\address Center for the Fundamental Laws of Nature, Harvard University,
Cambridge, MA 02138, USA
School of Natural Sciences, Institute for Advanced Study,
Princeton, NJ 08540, USA
The fourdimensional (4D) Lorentz group acts as the twodimensional (2D) global conformal group on the celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of 4D flat space amplitudes and 2D correlators on the conformal sphere are obscured by the fact that the former are usually expressed in terms of asymptotic wavefunctions which transform simply under spacetime translations rather than the Lorentz . In this paper we construct onshell massive scalar wavefunctions in 4D Minkowski space that transform as conformal primaries. Scattering amplitudes of these wavefunctions are covariant by construction. For certain mass relations, we show explicitly that their threepoint amplitude reduces to the known unique form of a 2D CFT primary threepoint function and compute the coefficient. The computation proceeds naturally via Wittenlike diagrams on a hyperbolic slicing of Minkowski space and has a holographic flavor.
Contents
1 Introduction
Quantum field theory (QFT) scattering amplitudes in fourdimensional (4D) Minkowski space are typically expressed in terms of asymptotic plane wave solutions to the free wave equation. Translation invariance is manifest because the plane waves simply acquire phases which cancel due to momentum conservation. The Lorentz invariance is more subtle as plane waves transform nontrivially into one another. In this paper we find a basis of primary solutions to the massive scalar wave equation and recast certain 4D scattering amplitudes in a manifestly covariant form. This form is very familiar from the the study of twodimensional (2D) conformal field theory (CFT), in which acts as the global conformal group. The appearance of the 2D conformal group is no coincidence, since the Lorentz group acts as the global conformal group on the celestial sphere, denoted by , at null infinity where the asymptotic states are specified.
Studies of the properties of scattering amplitudes date back to Dirac [1], but a new reason has arisen for interest in the topic. When gravity is coupled, it was conjectured in [2, 3, 4, 5] that the is enhanced to the full infinitedimensional
local conformal (or Virasoro) group. This conjecture was recently proven [6, 7, 8] to follow at tree level
This paper considers massive scalar threepoint functions, and recasts them in the standard form of 2D CFT correlators on the celestial sphere . To summarize the result, let be the flat coordinates on Minkowski space. A natural coordinate on where is
(1.1) 
Lorentz transformations act as the global conformal group on
(1.2) 
Here the complex parameters obey . We will construct a threeparameter family of solutions labelled by a point on and an weight (rather than the three components of spatial momenta which label plane waves) which transform as conformal primaries. We will find below they are naturally displayed in a hyperbolic slicing of Minkowski space, in harmony with the form of flat space holography advocated in [2]. then implies that the 4D tree amplitude takes the form
(1.3) 
where the ‘OPE coefficients’ depends on the masses, conformal weights, and cubic coupling of the three asymptotic scalars. An integral formula for involving Wittenlike diagrams on the hyperbolic slices is derived. In general this integral may not be computable in closed form, but it simplifies in the nearextremal case when the incoming particle is only slightly heavier than the sum of the outgoing particles and is explicitly given below.
Other tractable examples would be of great interest. In particular, the beautiful structure of amplitudes suggest they may take a particularly nice 2D form rewritten as correlators on . In [8] one contribution to such amplitudes (from the interior of the forward lightcone) was expressed as a Witten diagram, but to obtain the full amplitude additional hardertocompute contributions (from outside the lightcone) must be added in. This remains an outstanding open problem.
The utility of hyperbolic slicing was already noticed in a context with some similarity to the present one by Ashtekar and Romano in [18]. de Boer and Solodukhin initiated a program to understand flat space holography in terms of AdS holography via hyperbolic slicing in [2]. Soft theorems and aspects of scattering were hyberbolically studied in [19, 20, 21, 8]. In the context of the recent revival of the conformal bootstrap program, the linear realization of the conformal symmetry in the embedding Minkowski space has been used to simplify computations of, for example, conformal blocks and propagators in AdS [22, 23, 24, 25, 26].
The outline of the paper is as follows. In Section 2 we define and construct the massive and massless scalar wavefunctions that are conformal primaries of the Lorentz group . Our construction, equation (2.10) below, is a convolution of plane waves with the bulktoboundary propagator on the hyperbolic slice , and is evaluated in terms of Bessel functions. We also present an integral transform that takes a massive scalar scattering amplitude into an covariant correlation function. In Section 3 we compute the threepoint amplitude of massive conformal primary wavefunctions in the nearextremal limit. The main result is equation (3). In Appendix A we compute the KleinGordon inner product of these primary wavefunctions for a fixed mass.
