Analytic expansion ofMHV amplitude at strong coupling
Yuji Satoh(University of Tsukuba)

Based on collaboration w/ Y. Hatsuda, K. Ito and K. Sakai,in particular,Y. Hatsuda (DESY), K. Ito (TIT) and Y.S., JHEP 1302(2013)067

Introduction●

MHV amplitude of N=4 SYM at strong coupling⇒ null-polygonal minimal surface in AdS[ ≈ null-polygonal Wilson loop ]●[Alday-Maldacena ’07]

Area of min. surface [ in AdS3, AdS4 ]⇐ TBA eq. of homogeneous sine-Gordon (HSG) model[Alday, Gaiotto, Maldacena, Sever, Vieira ’09,’10
Hatsuda-Ito-Sakai-YS ’10]

Regular-polygonal/UV limit●Regular-polygonal/UV limit⇒ HSG model →SU(n)kU(1)n 1IRcoset CFTUVHSGCFT●●MHV amplitude around UV lim.
⇐ conformal perturbation theory (CPT) of HSG model
CPT : expansion by couplings of relevant op.

Expansion in terms of mass●

Mass-coupling rel. of HSG model ⇒●In special cases, TBA system

expansion by massesof particles of HSG[Ito-san’s talk]⇒ simpler ones for perturbed diagonal coset(W-)minimal model⇒ explicit mass-coupling relation

Expansion in terms of momenta●Solving TBA eq. ⇒ relation btw mass and momenta/Y-fn⇒ expansion by momenta
analytic expansion alongcertain trajectories

CFT/regular polygon-lim.

In the following, we focus on AdS4 casemomentaR1,2

Plan of talk
1. Introduction
2. MHV amplitude at strong coupling and TBA
3. Expansion around regular-polygonal/UV limit
4. Examples
5. Summary

MHV amplitude and TBA

MHV amplitude at strong coupling and TBA●MHV amplitude at strong coupling ⇐ min. surface in AdSM
x4x2(Area)

・ M : scalar part of MHV amplitude
k2k4e2・ λ : ’t Hooft coupling
・ null boundary at AdS boundaryx3x1µxi+1µxi = 2µki

(momentum of particle)

●

Thermodynamic Bethe ansatz (TBA) system [n-pt. fn.]log Y1,s ( ) =log Y2,s ( ) =ms cosh

K12ms cosh

K2

s

1K22

sK1ss

[s=1,...,n-5](1 + Y1,s )2,
s = ln(1+Y2,s 1 )(1+Y2,s+1 )(1 + Y2,s )s = 2 ln
(1+Y1,s 1 )(1+Y1,s+1 )1,K1 ( ) =2 cosh

2 coshcosh 2K2 ( ) =・ θ: spectral parameterµx・ Ya,s : (extended) cross-ratios of a [cusp coord.]11(0)=ue.g., 1 + Y2,2k2
k 1, k 1 ,x2i,j+1 x2i+1,jui,j = 2 2xi,j xi+1,j+1

・ ms : complex (mass) param. ≈ shape of surface ⇔ momenta

●Remainder fn. = (area) - (BDS formula)
Rn = Aperiods + Afree +ABDS

n 5Aperiods =s,s =1Kss ms m̄sK(n=6)

11(n=7)= , K=42 22112n 5dAfree =|ms | cosh log (1 + Ỹ1,s ( ))2 (1 + Ỹ2,s ( )) 22s=1[free energy of 2D model]
3
11(n=6)ABDS =Li2 1
4 i=1ui,i+37(n=7)
ABDS =

1log2 ui,i+34 i=1
1ui+2,i+5 ui+1,i+5log ui,i+3 log+ Li2 (12ui+3,i+6 ui,i+4ui,i+3 )UV Expansion

Homogeneous sine-Gordon model●

integrable deformation ofSHSG = SgWZNW +su(n 4)4[u(1)]n 5coset CFT2d x≈Φ : comb. of weight 0 adjoint op. [multi-param. deformation]n¯= =●4n,=M2(1)simplest case from su(2)k /u(1)⇒ complex sine-Gordon/minimal Ak 1 affine Toda

CPT for free energy●free energy around UV/CFT limit ⇐ CFT perturbation
Afree =6(k) 4k
fn l nbulk
cn + fn +k=2

cn=fnbulk=3(n4)(n5)/n[central charge]

Aperiodfn(k)k-pt. fn. ofl = ML0 ( ms = Ms L = M̃s M L )[ M: mass scale, L: length scale ]Y-, T- and g-functions●●●●ABDSui,jYa,sYa,sTa,sboundary entropy[T-function][g-function]

Ta,s+1 T4 a,s 1
Ya,s =Ta+1,s Ta 1,sg-fn. ≈(0)log G|↵

Z⇤ | ⌅ =|e RH | ⇥ =_⇥
⇤

k=0

⇥2(k)
G| ⌅ (l) e REk_L
R

Relevant boundaries●reflection factor should satisfy bd YB eq.,|a,s;C >Ra,s ( )Ra,s () = 1[unitarity]

Ra,s ( )Rā,s (ssi ) = Saa(2 )[crossing sym.]Rc̄,s ( ) = Ra,s ( + iūbac )Rb,s (●assuming ∃|b,r;CRa,sR rssiūabc )Sab
(2 + iūbac iūabc )[boundary bootstrap]|1 : id. boundary ⇒ set of ref. factors|1Ra,s ,|1|b,r;C= Ra,s /Za,s|b,r;CZa,s=FSabi
F+ C Sabki
Ck(a + bsinh 12(x) :=sinh 12min(a,b) 1FSab( )=j=012j) ,sr+ ik xik x●what boundary is |a, s; C?

