abelian gauge theories DUALITY N=8 SG N=4 SG

Transcription

1 Based on work with Bern, Broedel, Chiodaroli, Dennen, Dixon, Johansson, Gunaydin, Ferrara, Kallosh, Roiban, and Tseytlin Interplay with work by Aschieri, Bellucci, abelian gauge theories DUALITY Bergshoeff, Bossard, Coomans, Howe, Ivanov, Krivonos, Nicolai, Van Proeyen, Stelle, Shahbazi, Tourkine, Vanhove,... N=8 SG N=4 SG Stanford Institute for Theoretical Physics Breaking of supersymmetry and Ultraviolet Divergences in extended Supergravity (BUDS)

2 abelian gauge theories Status update on U(1) duality satisfaction of BIK N=2 Born-Infeld DUALITY action from 2001 JJMC, Kallosh 13 easy to see duality invariance of W^(10) term hints that hidden supersymmetry => duality invariance (see Toine, Eric, Renata talks) Bossard, Nicolai 11 Chemissany, Kallosh, Ortin 12 JJMC, Kallosh, Roiban 11 Broedel, JJMC, Ferrara, Kallosh, Roiban 12

3 DUALITY N=8 SG N=4 SG similarities and differences between N=4 SG and N=8 SG possibility of matter-couplings existence of anomalies goal: convince you calculation of amplitudes can help clarify JJMC, Chioradelli, Gunyadin, Roiban 12 JJMC, Kallosh, Roiban, Tseytlin 13

4 Gluons for (almost) nothing and gravitons for free! motivate calculating with color-kinematics & double copy (see also talks by Tristan and Henrik)

5 Bern, JJMC, Dixon, Kosower, Johansson, Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

6 Bern, JJMC, Dixon, Kosower, Johansson, Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

7 Bern, JJMC, Dixon, Kosower, Johansson, Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

8 Bern, JJMC, Dixon, Kosower, Johansson, Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

9 BCJ 08, 10

10 DECODING THE DNA OF GRAVITY BCJ 08, 10 (`i) L g n`22l Aloop = X G2cubic Z L Y l=1 d D p l 1 (2ı) D S(G) n(g)c(g) D(G) LOOP LEVEL DOUBLE COPY

11 DECODING THE DNA OF GRAVITY BCJ 08, 10 (`i) L g n`22l Aloop = X G2cubic Z L Y l=1 d D p l 1 (2ı) D S(G) n(g)c(g) D(G) LOOP LEVEL DOUBLE COPY (`i) L1 X (»=2) n`22lmloop = G2cubic Z L Y l=1 d D p l 1 (2ı) D S(G) n(g)~n(g) D(G)

12 BCJ (2010) Only need maximal cut information of (e) graph to build full amplitude!

13 Aside: on cuts (where do we get our data?)

14 TEXTBOOK APPROACH

15 TEXTBOOK APPROACH Simple graph rules for constructing scattering amplitudes

16

17 JUST THE GRAVITON... 3 loops ~10 20 TERMS 4 loops ~10 26 TERMS 5 loops ~10 31 TERMS BUT FINAL EXPRESSIONS ARE TRACTABLE

18 MOST SYMMETRIC 4D THEORY, N=8 SUGRA ~10 20 TERMS add all other particles (e) 4 / Z stum (0) 4 s (k 4 l 5 ) 2 2 d (e) (l 2 5 l2 6 l2 7 (k 1 l 5 ) 2 ) Bern, JJMC, Dixon, Kosower, Roiban 07

19 NECESSARY & SUFFICIENT

20 NECESSARY & SUFFICIENT

21 SUFFICIENT Bern, Dixon, Dunbar, and Kosower ( 94, 95) Bern, Dixon, and Kosower ( 96) Britto, Cachazo, and Feng ( 04)

22 TREE-LEVEL CUT X M (0) U c,1 M(0) U c,2 M(0) U c,m int states

23 2 8 = 256 massless states, ~ expansion of (xy) 8 Scared by state sums? SUSY 2 4 = 16 states ~ expansion of (xy) 4 P i dil i 2 M!=4 HA 3 H-l 1, k 1, -l 2 L A 3 Hl 1, -l 3, -l 4 L A 3 Hl 4, -l 5, k 3 L µ A 3 H-l 6, l 3, k 2 L A 3 Hl 6, l 2, -l 7 L A 3 Hl 7, k 4, l 5 LM = Hk 1 k 2 L 2

24 Don t be! 4D state-sums completely under control for N<=4 sym = Bern, JJMC, Ita, Johansson, Roiban 09

25 N=4 sym = N(state sums follow the gluons) S N =4 =(A B C...) 4 S N <4 =(A B C...) N (A 4 N B 4 N C 4 N...) Bern, JJMC, Ita, Johansson, Roiban 09

