Scattering amplitudes in Einstein-Yang-Mills and other supergravity theories

Transcription

1 Scattering amplitudes in Einstein-Yang-Mills and other supergravity theories Radu Roiban Pennsylvania State U. Based on work with M. Chiodaroli, M. Gunaydin and H. Johansson and Z. Bern, JJ Carrasco, W-M. Chen, Johansson

2 S-matrix one of the tools to exploring the UV behavior of gravity - What is the actual UV behavior of N=8 supergravity? - Is there a QFT of gravity that is UV-finite? Not philosophical quesaons, but rather technical ones By trying to answer them we are likely to learn a lot about the structure and properaes of gravity and supergravity theories

3 New mathemaacs New physics String theory General principles AdS/CFT & integrability QFT S matrix Color/kinemaAcs duality & Double copy Twistor and ambitwistor string(s) Duality Coffee ScaPering equaaons

4 New mathemaacs New physics String theory General principles AdS/CFT & integrability QFT S matrix Color/kinemaAcs duality & Double copy Twistor and ambitwistor string(s) Duality Coffee ScaPering equaaons

5 PerturbaAve gravity and supergravity have a long history : t Hoo^ & Veltman: 1-loop finiteness of pure gravity using Feynman rules : Howe & Lindstrom; Kallosh: symmetries vs counterterms : Goroff & Sagno`: 2-loop divergence of pure gravity using Feynman rules : Kawai, Lewellen, Tye: (KLT) relaaon between tree amplitudes of open and closed string theories maximal susy QFT relaaons : van de Ven: confirmaaon of 2-loop divergence in background field method : Bern, Dunbar, Shimada: string methods in perturbaave (super)gravity resurrecaon of the KLT relaaons : (generalized) unitarity + KLT relaaons: gauge theory cuts maximal supergravity cuts loops : Bern, Carrasco, Johansson: color/kinemaacs duality and a double-copy relaaon between N=4 sym and N=8 supergravity; clean & direct at loop level - since then: Many supergravity theories, with various amounts of supersymmetry have been shown to be related to pairs of gauge theories QuesAon(s): Are all (super)gravity theories related to a pair of gauge theories? If not all, why and which ones are (not)?

6 Plan - Color/kinemaAcs duality and the double-copy construcaon - Gravity symmetries from symmetries of (Yang-Mills)^2: - Infinite families: examples - all N=2 SGs with abelian vector fields and homogeneous scalar manifolds - some N=2 SGs with non-abelian vector fields; some all-mulaplicity amp s - What to do when the Gods look the other way - A biased outlook

7 ScaPering amplitudes and color/kinemaacs duality Textbook approach: scapering amplitudes from Feynman rules L = 1 Fµ a µ A A a µ +gf abc A b µa c 4 F µ F a µ a + matter i µ ab p 2 + i gf abc ( µ (k 1 k 2 ) + 2 more) General form of an L-loop amplitude A L m loop = i L g X Z Y L m 2+2L i2g 3 Example: 4-pt tree amp: A (0) 4 (1, 2, 3, 4) = g2 cs n s (p, ) s Color/kinemaAcs duality c s + c t + c u =0 l=1 + c tn t (p, ) t d D p l 1 (2 ) D S i n i C i Q i p 2 i n i = n i (p p, p,...) + c un u (p, ) u c s = f A 1A 2 B f BA 3A 4 etc. Bern, Carrasco, Johansson n s (p, )+n t (p, )+n u (p, ) =0 4

8 The general picture/conjecture: a duality between color and kinemaacs For (s)ym theories in any dimension with certain addiaonal maper A L m loop = i L g X Z Y L m 2+2L i2g 3 such that, when 3 l=1 C i + C j + C k =0 adjoint rep: Bern, Carrasco, Johansson non-adjoint rep: Chiodaroli, Jin, RR; Johansson, Ochirov Chiodaroli, Gunaydin, Johansson, RR d D p l 1 (2 ) D S i n i C i Q i p 2 i required by gauge inv., 3 n i = n i (p p, p,...) 3 n i + n j + n k = a d b c a d b c a d c b = m m m 3 Present in many theories: YM+maPer, QCD, Coulomb branch,, Z-theory, BLG, ABJM, as well as certain form factors and correlaaon fcts. Implies nontrivial relaaons btw amplitudes (L=0) and integrands (L>0)

