Soft Theorems: trees, loops and strings

Transcription

1 Soft Theorems: trees, loops and strings with Song He, Yu-tin Huang, CongKao Wen ( ) Roma Tor Vergata - INFN Talk at Joint ERC workshop Superfields, Self-completion and Strings & Gravity, Ludwig-Maximilians-University Munich, Arnold-Sommerfeld-Center

2 Foreword: Scattering Amplitudes... see e.g.[elvang, Huang] Spinor helicity formalism and SUSY [Grisaru, Pendelton; Van Nieuwenhuizen] Color Ordering and dual diagrams [Parke-Taylor; Mangano-Parke; Berends, Giele; Kleis, Kuijf] On-shell recursion relations [Britto, Cachazo, Feng, Witten] MHV expansion [Cachazo, Svrcek, Witten], Twistor Strings [Witten, Berkovits] NMHV and beyond, Momentum Twistors [Hodges; Boels, Mason, Skinner], all-loop formulae [Arkani-H, Bourjaily, Cachazo, Caron-Huot, Trnka] (Generalized) Unitary methods [Bern, Dixon, Dunbar, Kosower; Elvang, Freedman, Kiermeier] Gravity = (Gauge) 2 [Kawai, Lewellen, Tye; Bern, Dixon, Perelstein, Rozowsky; MB, Elvang, Freedman]

3 Foreword: Scattering Amplitudes... and more Amplitude - Wilson loop duality [Alday, Maldacena; Drummond, Henn, Korchemsky, Sokatchev; Brandhuber, Heslop, Travaglini; Eden, Beisert, Staudacher; Bern, Dixon, Smirnov; Broedel; Plefka;... Wen] Amplitude - Wilson loop - Correlator triality Drummond, Henn, Korchemsky, Sokatchev; Brandhuber, Heslop, Travaglini; Wen; MB] Colour-kinematics duality and double copy [Bern, Carrasco, Johanssen] Polylogs,... symbols [Goncharov; Bern, Del Duca, Schmidt; Spradlin, Volovich;... ] Leading singularities, Grassmannian [Arkani-H, Bourjaily, Cachazo, Trnka;... ] Bipartite FT [Franco, Galloni, Mariotti, Seong] Scattering Equations [Cachazo, He, Yuan; Dolan, Goddard] Soft Theorems and BMSV group [Bondi, Metzner, Sachs, Van der Burg; Barnich; Strominger; Casali; Cachazo... Schwab, Volovich]... Loop corrections? Higher-derivative corrections? Strings?

4 Plan Soft Theorems at tree level: leading, sub-leading, sub-sub-leading Loops see also [He, Huang, Wen ; Bern, Davies, Nohle] Soft theorems for Integrands in N = 4 and in N < 4 Finite loop amplitudes in gravity and gauge theories Conformal anomaly and integrated soft theorems Higher-derivative interactions F 3 R 3 and R 2 φ Strings Explicit results for n = 4, 5, 6 open and closed World-sheet analysis on the disk and the sphere Conclusions and Outlook

5 Soft theorems at tree level Soft behavior k s δk s, δ 0 from Ward identity [Low] (a) Leading term δ 1 from (a) : (b) i e i ɛs ki δk s k i A n 1 (..., k i + δk s,...) Sub-leading term from (a) and (b): universal operator acting on A n 1 QED [Low] A n = δ 1 S (0) A n 1 + S (1) A n 1 + O(δ 1 ) Gravity [Weinberg, Gross-Jackiw] M n = δ 1 S (0) M n 1 + S (1) M n 1 + δs (2) M n 1 + O(δ 2 )

