Celestial OPE in Self Dual Gravity

Celestial holography is a conjectured duality between quantum gravity in 4D asymptotically flat spacetime and a quantum field theory on the 2D celestial sphere Strominger:2017zoo ; Pasterski:2016qvg ; Pasterski:2017kqt . Symmetries play an important role in this conjectured duality. The Lorentz group in 4D acts on the 2D celestial sphere as the global conformal group. So the dual theory should be a conformal field theory. Motivated by this, a new basis was introduced Pasterski:2016qvg ; Pasterski:2017kqt ; Banerjee:2018gce in which the $S$-matrix elements transform like 2D conformal correlators. Besides the two dimensional global conformal symmetry, CCFT has various infinite dimensional current algebra symmetries Sachs:1962zza ; Strominger:2013jfa ; He ; Strominger:2014pwa ; Barnich:2009se ; Kapec:2016jld ; Kapec:2014opa ; He:2017fsb ; Donnay:2020guq ; Stieberger:2018onx ; Banerjee:2022wht ; Banerjee:2020zlg ; Banerjee:2021cly ; Gupta:2021cwo ; Guevara:2021abz ; Strominger:2021lvk ; Himwich:2021dau ; Ball:2021tmb ; Adamo:2021lrv ; Costello:2022wso ; Stieberger:2022zyk ; Stieberger:2023fju ; Ball:2023qim ; Melton:2022fsf ; Banerjee:2020kaa ; Costello:2022upu ; Costello:2023vyy ; Garner:2023izn ; Donnay:2021wrk ; Donnay:2022hkf .

Operator product expansion (OPE) is a central tool to study various aspects of any CFT. In the context of celestial CFT also, OPE played an important role in identifying new symmetries Guevara:2021abz ; Strominger:2021lvk ; Banerjee:2020zlg , null states Banerjee:2020zlg ; Banerjee:2021cly ; Banerjee:2019aoy ; Banerjee:2019tam ; Pano:2023slc ; Banerjee:2023zip ; Banerjee:2020vnt ; Banerjee:2023rni ; Adamo:2022wjo ; Ren:2023trv ; Hu:2021lrx ; Ebert:2020nqf ; Pasterski:2021fjn etc. It has also found applications in the Bootstrap program Atanasov:2021cje ; Ghosh:2022net . Based on the universal singular structure of the tree-level OPE between two positive helicity gravitons, it was shown in Guevara:2021abz that CCFT has an infinite tower of soft symmetries which close into $w_{1+\infty}$ algebra Strominger:2021lvk . Loop corrections to the tree level celestial OPEs have been studied in Bhardwaj:2022anh ; Krishna:2023ukw .

In a previous paper Banerjee:2023zip , we have studied the implications of the $w_{1+\infty}$ symmetry at the level of OPEs by using representation theory. By studying the subleading terms in the OPE between two positive helicity outgoing gravitons, we have shown that there should exist an infinite number of theories which are invariant under $w_{1+\infty}$ algebra.

In this paper we derive the OPE in one such theory, known as the quantum self dual gravity Ooguri:1991fp ; Chalmers:1996rq ; Bern:1998xc ; Bern:1998sv ; Krasnov:2021sf which was shown to be $w_{1+\infty}$ invariant in Penrose:1968pr ; Penrose:1976pr ; Boyer:1985pn . Here we do a collinear expansion of the known graviton scattering amplitudes in the self dual gravity theory and extract the celestial OPE from there. For simplicity, we analyse the $5$-point all plus amplitude in self dual gravity and factorize it in the collinear limit through a $4$ point amplitude. The results we obtain are consistent with what we proposed in Banerjee:2023zip based on the representtion theory of $w_{1+\infty}$. The rest of the paper is organised as follows.

In section 2 we introduce notations and conventions used in this paper. Section 3 briefly describes the $w_{1+\infty}$ algebra and how the whole tower of the $w_{1+\infty}$-currents can be generated using the two $sl_{2}(R)$ sub-algebras. In section 4 we briefly discuss about the scattering amplitudes in quantum self dual gravity. Section 5 discusses how to extract the OPE between two positive helicity outgoing gravitons from the 5-point one-loop self dual amplitude. We start by simplifying the 5-point amplitude in the momentum space and then Mellin transform it to get the celestial amplitudes. We then discuss how to factorize each term order by order in the OPE limit of the celestial amplitudes. The null states of the self dual gravity appearing at various orders of the OPE and the invariance of the OPE under $w_{1+\infty}$ algebra are discussed in Appendices I and J.