9 Gravity from gauge theory: Given two gauge theories with duality-saasfying m-point amplitudes, the scapering amplitudes of a supergravity is (the double-copy construcaon) M L m loop = i L+1 apple m 2+2L X Z Y L d D p l 1 n i ñ Q i 2 (2 ) D S i i p 2 i i2g 3 Expected to hold to all loop orders; paraal arguments available - Explicitly tested in various susy and non-susy theories w/ & w/o maper - At various loop orders (1 through 4 loops in N=4 and N=8 SG) - Captures subtle field theory effects, such as anomalies l=1 - Extended to classical soluaons of eqs of moaon Bern, Carrasco, Johansson N<4: Chiodaroli, Jin, RR; Johansson, Ochirov Chiodaroli, Gunaydin, Johansson, RR Spectrum of the (super)gravity theory: Tensor product of spectra of the two gauge theories such that the fields of gauge theories form a singlet under the gauge group Luna, Monteiro, Nicholson, O'Connell, White When duality-saasfying reps exist but are not available, double copy is sall surprisingly simple and structured Bern, Carrasco, Chen, Johansson, RR

10 Gravity symmetries from symmetries of (Yang-Mills)^2: Nonabelian gauge symmetry of YM theories is gone Gauge theory global symmetries survive and can either remain global (e.g. R-symmetry) or be gauged Enhancement: R 1 R 2! R : {T j i, T i 0 j0, Q i Qj 0, Qi 0 Q j0 } general discussion: Anastasiou, Borsten, Duff et al

11 Gravity symmetries from symmetries of (Yang-Mills)^2: Nonabelian gauge symmetry of YM theories is gone Gauge theory global symmetries survive and can either remain global (e.g. R-symmetry) or be gauged Enhancement: R 1 R 2! R : {T j i, T i 0 j0, Q i Qj 0, Qi 0 Q j0 } Emergent global symmetries e.g. duality symmetries U(1) : q = h L h R

12 Gravity symmetries from symmetries of (Yang-Mills)^2: Nonabelian gauge symmetry of YM theories is gone Gauge theory global symmetries survive and can either remain global (e.g. R-symmetry) or be gauged Enhancement: R 1 R 2! R {T j i, T i 0 j0, Q i Qj 0, Qi 0 Q j0 } Emergent global symmetries e.g. duality symmetries U(1) : q = h L h R But. What makes a theory of spin-2 paracles a theory gravity is diffeomorphism invariance All double-copy theories are diffeomorphism-invariant: M = X n ( 1 (p 1 ), 2,...)ñ ( 1 (p 1 ), 0 2,...) D µ (p) 7! (µ (p) 0 ) (p) 7! p (µ 0 ) (p)+p ( µ) (p) graviton double-copy polarizaaon vector linearized diffeomorphism Linearized diffeomorphisms related to YM linearized gauge transf s

13 Gravity symmetries from symmetries of (Yang-Mills)^2: But. What makes a theory of spin-2 paracles a theory gravity is diffeomorphism invariance All double-copy theories are diffeomorphism-invariant: M = X n ( 1 (p 1 ), 2,...)ñ ( 1 (p 1 ), 0 2,...) D µ (p) 7! (µ (p) 0 ) (p) 7! p (µ 0 ) (p)+p ( µ) (p) M = X n (p 1, 2,...)ñ ( 1 (p 1 ), 0 2,...) D linearized diffeomorphism +(n $ ñ) Linearized diffeomorphisms related to YM linearized gauge transf s 0 = X n (p 1, 2,...)c A = X n ( 1 (p 1 ), 2,...)c! D D -structure of n -Jacobi idenaaes for c n, ñ &c have the same properaes =) M =0 graviton double-copy polarizaaon vector

