A tale of two exponentiations in N = 8 supergravity Paolo Di Vecchia - PowerPoint PPT Presentation

A tale of two exponentiations in N = 8 supergravity Paolo Di Vecchia Niels Bohr Institute, Copenhagen and Nordita, Stockholm UCLA, December 11th, 2019 Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 1 / 31

This talk is based on two papers together with A. Luna, S. Naculich, R. Russo, G. Veneziano and C. White, 1908.05603. and S. Naculich, R. Russo, G. Veneziano and C. White, 1911.11716. Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 2 / 31

Plan of the talk Introduction 1 Two different kinds of exponentiation 2 Check of (and constraints from) the leading-eikonal 3 Exponentiation at the first subleading eikonal 4 Comparing the two exponentiations 5 The deflection angle 6 Conclusions and outlook 7 Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 3 / 31

Introduction ◮ High-energy scattering has been studied, both in field and string theories, since the end of the eighties ’ t Hooft; Amati, Ciafaloni and Veneziano; Muzinich and Soldate ◮ The scattering of 2 → 2 scalar massless particles at high energy is dominated by the graviton exchange: T ( s , t ) = 8 π G N s 2 ( − t ) ◮ Since the graviton couples to energy, T diverges at high energy. ◮ Then, at sufficiently high energy, unitarity is violated. ◮ The way to restore unitarity is by summing over the contribution of loop diagrams. ◮ In this way the divergent contribution exponentiates in a phase, called the eikonal. ◮ From the eikonal one can then compute classical quantities as the deflection angle and the Shapiro time delay. Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 4 / 31

◮ This is by now not just the way to solve a theoretical problem. ◮ It may also have important applications to the study of the dynamics of binary black holes at the initial state of their merging. ◮ Modern quantum field theory techniques may turn out to be very efficient for extracting classical quantities needed for the study of black hole merging. ◮ They have allowed to compute the classical potential and the deflection angle at 3PM Bern, Cheung, Roiban, Shen, Solon, Zeng (2019) ◮ In this talk I am going to discuss the scattering of four massless particles in N = 8 supergravity. ◮ In this case the scattering amplitude has been explicitly computed up to three loops Henn and Mistlberger (HM) (2019). ◮ Different from CGR but should share with it the most important large-distance (infrared) features. ◮ In the probe analysis, by using D6-branes, it was shown that all classical corrections to the leading eikonal are vanishing D’Appollonio, DV, Russo and Veneziano (2010). Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 5 / 31

◮ Also from one loop calculations in N = 8 supergravity with masses it has been shown that the triangle diagrams do not contribute Caron-Huot and Zakraee (2018). ◮ In this talk we will see that, at two loops, one gets an additional classical contribution. Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 6 / 31

Two different kinds of exponentiation ◮ The UV properties of N = 8 supergravity have been studied to high loop order Bern, Carrasco, Chen, Edison, Johansson, Parra-Martinez, Roiban and Zeng (2018) ◮ Here we are interested in a complementary aspect: the high-energy, small angle (Regge) regime. ◮ In terms of the three Mandelstam variables: s = − ( k 1 + k 2 ) 2 ; t = − ( k 2 + k 3 ) 2 ; u = − ( k 1 + k 3 ) 2 s + t + u = 0 we work in the s -channel physical region ( s > 0 ; t , u < 0 ) and focus on the near-forward regime | t | << s . Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 7 / 31

◮ In N = 8 supergravity the full amplitude can be written as follows: � � ∞ ∞ � � A ( ℓ ) ( k i , . . . ) = A ( 0 ) ( k i , . . . ) α ℓ G A ( ℓ ) ( t , s ) A ( k i ) = 1 + . ℓ = 0 ℓ = 1 ◮ A ( 0 ) ( k i , . . . ) is the tree level amplitude and A ( ℓ ) is the ℓ -loop amplitude. Dots stand for the dependence on polarizations and flavors of external states. ◮ A ( ℓ ) is its “stripped" counterpart, and B ( ǫ ) ≡ Γ 2 ( 1 − ǫ )Γ( 1 + ǫ ) α G ≡ G π � ( 4 π � 2 ) ǫ B ( ǫ ) ; , Γ( 1 − 2 ǫ ) G is the Newton’s constant in D = 4 − 2 ǫ dimensions. ◮ A ( ℓ ) is infrared divergent, but all IR divergences are contained in the exponentiation of one loop amplitude. ◮ Therefore, it is convenient to write it in the form: � ∞ � � � � A ( k i ) = A ( 0 ) ( k i ) exp α G A ( 1 ) ( t , s , ǫ ) α ℓ G F ( ℓ ) ( t , s , ǫ ) exp . ℓ = 2 All remainder functions F ( ℓ ) are finite in the limit of ǫ → 0. Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 8 / 31

