Multi-Gluon Scattering and MHV Amplitudes in Superstring Theory
Stephan Stieberger, MPP M¨unchen
Workshop: Twistors, perturbative gauge theories, supergravity and superstrings Ludwig–Maximilians–Universit¨at M¨unchen, June 18 -22, 2007 Based on Work:
St. St. and T.R. Taylor: • “Amplitude for N–Gluon Superstring Scattering”, hep-th/0607184, Phys. Rev. Lett. 97 (2006) 211601;
“Multi–Gluon Scattering in Open Superstring Theory”, hep-th/0609175, Phys. Rev. D 74 (2006) 126007;
“All Order α′ MHV Amplitudes ”, to appear.
D. Oprisa and St. St.: • “Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler–Zagier Sums ”, hep-th/0509042; Effective gauge interactions: Dp–branes
A variety of string theories contain gauge theories in their α 0 limits. ′ → E.g.: Type II with N D–branes gives U(N) gauge group
2 At tree–level effective gauge interactions are described by (valid up to α′ ):
“Non–Abelian Born–Infeld action”
2 π2 2 4 1 2 2 5 eff. = STr F 2 α′ [ F + 4 (F ) ] + (F ) L − O with Tseytlin’s symmetrized trace : 1 STr(A1 ...AN ) := N! Aπ(1) ...Aπ(N) . π (1,...,N) ∈ P 2 2 1 eff. = Tr Fab 8 α′ ζ(2) FabFbdFcaFdc + FabFbcFcdFda L − 2 1 1 3 FabFbaFcdFdc FabFcdFbaFdc + (α′ ) − 4 − 8 O Open questions
Higher orders in α , i.e. α 4 and beyond ? • ′ ′
Derivative terms D F ? •
Systematics ? •
Use ? •
Probe gluon interactions by string amplitudes and find interesting insights Gluon string S–matrix
N–gluon amplitude (cf. QCD)
(k , ξ , a ; ... ; k , ξ , a ; α )= AN 1 1 1 N N N ′ π(1) π(2) π(N) Tr(T T ...T ) A(π(1),π(2),...,π(N); α′) π (1X,2,...,N) ∈
π: all (N 1)! cyclically inequivalent orders −
E.g.: N =3: Tr(T 1T 2T 3) = Tr(T 2T 3T 1) = Tr(T 3T 1T 2)
Tr(T 1T 3T 2) = Tr(T 3T 2T 1) = Tr(T 2T 1T 3)
Important: A enjoys important properties: cyclic invariance, soft–gluon factorization, etc ! Non–trivial Properties of the N–Gluon String S–matrix
Consider partial amplitude A(1,...,N; α ) A( ξ , k ,..., ξ , k ; α ) ′ ≡ { 1 1} { N N } ′
cyclic invariance: • 1 2, 2 3,...,N 1 → → →
soft–boson limit: • k 0 for one i 1,...,N i −→ ∈{ }
factorizing into collinear limits: • k k , with i +1 mod N i || i+1 Four gluon open string S–matrix
N = 4 : Green, Schwarz, 1982
k1 k3 k1 k3 k1 k3 k2 k3
0 ∂A ∂A α ∂A t ′ s u
¡
k2 ¡ ¢ k4 k2 £ ¤ k4 k2 ¥ k4 k1 k4
reducible diagrams constructed out of Feynman rules from (basic Yang–Mills diagrams)
2 F = (∂µAν ∂νAµ + [Aµ,Aν]) (∂µAν ∂νAµ + [Aµ,Aν]) − −
k2 k3 k3 A
[A,A][A,A] k1 ∂A[A,A] ∂A
¡ ¡ A k1 k4 k2 2 Irreducible contact interaction at α′
α 2 : (∂A)4 ζ(2) F 4 ′ →
¡
New interaction term to be written into 2 the low–energy effective action at α′ . α –correction to YM theory. ⇒ ′
