The ultraviolet structure of N=8 supergravity at higher loops

Transcription

1 The ultraviolet structure of N=8 supergravity at higher loops based on work with Z. Bern, J.J. Carrasco, L. Dixon, H. Johanson Miami 2009

2 Discuss evidence of nontrivial cancella;ons in N=8 supergravity sugges;ng that it may be a UV- finite theory Use 4- point scaoering amplitudes as probes of UV behavior - higher- point divergences visible in higher- loop 4- point amp s - if finite to all orders, higher- point amplitudes are also finite We are bound to learn a lot if N=8 supergravity is indeed finite e.g. what symmetry forbids infinitely many counterterms; perhaps other gravity theories can have similar proper;es If perturba;ve finiteness is a reality, non- peturba;ve ques;ons become interes;ng: e.g. resumma;on of perturba;on theory (zero radius of convergence; perhaps not Borel summable); UV comple;on; nonperturba;ve states and where they belong in the spectrum, etc

3 Discuss evidence of nontrivial cancella;ons in N=8 supergravity sugges;ng that it may be a UV- finite theory Outline: UV divergences in quantum gravity: conven;onal wisdom Evidence of all- loop improved UV behavior Less pain more gain or, higher loops in supergravity SYM vs. supergravity amplitudes Generalized unitarity Three- and four- loop explicit calcula;ons

4 N=8 supergravity: [Cremer, Julia] maximally extended supergravity theory in four dimensions 256 states related to each other by symmetries Tensor product of two maximal SYM mul;plets E 7(7) 32 supercharges, R symmetry, duality symmetry add your name here symmetry Why this? - essen;ally unique - high degree of symmetry technical simplicity - high number of supersymmetries expect beoer UV behavior

5 Power- coun;ng in (super)gravity SYM: dimensionless coupling g YM p µ i d D p i p µ +... (2π) D propagators more momenta in numerator gravity: κ = 32π 2 G N dimensionful coupling Renormalizable by power- coun;ng κ p µ p ν i d D p i p µ p ν +... (2π) D propagators Badly- behaved integrals non- renormalizable by power- coun;ng supersymmetry: improvements but upper limit on loop order; essen;ally the same conclusion

6 So where are the divergences? Opinions over the years in D=4 compila;on by L. Dixon & Z. Bern No divergence was demonstrated To seole the issue explicit calcula;ons are necessary

7 New hopes The no- triangle property of N=8 supergravity - discovery - proof [Bern, Dixon, Perelstein, Rozowsky] [ Bern, Bjerrum- Bohr, Dunbar] [Bjerrum- Bohr, Dunbar, Ita, Perkins, Risager] [Bjerrum- Bohr, Vanhove; Arkani- Hamed, Cachazo, Kaplan] All explicit calcula;ons in N=8 exhibit the same degree of convergence as those in N=4 super- Yang- Mills [Green, Schwarz, Brink] [Bern, Dixon, Perelstein, Rozowsky;, Bjerrum- Bohr, Dunbar, Ita, Perkins, Risager] [Bern, Carrasco, Dixon, Johanson, RR] Hints from string duali;es - duali;es restrict the form of 10d effec;ve ac;on of string theory and can bootstrap low order informa;on to all loop orders [Chalmers; Green, Russo, Vanhove] Non- renormaliza;on theorems due to Berkovits suggest that divergences are delayed to 9- loops Analogous direct QFT deriva;on (e.g. in D=4)? Beyond L=9? [Chalmers; Green, Russo, Vanhove]

8 All- loop improved UV behavior No- triangle property implies all- order cancella;ons: [Bern, Dixon, RR] any 1- loop sub- amplitude must exhibit the no- triangle degree of convergence and the no- triangle property (not manifestly) Any- point L- loop sub- amplitude must exhibit the degree of convergence of L- loop 4- point amplitude Explicit calcula;ons up to 4 loops yield the same degree D<4+ 6 of convergence as N=4 super- Yang- Mills amplitudes L Origin of observed cancella;ons might have been iden;fied in a certain scaling of tree level ampl s p µ p ν 1 p 2 [Bern, Carrasco, Dixon, Johanson, Kosower, RR] [Bern, Carrasco, Ita, Johanson, Forde]