6 Soft operators at tree level Gauge theory (color ordered) A n = δ 1 S (0) YM A n 1 + S (1) YM A n 1 + O(δ 1 ) where S (0) SYM = ɛ s k s+1 ɛ s k s 1 k s k s+1 k s k s 1 [ ] J S (1) s+1 SYM = (kµ s ɛ ν s k ν s ɛ µ µν s ) Js 1 µν k s k s+1 k s k s 1 Gravity M n = δ 1 S (0) G M n 1 + S (1) G M n 1 + δs (2) G M n 1 + O(δ 2 ) where S (0) Grav = i s S (1) Grav = i s S (2) Grav = i s k µ i Es µνk ν i k s k i k µ i Es µνj νλ i k sλ k s k i k s J i E s J i k s k s k i

7 Holomorphic soft limit from BCFW Tree-level (holomorphic) soft limit via recursion relations, helicity spinors (k 2 = 0 k α α = λ α λ α in D = 4) [Strominger, Cachazo; Casali] λ s δλ s + zλ n, λ n λ n zδλ s i s = M 3 (ŝ +, i, ˆK is ) 1 K 2 M n ( ˆK is,..., ˆn) + R fin mass 1 i<n i,s M 3 (ŝ +, i, ˆK is ) 1 K 2 M n ( ˆK is,..., ˆn) = 1 ( ) s n is δ 3 S0 s,im n λ i + δ i n λ s i s, λ n + δ n i λ s with inverse soft function S (0) s,i = ( ni 2 / ns 2 )([is]/ is ) independent of helicity h i. Exponentiation [S.He, Y-t Huang, C. Wen ] ( = δ 3 S (0) s n s,i exp δ i n λ ) s i s + λ i n i λ s M n λ n

8 Further developments [S.He, Y-t Huang, C. Wen ] Using helicity spinors in D = 4 Soft theorems for MHV Double Copy S (k) Grav = i K is [S (0) YM (i)ej/2 ] 2 Supersymmetric version: [ sn J = in ( λ s λ i + η s η i ) + si ( ni Sub-leading YM soft operator (s = n + 1) S (1) YM = 1 ns s1 [1n] 2 p α α s )] λ s + η s λ n η n [ ] p nα β (M β 1 α δ β α (D 1 h 1 )) (n 1),

9 Kinematic simplification Kinematics and holomorphic shift in the soft limit λ s δ λ s A n+1 = ( ) S (0) YM + δs(1) YM A n + O(δ 2 ) [ S (1) sn YM = S(0) YM in λ s + s1 ] λ 1 n1 λ s λ n Solve momentum conservation i p i = 0 with n 1 1] = i=2 ni [i n1, n 1 n] = i=2 1i [i 1n, Sub-leading term vanishes S (1) YM A n = 0 A n+1 = S (0) YM A n + O(δ 2 )

10 1. What happens at loop level? IR-divergences [Bern, Davies, Nohle] IR-finite rational terms [Bern, Davies, Nohle; S. He, C. Wen, Y-t Huang] A n+1 ( ) A n ( ) (Exact) A n+1 ( ) A n ( ) (Exact) A n+1 ( ) A n ( ) (Corrected) S (1) (0) [n n+1] A n ( ) = S n n+1 A(0) n (1, 2 +,..., (n 1) +, n ) For pure gravity M n, Soft operators corrected at loop-level Soft theorems for integrands? Do ɛ 0 and δ 0 commute? [Cachazo, Yuan] dl 4 2ɛ 1 (l + δs si ) 2 l hard wrt δk s?

11 Manifest soft integrands Is there a representation of loop integrands well-behaved in the soft limit? I n+1 = S (0) YM I n + O(δ 2 ) Use super momentum twistors [Hodges; Boels, Mason, Skinner] Z = {Z I χ A } = {λ α, µ α χ A } to trivialize (super) momentum conservation λ a = µ a 1 a, a+1 + a 1, a+1 µ a a 1, a a, a+1 + µ a+1 a 1, a same for η A a and χ A a A = 1,... N [Drummond, Henn] Anti-holomorphic soft limit: Z n αz 1 + βz n 1 + δz s, since λ n = δ n 1, 1 µ s + 1, s µ n 1 + s, n 1 µ 1 1, n 1 2 αβ = δˆλ s

