Antenna on Null Polygon ABJM Wilson Loop

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1 Antenna on Null Polygon ABJM Wilson Loop Soo-Jong Rey Seoul National University, Seoul KOREA Institute of Basic Sciences, Daejeon KOREA Scattering Amplitudes at Hong Kong IAS November 21, 2014 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

2 What Are We Studying? 3d ABJM theory null hexagon Wilson loop at planar, 2 loop t Hooft coupling null tetragon [Henn Plefka Wiegandt] antenna (collinear/soft factorization) function of null polygons tetragon / hexagon antenna recurrence relation polygon circular Wilson loop check against exact SUSY localization results [SJR Suyama Yamaguchi, Marino Putrov] Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

3 Why Interesting? ABJ(M) = 3d OSp(6 4) SCFT with 2 couplings k, N k = 1, 2: nonperturbative enhancement to OSp(8 4) curious parallels with 4d SU(4 4) SYM theory details differ: scattering amplitude X wilson loop correlators string theory & M-theory quest: how closely parallel? exact / precision results are available for comparative study SUSY localization techniques conformal bootstrap approach Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

4 Previous Works SU(4 4) SYM: MHV amplitudes null polygon Wilson loop [Drummond, Henn, Korchemsky, Sokatchev] ABJM: 1-loop, 4-pt amplitude =0; via AdS/CFT, relation to 4-pt SYM amplitude [Argawal, Beisert, McLoughlin] ABJM: 1-loop, Wilson loop = 0 [Drukker Plefka Young, SJR Suyama Yamaguchi, Chen Wu] ABJM: 1-loop, Wilson loop = correlator [Bianchi, Leoni, Mauri, Penati, Ratti, Santambrogio] ABJM: 2-loop, tetragon Wilson loop [Henn, Plefka, Wiegandt] ABJM: 2-loop, 4-pt amplitude = tetragon Wilson loop [Chen Huang; Bianchi Leoni Mauri Penati Santambrogio] ABJM: 2-loop, polygon Wilson loop partial results [Wiegandt] ABJM: 3-loop, 4-pt amplitude via DCI integrals + 2 particle cuts [Bianchi Leoni] Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

5 Results Input Data hexagon Wilson loop cf. tetragon Wilson loop [Henn, Plefka, Wienhardt] Recursion via Soft / Collinear Factorization antenna function for n-gon to (n 2)-gon factorization polygon Wilson loop recursion relation via antenna function Polygon Wilson loop W n = 1 2 n (xi,i+2 2 µ2 ) 2ɛ [ ( (2ɛ) 2 + BDS (2) π 2 n (x) + n ) ] 4 log2 (2) π2 6 i=1 n limit = circular Wilson loop = exact localization result Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

6 4d from 3d Matter CS coupled to matter induces nonvanishing 1-loop effect [SJR] L (1) λ2 4 TrF 1 mn F mn D 2 3d counterpart of 4d beta function contribution finite coordinate space propagator η mn D mn = λ2 N π 1+2ɛ Γ(1/2 ɛ) ( x 2 ) 1 2ɛ electric flux energy = scaling dimension of twist-1, high spin operators [Maldacena SJR] Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

7 Tetragon [Henn, Plefka, Wiegandt] Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

8 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

9 W (2) 2 4,mat = ( x 13 4πeγ E µ 2 ) 2ɛ (2ɛ) 2 + (13 24) Log2( x13 2 ) + Rem (2) 4 (u) mat W (2) 4,CS = log(2) 2 4 ( x i,i+2 πe γ E µ 2 ) 2ɛ i=1 2ɛ x Rem (2) 4 (u) CS Rem (2) n (u) := IR finite part modulo the BDS finite part. Rem (2) 4 = π2 mat 4 ; Rem(2) 4 = 5π2 CS 12 2Log2 (2); µ 2 = 8πe γ E µ 2 W (2) 2 4,ABJM = ( x 13 µ2 ) 2ɛ (2ɛ) 2 + (13 24) Log2 ( x 2 13 x 2 24 ) + Rem (2) 4 (u) Rem (2) 4 (u) = Rem 4(u) + Rem 4 (u) + 5 Log 2 (2) = 3 Log 2 (2) + 2π2 mat CS 3 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