◮ This is the first exponentiation and it is done in momentum space. ◮ The leading contribution to the ℓ -loop amplitude A ( ℓ ) scales as s ℓ + 2 (for large s ) with subleading contributions having, modulo logarithms, lower powers of s and higher powers of t . ◮ At sufficiently high s unitarity is violated. ◮ To recover it we need another exponentiation, this time in impact parameter space b ∼ 2 J √ s rather than in momentum space. ◮ Let us see how that happens in the case of leading eikonal. ◮ The leading high energy behavior of the tree amplitude is given by = 8 π � Gs 2 ; q 2 = − t A ( 0 ) L q 2 where, at high energy, q is along D − 2 transverse directions. ◮ Then go to impact parameter space by ( 2 π � ) D − 2 e ibq / � iA ( 0 ) d D − 2 q � = − iGs ǫ � Γ( 1 − ǫ )( π b 2 ) ǫ . L 2 i δ 0 ( s , b ) = 2 s Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 9 / 31

◮ At one loop, we have for the leading term in s � − i π s � A ( 1 ) = A ( 0 ) α G A ( 1 ) − → A ( 0 ) ≡ A ( 1 ) L α G , L ǫ ( q 2 ) ǫ ◮ By going to impact parameter space one gets: � iA ( 1 ) � iA ( 0 ) d D − 2 q d D − 2 q � � ǫ ( q 2 ) ǫ = − 1 − i π s ibq ibq 2 ( 2 δ 0 ) 2 . L L = 2 s α G ( 2 π � ) D − 2 e ( 2 π � ) D − 2 e 2 s ◮ Summing the two iA ( 0 ) 2 s + iA ( 1 ) � � d D − 2 q � ( 2 π � ) D − 2 e ibq / � L L 2 s + . . . = 2 i δ 0 − 1 2 ( 2 δ 0 ) 2 + . . . = e 2 i δ 0 ( s , b ) − 1 . we see that they start to exponentiate in impact parameter space. Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 10 / 31

◮ Introduce a quantity related to the Schwarzschild radius: √ √ 1 s ≡ R D − 3 , 1 − 2 ǫ , i . e . G R ≡ ( G s ) ◮ Express the scaling of different terms at a given loop order in terms of the classical quantities as b and R . ◮ The Fourier transform of the leading energy contribution to the ℓ -loop amplitude scales as ( A ( ℓ ) ∼ G ℓ + 1 s ℓ + 2 ): L q 2 � − 2 ǫ R √ s � ℓ + 1 ( 2 π � ) D − 2 e ibq / � iA ( ℓ ) �� R d D − 2 q � L ∼ 2 s b � ◮ precisely as the ( ℓ + 1 ) th power of the leading eikonal phase δ 0 δ 0 ∼ R √ s ∼ b √ s � − 2 ǫ � 1 − 2 ǫ � R � R , b b � � ◮ This confirms that the leading eikonal resums arbitrarily high powers of � − 1 into an O ( � − 1 ) phase provided we consider R and b as classical quantities (as in CGR). Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 11 / 31

◮ Let us now consider the subleading energy contributions. A ( ℓ ) 2 s consists of a sum of terms having powers of s all the way up ◮ to the leading power ℓ + 1. ◮ Each of these terms behaves in impact parameter space as follows (again neglecting possible logarithmic enhancements): ( 2 π � ) D − 2 e ibq / � iA ( ℓ ) d D − 2 q � � G ℓ + 1 s ℓ + 1 − m b 2 ǫ ( ℓ + 1 ) − 2 m ∼ 2 s m = 0 � 2 m − 2 ǫ ( ℓ + 1 ) � R √ s � ℓ + 1 − 2 m � R � = . b � m = 0 ◮ In the massless case, and in D = 4, the amplitude A ( ℓ ) cannot depend on fractional powers of s . ◮ Therefore the expansion above is only in terms of even powers 1 / b 2 m . Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 12 / 31

◮ At each even order A ( 2 ℓ ) we get a new contribution to the classical eikonal for m = ℓ 2 and to the classical deflection angle. ◮ The odd-loop orders A ( 2 ℓ + 1 ) do not contribute directly to the classical phase or deflection angle. ◮ However, they still take part in the exponentiation. Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 13 / 31

◮ On the basis of the previous considerations we propose the following extension of the leading eikonal to include also subleading contributions: iA ( k i ) � d D − 2 b e − ibq / � �� � e 2 i δ ( s , b ) − 1 � ≃ ˆ A ( 0 ) ( k i ) 1 + 2 i ∆( s , b ) , 2 s ◮ All the terms appearing in e 2 i δ ( s , b ) are proportional to � − 1 . ◮ Those present in the prefactor ∆ contain the contributions with non-negative powers of � . ◮ Above identity is restricted to non-analytic terms as q → 0 that capture long-range effects in impact parameter space. Paolo Di Vecchia (NBI+NO) N=8 supergravity UCLA, 2019 14 / 31