETER INTEGRALS

In this part of the Appendix, we give a list of some of the results for the Feynman parameter integrals which we use in Chapter 4.

\begin{equation}
\label{1}
\int \frac{d^{4-\epsilon}q}{(2\pi)^{4}}\frac{1}{(q^2+D)^2}=\frac{i}{16\pi^2}\frac{\Gamma(\epsilon/2)}{\Gamma(2)}\frac{1}{(-D)^{\frac{\epsilon}{2}}},
\end{equation}

\begin{equation}
\label{2}
\int \frac{d^{4-\epsilon}q}{(2\pi)^4}\frac{q^2}{(q^2+D)^2}=\frac{-2i}{16\pi^2}\frac{\Gamma(\epsilon/2-1)}{\Gamma(2)}\frac{1}{(-D)^{\frac{\epsilon}{2}-1}},
\end{equation}

\begin{equation}
\label{3}
\int \frac{d^{4-\epsilon}q}{(2\pi)^{4}}\frac{q^2}{(q^2+D)^3}=\frac{2i}{16\pi^2}\frac{\Gamma(\epsilon/2)}{\Gamma(3)}\frac{1}{(-D)^{\frac{\epsilon}{2}}},
\end{equation}

\begin{equation}
\label{4}
\int \frac{d^{4-\epsilon}q}{(2\pi)^4}\frac{1}{(q^2+D)^3}=\frac{2i}{16\pi^2}\frac{\Gamma(\epsilon/2)}{\Gamma(3)}\frac{1}{(-D)^{\frac{\epsilon}{2}}},
\end{equation}

\begin{equation}
\frac{1}{(-D)^{\epsilon/2}}=1-(\epsilon/2)log(-D)+.....
\end{equation}

The Gamma function can be expanded near its poles as follows:

CHAPTER: EXTRACTING THE SHIFT

In this part, we extract the shift using standard technology, but in spinor notations. We use this result in Chapter 5.
Let us consider the difference of two one dimensional quadratically divergent integrals of the form
J(b)-J(0)≡∫_-∞^+∞dy [f(y+b)-f(y)],
where $f(\pm\infty)=d_{\pm}$ are finite constants. If the integral were convergent, it is easy to see that $J(b)-J(0)$ would vanish by a quick change of integration variables $y+b\rightarrow y'$. In the present case, however, let us Taylor expand $f(y+b)$ in powers of $b$. Note that second and higher derivatives of $f(y)$ vanishes at $y=\pm\infty$.
We see that the difference gives a finite value in this case.

CHAPTER: CHECK OF REGION MOMENTA INDEPENDENCE IN SDYM

In this part of the Appendix, we explicitly show that the four and five point amplitudes are region momenta independent. Thus, we use the shift symmetry of region momenta and write all region momenta in terms of one of them, say $x$. We then show that the terms containing $x$ vanish identically. We first do it for the four point case and subsequently for the five point case.

§ FOUR POINT CASE

We demonstrate that the 4-point amplitude is region momentum independent. Let us start by analysing the terms linear in $x$. For instance, there is such term coming from $\langle q|p_3\circ p_2|q\rangle^2$. Indeed, we have
⟨q|p_3∘p_2|q⟩^2= ⟨q|(2 + x)∘3|q⟩^2
= ⟨q|2∘3|q⟩^2 + 2⟨q|2∘3|q⟩⟨q|x∘3|q⟩
+ ⟨q|x∘3|q⟩^2.
The second term on the right-hand side is linear in $x$. Keeping only such terms we get the linear in $x$
part of the amplitude

𝒜^x_4   =2[23]⟨q|x|3]/⟨q4⟩⟨41⟩⟨12⟩-2[14]⟨q|x|4]/⟨q3⟩⟨23⟩⟨12⟩+2[12](⟨q|x|3]⟨3q⟩+⟨q|x|4]⟨4q⟩)/⟨q3⟩⟨34⟩⟨12⟩⟨4q⟩

The last equality follows by momentum conservation as the terms in brackets are zero.
We now analyse the terms quadratic in $x$. There are such terms coming from all the terms in (<ref>).
We get

𝒜_4^x^2   =⟨q|x|1]^2/⟨q2⟩⟨23⟩34⟩⟨q4⟩+

We want to demonstrate that this vanishes by momentum conservation and Schouten identities. Momentum conservation gives us the following 4 relations (not all independent)

⟨q|x|2]⟨21⟩+ ⟨q|x|3]⟨31⟩+ ⟨q|x|4]⟨41⟩= 0,

⟨q|x|1]⟨12⟩+ ⟨q|x|3]⟨32⟩+ ⟨q|x|4]⟨42⟩= 0,

⟨q|x|1]⟨13⟩+ ⟨q|x|2]⟨23⟩+ ⟨q|x|4]⟨43⟩= 0,

⟨q|x|1]⟨14⟩+ ⟨q|x|2]⟨24⟩+ ⟨q|x|3]⟨34⟩= 0.