9 Higher- loop calcula;ons in super- Yang- Mills and supergravity Feynman graphs are not an efficient way of doing (higher- )loop calcula;ons in any theory with a gauge invariance E.g. has about 100 terms; 1- loop 4- graviton amplitude contains 4 of them. Yet, it is very simple: [Green, Schwarz, Brink] On- shell: which factorize as the product of 2 YM 3- ver;ces Message: 1) stay on- shell! 2) use factoriza;on of the 3- vertex and its generaliza;ons

10 Higher- loop calcula;ons in super- Yang- Mills and supergravity generalized unitarity (mul;ple cut condi;ons) and improvements (solve cut condi;ons) and varia;ons (maximal cuts) and extensions (leading singularity) [Bern, Dixon, Dunbar, Kosower] [BriOo, Cachazo, Feng; Forde] [Cachazo; Cachazo, Buchbinder] [Bern, Carrasco, Johanson, Kosower] [Arkani- Hamed, Cachazo] - if one cut is good, then many cuts must be even beoer

11 Higher- loop calcula;ons in super- Yang- Mills and supergravity Strategy: 1) No;ce that, at each loop order, amplitudes have the structure A = c i I i constructed from Feynman propagators S n i numerical coefficients (sufficiently many) cut condi;ons isolate a small piece of amplitude 2) Evaluate all generalized cuts (products of tree amplitudes) 3) Reconstruct func;on exhibi;ng these cuts

12 Higher- loop calcula;ons in super- Yang- Mills and supergravity Strategy: 1) No;ce that, at each loop order, amplitudes have the structure A = c i I i constructed from Feynman propagators S n i numerical coefficients (sufficiently many) cut condi;ons isolate a small piece of amplitude 2) Evaluate all generalized cuts 3) Reconstruct func;on exhibi;ng these cuts Some comments: only tree- level amplitudes are necessary symmetries manifestly preserved by tree amplitudes carry over to loops improved tree amplitudes improved loop expressions typically, fewer than all cuts are sufficient; argument necessary

13 KLT rela;ons [Kawai, Lewellen, Tye] Derived from low energy limit of string theory tree amplitudes and the observa;on that closed string states are created by bilinears in operators crea;ng open string states; zero modes are different compac;fica;on N=8 trees from N=4 trees for all states M tr 4 (1, 2, 3, 4) = is 12 A tr 4 (1, 2, 3, 4)A tr 4 (1, 2, 4, 3) M tr 5 (1, 2, 3, 4, 5) = is 12 s 34 A tr 5 (1, 2, 3, 4, 5)A tr 5 (2, 1, 4, 3, 5) + (2 3) M6 tr = 12 terms of the type s 3 A 6 A 6 A tr n = non cyc color- stripped amplitudes Tr[T a i 1 T a i2...t a i n ]A tr (i 1,i 2,...,i n ) Key observa;on: in conjunc;on with the generalized unitarity method, reduce supergravity (generalized) cuts to SYM (generalized) cuts!