2 Conformal Primary Wavefunctions
In this section we construct scalar wavefunctions that are conformal primaries of the Lorentz group . A scalar conformal primary wavefunction of mass and conformal dimension is defined by the following two properties:

It is a solution to the KleinGordon equation of mass ,
^{4} (2.1) 
It transforms covariantly as a conformal (quasi)primary operator in two dimensions under an Lorentz transformation,
(2.2) where with and is its associated group element in the fourdimensional representation.
^{5}
Note that, in contrast to the situation in AdS/CFT, the mass and the conformal dimension are not related.
2.1 Integral Representation
Conformal primary wavefunctions for a massive scalar field can be constructed from the Fourier transform of the bulktoboundary propagator in threedimensional hyperbolic space . Let be the Poincaré coordinates of the with metric,
(2.3) 
Here and is the boundary of the . This geometry has an isometry that acts as
(2.4) 
with and . The can be embedded into as either one of the two branches ( or ) of the unit hyperboloid
(2.5) 
More explicitly, we can choose this embedding map for the upper hyperboloid, corresponding to an outgoing particle, to be
(2.6) 
This implies the useful relation
(2.7) 
Let be the scalar bulktoboundary propagator of conformal dimension in [27],
(2.8) 
This transforms covariantly under the transformation (2.1),
(2.9) 
where and .
The conformal primary wavefunction for a massive scalar is
(2.10) 
where we pick the minus (plus) sign for an incoming (outgoing) particle. In the next subsection we will see that potentially divergent integrals can be regulated in an covariant manner by complexifying the mass and (2.10) is expressed in terms of Bessel functions.
It is trivial to check that (2.10) is indeed a conformal primary wavefunction. First, it satisfies the massive KleinGordon equation because each plane wave factor does. Second, it is a conformal quasiprimary (in the sense of (2.2)) because of the covariance (2.9) of the bulktoboundary propagator in . Our definition and formula for the conformal primary wavefunction (2.10) can be readily generalized to Minkowski space of any dimension and it would transform covariantly under the Euclidean dimensional conformal group .
The is embedded, via the map (2.6), into the hyperboloid in the momentum space, rather than position space and the boundary point might seem to live at the boundary of momentum space rather than Minkowski space. However, these spaces are canonically identified. The trajectory of a free massive particle is
(2.11) 
At late times, and
(2.12) 
That is, massive particles asymptote to a fixed position on the hyperbolic slices of Minkowski determined by their fourmomenta. Hence can be interpreted as a boundary coordinate of the latetime asymptotic slice.
Although we are far from constructing any example of such, the authors of [2] speculate on a boundary 2D CFT (of some exotic variety) on a subset of whose correlation functions are the 4D bulk Minkowski scattering amplitudes. Every bulk field would be dual to a continuum of operators labelled by their conformal weights. In this putative 2D CFT, the scalar bulk field mode (2.10) would be holographically dual to a local scalar boundary operator of dimension .
The covariance of the conformal primary wavefunction implies the covariance of any scattering amplitudes constructed from them. Let be the onshell momenta of massive scalars of masses (). Given any Lorentz invariant point momentumspace scattering amplitude of these massive scalars (including the momentum conservation delta function ), the conformal primary amplitudes are
(2.13) 
where is given by (2.6) satisfying . By construction transforms covariantly under ,
(2.14) 
2.2 Analytic Continuation and the Massless Wavefunction
The integral expression (2.10) is only a formal definition for the conformal primary wavefunction since the integral is divergent for real mass . More rigorously, we should define our conformal primary wavefunction by analytic continuation of the integral expression (2.10) from an unphysical region. When the mass is purely imaginary and lies inside the future lightcone, the outgoing wavefunction (2.10) is convergent and can be evaluated as
(2.15) 
where is a null vector in defined as
(2.16) 
After landing on the Bessel function expression (2.2), we can then analytically continue it to real mass and other regions in ,
(2.17) 
We have introduced an prescription to regularize the integral (2.10) in the case of real mass . In practice, the integral representation (2.10) will however prove to be more convenient to compute the scattering amplitudes of these conformal primary wavefunctions.
We note that at late times inside the future lightcone, the wave equation takes the asymptotic form
(2.18) 
This is solved to leading nontrivial order at large by
(2.19) 
where is function on . One may check that (2.17) with takes this form. On the other hand, outside the lightcone near spatial infinity we have
(2.20) 
This is solved to leading nontrivial order at large by
(2.21) 
with any function on dS. One may verify that (2.17) with takes this form.
3 Massive ThreePoint Amplitude
In this section we will consider the treelevel threepoint amplitude of the conformal primary wavefunction (2.10) , interacting through a local cubic vertex in ,
(3.1) 
The three point scattering amplitude for plane waves is then simply
(3.2) 
For conformal primary wavefunctions we have
(3.3) 
where we take the first particle to be incoming and the other two be outgoing. The threepoint amplitude is fixed by the covariance (2.2) to be proportional to the standard threepoint function in a twodimensional CFT:
(3.4) 
but it is nevertheless satisfying to see this formula explicitly appear in a 4D scattering amplitude. We wish to determine the finite proportionality constant which is a function of the masses and the conformal dimensions . In general these are given by the integral expression (3.3) which may not be possible to analytically evaluate. We will compute this constant explicitly in the nearextremal case when the mass of the first particle is slightly heavier than the sum of those of the other two. In this case the integral simplifies considerably and the threepoint amplitude reduces to the treelevel threepoint Witten diagram in .