These can be used to express two of the quantities $\langle q|x|i]$ in terms of the other two. For example, we
can decide to express $\langle q|x|1],\langle q|x|2]$ in terms of $\langle q|x|3],\langle q|x|4]$. We have
⟨q|x|1]⟨12⟩= ⟨q|x|3]⟨23⟩+ ⟨q|x|4]⟨24⟩,
⟨q|x|2]⟨12⟩= ⟨q|x|3]⟨31⟩+ ⟨q|x|4]⟨41⟩.
Using this, we can collect coefficients in front of $\langle q|x|3]^2$, $\langle q|x|4]^2$ and $\langle q|x|3]\langle q|x|4]$. For example, the
coefficient in front of $\langle q|x|4]^2$

This is further transformed using Schouten identities. Writing everything with the common denominator $\langle q1\rangle\langle q2\rangle\langle q3\rangle\langle q4\rangle\langle 23\rangle\langle 34\rangle\langle 12\rangle^2$ we get the following expression

⟨q1⟩⟨q3⟩+ ⟨23⟩⟨41⟩⟨q2⟩⟨q4⟩+

We can transform the first term using
⟨q1⟩⟨24⟩=⟨q2⟩⟨14⟩-⟨q4⟩⟨12⟩,   ⟨q3⟩⟨24⟩=⟨q2⟩⟨34⟩-⟨q4⟩⟨32⟩.
This gives four terms that cancel the remaining terms in (<ref>). The coefficients in front of $\langle q|x|3]^2$ and
$\langle q|x|3]\langle q|x|4]$ are checked to vanish analogously.

§ FIVE POINT CASE

Let us now show that the 5-point amplitude is region momenta independent. To this end, we parametrise the region momenta in terms of one of them and the external momenta
p_1=x,  p_2=2+x,  p_3=3+2+x,  p_4=4+3+2+x,  p_5=x-1.
Let us start by analysing the terms linear in $x$. We exclude terms containing the region momenta $p_1$ because such terms are always quadratic in $x$. We extract the linear terms in $x$ from the remaining ones. For example, there is such term coming from
$\langle q|p_4\circ p_3|q\rangle^2$. Indeed, we have

We can see that the second term on right hand side of the above equation is linear in $x$. Collecting such terms, we have

Let us compute the bracketed terms for each of the coefficients $\langle q|3\circ x|q\rangle, \langle q|4\circ x|q\rangle, \langle q|5\circ x|q\rangle$. After bringing to a common denominator, we have for the coefficient of $\langle q|3\circ x|q\rangle$
Using Schouten identity, we can write the first and fourth terms in the numerator as

We next use momentum conservation to write the second and third terms in the numerator as

Clearly, the computed terms in (<ref>) and (<ref>) cancel each other. Thus, the coefficient of $\langle q|3 \circ x|q\rangle$ in (<ref>) vanishes.Let us now compute the coefficients of $\langle q|4\circ x|q\rangle$. Putting the coefficients in (<ref>) over a common denominator, we have

First, let us apply momentum conservation on the fifth and sixth terms in the above. Thus we have

Let us then combine the term on the right hand side in (<ref>) with the last term in (<ref>) using Schouten identity. We have

[15]⟨35⟩⟨51⟩⟨4q⟩+[51]⟨51⟩⟨34⟩⟨5q⟩   =[15]⟨51⟩(⟨35⟩⟨4q⟩-⟨34⟩⟨5q⟩)

We also apply Schouten identity to combine the second and third terms in (<ref>)

[42]⟨45⟩⟨34⟩⟨2q⟩+[42]⟨45⟩⟨23⟩⟨4q⟩   =[42]⟨45⟩(⟨34⟩⟨2q⟩+⟨23⟩⟨4q⟩)

All in all, we finally reduced the numerator in (<ref>) to the following sum of terms