u 1 ut E.g.: (e e+ γ γ )= 2 g2 1+ ζ(2) + ... L− R L R string 4 A → − r t 2 Mstring Production of Regge–excitations Cullen, Perelstein, Peskin, 2000 N gluon open string S–matrix
Tree–level scattering of N open strings
A I A I
CFT
A I A I Disk N open strings (Tree−Level)
+ C
z1 z 2 z N−1 z N Generalized Euler integrals
N 1 A(1, 2,...,N; α′)= VCKG− dzr VAa1(z1) VAa2(z2) ... VAaN (zN ) r=1 h i z1 I na Boils down to: A(1, 2,...,N; α′)= F I kinematics KI "nab# X KI Generalized Euler integrals: b+2 2α k k + k I 1 1 N 3 N 3 ′ b+3 1 j na − 1+a N+na − j=a+3 ! F I = dx1 ... dxN 3 xa − xa P "n # 0 0 − ab Z Z aY=1 bY=a 2α k2+ak3+b+nab b ′ b a = 1, 2,...,N 3 , 1 xj , ≥ − × − na,nab = 0, 1 jY=a ± N=4: Veneziano–amplitude E.g.: N = 4 1 2α k k 1 2α k k 1 Γ(1 + s1) Γ(1 + s2) dx x ′ 1 2− (1 x) ′ 2 3 = Z0 − s1 Γ[1 + s1 + s2] 1 s1 = 2α′ k1k2 = s2 ζ(2) + (s1 + s2) s2 ζ(3) + ... , s1 − s2 = 2α′ k2k3 Beta–function relevant for scattering of four open strings Veneziano–amplitude N(N 3) 2− Laurent polynomials in xa = Number of independent kinematic invariants (Mandelstamm variables) N(N 3) Integer powers na, nab control the physical poles of the amplitude in 2− N invariant masses of dual resonance channels involving 2, 3,..., [ 2 ] external particles: [[i]] = α (k + k + + k )2 , i + N i n ′ i i+1 ··· i+n ≡ E.g.: 2 N = 4 : [[1]]1 = α′ (k1 + k2) = s1 2 2 invariants [[2]]1 = α′ (k2 + k3) = s2 2 2 N = 6 : [[1]]1 = α′ (k1 + k2) = s1 , [[1]]2 = α′ (k1 + k2 + k3) = t1 . , [[2]] = α (k + k + k )2 = t 9 2 ′ 2 3 4 2 inv. 2 2 [[6]]1 = α′ (k6 + k1) = s6 , [[3]]2 = α′ (k3 + k4 + k5) = t3 Multiple Gaussian hypergeometric functions Hypergeometric function N = 4 : F n1 = F ... ; 1 n11 2 1 ... ⇒ a ,...,ap h i h i pFq 1 ; u = 1 n1, n2, n3 ... b1,...,bq N = 5 : F = 3F2 ; 1 n11, n12, n22 ⇒ ... of one variable h i h i N 6 = generalized Kamp´ede F´eriet functions F A:B ≥ ⇒ C:D Multiple Gaussian hypergeometric functions (depending on several variables ui) E.g.: Triple n1, n2, n3 (3) N = 6 : F = F (si, tj) hypergeometric n11, n12, n22, n13, n23, n33 ⇒ h i o function α′–Expansion of the Euler integrals Problem: α′–expansion of integrals or multiple hypergeometric functions = ζ–functions of various kinds appear: ⇒ 1 1 1 1 E.g.: dx dy dz = ζ(2) (Euler) 0 0 0 (1 xyz)2 Z Z Z − More general: k s 1 k sj 1 ( 1) j− 1 1 k j (ln xj) − ζ(s ,...,s )= − dx ... dx x − 1 k 1 k j j j=1 Γ(sj) Z0 Z0 j=1 Y Y 1 xi − i=1 They integrate to multiple zeta values of length k: Q k k k sj 1 ∞ − ζ(s1,...,sk)= s = ni , n j n1>...