Conformal supergravity tree amplitudes from open twistor string theory

Abstract

We display the vertex operators for all states in the conformal supergravity sector of the twistor string, as outlined by Berkovits and Witten. These include 'dipole' states, which are pairs of supergravitons that do not diagonalize the translation generators. We use canonical quantization of the open string version of Berkovits, and compute N -point tree level scattering amplitudes for gravitons, gluons and scalars. We reproduce the Berkovits-Witten formula for maximal helicity violating (MHV) amplitudes (which they derived using path integrals), and extend their results to the dipole pairs. We compare these trees with those of Einstein gravity field theory. email: ldolan, jihry@physics.unc.edu IntroductionWe pursue the tree amplitudes for graviton scattering in conformal gravity, described by twistor string theory. The twistor string Conformal gravity field theories [4,5] provided early examples of finite field theories of gravity [6,7]. They are not unitary theories, but have interesting structure and continue to provoke comments about possible uses [8]. Of course the conformal supergraviton states have zero norm, due to the lack of unitarity [9]. Nevertheless, the equivalence of the twistor string with this field theory system can be exploited to derive conformal gravity tree level scattering amplitudes hard to access in the field theory. We compute the gravity trees as a step toward learning how to decouple them in the twistor string. This would result in a perturbative string theory for super Yang-Mills (with no tower of massive states), and the computational advantage one hopes for in a string theory vs. field theory description.Various efforts towards a QCD string are discussed in [10].We work in a spinor helicity basis [11]- [13], and compare the conformal gravity tree amplitudes with those of Einstein gravity [14]- [19]. The conformal gravity trees have fewer poles.We compute the conformal couplings in detail, as they should be important in further study of the loop calculation [20].Computation is done in the Berkovits open string version [2]. We use the twistor string canonical quantization described in [20,21] and follow their notation. In section 2, we give the vertex operators for all states in the conformal supergravity multiplets, as outlined by Berkovits and Witten [3]. These include the dipole states, which form pairs of supergravitons, where one state in each pair does not diagonalize the translation generators, and is not a momentum eigenstate. We show all supergraviton states have zero norm in our basis.In section 3, three-point scattering amplitudes for gluons and gravitons and scalars, with both one and two negative helicities are calculated. We include cases for both members the dipoles, and find a momentum derivative appearing in amplitudes for states that do not diagonalize the translations generators. These amplitudes still have translational invariance.In section 4, we extend our results to N -point tree level amplitudes for these dipole pairs. We reproduce the Berkovits-Witte...

Double copy structure of CFT correlators

“…Hence, A 220 φR 2 is the only 3-point amplitude which can be obtained from combining Yang-Mills and the F 3 gluon amplitudes in four dimensions. Generalizations to higher points in the form of worldsheet formulas can be found in [52][53][54][55].…”

On triviality of S-matrix in conformal higher spin theory

“…The propagator here is symbolically 1 ap 2 +bp 4 [34] (reducing to the Weyl graviton propagator for a → 0 or to the Einstein propagator for b → 0) so as long as the asymptotic states are chosen to be massless helicity ±2 gravitons the resulting amplitude interpolates smoothly between the standard Einstein 4-graviton one and zero in the Weyl theory. 30 30 Let us also mention that the conformal graviton amplitudes in flat space were computed in [35] starting with the twistor string theory of [36]. The latter should be related to "non-minimal" conformal supergravity containing extra dimension 0 scalar coupling to Weyl squared term, φ 2 φ + (1 + k φ + ...)C 2 + ....…”

“…To check our conjecture that all 4-point CHS amplitudes should vanish we may (i) first make a guess for the CHS spin j 4-point exchange amplitude generalising the expressions for the 11→11 and 22→22 amplitudes explicitly computed in (3.18),(3.11) and (5.2),(5.3) being guided by the expected structure of spin J ≥ 2j exchange amplitude in (4.1),(4.5), (4.14) and (ii) then check its vanishing at a special kinematical point. 35 Then the total amplitude is expected to be given as in (3.19),(3.25) and (5.6),(5.9) by the sum of the t-channel and u-channel exchanges of even spin s CHS states Let us now show the vanishing of the sum σ(x) + σ(−1 − x) at u = 0 or x = t s = −1, i.e. σ(0) = −σ(−1).…”

Perturbative gauge theory at null infinity

“…Low-point amplitudes of non-minimal conformal supergravity have been calculated in the context of twistor-string theory [51,[54][55][56], and the structure of the vertex operators makes it clear that our model will reproduce those amplitudes in a parity-symmetric form, analogous to the gauge theory calculation above. For instance, the n = 3, d = 0 correlator…”