However, momentum conservation gives us
Thus the terms in (<ref>) add up to zero. Therefore we find that the coefficient of $\langle q|4\circ x|q\rangle)$ vanishes. Next we compute the coefficient of $\langle q|5\circ x|q\rangle$. Once again, we put the relevant coefficient in (<ref>) over a common denominator. We have

Consider the numerator in (<ref>). Using Schouten identity for the second and fourth terms, we write it as

We can then write the numerator in (<ref>) as

Thus we have shown that all the linear terms in $x$ vanish, i.e
Consider now the quadratic terms in $x$. The relevant ones are

𝒜^x^2_5   =⟨q|1∘x|q⟩^2(J(1)J(2,3,4,5)+J(5,1)J(2,3,4)+J(1,2)J(3,4,5))
⟨q|2∘x|q⟩^2 (J(2,3)J(4,5,1)+J(1,2)J(3,4,5)+J(2)J(3,4,5,1))

There are two kinds of terms. One is of the form $\langle q|i\circ x|q\rangle^2, i=1,2,..5$ and another of the form $\langle q|i\circ x|q\rangle\langle q|j\circ x|q\rangle$.
We want to demonstrate $\mathcal{A}^{x^2}_5$ vanishes using momentum conservation and Schouten identity. First, we have the following set of relations on using momentum conservation identities

Using this, we can express any two of the quantities $\langle q|x|i]$ in terms of the other three. Thus we express

We next collect coefficients of terms like $\langle q|3\circ x|q\rangle^2, \langle q|4\circ x|q\rangle^2, \langle q|4\circ x|q\rangle\langle|q|5\circ x|q\rangle$, etc. For instance, the coefficient of $\langle q|4\circ x|q\rangle\langle|q|5\circ x|q\rangle$ is given by

+J(4,5)J(1,2,3)+⟨24⟩⟨1q⟩/⟨4q⟩⟨12⟩ J(5,1)J(2,3,4).

Let us simplify the above terms. Note that the numerator of the coefficients of $J(1,2)J(3,4,5)$ in the second and third bracketed terms can be expanded using the  Schouten identity. We have

Some of the terms in the right hand side now cancel the terms of the first bracketed expression in (<ref>). In a similar way we combine the coefficients of $J(5,1)J(2,3,4)$ using Schouten identity.
Then we are left with the following terms

+J(5,1)J(2,3,4)+ J(1,2)J(3,4,5).

Putting everything over the common denominator $\langle 1q\rangle^2\langle 2q\rangle^2\langle 4q\rangle^2\langle 5q\rangle^2\langle 3q\rangle\langle 12\rangle^2\langle 23\rangle\langle34\rangle\langle45\rangle\langle51\rangle$ we get the following terms in the numerator

We next show that all these terms add up to zero by repeated use of Schouten identities. First, consider the term $\langle 14\rangle\langle 15\rangle\langle 34\rangle\langle 2q\rangle^3\langle 1q\rangle\langle 5q\rangle$. Using Schouten identity for the pair $\langle 14\rangle\langle 2q\rangle$, we can expand it as
We again transform the second term on the right hand side to get

The first and the third terms on the right hand side of (<ref>) cancel the fourth and the fifth terms in (<ref>) respectively. We are then left with the following terms

Let us now expand the term $\langle 12\rangle\langle 23\rangle\langle 51\rangle\langle 1q\rangle\langle 2q\rangle\langle 4q\rangle^2\langle 5q\rangle$ using Schouten identity for the pair $\langle 12\rangle\langle 4q\rangle$. We have

The first term on the right hand side in (<ref>) cancels the second term in (<ref>). Finally, we have the remaining terms
Taking a factor of $\langle 24\rangle\langle 51\rangle\langle 1q\rangle^2\langle 4q\rangle\langle 2q\rangle$, we can write the above as
The bracketed terms then vanish due to the Schouten identity. Thus we have shown that the coefficient of $\langle q|4\circ x|q\rangle\langle q|5\circ x|q\rangle$ vanish completely. Analogously, it is checked that all the coefficients of the other quadratic terms vanish similarly. Thus the linear and quadratic terms in $x$ for the five point amplitude all vanish, establishing that it is region momenta independent.

CHAPTER: BRST CLOSURE

In this part, we verify that the BRST transformations outlined in (<ref>) are nilpotent. We already described this for the trivial set of transformations in the main text. Here, we detail out the rest of the non-trivial sets.