⇒ primary w/ Dynkin labelj=()=a(a,s)(a,s) jsIntegral equations for g-/T-function●TBA eqs. ⇒log(Ta,s e) = Kaba,s coshlog(1 + Yb,s )a/ sin, ms = In 5a,s = s sinkkK( ) =●

K2 ( ) K1 ( )2K1 ( ) K2 ( )k· 1 sr r

k = 4 , IN : incidence mat.

,Eqs. for g-fn.
g|log
g|

2 cos

[Dorey-Lishman-Rim-Tateo ’05 ]

1=4|

R

1:=id|( )log R| ( )|( ) log 1 + Y ( )●comparing integral eqs. for T- and g-fn.

⇒(0)
G|a,s;C(0)G|1= Ta,siCk

CPT for g-/T-function●g-fn. ≈ boundary contribution to free energy[Dorey-Runkel-Tateo-Watts ’99;Dorey-Lishman-Rim-Tateo ’05]

⇐ CPT w/ boundary●T-system+Ta,s T4 a,s = T4 a,s+1 Ta,s 1 + Ta+1,s Ta 1,s⇒ periodicity⇒ expansion

Ta,sn+i = Ta,N s ( )k(1t(p,2q)
la,sTa,s ( ) =)(p+2q)p,q=0(p,2q)(p,2q)

ta,N s = ( 1)p ta,skpcoshn

(k)e.g.)t(0,0)
a,s =(2,0)
ta,s=
(0,0)
ta,s

S (a,s) 0(k)S00

B(1 2 , )n G(M̃s ) ·2(2 )1 2(k)S µ(k)

S (a,s) adj(k)S (a,s) 0: modular S-matrix for SU(n⇥ G2 (M̃s )(z) (0) =|z|4●

T-system :(0,4)
ta,s(2,0)ta,sExpansion in terms of coupling(k)
S0 adj(k)
S00(k)
S0 adjnG(k)S004)k●●So far, masses are assumed to be real
to keep boundary integrability etc
Phases are recovered so as to maintain Zn-symmetry[renaming cusp points]●

mass-coupling relation msn⇐ data from simplified TBA system[diagonal coset, W-minimal model]

・ n=6 :

・ n=7 :

6 G =✓ ◆✓ ◆ 4/3
p
1 1/2 13⇡,2⇡64

7 G(M̃1 , M̃2 ) =7 F11 =G(M̃1 ) = 1(x) = (x)/ (1x)

8/7
8/7
4/7
4/7+ M̃2 + B M̃1 M̃2 )7 F11 (M̃1 ✓ ◆ ✓ ◆ 1/2 2243p3⇡
774 2✓ ◆ 8/7324✓ 11 8 ◆ 141  ✓ ◆ ✓ ◆ ✓ ◆ 1/2 ( 78 )42 ⇡61
B = 327714( 34 ) ( 58 )Expansion in terms of masses

872

Examples6-point function●Y-fn. :(0,0) 2(0,0) (2,0) 4/3
12Y2,1 = (T1 ) = (t1 ) + 2t1 t1 l cosh43+ O(l8/3)⇒ agrees w/ our previous results [numerics]●Remainder fn. :(0)(4)
R6 = R6 + R6 · l16/6 + O(l24/6 )3+ Li2 ( 3)
6 4
1(4)
(2)
(2,0) 2R6 =C6(3 + 2 log 2)(t̃1,1 ) ( 6 G)2
616(0)
R6 =Cn(2) = 3(22 16/n 2)

14
n8
n

1 ,(2,0)t̃(2,0):=ta,sa,s / n G●Rescaled remainder fn. :Rn Rn,UVR̄n :=
Rn,UV Rn,IR●

Comparison w/ 2-loop results
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246

l

810

numerics for real m = l+ : 2 loops* : strong coupling
leading analytic expansion: 2 loops: strong coupling7-point function●Remainder fn. :(0)
(4)
16/7
24/7
R7 = R 7 + R 7 · l+ O(l)

74

3(0)
R7 =
7(4)
R7 =(4)
D7 =

6



log

2

⇣

⌘

⇡
2 cos + 1 + Li27(2)(4) (2,0) 2
C7 + D7 (t̃1,1 )7 s0 (3✓◆⇡2 cos 72 cos ⇡7 + 1( 7 G)24s0 ) + (1 s0 )(116s0 (15 + 5s04s0 ) log(324s20 )4s20 ), s0 = sin( /14)

1.80
1.82
1.84
1.86
1.88
1.90
0.5

1.0l

1.5

2.0m1 = e

40 il

m2 = e20 il

Summary●

MHV amplitude of N=4 SYM at strong coupling⇒ null-polygonal minimal surface in AdS⇒ TBA eq. of HSG model●Regular polygonal/UV lim. ⇐ CPT of HSG model[ AdS3, AdS4 ]●General mass-coupling relation ?

●UV limit for AdS5 : not identified yet●Amplitude for ∀ coupling[Basso-Sever-Vieira ’13]