12 Planar N = 4 SYM A (L) n,k = AMHV 0 d D l 1... d D l L R (L) n,k with A MHV 0 = δ4 2N (λ a λ a η A a λ a ) 1, 2... n, 1 All-loop recursion relation [Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka] R (L) n,k = R (L) n 1,k + [1, A, B, n 1, n]r (L 1) n+2,k+1 (1,..., ˆn, A, ˆB) + L,k,i GL(2) R (L ) i,k (1,..., i 1, I i)[1, i 1, i, n 1, n]r (L L ) n+2 i,k 1 k (I i, i,..., ˆn i ) where ˆn i = (n 1n) (1i 1i), I i = (i 1i) (1n 1n), ˆn = (n 1n) (1AB), ˆB = (AB) (1n 1n) with overlap : (ab) (ijk) = Z a bijk Z b aijk 4-pt invariant: abcd = ɛ IJKL Z I az J b ZK c Z L d and 5-pt R-invariant: [a, b, c, d, e] δ 0 4 (χ A a bcde + cyc) abcd bcde cdea deab eabc.

13 Planar N = 4 SYM: Soft limit (I) First consider factorisation terms Z n αz n 1 + βz 1 + δz s R (L) n,k = R (L) n 1,k + R (L ) i,k (1,..., i 1, I i)[1, i 1, i, n 1, n]r (L L ) n+2 i,k 1 k (I i, i,..., ˆn i ) + L,k,i I i = δ(i 1i) (1n 1s), Zˆni = Z 1 +O(δ) [1, A, B, n 1, n]r (L 1) n+2,k+1 (1,..., ˆn, A, ˆB) GL(2) [1, i 1, i, n 1, n] δ4 δ 0 4 (χ [i 1 i], n 1, s, 1 + χ s 1, i 1, i, n ) α β δ 2 1, i 1, i, n 1 3 n 1, s, 1, i 1 n 1, s, 1, i + O(δ3 )

14 Planar N = 4 SYM: Soft limit (II) Then consider forward-limit term Z n αz n 1 + βz 1 + δz s R (L) n,k = R (L) n 1,k + R (L ) i,k (1,..., i 1, I i)[1, i 1, i, n 1, n]r (L L ) n+2 i,k 1 k (I i, i,..., ˆn i ) where + L,k,i O(δ 2 ) [1, A, B, n 1, n]r (L 1) n+2,k+1 (1,..., ˆn, A, ˆB) GL(2) Zˆn = Z 1 +O(δ), [1, A, B, n 1, n] Z ˆB = δ(ab) (1n 1s) δ 2 δ 0 4 (χ [A B], n 1, s, 1 + χ s 1, A, B, n ) αβ 1, A, B, n 1 A, B, n 1, 1 1, A, n 1, s 1, B, n 1, s n 1, 1, A, B

15 Planar N = 4 SYM: Soft limit (III) To any loop order in planar N = 4 SYM R (L) n,k R(L) n 1,k + O(δ2 ) then I n+1 = ( ) S (0) YM + δs(1) YM I n + O(δ 2 ) Universal soft behaviour at all loops at integrand level! Integrands with manifest soft behavior have interesting properties... not explicit... soft guidance as how to find them?

16 N < 4 SYM SYM theories with lower susy N < 4 SYM = N = 4 SYM (4 N ) Chiral Use CSW expansion [Brandhuber, Spence, Travaglini] + R (1),chiral n,2 = R (1),chiral n 1,2 a ˆB b ˆB AB 2 AB1n 1 ABn 1n AB1n a<i b ai i bi i AB1i 1 ABi 1i AB1i Thus ZˆB = δ(ab) (1n 1s) R (1) n,2 R(1) n 1,2 + O(δ2 ) Soft theorems are manifest for planar integrands in D = 4 2ɛ dimensions