14 The reverse implicaaon is far from obvious; consider

15 The reverse implicaaon is far from obvious; consider proof by exhausaon 1) Pick a class of supergraviaes 2) Construct their scapering amplitudes and compare 3) Repeat A suitable class: N=2 supergraviaes - The double-copy always gives SG, but which one? - Nontrivial: maper content does not uniquely determine the theory - Lagrangians are known! explicit comparison can be carried out - Their 5D origin uniquely idenafies them by their three-point amplitudes - Some of them have global symmetries; can be gauged - Maxwell-Einstein vs Yang-Mills-Einstein - Different construcaons are useful for different purposes General N=2 5D Maxwell-Einstein Lagrangian (suppress fermions): e 1 L = R 2 Scalar manifold: graviphoton + maper 1 IJF I 4å µ F Jµ 1 2 g xy@ µ ' µ ' y + e 1 6 p 6 C IJK µ Fµ F I J A K funcaons of ' 2 3/2CIJK V( ) = I J K =1 3 Gunaydin, Sierra, Townsend Canonical basis: C 000 =1,C 00i =0 C 0ij = 1 2 ij C ijk = arbitrary

16 ConstrucAons of N=2 SGs: (N=2) = (N=2) x (N=0) or (N=2) = (N=1) x (N=1) Each offers several opaons; focus (for now) on a paracular case of the former - Gauge theory 1: N=2 sym with a single ½-hypermulAplet in pseudo-real rep R - Obeys color-kinemaacs duality - Gauge theory 2: YM w/ (q+2) adj. scalars and r fermions in pseudo-real rep R L = 1 4 F âµ F âµ (D µ a )â(d µ a )â + i 2 + g 2 aâ a - Color-kinemaAcs duality requires: { a, Dµ µ 5T â g 2 4 f âˆbê f ĉ ˆdê aâ bˆb aĉ b ˆd Chiodaroli, Gunaydin, Johansson, RR b } =2 ab - Why these? - Simple gauge theory maper content - Contains the two possible realizaaon of vector fields - PrePy large manifest global symmetry SO(q + 2) - More reasons related to the spectrum

17 ConstrucAons of N=2 SGs: (N=2) = (N=2) x (N=0) or (N=2) = (N=1) x (N=1) Each offers several opaons; focus (for now) on a paracular case of the former - Gauge theory 1: N=2 sym with a single ½-hypermulAplet in pseudo-real rep R Aâ+, â+, â - Obeys color-kinemaacs duality G Aâ, â, â - Gauge theory 2: YM w/ (q+2) adj. scalars and r fermions in pseudoreal-real rep R - Color-kinemaAcs duality requires: { a, G +, ' 1, ' 2, L = 1 4 F âµ F âµ (D µ a )â(d µ a )â + i 2 + g 2 aâ a b } =2 ab - The 4D bosonic spectrum: A 1 = A, h = A A, A 0 = A, i z 0 = A + A, A a = A a, i z a = a, A = (U ), i z = + (U ), - RepresentaAons of the gauge group forbid other products R Dµ µ 5T â g 2 4 f âˆbê f ĉ ˆdê aâ bˆb aĉ b ˆd Chiodaroli, Gunaydin, Johansson, RR