26 Higher dimensional cuts also important! Workhorse: N=1 in 10D (as tree multiplicity increases expressions can be unwieldy) Also very useful: N=2 in 6D Cheung, O Connell; Dennen, Huang, Siegel; Boels; Bern, JJMC, Dennen, Huang, Ita Best: Recycling known D-dimensional amplitudes EVERYONE

27 Back to Color-Kinematics / Double-copy

28 How does color-kinematics help us with less SUSY? N=4 SYM N<4 SYM Look at N=4 Super Yang-Mills 4-pt1-loop Green, Schwarz, Brink =1

29 How does color-kinematics help us with less SUSY? N=4 SYM N<4 SYM Look at N=4 Super Yang-Mills 4-pt1-loop Symmetric Green, Schwarz, Brink Numerator ansatz by powercounting? a constant: =1

30 JJMC, Chiodaroli, Gunaydin, Roiban 12 N<4 SYM N=4 N<4

31 JJMC, Chiodaroli, Gunaydin, Roiban 12 N<4 SYM N=4 N<4

32 N<4 SYM N=4 N<4 Asymmetric but problem! local gauge-inv. numerators don t work JJMC, Chiodaroli, Gunaydin, Roiban 12

33 JJMC, Chiodaroli, Gunaydin, Roiban 12 N<4 SYM N=4 N<4 Asymmetric but problem! local gauge-inv. numerators don t work ( ) Lessons from N=4 5-pt multiloop, and JJMC, Johansson symmetric tree through 6-pt: Broedel, JJMC divide by [Gram Det]^k

34 JJMC, Chiodaroli, Gunaydin, Roiban 12 N<4 SYM N=4 N<4 Asymmetric gauge invariant numerators, requires non-locality in ext legs Parity odd components! Numerator ansatz for N 1 SYM N(g) = 1 (stu) 2 P g;(6,2) ( l,k1, l,k2, l,k3 ; s, t) 4i"(k 1,k 2,k 3,l) P g;(4,0) ( l,k1, l,k2, l,k3 ; s, t)

35 Universal structure for N 1 SYM N N i = N N =4 i (4 N )N chiral i N N =4 1,2,3 =1 N chiral 1 = 1 s ( l,k 1 l,k2 ) 2 stu (s l,k 1 l,k2 u l,k1 l,k3 t l,k2 l,k3 ) N chiral 2 = 1 s 2 (t l,k 2 s l,k3 u l,k1 ) 2 stu (s l,k 1 l,k2 u l,k1 l,k3 t l,k2 l,k3 ) 2i s 2 " k 1,k 2,k 3,l N chiral 3 = 1 u ( l,k 3 l,k1 ) 2 stu (s l,k 1 l,k2 u l,k1 l,k3 t l,k2 l,k3 ) others given by Jacobi JJMC, Chiodaroli, Gunaydin, Roiban 12

36 Universal structure for N 1 SYM N N i = N N =4 i (4 N )N chiral i N N =4 1,2,3 =1 N chiral 1 = 1 s ( l,k 1 l,k2 ) 2 stu (s l,k 1 l,k2 u l,k1 l,k3 t l,k2 l,k3 ) N chiral 2 = 1 s 2 (t l,k 2 s l,k3 u l,k1 ) 2 stu (s l,k 1 l,k2 u l,k1 l,k3 t l,k2 l,k3 ) 2i s 2 " k 1,k 2,k 3,l N chiral 3 = 1 u ( l,k 3 l,k1 ) 2 stu (s l,k 1 l,k2 u l,k1 l,k3 t l,k2 l,k3 ) others given by Jacobi JJMC, Chiodaroli, Gunaydin, Roiban 12

37 Universal structure for factorizable N 1SG N N i = N N =4 i (4 N )N chiral i N N =4 1,2,3 =1 (`i) L1 X (»=2) n`22lmloop = G2cubic Z L Y l=1 d D p l 1 (2ı) D S(G) n(g)~n(g) D(G) JJMC, Chiodaroli, Gunaydin, Roiban 12

38 M (1) 4,N =1 N =1 = M(1) 4,N =8 i apple (1 f)tu s 2 (t u)(i 2(u) I 2 (t)) h Mtree 6 tu 3(1 f) s 2 (s2 tu) I4 D=6 2 (t, u) 4 1(3 D)f i (I 2 (u)i 2 (t)) D 2 f!1 M (1) 4,N =2 N =1 = M(1) 4,N =8 i apple (1 f)tu s 2 (t u)(i 2(u) I 2 (t)) 3 M (1) 4,N =2 N =2 = M(1) 4,N =8 i apple 2 (1f) tu s 2 (t u)(i 2(u) I 2 (t)) 2 4 h tu Mtree s 2 (5s2 D=6 2 3(1 f)tu) I4 (t, u) 1(3 D)f i (I 2 (u)i 2 (t)) D 2 f!1 4 h Mtree 4 tu 1f s 2 (s2 tu) I4 D=6 2 (t, u) 2 1(3 D)f i (I 2 (u)i 2 (t)) D 2 f!1 (same as ) M (1) 4,N =4 N =02 JJMC, Chiodaroli, Gunaydin, Roiban 12