For the sake of completeness of the paper, we give a brief review of the celestial amplitude in Appendix A. In Appendix B, we discuss the parametrization of the $4$ and $5$-point delta functions which are useful in our context of the OPE expansion. Appendices C and D discuss how to simplify the $4$ and $5$-point amplitudes in momentum space using momentum conserving delta functions and various identities of the spinor-helicity brackets. These simplifications are done keeping in mind the fact that we want to factorize the $5$-point amplitude in terms of the $4$-point amplitude in the OPE expansion. Appendix E deals with the Mellin transformation of the $5$-point amplitude. In Appendix F we discuss the conditions on the graviton primary operators under the $w_{1+\infty}$ algebra which follow from the universal structure of the OPE. In Appendix G, we list the transformation properties of all the MHV null states under the action of ${sl_{2}(R)}_{V}$ and $\overline{sl_{2}(R)}$ algebras which are required to show the $w_{1+\infty}$-invariance of the self dual OPE. Appendix H briefly reviews the construction of a general $w_{1+\infty}$-algebra invariant OPE and how one can obtain an infinite family of $w_{1+\infty}$-algebra invariant theories.

In this paper, we will work in the $(2,2)$ signature space-time, which is also known as Klein space. The null momentum $p^{\mu}$ of a massless particle, satisfying the onshell condition $p^{2}=0$, is parametrized as,

where $\epsilon=\pm 1$ for outgoing and incoming particles respectively, $(z,\bar{z})$ are two independent real variables and $\omega$ is any positive number interpreted as the energy of the particle. In Klein space the null infinity takes the form of a Lorentzian torus (known as the celestial torus) times a null line. The Lorentz group in $(2,2)$ signature is given by $SO(2,2)\simeq\frac{SL(2,\mathbb{R})_{L}\times SL(2,\mathbb{R})_{R}}{\mathbb{Z}_{2}}$ and acts as the group of conformal transformations on the celestial torus:

In our conventions the spinor-helicity variables are given by,

where $z_{ij}=z_{i}-z_{j}$ and we also have $2p_{i}\cdot p_{j}=-\left<ij\right>[ij]$.

We start by reviewing the $w_{1+\infty}$ algebra which follows from the universal singular terms in the OPE between two positive helicity outgoing gravitons. Let $G^{+}_{\Delta}(z,\bar{z})$ denote the positive helicity graviton conformal primary operator of dimension $\Delta$ at the point $(z,\bar{z})$ on the celestial torus. The universal singular terms in the OPE are given by

Let us define an infinite family of positive helicity conformally soft Donnay:2018neh ; Pate:2019mfs ; Fan:2019emx ; Nandan:2019jas ; Adamo:2019ipt ; Puhm:2019zbl ; Guevara:2019ypd gravitons Guevara:2021abz as,

with weights $\left(\frac{k+2}{2},\frac{k-2}{2}\right)$. It follows from the OPE (4) that we can introduce the following truncated mode expansion

and the modes $H^{k}_{m}(z)$ are the conserved holomorphic currents. The currents $H^{k}_{m}(z)$ can be further mode expanded in the $z$-variable to get,

and one can show Guevara:2021abz that the modes $H^{k}_{\alpha,m}$ satisfy the algebra¹¹1Here we are assuming that $\kappa=\sqrt{32\pi G_{N}}=2$.

This is called the Holographic Symmetry Algebra (HSA). Now if we make the following redefinition (or discrete light transformation)Strominger:2021lvk

then (8) turns into the $w_{1+\infty}$ algebra²²2This is the wedge subalgebra of $w_{1+\infty}$.

For our purpose it is more convenient to work with the HSA (8) rather than the $w_{1+\infty}$ algebra. However, we continue to refer to the HSA as the $w$ algebra.