17 All Plus Rational terms Why do the two limits commute for all-plus? [Bern, Morgan] ( A YM 2 5 (+, +, +, +, +) = 5 i=1 ii [ µ 4 ] s 12 s 23 + cyclic 2 d 1 d 2 d 3 d 5 I m [µ 2r ] = ɛ(1 ɛ)... (r 1 ɛ)(4π) r I D=4+2r 2ɛ m, Dimension shifting: [Bern, Dixon, Perelstein, Rozowsky] i s R YM n (+,..., +) = µ 4 R n (N = 4 SYM) R Grav n (+,..., +) = µ 8 R n (N = 8 Sugra) Non-commutativity stems from scalar bubbles In susy theories A n (+, +,... +) = 0 = A n (, +,... +) 2 ) + 4iµ6 ɛ(1234) d 1 d 2 d 3 d 4 d 5

18 Soft corrections from anomalies Under conformal boosts n K = A n+1 ({λ s, λ s }, δλ s, λ s ) = 1 δ 2 S(0) A n + 1 δ S(1) A n i=1 2 λ i λ i + 2 δ λ s λ s = K δ K s neglecting anomalies KA n+1 = 0: conformal invariance implies a differential equation for soft functions [Larkoski] S (0) = 1n 1s sn ( O(δ 2 ) : K 0 S (0) A n + K s S (1) λn A n = ns 2 + (K s S (1) )A n = 0 λ n + λ 1 1s 2 λ 1 ) A n

19 Conformal anomalies Tree-level soft functions are homogenous solutions to CDE Loop level corrections must arise from conformal anomalies (K δ K s)a n+1 ({λ i, λ i }, δλ s, λ s ) = r a (r) n+1 δr All-plus amplitude A n (+, +,..., +) conf-invariant (Self-dual) Single-minus amplitude A n (, +,..., +) not conformally invariant, only scale invariant. Depending on h = ± of soft leg A n+1 ( ) a (0) n A n+1( ) 1 δ 2 a( 2) n O(δ 2 ) K 0 S (0) A n + K s (S (1) + (1) )A n = a ( 2) n+1. (1) A n = λȧsλ a s(s tree(0) K 0 A n a ( 2) n+1 ). Universality would imply a ( 2) n+1 = OA n

20 2. Are soft theorems universal? Effective field theories with F 3, R 3 (Forbidden by susy, appear in bosonic strings) Effective field theories with R 2 φ (appears in SUGRA due to U(1) anomalies and in bosonic strings) [Carrasco, Kallosh, Roiban, Tseytlin] String theory at finite α [MB, He, Huang, Wen; Bern, Di Vecchia, Nohle;...]

21 F 3 coupling CSW representation, two basic vertices [Broedel, Dixon] k j l : jk 2 kl 2 lj 2 n (new i=1 ii + 1 F 3 SD) k j : jk 4 n (usual i=1 ii + 1 Consider diagrams with only one white vertex MHV) m 4 i m2, (b) : i m2 (a) : m 1 m3 m 1 m3 j j m 4

22 F 3 amplitudes CSW-like expansion with two MHV vertices m 4 i m2, (b) : i m2 (a) : m 1 m3 m 1 m3 j j m 4 (a) : (b) : 1 n l=1 ll+1 m 1m 4 4 [ii 1jj 1 ] m 2 m 3 2 m 3 î 1 2 î 1m n l=1 ll+1 m 1î 1 4 [ii 1jj 1 ] m 2 m 3 2 m 3 m 4 2 m 4 m 2 2, where denotes reference twistor Z = (0, µ, 0) and î 1 = (ii 1) ( jj 1). For instance R F 3 1, n = [ii 1jj 1 ]( m 1 m 4 4 m 2 m 3 2 m 3 î 1 2 î 1m )