18 E.g. of 3-point amplitudes and properaes of higher-point amplitudes: N=2 factor: N=0 factor: The double-copy: Which supergraviaes are these? A (0) 3 1Aâ, 2, 3 + = ig h12i2 h23i T âv 1 A (0) 3 1 âa, 2, 3 = ig M (0) 3 1A a, 2A, 3 z = General properaes of amplitudes of this double-copy: lim p n!0 M(0) n...,nz =0= lim p n!0 M(0) n Scalars take values in a (locally-) homogeneous space! Given by the prepotenaal (not in canonical form) p h23i( a C 1 )(T âv 1 ) 2 apple 2 p h12i 2 (U t a CU) 2...,n z V( ) = p 2 0 ( 1 ) 2 0 ( i ) ( ) 2 + i i - 3-point amplitudes are reproduces correctly (nontrivial choice of U) - (U t C a U)= 1, i i Chiodaroli, Gunaydin, Johansson, RR de Wit, van Proeyen Through double-copy can compute loop amplitudes; no 1-loop finite theories Can bypass use of manifest c/k-saasfying reps. Bern, Carrasco, Chen,, Johansson, RR

19 The current perspecave on how to gauge a global (non-r) symmetry of SG 1. The symm. to be gauged -- visible in one gauge theories: A A µ A µ A 2. Minimal couplings must have a double-copy structure Minimal couplings with spin-0 and spin-1/2 fields std 3-point S-matrix elements (same as in a gauge theory) - Require that this can be factorized and that are Lorentz-invariant from dim.-3 operator (4d counang) from standard dim.-4 operator (4d counang) Unique local opaon: trilinear scalar operator f ABC fâˆbĉ Aâ Bˆb Cĉ

20 Homogeneous theories: vector of so(q + 2) A 1 = A, A 0 = A A a = A a, A = (U ) - Adjoint rep. of the gauge group must fit inside the vector rep. of so(q+2) - Many spectator vector fields + more economical: truncate away rep R fermions generic Jordan family! Lagrangian: V( ) = p 2 0 ( 1 ) 2 0 ( i ) 2 - Gauge theory 1: N=2 super-yang-mills theory - Obeys color-kinemaacs duality - Gauge theory 2: Yang-Mills theory with n V adjoint scalars L = 1 4 F âµ F âµ (D µ a )â(d µ a )â + g 3 F abcfâˆbĉ aâ bˆb cĉ g 2 4 f âˆbê f ĉ ˆdê aâ bˆb aĉ b ˆd - Color-kinemaAcs duality demands that F abc obeys the Jacobi idenaty

21 Gauged generic Jordan family SGs: V( ) = p 2 0 ( 1 ) 2 0 ( i ) 2 M 4 = SO(n v, 2) SO(n v ) SO(2) SU(1, 1) U(1) - Gauge theory 1: N=2 super-yang-mills theory - Obeys color-kinemaacs duality + Covariant derivaaves + extra terms - Gauge theory 2: Yang-Mills theory with n V adjoint scalars L = 1 4 F âµ F âµ (D µ a )â(d µ a )â + g 3 F abcfâˆbĉ aâ bˆb cĉ g 2 4 f âˆbê f ĉ ˆdê aâ bˆb aĉ b ˆd - Color-kinemaAcs duality demands that F abc obeys the Jacobi idenaty Comparison with Lagrangian: - precise field map - precise parameter map F abc =2ig sg f abc 6=0i a, b, c =2,...,n V - 3- and 4-point amplitudes explicitly checked

22 Gauged generic Jordan family SGs: V( ) = p 2 0 ( 1 ) 2 0 ( i ) 2 M 4 = SO(n v, 2) SO(n v ) SO(2) SU(1, 1) U(1) - Gauge theory 1: N=2 super-yang-mills theory - Obeys color-kinemaacs duality + Covariant derivaaves + extra terms - Gauge theory 2: Yang-Mills theory with n V adjoint scalars L = 1 4 F âµ F âµ (D µ a )â(d µ a )â + g 3 F abcfâˆbĉ aâ bˆb cĉ g 2 4 f âˆbê f ĉ ˆdê aâ bˆb aĉ b ˆd - Color-kinemaAcs duality demands that F abc obeys the Jacobi idenaty Comparison with Lagrangian: - precise field map - precise parameter map F abc =2ig sg f abc 6=0i a, b, c =2,...,n V - 3- and 4-point amplitudes explicitly checked With 3-scalar coupling, global sym. of the N=0 factor becomes local SG symmetry Similar tree-level construcaon from scapering equaaon POV Cachazo, He, Yuan Through double-copy can compute loop amplitudes; same UV prop s as MESGTs