Now, in Banerjee:2023zip , it was shown that the whole tower of the $w$-currents can be generated using the two $sl_{2}(R)$ sub-algebras. One of them is $sl_{2}(R)_{V}$³³3Here V stands for vertical. Please see Fig.1 for an explanation. generated by the operators $\{H^{1}_{-\frac{1}{2},-\frac{1}{2}},\ H^{0}_{0,0},\ H^{-1}_{\frac{1}{2},\frac{1}{2}}\}$,

The other $sl_{2}(R)$ sub-algebra is generated by the global (Lorentz) conformal transformations $\{H^{0}_{0,1},H^{0}_{0,0},H^{0}_{0,-1}\}$. We call this $\overline{sl_{2}(R)}$ because this acts only on the $\bar{z}$ coordinate. Now the $w$ symmetry is generated by the infinite number of soft currents $\{H^{k}_{p}(z)\}$ where $k=1,0,-1,-2,...$ is the dimension $(\Delta)$ of the soft operator and $\frac{k-2}{2}\leq p\leq-\frac{k-2}{2}$. For a fixed $k$, the soft currents $\{H^{k}_{-\frac{k-2}{2}}(z),...,H^{k}_{\frac{k-2}{2}}(z)\}$ transform in a spin $\frac{2-k}{2}$ representation of the $\overline{sl_{2}(R)}$.

Now let us consider the currents $\{H^{1}_{\frac{1}{2}},H^{0}_{1},...,H^{k}_{\frac{2-k}{2}},...\}$ with the lowest $\overline{sl_{2}(R)}$ weights. These currents transform in an irreducible highest weight representation of the $sl_{2}(R)_{V}$. This can be seen from the following commutation relations following from (8),

Therefore, starting from the current $H^{1}_{\frac{1}{2}}(z)$ we can generate any other $w$ current by the combined action of the $\overline{sl_{2}(R)}$ and $sl_{2}(R)_{V}$ (Fig.1).

In this section, following Bern:1998xc we briefly review the all-plus helicity scattering amplitudes in quantum self dual gravity. In $(2,2)$ signature, self duality translates into the following condition on the Riemann tensor,

where $\varepsilon^{\mu\nu\alpha\beta}$ is the completely antisymmetric tensor with $\varepsilon^{0123}=+1$. In order to maintain the reality condition on the fields, the self dual gravity is described in either $(2,2)$ or $(0,4)$ signature. In Lorentzian $(1,3)$ signature, the condition (13) acquires an extra factor of $i$ and contain no real solutions. At the classical level, the linearized self dual solutions consist of positive helicity plane waves.

In this paper, we are interested in the collinear behaviour of gravitons in the self dual gravity. At the tree level, we have only one non-trivial amplitude: the three point $\overline{MHV}$ amplitude where, only one external graviton has negative helicity and other two have positive helicity. The appearance of the negative helicity graviton in the three point $\overline{MHV}$ amplitude can be explained from the fact that the action contains a Lagrange multiplier which is physically interpreted as the negative helicity graviton.

At the one loop level, the only non-zero amplitudes are the ones with all plus helicity gravitons with the minimum number of gravitons being four. These amplitudes are both UV and IR finite. The only divergences of these amplitudes are collinear and soft divergences. Our interest in this theory stems from the fact that this is a non-trivial quantum theory which is known to be $w$ invariant.

The one loop all-plus $n$-graviton stripped amplitude in self dual gravity is given by Bern:1998xc ,

where $a$ and $b$ are the external legs and $M$ and $N$ are two sets such that $M\bigcup N={1,...a-1,a+1,...b-1,b+1,...n}$ and $M\bigcap N=\phi$. The sum is over all possible $(a,b)$ and $(M,N)$, where $(M,N)$ and $(N,M)$ are not distinguished. The trace is defined as,

where $K_{M}=\sum_{i\in M}k_{i}$. The “half-soft” function $h$ is given by,

where $K_{1,m}=\sum_{i=1}^{m}k_{i}$ and $\mathcal{P}(2,3,\cdots,n)$ represents all permutations keeping the first leg fixed. Throughout this paper we will set $\kappa=2$.

In this section, we take the $4$ and $5$ point all plus amplitudes and express them in the conformal primary basis by (modified) Mellin transformation. Then we take the (collinear) OPE limit $(z_{45}\to 0,\bar{z}_{45}\to 0)$ in the 5-point amplitude with the aim of factorizing it into some differential operators acting on the 4-point amplitude at every order in the $(z_{45},\bar{z}_{45})$ expansion. Let us now closely look at the 4-point amplitude, first in momentum space and then in Mellin space.⁴⁴4For the sake of convenience of the reader we have moved some of the intermediate steps in the calculations to an Appendix. We have refereed to the Appendix in the main text whenever necessary.