>nX k>0 jY=1 j n1,...,nXk=1 jY=1 iX=j with s1 2 , s2,...,s 1 ≥ k ≥ Multiple harmonic sums and multiple Euler–Zagier sums m 1 Multiple harmonic sums: sums involving the harmonic number Hm = n nX=1 H2 17 E.g.: ∞ n = ζ(4) n2 4 nX=1 Multiple Euler–Zagier sums: generalized multiple harmonic sums ∞ 1 E.g.: W (a, b, c)= a b c (Witten ζ–function) m,n=1 m n (m+n) P current research in analytic Number Theory E.g. N=7: 1 1 1 1 xs2 (1 x)s3 yt2 (1 y)s4 zt6 (1 z)s5 ws7 (1 w)s6 s1 t1+t4 t7 dx dy dz dw − − − − (1 wxyz) − − (1 xy) (1 wz) (1 yz) − Z0 Z0 Z0 Z0 − − − s3 s4+t3 s5 s6+t5 s4 s5+t4 s5+t1 t4 t5 s4 t3 t4+t7 (1 xy)− − (1 wz)− − (1 yz)− − (1 wyz) − − (1 xyz) − − × − − − − − 2 = 0 + 1a (s1 + s2 + s3 + s4 + s5 + s6 + s7) + 1 (t1 + t2 + t3 + t4 + t5 + t6 + t7) + (α′ ) I I I b O with Euler–Zagier sums , , : I0 I1a I1b 1 ∞ 1 27 0 = = = ζ(4) I (1 xy)(1 wz)(1 yz) n3 (1 + n1) (n1 + n2 +1) (n2 + n3) 4 Z − − − n1,n2=0 Xn3=1 ln w ∞ 1 7 1a = = 2 = ζ(5) 4ζ(2)ζ(3) I (1 xy)(1 wz)(1 yz) n1 n (n1 + n2) (n2 + n3) 2 − Z − − − n1,n3=1 3 Xn2=0 ln y ∞ 1 9 = = = ζ(5) + ζ(2)ζ(3) 1b 2 I (1 xy)(1 wz)(1 yz) n1 n2 (n2 + n3) (n1 + n3) −2 Z − − − n1,n2=1 Xn3=0 Six gluon open string S–matrix N = 6 : (1, 2, 3, 4, 5, 6,α ) Oprisa, St. St., 2005 A6 ′ comprises reducible and irreducible diagrams with six external legs A6 ¡ ¢ £ ¤ ¥ ¡ In particular at α 4 amplitude probes new contact terms: ′ A6 ζ(4) F 6 4 4 new interaction terms ζ(4) ζ(4) D F ⇒ ∼ ζ(4) D2F 5 ¡ N–gluon string S–matrix In general: nN j n4 = 1, n5 = 2, A(1,...,N; α′)= (krks) Φj(α′) , P n6 = 6 ,n7 = 24,... jX=1 j= pole structure (Yang–Mills) P Φj = α′–dependence, ordered according to ζ–values (St. St., Taylor, work in progress) Higher orders in the field strength F (i.e. α′) Series of higher derivative terms (α′–corrections): Dp ′ 1 1 2n+m 2 n m effective = Tr α′ − ζ( n + m 2) D F L m 4 2 − nX≥0 ≥ non–constant field strength F : ∂F = 0 6 0 F2 α′ 1 3 2 2 α′ 0 F D F 2 F4 2 3 4 2 α′ ζ(2) D F D F 3 F5 D2F4 6 2 α′ ζ(3) D F 4 F6 D4F4 D2F5 α′ ζ(4) 5 F7 D6F4 D4F5 D2F6 α′ ζ(5) ...... Number Theory String Theory Geometry ↔ ↔ Multiple Gaussian Tree−level string amplitude GKZ hypergeometric structures hypergeometric functions (string S−matrix) Polytope, Fan e.g.: Lauricella, Kampe de Feriet Generalized Euler integrals Affine toric variety X Number theory: Instanton counting on X Generalized zeta−functions α′ − expansion Multiple Euler−Zagier sums Periods e.g.: Witten zeta−functions Spinors (λ, ˜λ) and spinor products 2 ˜ Any null–vector ki = 0 may be written in terms of two spinors (λ, λ) u (k ) Momentum kµ Dirac spinor + i α (λi)α i u (ki)α˙ (λ˜i)α˙ −→ − ≡ 1 u(k) = Dirac spinor , helicity states : u (k)= (1 γ5) u(k) , ± 2 ± √ + √ iϕ k k− e− 0 3 iϕ + k± = k k 1 √k− e 1 √k ± with choice: u+(k)= , u (k)= , and √2 + √2 − iϕ k1 ik2 √k − √k− e− e iϕ = ± √±+ iϕ + k k− √k− e √k Define: i± = u (ki) , i± = u (ki) | i ± h | ± Spinor products: ij := i j+ = u (k ) u (k ) ǫαβ (λ ) (λ ) = k k eiφij , ij [ji] − i + j i α j β i j h i h i h | i − ≡ ˙ + α˙ β ˜ ˜ q iφij [ij] := i j− = u+(ki) u (kj) ǫ (λi)α˙ (λj) ˙ = kikj e− = kikj h | i − ≡ β − q Spinor representation for the polarization ξ of a massless gauge boson 1 q∓ γµ k∓ q = reference momentum ξ±(k, q)= h | | i , µ ±√2 q k (gauge invariance) h ∓| ±i = ξ (k, q) k = 0 ⇒ ± + 12 [35] ξ (k , k ) ξ (k , k )= h i − 2 5 3 1 − 13 [25] E.g.: h i 1 [42] 21 ξ (k , k ) k = h i − 1 4 2 −√2 [41] = gluon (k, ξ) specified by λ and helicity ⇒ ˜λ MHV amplitudes = N–Gluon amplitude becomes function of inner products of spinors ⇒ of positive and negative helicity, i.e. function on E.g. N=4: k1 k3 k1 k3 k1 k3 k2 k3 ∂A ∂A + + ∂A t A(1 , 2 , 3 , 4 ) =s + − − u ¡ k2 ¡ ¢ k4 k2 £ ¤ k4 k2 ¥ k4 k1 k4 12 4 = i h i 12 23 34 41 h ih ih ih i with: q1 = q2 = k4 , q3 = q4 = k1 MHV amplitudes More efficient way to see that many QCD amplitudes are much simpler Not easy to prove with ordinary Feynman rules: + + A(1±, 2 ,...,N ) = 0 4 λrλs A(1+, 2+,...,r ,...,s ,...,N +) = Tr(T 1 ...T N ) h i − − N λkλk+1 k=1h i Q Parke, Taylor 1986; Berends, Giele 1989 α′–Corrections to MHV amplitudes Question: What is + + A(1−, 2−, 3 ,...,N , α′) ? α′–Corrections to MHV amplitudes + + Γ(1 s1) Γ(1 s2) + + N = 4 : A(1−, 2−, 3 , 4 , α′)= − − A(1−, 2−, 3 , 4 ) Γ(1 s s ) − 1 − 2 2 2 with: s1 = α′ (k1 + k2) , s2 = α′ (k2 + k3) A(1 , 2 , 3+, 4+, 5+, α ) = V (5)(s ) 2i P (5)(s ) ǫ(1, 2, 3, 4) − − ′ i − i N = 5 : h i A(1 , 2 , 3+, 4+, 5+) × − − with: 2 α β µ ν ǫ(i,j,m,n)= α′ ǫαβµν ki kj km kn 1 ij [jl] lm [mi]= 2 iǫ(i,j,l,m)+ k k k k + k k h i h i − 2 ij lm − il jm im jl α′–Corrections to MHV amplitudes N = 6 : 5 A(1 , 2 , 3+, 4+, 5+, 6+, α ) = V (6)(s , t ) 2i ǫ P (6)(s , t ) − − ′ i i − k k i i kX=1 A(1 , 2 , 3+, 4+, 5+, 6+) × − − with : ǫ1 = ǫ(2, 3, 4, 5) , ǫ2 = ǫ(1, 3, 4, 5) , ǫ3 = ǫ(1, 2, 4, 5) , ǫ4 = ǫ(1, 2, 3, 5) , ǫ5 = ǫ(1, 2, 3, 4) . Non–trivial consistency: Soft–gluon factorization, cyclic invariance, etc. Note: ǫ s = 0 , s = 5 5 Gram matrix × α′–Corrections to MHV amplitudes Arbitrary N: + + (N) + + A(1−, 2−, 3 ,...,N , α′)= 1+ ζ(2) F A(1−, 2−, 3 ,...,N ) with: E( N 1) E( N 1) 2 − 2 − (N) (N) F = [[1]]k[[2]]k [[1]]k[[2]]k 2 + C + i ǫ(k,l,m,n) { }− { − } k=1 k=3 k N [[1]] N 2[[ 2 + 1]] N 2 N even −{ 2 − 2 − } C(N) = N+1 [[1]] N 5 [[ ]] N 3 N odd − 2 − −{ 2 2 } Non–trivial consistency: Soft–gluon factorization, cyclic invariance, etc. One–loop YM amplitude tree-level α 2–correction ≃ ′ N–gluon one–loop amplitude with maximal helicity