39 Many interesting lessons from this: observe a unified structure in gauge and gravity amplitudes four-graviton amplitudes are insensitive to the precise nature of the matter couplings everywhere we look two different presentations of N=4 SG are the same just because something integrates to zero for gauge theories doesn t mean we can ignore it in doublecopying to gravity

40 N=8 SG N=4 SG difference structurally between N>4 SG and N=4 SG

41 N=8 SG N=4 SG N=4 SG first time double copied from N=0 YM

42 something new with N=0 YM amplitudes? YES! FRIENDS = 0 FRIENDS = finite contribution

43 This means something new at N=4 SG N=0 YM N=4 SYM

44 N=0 YM N=4 SYM s N=4 SG: s N=4 SG

45 DUALITY N=8 SG N=4 SG New stuff in N=4 SG s s N=4 Is there a way to line this up and identify / clarify known U(1) anomaly in N=4 SG?

46 DUALITY N=8 SG N=4 SG New stuff in N=4 SG s s N=4 Is there a way to line this up and identify / clarify known U(1) anomaly in N=4 SG? YES JJMC, Kallosh, Roiban, Tseytlin

47 N=4 CSG VM U(1) Anomaly very nice story we may hear from Renata N=4 (P)SG U(1) Anomaly JJMC, Kallosh, Roiban, Tseytlin

48 DUALITY N=8 SG N=4 SG N N=8: SU(8) SU(4) SU(4) U(1). 1 = (1, 1) 0 8 = (4, 1) q (1, 4) q 28 = (6, 1) 2q (1, 6) 2q (4, 4) 0 56 = ( 4, 1) 3q (1, 4) 3q (6, 4) q (4, 6) q 70 = (1, 1) 4q (1, 1) 4q ( 4, 4) 2q (4, 4) 2q (6, 6) 0. JJMC, Kallosh, Roiban, Tseytlin

49 DUALITY N=8 SG N=4 SG N N=8: SU(8) SU(4) SU(4) U(1). 1 = (1, 1) 0 8 = (4, 1) q (1, 4) q 28 = (6, 1) 2q (1, 6) 2q (4, 4) 0 56 = ( 4, 1) 3q (1, 4) 3q (6, 4) q (4, 6) q 70 = (1, 1) 4q (1, 1) 4q ( 4, 4) 2q (4, 4) 2q (6, 6) 0. N=4 SG mult. JJMC, Kallosh, Roiban, Tseytlin

50 DUALITY N=8 SG N=4 SG N N=8: SU(8) SU(4) SU(4) U(1). 1 = (1, 1) 0 8 = (4, 1) q (1, 4) q U(1) identified w/ U(1) symmetry of N=4 SG 28 = (6, 1) 2q (1, 6) 2q (4, 4) 0 56 = ( 4, 1) 3q (1, 4) 3q (6, 4) q (4, 6) q 70 = (1, 1) 4q (1, 1) 4q ( 4, 4) 2q (4, 4) 2q (6, 6) 0. N=4 SG mult. JJMC, Kallosh, Roiban, Tseytlin

51 Correct choice of charge q under U(1) leads to anomalous part of effective action N =4,n v = 1 an 2 (2 n v) N =4,n v=0 an = 2n v 4(4 ) 2 Z RR r 2 r µ a µ Amplitude considerations reproduces this exactly (n_v corresponding to including n_v scalars in N=0 YM ) JJMC, Kallosh, Roiban, Tseytlin

52 s Preview (anomalous amplitudes): (1);N =4 M (1, 2, 3) = i apple 3 (4 ) 2 2 (1);N =4 M (1, 2, 3, 4) = i apple 4 (4 ) (8) ( (1);N =4 M (1, 2, 3, 4, 5) = 2i apple 5 (4 ) 2 2 3X i=1 4 (8) ( A i i) 4X i=1 5 (8) ( s A i i) 5X i=1 A i i) N=4 SG related by soft-limits M (L) n (1, 2,...n 1,n) k n!0! apple 2 S(0) n M (L) (1, 2,...n 1) n 1...more very soon... JJMC, Kallosh, Roiban, Tseytlin

53 abelian gauge theories DUALITY 2001 BIK derives nice recursive formula for N=2 Born-Infeld action gives to order W^10 open question if this last term satisfied duality... JJMC, Kallosh 13