14 gravity sts N=8 cuts are sums of products of N=4 cuts SYM sts Comments: M n1 (...l 1,l 2,l 3...)M n2 (...l 1,l 2,l 3...) = A n1 (...l 1,l 2,l 3...)A n2 (...l 1,l 2,l 3...) KLT terms SYM sts (s ij factors) A n 1 (...l 1,l 2,l 3...)A n 2 (...l 1,l 2,l 3...) Knowledge of SYM loop amplitudes simplifies supergravity calcula;ons - extract cuts from cleaned- up SYM amplitudes Use SYM cuts which break up amplitude into product of trees, e.g. Tricks for evalua;on of sum over states crossing generalized cuts Use complex momenta

15 Algebraic and pictorial methods for wri;ng down the simplified answer [Bern, Carrasco, Ita, Johanson, RR] E.g. (12[23][l 3 l 4 ][l 5 l 7 ]) 8 max susy sum of 4096 states (12[l 7 2][l 6 3][14]) 8 max susy Cuts w/ only MHV trees are easy; others, use MHV vertex expansion Not restricted to 4- points Treat planar and non- planar on equal foo;ng

16 All this leads to: Two- loops (ini;ally derived following a different strategy) [Bern, Dixon, Perelstein, Rozowsky] - Valid in D dimensions - Exhibits SYM convergence D c =4+ 6 L i.e. more cancella;ons than needed for finiteness

17 Three loops M (3) 4 = κ8 2 8 stumtree 4 S [Bern, Carrasco, Dixon, Johanson, Kosower, RR] I (a) + I (b) + 12 I (c) + 14 [Bern, Carrasco, Dixon, I (d) +2I (e) +2I (f) +4I (g) + 12 Johanson, RR] I (h) +2I (i) 2 3 valid in D dimensions s (a) 4 3 s 4 1 (b) more cancella;ons than needed for finiteness s 4 1 (c) s 4 1 (d) same UV as N=4 SYM s 2 τ 35 τ 46 s 2 τ 35 τ 46 s 2 τ 35 τ 46 D c =4+ 6 L not beoer: UV div. in D=6 presenta;on ambiguity (algebraic): contact terms may be moved around 1 6 (e) 4 (s(τ 26 + τ 36 )+t(τ 15 + τ 25 )+st) 2 +(s 2 (τ 26 + τ 36 ) t 2 (τ 15 + τ 25 ))(τ 17 + τ 28 + τ 39 + τ 4,10 ) +s 2 (τ 17 τ 28 + τ 39 τ 4,10 )+t 2 (τ 28 τ 39 + τ 17 τ 4,10 ) +u 2 (τ 17 τ 39 + τ 28 τ 4,10 ) (sτ 45 tτ 46 ) 2 τ 27 (s 2 τ 45 + t 2 τ 46 ) 1 6 (f) τ 15 (s 2 τ 47 + u 2 τ 46 ) τ 36 (t 2 τ 47 + u 2 τ 45 ) +l 2 5s 2 t + l 2 6st l2 7stu (g) 6 (h) (i) 4 5 4

18 illustrates consequences of no- triangle behavior In some (natural?) presenta;on it contains integrals of the type [Bern, Dixon, Perelstein, Rozowsky] Upon integral reduc;on top loop yields triangle and bubble integrals; 1- loop no- triangle & unitarity requires them to cancel [Bern, Dixon, RR] at 3- loops we see this cancella;on explicitly (and manifestly)

19 Four loops First SYM: 50 featuring integrals, obtained by aoaching external legs to - (0, 3, 8, 28, 13) possible contribu;ons with only cubic ver;ces - (0, 1, 1, 4, 0) planar ones; known numerators and coefficients; contact terms known (dual conformal integrals) - Make ansatz for the rest, fix it from cuts breaking up the amplitude into product of trees and test result against more restric;ve cuts [Bern, Czakon, Dixon, Kosower, Smirnov]