§ DIFFEOMORPHISM GHOST

Let us start with the second transformation in (<ref>), which is that of the diffeomorphism ghost. We have

s^2c_μ   =sc^λ∇^T_λc_μ-c^λ∇^T_λsc^μ

=-1/2c^λc^νR^μ_ ανλc^α

=-1/2c^αc^λc^νR^μ_ ανλ

=-1/6c^αc^λc^ν(R^μ_ ανλ+R^μ_ νλα+R^μ_ λαν)=0.

where we have used the sign carefully to take care of the anti-commutativity of the ghosts. In the fourth line of the above, we used this property to write the single term as a combination of two terms, which then becomes the commutator of the covariant derivatives. Finally, we express this commutator in terms of the Riemann tensor and used the symmetry properties of it to deduce that the expression vanishes due to the first Bianchi identity.

§ TETRAD

For the transformation of the tetrad, we have

s^2h^i_μ   =sc^ν∇_νh^i_μ(x)-c^ν∇_νsh^i_μ(x)+sh^i_ν(x)∇_μc^ν+h^i_ν(x)∇_μsc^ν

=c_γ∇^γc^δ∇_δ h^i_μ-c^ν∇_νc^δ∇_δh^i_μ-c^νc^δ∇_δ∇_νh^i_μ

=h^i_νc^γ R^ν_ αγμc^α+1/2c^δc^γR^β_ μγδh^i_β
=h^i_νc^γc^α R^ν_ αγμ+1/2c^δc^γR^β_ μγδh^i_β

=1/2h^i_ν(c^γc^α-c^αc^γ)R^ν_ αγμ+1/2c^δc^γR^β_ μγδh^i_β

=1/2h^i_νc^γc^α R^ν_ αγμ+1/2h^i_νc^γc^α R^ν_ γμα+1/2c^δc^γR^β_ μγδh^i_β

=1/2h^i_νc^γc^α (R^ν_ αγμ+R^ν_ γμα+R^ν_ μαγ)=0.

In the second line of the above, many terms get cancelled and we are left with only three terms. These terms are rearranged in the form of the commutator of covariant derivatives which are further expressed again in terms of the Riemann tensor. We then used the anti-commutativity of the ghosts to expand one of the terms. The three terms now have the same form and thus we used the first Bianchi identity for the Riemann tensor in the last expression to yield a vanishing result.

§ LORENTZ GHOST

For the transformation of the Lorentz ghost field, we find

s^2b^ij   =-1/2s[b,b]^ij+1/2s(c^μc^νF^ij_  μν)

=-sb^i_ kb^kj+b^i_ ksb^kj+1/2sc^μc^νF^ij_  μν

-1/2c^μsc^νF^ij_  μν+1/2c^μc^νsF^ij_  μν

=b^i_ mb^m_ kb^kj-1/2c^μc^νF^i_ kμνb^kj-b^i_ kb^k_ mb^mj

+1/2b^i_ kc^μc^νF^kj_  μν+1/2c^λ∇^T_λc^μc^νF^ij_  μν
-1/2c^μc^λ∇^T_λc^νF^ij_  μν

+1/2c^μc^νc^λ∇^T_λF^ij_  μν+1/2c^μc^νF^ij_  μλ∇^T_νc^λ+1/2c^μc^νF^ij_  λν∇^T_μc^λ+1/2c^μc^ν[F_μν,b]^ij

=1/2c^μc^νc^λ∇^T_λF^ij_  μν
=1/6c^μc^νc^λ(∇^T_λF^ij_  μν+∇^T_μF^ij_  νλ+∇^T_νF^ij_  λμ)

where in the last line, we used the Bianchi identity and the fact that the Lorentz ghost fields anti commute and hence can be appropriately permuted to give rise to three such terms.

§ CONNECTION

Let us now check the closure for the connection field. We have

s^2ω^ij_ μ   =∇^T_μsb^ij+sc^νF^ij_  νμ-c^νsF^ij_  νμ

=-∇^T_μ(b^i_ kb^kj)+1/2∇^T_μ(c^μc^νF^ij_  μν)+c^k∇^T_kc^νF^ij_  νμ-c^ν[F_νμ,b]^ij