23 Explicit tree-level EYM amplitudes from double-copy One graviton, (m-1) gluons, single-trace Two gravitons, (m-2) gluons, single-trace Three gravitons, (m-3) gluons, single-trace Simpler expressions than from other techniques Nandan, Ple a, SchloPerer, Wen Four & five gravitons not too bad either

24 A panoramic view of N=2 x N=0 theories Chiodaroli, Gunaydin, Johansson, RR Straigh}orward extension to N=4 SG and certain N=1 and N=0 (S)Gs

25 What to do when the Gods look the other way or -- Double-copy w/o explicit color/kinemaacs Use only gauge theory data In progress Bern, Chen, Carrasco, Johansson, RR 1. Start with a pair of gauge theory amplitudes (this is the data) 2. Construct naïve double-copy; proceed to correct it 3. Find cuts of naïve double-copy 4. Find cuts of the desired SG amplitude using KLT 5. Compare with 3. and shake the difference to a graph-like presentaaon 6. IdenAfy missing terms (contact terms); step through all cuts A miracle occurs: contact terms are simple and are built from the violaaon of the kinemaac Jacobi relaaons by the iniaal amplitudes: skip steps 2/ a b a b BCJ discrepancy funcaon: d c d c a d c b =0= J m m m c 1 + c 2 + c 3 =0 n 1 + n 2 + n 3 = J

26 Example: one loop or any generalized cut made up of two 4-point amplitude factors 2 3 l 1 1 l 2 4 Gauge theory cut: TransformaAon relaang it to c/k-saasfying one: C 4 4 YM = X n i1 i 2 c i1 i 2 i 1,i 2 d (1) i 1 d (2) i 1 i 2 n i1 i 2 n BCJ i 1,i 2 = d (1) i 1 k (2) (i 2 )+d (2) i 2 k (1) (i 1 ) i 2 ProperAes of gauge parameters: X i 1,i 2 i 1 i 2 c i1 i 2 d (1) i 1 d (2) i 2 =0= X i 1,i 2 i 1 i 2 n BCJ i 1 i 2 d (1) i 1 d (2) i 2 BCJ discrepancy J,i2 X funcaons: X n i1 i 2 = d i2 k (1) (i 1 ) J i1, i 1 i 1 Supergravity cut (there are several equivalent variants): X i 2 n i1 i 2 = d i1 X i 2 k (2) (i 2 ) C 4 4 SG = X n BCJ i 1 i 2 n 0BCJ i 1 i 2 i 1,i 2 d (1) i 1 d (2) i 2 = n i1i2 n0 i 1 i 2 d (1) i 1 d (2) 1 i 2 d (1) 1 d(2) 1 J,1 J 0 1, + J 1, J 0,1

27 Series of formulae for all types of generalized cuts (derivaaon/proof in pro Works for asymmetric double-copies StarAng point can be any representaaon of amplitudes, including Feynman diagrams Novel way to find gravity tree-level amplitudes adapted to cubic graphs Stay tuned for applicaaons to N=8 SG at 5 loops

28 A biased outlook - Discussed double-copy construcaons of large classes of ME & YMESGTs needs massive maper and maper in non-adjoint rep - QuesAon remains: Do all N=2 SGs have a double-copy construcaon? If not, why not? - IdenAfy double-copy construcaons of other, more general, families of SGs - SystemaAc study of amplitudes in N = 2 theories at one loop and beyond Are there any finite theories? (hypermulaplets necessary) - R-symmetry gauging and applicaaons to gauge/string duality Do N=4 sym correlaaon fcts have a double-copy structure? PerturbaAon theory in curved space?