Double logarithms in $$ \mathcal{N} $$ ≥ 4 supergravity: weak gravity and Shapiro’s time delay

Abstract

A study of the double logarithmic in the center-of-mass energy, s, contributions to the four-graviton scattering amplitude is presented for four-dimensional N ≥ 4 supergravities. This includes a novel representation for the coefficients of the perturbative expansion based on exactly solvable recurrences. A review is given of the structure in the complex angular momentum plane for the t-channel partial wave singularities of the different amplitudes. Working in impact parameter representation, ρ, it is shown that the resummation of double logarithms makes gravity weaker in regions of small ρ and large s. This screening of the gravitational interaction at short distances in the double logarithmic sector of the amplitudes is more acute as the number of gravitinos in the theory increases. It brings corrections to the eikonal phase which can change the sign of the graviton's deflection angle and generate regions with repulsive interaction. For very small impact parameters there appears a constant negative shift in both the eikonal phase and Shapiro's time delay which is not large enough to generate causality violation.The AdS/CFT correspondence [1-3] and the physics program at the Large Hadron Collider (LHC) have revamped the interest in the calculation of scattering amplitudes. Being fundamental for the former and an interesting simplified model for the latter, N = 4 supersymmetric Yang-Mills theory has played a pivotal role in recent developments. A very interesting finding [4] has been the double-copy structure linking this theory to N = 8 supergravity [5,6].An interesting limit where to investigate scattering amplitudes in gravitational theories, of relevance in the work here presented, is that of multi-Regge kinematics. In this case the center-of-mass energy squared, s, is asymptotically larger than all other Mandelstam invariants and amplitudes factorize in terms of impact factors and reggeized gravitons exchanged in the t-channel [7,8]. For gravity, eikonal contributions are also relevant together with double-logarithmic in s (DL) terms [9][10][11]. Both can be studied using the high energy effective action derived by Lipatov [12]. In this framework the leading contributions to scattering amplitudes are generated by bunches of produced gravitons well separated in rapidity from each other. The regions in rapidity without radiation have their origin in the exchange of reggeized gravitons. These are created (annihilated) in the t-channel by A ++ (A −− ) fields, which are subject to the constraints ∂ ± A ±± = 0 in order to generate the rapidity gaps. The local and non-local (in rapidity) effective interactions which appear in this context are described by the action given in [12]. It contains the Einstein-Hilbert action together with a kinetic term for the reggeon fields plus induced contributions. Within this setup it is possible to calculate the graviton Regge trajectory and those effective vertices needed to produce inelastic amplitudes.Connecting with the double-copy structure of gravity [13], it is remarkable t...

Multi-Regge limit of the two-loop five-point amplitudes in $$ \mathcal{N} $$ = 4 super Yang-Mills and $$ \mathcal{N} $$ = 8 supergravity

“…The conceptual progress in understanding the Regge limit in quantum field theory [15][16][17] lead to predictions that were successfully compared against explicit three-loop results for the full-colour four-gluon amplitudes in N = 4 super Yang-Mills [18]. Moreover, there is recent work on understanding certain terms in the Regge limit in supergravity theories [19][20][21][22][23], and perturbative data for the four-graviton amplitude is available up to three loops [24][25][26][27].…”

Enumeration of multi-rooted plane trees

“…To summarize this section, the recursion of Theorem 1 applied to expressions (13) for T 3 implies (33), which is a nontrivial condition on the sum S 5 . We have been led to conjecture a different form for the sum S 5 (the right hand side of (34)), which satisfies (33). Our conjecture can also be expressed as an equality between two hypergeometric functions.…”

“…The counting of rooted maps is of interest in the study of other quantum field theories, some of which having supersymmetry, see for example [6,7,8,21,26,32,33,34].…”