23 Soft limit of F 3 amplitudes m i 4 m2, (b) : i m2 (a) : m 1 m3 m 1 m3 j j m 4 (a) : (b) : 1 n l=1 ll+1 m 1m 4 4 [ii 1jj 1 ] m 2 m 3 2 m 3 î 1 2 î 1m n l=1 ll+1 m 1î 1 4 [ii 1jj 1 ] m 2 m 3 2 m 3 m 4 2 m 4 m 2 2, Recursion relation à la BCFW and soft limit R F3 k, n = R F3 k, n 1 + j [n 1, n, 1, j 1, j]r F3 k, jr F2 k 1 k, n+2 j + (F 3 F 2 ) R F 3 k, n = RF 3 k, n 1 + O(δ2 ) A n+1 = S (0) YM A n + O(δ 2 ) The presence of F 3 does not spoil S (1) F 3 = S (1) YM

24 R 3 /R 2 φ amplitudes For gravity, use KLT-like formulae M(1, 2, 3, 4, 5 + ) = is 12 s 34 A F3 (1, 2, 3, 4, 5 + )A F3 (2, 1, 4, 3, 5 + ) + P(2, 3) There is a mixing of operators (α 2 ) R 3, (α 1 ) φr 2 M n+1 1 δ 3 S(0) M n + 1 δ 2 S(1) M n + 1 δ S(2) M n + (2) + O(δ 0 ) (2) = j 2 1j 3 [1j] M n(φ, i 1, i 2,..., i n 2, n+ ) For N 4 there are U(1) anomalies: R 2 φ generated at one-loop [Carrasco, Kallosh, Roiban, Tseytlin] Same for bosonic string already at tree level

25 Open superstring amplitudes on the disk Use KLT-like representation [Mafra, Schlotterer, Stieberger] A n = F (2σ,...,(n 2)σ) A YM (1, 2 σ,..., (n 2) σ, n 1, n) σ S n 3 For n = 4, (n 3)! = 1, setting s ij = 2α k i k j For n = 5, (n 3)! = 2 A 4 = F (2) A YM = s dz 2 z s (1 z 2 ) s23 A YM A 5 = F (2,3) A YM (1, 2, 3, 4, 5) + F (3,2) A YM (1, 3, 2, 4, 5) 1 1 F (2,3) = s 12 s 34 dz F (3,2) = s 13 s 24 dz z 2 z 2 dz 3 z s z s13 3 zs23 32 (1 z 2) s24 (1 z 3 ) s34 1, dz 3 z s12 2 zs z s23 32 (1 z 2) s24 1 (1 z 3 ) s34

26 Soft limit of 5-pt amplitude Integrate over z 3 and keep terms up to sub-leading order in δ F (2,3) s dz 2 z s (1 z 2 ) s 24 [1 + δ(s23 + s 34 + k 2 p 4 ) log(1 z 2 )]. Fix D = 4 kinematics: k 4 = 4 5 (1+2) 45, p 4 = , k 5 = 5 4 (1+2) 54, p 5 = = [31] 1 s S (1) F (2,3) F (3,2) S (1) = s 13 s dz 2 z s (1 z 2 ) s 24 log(1 z2 ) dz 2 z s12 2 (1 z 2) s 24 1 log(z 2 ) ( λ S (1) YM (234)A(1, 2, 4, 5) = λ ) 3 A(1, 2, 4, 5) λ 2 34 λ 4 Soft expansion is valid for finite α. Explicit check for 6-pt amplitude

27 Closed superstring amplitudes Use (generalized) KLT formulae [Kawai, Lewellen, Tye] M n = A n (1, 2,..., n)[ {i},{j} f(i 1,..., i n 2 1 ) f(j 1,..., j n 2 2 ) A n ({i}, 1, n 1, {j}, n) + Perm(2,..., n 2)] with momentum kernels [Bjerrum-Bohr, Vanhove] m 1 f(i 1,..., i m ) = sin(πs 1,im ) sin π f(j 1,..., j m ) = sin(πs j1,n 1) k=1 ( ( m sin π k=2 s 1,ik + m l=k+1 k 1 s jk,n 1+ After some algebra... get expected soft behaviour S (k) k = 0, 1, 2 g(i k, i l ) ) ) g(j l, j k ), l=1 closed = S(k) Grav for,