10 Strategy Define functional relation W (2) (1) n,abjm := W n,sym + Rem(2) n Remainder function Rem (2) n := Rem (2) n + Rem (2) + 5 CS mat 4 n Log2 (2) divide & conquer: matter part easier + CS part complicated n Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

11 matter 1-loop contribution 1-loop in SYM Wilson loop Remainder from matter: Rem (2) n,mat = 1 16 nπ2 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

12 CS 2 loops 2-loop in SYM Wilson loop (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Remainder from CS: Rem (2) n,cs = I CS + n 2 Log(2) Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

13 Mandelstam & Gram moduli space of null polygon vertices : x x n = 0 edges : y y n = 0 M[C n ] = 3n 15 null edge conditions = nonzero Mandelstam invariants dimm[c n ] = 1 n(n 3) 2 projection to Gram condition subspace dimπ G M[C n ] = dimm[c n ] 1 (n d)(n d 1) 2 = n(d 1) 1 d(d + 1) 2 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

14 Euclid and Signs For Mellin-Barnes evaluation w/o spurious poles, choose all Mandelstam variables of equal sign In 4d, this is always possible. In 3d, this requires to have n = 2Z purely kinematical and computational; nothing to do with ABJM Conformal null polygon dimπ G M c [C n ] = (d 1)n 1 (d + 1)(d + 2) 2 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

15 hexagon wilson loop Mellin-Barnes + PSLQ analytic result: CS contribution Rem 6,CS = := 17 4 ζ(2) + 3 Log(2) + 3 Log2 (2) = adding matter contribution ABJM remainder function reads W (2) 6,ABJM = 3 i=1 Rem 6,mat = π2 (x 2 i,i+2 µ2 ) 2ɛ (2ɛ) 2 + BDS 6 (x) + ( ) 9 2 Log2 (2) + π2 3 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

16 ABJM Antenna Function To obtain remainder function for general polygon, look for recursive relation The soft or collinear limits of null segments relates different n s The 3d kinematics (Euclid for MB) requres to take simultaneous collinear+soft+collinear for n (n 2) Use the recurion relation with n = 4, 6 as inputs to solve for n 8 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

17 Analogous to the splitting function in (S)YM: k i k i + + k i+1 k i+1 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

18 compute antenna function for (collinear)+(soft)+(collinear) factorization: Matter contribution (a) (b) (c) (d) (e) (f) Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

19 CS contribution Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

20 Result for Antenna Function [ ] Ant (2) [C n ] Ant (0) [C n ] CS = Log(2) [ 1 + 2ɛ 2 log(2)log(h 1) Log(2) Log(h 3) Log(2) Log(x 24) ] 2 Log(2) Log(x 35) 2 [ ] Ant (2) [C n ] Ant (0) [C n ] mat 7π Log2 (2) = 1 4ɛ [ Log(h 1 ) + Log(h 3 ) + Log(x 2 4ɛ 24) ] + Log(x35) Log(h 1) Log(x24) Log(h 3) Log(x 2 35) Log(x 2 35) Log(x 2 24) 1 ] 2 Log(h 1) Log(h 3 ) π2 6 Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

21 Recursion Relation Result The recursion relation implied by the antenna function yields Rem (2) n,cs n 2 Log(2) = Rem(2) n 2,CS n 2 2 = Rem (2) n 2,CS n 2 = = Rem (2) 6,CS 3 Log(2) + n = [ 1 2 Log2 (2) 7π2 48 Log(2) + Ant (2) CS [C n] finite Log(2) 7 4 ζ(2) + Log2 (2) ( 7 4 ζ(2) + Log2 (2) ) ] π 2 n + 6. Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

22 Circular Wilson Loop We also considered special Euclid configuration that asymptotes to a circle: t t n 2sin π 2n y y x x Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23

23 The result is compared with the exact result from SUSY localization, whose weak coupling expansion yields W circle = 1 + 5π2 6 λ2 + Soo-Jong Rey (SNUIBS) Antenna on ABJM Wilson Loop November 21, / 23