Let the mass of the first particle be with and the masses of the other two particles be . Evaluating the integral, we arrive at the following expression for the scalar threepoint amplitude,
(3.5) 
where as defined in (2.6). Note the integral does not take the form of a treelevel threepoint Witten diagram in for general .
We now perform the integrals to get rid of three delta functions. As a result, we have
(3.6) 
where we have replaced
(3.7) 
The overall factor 2 is due to the Jacobian coming from rearranging the arguments of the delta functions. Now let us perform a change of variables from to ,
(3.8) 
The upper bound of is given by
(3.9) 
which comes from the constraint . We can then rewrite the threepoint amplitude as
(3.10) 
with and replaced by (3.8) and (3.7). The delta function has support on
(3.11) 
So far we have not taken any limit on the masses of the three particles. Now let us consider the near extremal limit . In this limit the three momenta become collinear and the corresponding points in become coincident. To leading order in , the solution of can be approximated by
(3.12) 
In the near extremal limit we have , hence the three bulk points in become coincident as commented above. We can then bring the three bulktoboundary propagators outside the integral. Next, by a simple power counting of , we find that the threepoint amplitude of the conformal primary wavefunction is zero when , which is related to the fact that the the phase space vanishes for marginal decay process. We should proceed to the subleading order in the near extremal expansion to obtain a nonzero answer, which is
(3.13)  
where the term in the parentheses in the second to last line is precisely the treelevel threepoint Witten diagram in , which was evaluated in [28]. Hence the near extremal massive threepoint amplitude takes the form of the threepoint function of scalar primaries with conformal dimensions in a twodimensional CFT.
Acknowledgements
We are grateful to T. Dumitrescu, P. Mitra, M. Pate, B. Schwab, D. SimmonsDuffin, and A. Zhiboedov for useful conversations. This work was supported in part by NSF grant 1205550. S.P. is supported by the NSF and by the Hertz Foundation through a Harold and Ruth Newman Fellowship. S.H.S. is supported by the National Science Foundation grant PHY1606531.
Appendix A KleinGordon Inner Product
In this section we evaluate the KleinGordon inner product between two conformal primary solutions with the same mass and generic complex weights . implies this must vanish for , while some kind of delta function is expected at .
The KleinGordon inner product between two outgoing wavefunctions and evaluated on the slice is
(A.1) 
Using the Feynman trick,
(A.2) 
we can perform the integrals in to obtain
(A.3) 
Here is the complex conjugate of . If we let ,
(A.4) 
This is divergent if is real, and equals to if is pure imaginary. Therefore, in order to have a delta function normalizable inner product, we require ’s to be complex numbers with the same real part, , (). The KleinGordon inner product for complex conformal dimensions, equal mass conformal primary wavefunction is,
(A.5) 
Footnotes
 The subleading soft theorem has a oneloop exact anomaly [9, 10, 11, 12] whose effects remain to be understood but are recently discussed in [13, 14].
 One may hope that ultimately 4D quantum gravity scattering amplitudes are found to have a dual holographic representation as some exotic 2D CFT on , but at present there are no proposals for such a construction.
 This generalizes the well known constraint that any QFT that can be coupled to gravity must have a local conserved stress tensor.
 We will use the convention for the signature in .
 There is no canonical way to embed the celestial sphere into the lightcone in Minkowski space. It follows that there is also no canonical way to associate a Möbius action on with an element in the fourdimensional representation. In fact, any two ’s that differ by an conjugation are equally good for our definition. Below we will make a choice of the map by fixing a reference frame for in (2.6). More explicitly, is the transformation matrix acting on given by plugging (2.1) into (2.6).
 One should take the square root in (2.17) corresponding to the decaying exponent when is outside the lightcone.
 Here we have dropped an overall constant factor compared to the massless limit of (2.17).
 We will use the integral representation (2.10) of the conformal primary wavefunction to simplify the calculation of the amplitude and eventually make contact with the Witten diagram in in the near extremal limit. However, as discussed at the end of Section 2, the integral representation of our the conformal primary wavefunction is divergent for real mass and the proper definition requires an analytic continuation from the Bessel function expression (2.2). Nonetheless, we will see that the threepoint amplitude computed using the integral representation turns out to be finite.
 Note that in the near extremal limit , the upper bound of the angular coordinate becomes on the support of the delta function.
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