54 S BI = S free S int,10 Bellucci, Ivanov, and Krivonos 01 Z S int,10 = 4 3 ( 1 d 12 Z 8 W2 W W4 hw 4ii h 72 W3 W 3i 1 16 W2 W 2 D 4 W 2 1 h 16 W2 W 2 D 4 W 2i 1 h 48 W3 W 3 D 4 W 2 i 1 h h 48 W3 W 3 D 4 W 2ii 1 32 W2 W 2 D 4 W W2 W 2 D 4 W 2 h D 4 W 2i 1 h 32 W2 W 2 D 4 W 2i 2 0 h W5 hw 5iii apple h 48 W3 W 3 D 4 W 2i W3 1 h h h 576 W4 W 4 D 4 W 2iii W 4 h W 4 D 4 W 2 i 1 48 W3 hw 3 D 4 W 2 i 2 h W 3 D 4 h W 2ii D 4 W W2 W 2 D 4 W W4 W 3 D 4 W W2 W 2 D 4 W 2 2 D 4 h W 2i 1 32 W2 W 2 D 4 h W 2 D 4 h W 2ii D 4 h W 2i 1 16 W2 W 2 D 4 W 2 D 4 h W 2i W2 W 2 D 4 h W 2i W4 h W 3i h h D 4 W 3ii 1 32 W2 W 2 D 4 W 2 h D 4 W 2 D 4 ) W 2 i W Z Z (1.2)

55 S BI = S free S int,10 Bellucci, Ivanov, and Krivonos 01 Z S int,10 = 4 3 ( 1 d 12 Z 8 W2 W W4 hw 4ii h 72 W3 W 3i 1 16 W2 W 2 D 4 W 2 1 h 16 W2 W 2 D 4 W 2i 1 h 48 W3 W 3 D 4 W 2 i 1 h h 48 W3 W 3 D 4 W 2ii 1 32 W2 W 2 D 4 W W2 W 2 D 4 W 2 h D 4 W 2i 1 h 32 W2 W 2 D 4 W 2i 2 0 h W5 hw 5iii apple h 48 W3 W 3 D 4 W 2i W3 1 h h h 576 W4 W 4 D 4 W 2iii W 4 h W 4 D 4 W 2 i 1 48 W3 hw 3 D 4 W 2 i 2 h W 3 D 4 h W 2ii D 4 W W2 W 2 D 4 W W4 W 3 D 4 W W2 W 2 D 4 W 2 2 D 4 h W 2i 1 32 W2 W 2 D 4 h W 2 D 4 h W 2ii D 4 h W 2i 1 16 W2 W 2 D 4 W 2 D 4 h W 2i W2 W 2 D 4 h W 2i W4 h W 3i h h D 4 W 3ii 1 32 W2 W 2 D 4 W 2 h D 4 W 2 D 4 ) W 2 i W Z Z (1.2)

56 Broedel, JJMC, Ferrara, Kallosh, Roiban 12 duality satisfied if Z d 8 Z W 2 M 2 Z R Z Z d 8 Z W W 2 M 2 =0, R R i M 4 d R dw R R S[W, W]. R R 0= 1 3 I I 3a 1 2 I 3b 2I 3c I S4a 1 12 I 4b 1 12 I 4c 1 6 I 4d 1 6 I 4e 1 18 I 4f 1 18 I 4g 1 18 I 4h 1 2 I 4i 1 2 I 4j JJMC, Kallosh 13

57 somewhat tedious, but straightforward integration by parts identities shows all terms vanishes... I 4i = 4 Z Z d 12 Z 7W 2 2 D 4 W 2 h 2 4 D W 2i 4W 2 W 2 D 4 2 h Z W W D W h i D W W W D 4 W 2 4 h D W 2ii h h ii h i W D h h 3W 2 W 2 D 4 W 2 D 4 W 2 D 4 W 2 ii h ii 3W 2 W 2 D 4 W 2 h D 4 W 2 D 4 W W 2 i h 3W 2 W 2 D 4 W 2 D 4 W 2 i 2 13 others exists much more satisfying constructive proof... JJMC, Kallosh 13

58 abelian gauge theories DUALITY find duality-conserving sources of deformation... JJMC, Kallosh 13

59 through W^8 I(T, T )= Z d 12 Z a 0 (T ) 2 (T ) 2 abelian gauge theories 2 a 1 (T ) 3 (T ) 3 3 a 2 (T ) 4 2 (T ) 4 DUALITY 3 a 3 (T ) 2 (T ) 2 D 4 ((T ) 2 )D 4 ((T ) 2 )! O( 4 ) I 4(T, T )= 4 Z Broedel, JJMC, Ferrara, Kallosh, Roiban 12 d 12 Z a 4 (T ) 5 3 (T ) 5! a 5 (T ) 3 D 4 ((T ) 2 ) (T ) 3 D 4 ((T ) 2 ). JJMC, Kallosh 13

60 ASK THE RIGHT QUESTIONS shoals of understanding when in doubt, calculate beauty that trivializes calculations is very special