28 World-sheet analysis Open superstring amplitudes on the disk A(1, 2,..., n) = ig n 2 s dz 2... dz n 2 cv(1)v(2)... cv(n 1)cV(n) Take two vertices in the q = 1 picture and the remaining n 2 in the q = 0 picture: V (0) A V ( 1) A = (ɛ ψ)e ϕ e ikx, V (0) A = (ɛ X + ik ψɛ ψ)eikx total derivative as k 0, only contribution from the two ends Use OPE V (0) A (z s)v ( 1) A (z s±1 ) z s z s±1 2ksks±1 1 e ϕ(zs±1) e i(ks+ks±1)x(zs±1) {ɛ s k s±1 ɛ s±1 ψ + ɛ s±1 k s ɛ s ψ ɛ s±1 ɛ s k s ψ + ɛ s k s±1 k s X s ɛ s±1 ψ}(z s±1 ) +...

29 Soft limit of disk amplitudes Integrate around the singularity z s z dz s z s z 2ks k 1 = 1/k s k Combine with first term, as expected, get S (0) open = S (0) SYM = ɛ s k s+1 k s k s+1 ɛ s k s 1 k s k s 1 Next consider sub-leading terms (in k s ), including expansion of e iks X Keep only BRST invariant operators (i[k s Xɛ s k s±1 ɛ s Xk s k s±1 ]ɛ s±1 ψ + [ɛ s±1 k s ɛ s ψ ɛ s±1 ɛ s k s ψ])e ϕ e iks±1 X [ ] = (k µ s ɛ ν s k ν s ɛ µ s ) k µ k + ɛ ν µ ɛ ν V ( 1) (k à s±1 ) Again S (1) open = S (1) SYM = (kµ s ɛ ν s k ν s ɛ µ s ) [ J s+1 µν k s k s+1 ] Js 1 µν k s k s 1

30 Closed strings on the sphere Graviton vertex operators in the ( 1, 1) and (0, 0) pictures V (0,0) G V ( 1, 1) G = E µν ψ µ ψ ν e ϕ ϕ e ik X = E µν ( X µ + ik ψψ µ )( X ν + ik ψ ψ ν )e ikx with E µν = ɛ µ ɛ ν to factorise Left- and Right-movers Use OPE from open strings for each of the n 1 hard gravitons, get S (0) closed = S(0) Grav = i s S (1) closed = S(1) Grav = i s S (2) closed = S(2) Grav = i s k µ i Es µνk ν i k s k i k µ i Es µνj νλ i k sλ k s k i k s J i E s J i k s k s k i

31 Other massless states Soft dilaton and soft Kalb-Ramond decouple at leading δ 1 order since k 2 i = 0 and k µ i kν i B [µν] = 0 Yet non vanishing sub-leading δ 0 behaviour as for pions [Adler] Derivative wrt to moduli fields, (not) puzzling: single soft limit 0, yet double soft limit non-vanishing! [Arkani-H, Cachazo, Kaplan; MB, Elvang, Freedman; Kallosh;... MB, Consoli, Di Vecchia, Guerrieri, Marotta]

32 Conclusions and outlook Conclusions Outlook Leading and sub-leading soft functions universal (at tree level!) Sub-sub-leading corrections are sensitive to scalar couplings F 2 φ, R 2 φ (even at tree level!) Loops: manifest soft integrands exists for planar N 4 SYM Loop corrections are intimately tied to anomalies Open and closed superstrings same soft behavior as SYM and SuGra at tree level IR-div corrections are known and physical... The tree-level YM soft functions are homogenous solution to conformal symmetry. Gravity? String theory soft theorems simply from OPE. Higher genus? Soft dilaton/moduli...