-c^νc^λ∇^T_λF^ij_  νμ-c^νF^ij_  νλ∇^T_μc^λ-c^νF^ij_  λμ∇^T_νc^λ

=-∇^T_μ(b^i_ kb^kj)-1/2c^ν∇^T_μc^μF^ij_  μν+1/2c^μ∇^T_μc^νF^ij_  μν
+1/2c^μc^ν∇^T_μF^ij_  μν+c^k∇^T_kc^νF^ij_  νμ-c^νc^λ∇^T_λF^ij_  νμ
-c^νF^ij_  νλ∇^T_μc^λ-c^νF^ij_  λμ∇^T_νc^λ-c^ν[F_νμ,b]^ij

=-∇^T_μ(b^i_ kb^kj)-c^ν[F_νμ,b]^ij

-1/2c^νc^λ(∇^T_λF^ij_  νμ+∇^T_νF^ij_  μλ+∇^T_μF^ij_  λν)

where in the above, we used the Bianchi identity for the curvature, interchanged dummy indices and used the anti-commutative nature of the ghost fields.
This completes the check of the nilpotency of the BRST operator.

CHAPTER: SELF ENERGY COMPUTATION IN GRAVITY

§ SHIFT COMPUTATION

§.§ Linear part

The linear part of the shift is given by
The non-zero contribution can only come from the quadratic and quartic in $x$ terms of this shift, after taking an average over all directions. Note, the $x^6$ term do not contribute since it vanishes after the possible contractions. The quadratic term is
This gives the shift

The quartic part of the shift is given by
The integral is computed using

The $x^6$ contractions vanish while the $x^4$ contractions give

§.§ Quadratic part

For the quadratic part of the shift, the integral is given by

When the derivative hits the denominator, it produces a factor proportional to
The quartic part of the numerator is the only one which contributes. Using (<ref>) this gives
When the derivative in (<ref>) hits the numerator, in the large $x$ limit, we get
Only the quadratic in $x$ terms in the numerator contribute, while the others vanish. Using the relevant contraction, this gives

§.§ Cubic part

For the cubic part of the shift, the integral is given by

In the quadratic part of the shift, we evaluated the action of the derivative on the function in two parts, one in which the derivative acts on the numerator and another in which it acts on the denominator. Here, we will take these parts separately and act the second derivative on each part. Let us first consider the part which comes as a result of the action of the derivative on the denominator of the function. It is given by

When the derivative hits the denominator, in the large $x$-limit, it gives

Only the $x^6$ part in the numerator contributes. Using
lim_x→∞∫dΩ/(2π)^4x_μ x_νx_αx_βx_γx_δ/x^6=1/32.48π^2(η_μνη_αβη_γδ+perm )
the only non vanishing contractions are six in total, giving
When the derivative in (<ref>) hits the numerator, in the large $x$-limit, it gives

Using the relevant contraction, this gives
Next consider the part which comes as a result of the action of the first derivative on the numerator of the function in (<ref>). Only the $x^4$ parts in the numerator contributes, giving
When the derivative hits the numerator,
only the $x^2$ terms in the numerator contribute, giving

Simplifying a bit, we get

§.§ Quartic part

For the quartic part of the shift, the integral is given by

We use the same evaluation technique for the shift as before. For the cubic part of the shift, we evaluated the action of the second derivative on the resulting functions from the quadratic part in two parts, one in which the derivative acts on the numerator and another in which it acts on the denominator. Here, we will take the corresponding functions from the cubic part and act the third derivative on each part. Let us first consider the part which comes as a result of the action of the derivative on the denominator of the function in (<ref>). It is given by
Only the $x^8$ factor in the numerator contributes, with sixty non-vanishing contractions, giving

Next, when the third derivative in (<ref>) hits the numerator of (<ref>), we have
Only the $x^6$ terms in the numerator contribute with six non-vanishing contractions, giving

We now consider the action of the third derivative on the next function, in (<ref>). When the derivative hits the denominator of the function, it gives

Terms with only $x^6$ factor in the numerator contribute, giving -i/32.3π^2⟨q|s'|p]^4.
When the derivative hits the numerator of the function, it gives in the large $x$-limit

The $x^4$ terms in the numerator contributes, giving
Next consider the function in (<ref>). When the third derivative hits the denominator in the large $x$-limit, it gives

The $x^6$ terms in the numerator contributes here, giving
When the derivative hits the numerator of the function in (<ref>), it gives

Only the $x^4$ terms in the numerator survive, giving
Next consider integral in (<ref>). When the third derivative acts on the numerator of this function, it gives

Evaluating the derivative and simplifying, we have

This gives

§.§ Result

Adding all the contributions, we have for the SDGR self energy bubble, upto numerical factors

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