20 and out comes e.g. s 12 (s 2,10 s 39 s 47 s 18 + s 2,10 s 59 + s 39 s 6,10 + s 23 s 6,11 ) s 23 s 57 s 68 s 13 s 59 s 6,10 +l6(s 2 12 s 35 + s 12 s 4,12 s 23 s 59 )+l5(s 2 12 s 26 + s 12 s 1,11 s 23 s 6,10 ) +l9 2(s 12s 12,13 s 13 s 10,11 )+l10 2 (s 12s 11,14 s 13 s 9,12 ) l13 2 s 12s 11,14 l14 2 s 12s 12,13 +(s 13 2s 12 )l9 2l2 10 +s 23 (l5l l7l l6l l5l 2 8)+s 2 12 l13l s 12 l5l s 12 ( l5l l5l l5l l5l l9l 2 15) 2 +s 12 ( l6 2l2 7 + l2 6 l2 10 l2 6 l2 12 l2 6 l2 16 l2 10 l2 16 ) +s 23 (l9l l10l l7l l8l 2 10)+s 2 13 (l9l l10l 2 12) 2 to appear [Bern, Carrasco, Dixon, Johanson, RR] (48), (51) 4 s 12 (s 47 s 5,12 s 19 s 36 s 48 s 36 )+s 23 (s 48 s 6,11 s 15 s 3,10 s 15 s 47 ) s 12 s 23 s 11,12 +l5(s 2 23 s 7,12 s 23 s 4,15 s 13 s 10,11 )+l6(s 2 12 s 8,11 s 12 s 4,15 s 13 s 9,12 ) +l 2 9(s 23 s 3,15 s 12 s 38 + s 23 s 6,10 )+l 2 10(s 12 s 1,15 s 23 s 17 + s 12 s 59 ) +l 2 13 (s 12s 23 + s 12 s 38 s 23 s 6,11 )+l 2 14 (s 23s 12 + s 23 s 17 s 12 s 5,12 ) +l 2 11 s 23(s 4,12 s 6,10 )+l 2 12 s 12(s 4,11 s 59 ) +s 13 (l 2 7l l 2 5l l 2 6l l 2 11l l 2 10l l 2 9l 2 17 l 2 9l 2 12 l 2 10l 2 11) +s 12 ( l 2 5 l l2 6 (l l2 13 l2 10 )+l2 12 (l2 9 l2 5 l2 7 + l2 14 )+l2 8 (l2 9 + l2 16 )) +s 23 ( l 2 6 l2 9 + l2 5 (l l2 14 l2 9 )+l2 11 (l2 10 l2 6 l2 8 + l2 13 )+l2 7 (l l2 17 )) (49), (52) +s 12 (l 2 12l 2 13 l 2 8l 2 13 l 2 10l 2 13 l 2 10l 2 14 l 2 13l 2 17)+s 23 (l 2 11l 2 14 l 2 7l 2 14 l 2 9l 2 14 l 2 9l 2 13 l 2 14l 2 16) 12 s 12 s 28 s 4,12 s 12 s 37 s 1,11 s 23 s 16 s 3,10 +s 23 s 25 s s 12s 23 (s 13,15 s 13,14 ) +s 12 (l 2 6l 2 10 l 2 5l 2 9)+s 23 (l 2 7l 2 11 l 2 8l 2 12) (50)

21 then supergravity hop://up.aip.org/epaps/phys_rev_leo/e- PRLTAO / Some features: same set of integrals and symmetry factors; different numerators N i = N (8) i + N (7) i + N (6) i + N (5) i + + N (0) i number of loop momenta Degree of divergence of N (m) i : ω(d, m) =4D + m 26 Extract UV divergence: expand in external momenta extra care to eliminate IR divergences and their byproducts l 8 and l 7 cancel in any dimension cancels upon use of D =5specific integral iden;;es l 5 N=8 supergravity is finite for D < 11 2

22 Summary (of 4- point results)

23 On- shell techniques are extremely powerful; they allow high- loop (and high- point) explicit calcula;ons N=8 supergravity exhibits cancella;ons w/o susy explana;on - The 1- loop no- triangle behavior implies, through unitarity, all- order cancella;ons in large classes of terms - Through 4 loops N=8 supergravity has the same power- count as N=4 SYM; all cancella;ons are complete N=8 supergravity may well be the first finite QFT of gravity; demonstra;ng it remains a challenge