violation in YM: (N) i Np F A(1+, 2+,...,N +)= 192 π2 N λkλk+1 k=1h i Q Bern, Chalmers, Dixon, Kosower, 1994 2 Agrees with α′ –correction of gluon amplitude in superstring theory ! Heterotic - Type I duality in D = 10 Heterotic-type I duality in the effective action, (Taylor, St.St., 2002): heterotic g loop − z g tree 1-loop }|2-loop 3-loop 4-loop { F 2g+2 F 2 F 4 F 6 F 8 F 10 type I tree level = Born Infeld − − | {z } E.g.: F 6 heterotic 2-loop F 6 Type I tree-level ≃ Heterotic - Type I duality in D = 10 Consider certain “BPS –kinematics” ( F 6 superinvariants) d3Ω d3Ω Heterotic: ∆2 loop = = ζ(4) F−6 Z det Im(Ω) F2 1 1 1 (1 xy)α′ Typ I : ∆Tree = dx dy dz − = ζ(4) F 6 Z0 Z0 Z0 (1 xyz) 2 − α′ Non–trivial test of heterotic-type I duality ! Supersymmetry and on–shell Ward identitites 6–gluon MHV amplitude at tree–level: In field–theory (by supersymmetry or on–shell Ward identitites): + + + + s1 + + A(g1−,g2−,g3 ,g4 ,g5 ,g6 )= A(Φ1−, Φ2−, Φ3∗−, Φ4∗−,g5 ,g6 ) s4 Parke, Taylor, 1985 To all orders in α′: + + + + s1 + + A(g1−,g2−,g3 ,g4 ,g5 ,g6 , α′)= A(Φ1−, Φ2−, Φ3∗−, Φ4∗−,g5 ,g6 , α′) s4 + + with an intriguing short expression for A(Φ1−, Φ2−, Φ3∗−, Φ4∗−,g5 ,g6 ,α′). T.R. Taylor, St.St., to appear α′–Corrections to MHV amplitudes Where is the effective action: Born–Infeld, etc. ? It is ambigous. Anyhow we don’t need it Scattering amplitudes are the relevant physical objects ! We compute scattering amplitudes directly from their analytic properties without the need for a Lagrangian Tree–level on–shell recursion relations [ cf. type II topological string amplitudes (partition function) Fg or Fg,h ] α′–Corrections to MHV amplitudes = Witten’s duality: ⇒ Perturbative gauge theory Twistor string theory ⇐⇒ (D–instanton calculations) = Generalization of Witten’s duality: ⇒ α –corrected gauge theory α –corrected twistor string theory ′ ⇐⇒ ′ (D–instanton calculations including α′–corrections) Is there any room for α′–corrections in twistor string theory to accomondate superstring corrections to YM scattering amplitudes ? Important developments in perturbative gauge theory & gravity string theory−relation N=4 super Yang−Mills N=8 Supergravity KLT: open <−> closed (UV finite) (UV finite) Type II on T6 with (Bern, Dixon, Kosower, Roiban) D−branes α ’ − states String amplitudes in gluon−amplitudes hidden symmetries in graviton−amplitudes hidden symmetries ( CFT ) α ( ’ and gstring ) ( revealed by twistor space ) ( revealed by twistor space ) Topological Topological Topological twistor string theoryfor SYM twistor string theory ( D−instanton calculation ) String Theory for gravity ? ( MHV−vertices, CSW )