Remainder function of N = 4 SYM
and massless TBA
Yuji SatohUniv. of Tsukuba

based on work with
K. Ito (TIT), Y. Hatsuda (DESY) and J. Suzuki (Shizuoka U)
to appearIntroduction●

Stimulated by duality N=4 SYM ⟷ string on AdS5 x S5
one may hope planar N=4 SYM can be solved

・ spectrum (scaling dim.) at ∀= gY2 M Nc [’t Hooft coupling]・ scattering amplitudes/Wilson loops..
.

In this talk,●●we derive analytic form of (AdS5) 6-pt amplitudes
at strong coupling around certain kinematic pt.
take new approach・ massless (not massive) thermodynamic Bethe ansatz (TBA)・ quantum Wronskian relation

Plan of talk
1. Introduction
2. Amplitudes/Wilson loops in N=4 SYM
3. 6pt remainder fn. at strong coupling
4. Analytic expansion from massless TBA
5. Comparison with perturbative results
6. Summary

Amplitudes/Wilson loops in N=4 SYM●

Amplitudes are equivalent to null polygonal Wilson loops
k4e.g.)

k3x4
k4

k3

x1

x3
k1

k1

●

k2x2

k2

Dual conformal symmetry implies for n-pt fn.
HVBDSlog MM=A+nnkkRn(k)[max. helicity violating amp.]

ABDS: Bern-Dixon-Smirnov formula; all order in λnRn(k) :remainder fn.; fn. of cross-ratios of cusp coord.

●1 loop: BDS formula is exact(2,3,4)● Analytic result of 6-pt remainder fn. at 2,3,4 loops R
6[Del Duca, Duhr, Smirnov ’09, ’10; Goncharov et al ’10, Dixon et al ’13, ’14]

●Strong coupling: area of null polygonal minimal surface[Alday, Maldacena ’07;
Itoyama, Mironov, Morozov ’07; Astefanesei, Dobashi, Ito, Nastase ’07, ... ]⇑ thermodynamic Bethe ansatz (TBA) of 2D integrable model[Alday, Gaiotto, Maldacena, Sever, Vieira, ’09,’10; Ito, Hatsuda, Sakai, YS ’10]●

Finite (any) coupling: OPE method based on integrability[Alday, Gaiotto, Maldacena, Sever, Vieira ’10; Basso, Sever, Vieira ’13]

In the following,
we focus on 6-pt remainder fn. at strong coupling●●

Detailed analysis: yet to be done
More analytic data at strong coupling may be of some use
as well as perturbative ones

6pt remainder fn. at strong coupling
TBA equation●To get 6-pt remainder fn. at strong coupling, first solvelog Ỹ1 ( ) = m cosh + K2log Ỹ2 ( ) =L2 + K12 m cosh + 2K1

L13

L2 + K2

L13

L2 = log(1 + Ỹ2 ) , L13 = log(1 + µỸ1 )(1 + µ
111Ỹ1 )
1K1 = 1/ cosh , K2 = 2 2 cosh / cosh 2

Ỹa ( ) := Ya ( + i )TBA of twisted (μ) Z4 integrable model [m : mass parameter]●3 param. (m , µ , ) give Y-fn. ( Ya )[cf. μ= 0 : AdS4]

●

Y-fn. give basis of indep. cross-ratios
1 + Y2ki = xi+1xi (i = 1, ..., 6 mod 6)

xij = xi

xj
(m , µ , )Remainder fn. at strong coupling is(j = 1, 2, 3)

→→

1
AL =
Li2 (14 j=1

1
Afree =2uj ) ,1

[implicit]

uj[cross-ratios]

strong
R6= AL + Aperiod + Afree3strong
R6

↭

●22xx
(2j + 1)
j+2,j 1 j,j+3
1
i = uj := 24xj,j+2 x2j+3,j 1

1 2
Aperiod = m4d · m cosh

L13 ( ) +2L2 ( )[free energy of Z4 int. model]

Y/T-system●Y-fn. satisfy Y-system
Y1+ Y1 = 1 + Y2

Y2+ Y2 = (1 + µY1 )(1 + µ 1 Y1 )±[±1]
Ya = Ya ,●[k]
Ya = Ya (i+ k)
4

Y-system is rewritten as T-system

Ta+ Ta = 1 + Ta 1 Ta+1Ta = Ya

(a = 1, 2)(a = 1, 2)

T3 = T1 + µ + µ1, T0 = 1

●From Y/T-system, one finds[6]
Ya = Ya (●3i) = Ya ( )+2This periodicity, small m behavior,conformal perturbation theory (CPT) [ for AdS4 case ] imply

4p(p,q) 43 (p+q)
ya mcoshỸa ( ) =p,q=03[similarly for T-fn. ]●m → 0 : UV/regular polygonal limit

UV expansion from massless TBA●AdS3, AdS4 case : remainder fn. are expanded around m=0
by ・ massive TBA of homogeneous sine-Gordon model・ conformal perturbation theory (CPT) for Afree・ boundary CPT for Y/T-fn.
[Ito, Hatsuda, Sakai, YS, ’11; Ito, Hatsuda, YS, ’11, ’12]

UV expansion from massless TBA●

AdS3, AdS4 case : remainder fn. are expanded around m=0
by ・ massive TBA of homogeneous sine-Gordon model・ conformal perturbation theory (CPT) for Afree・ boundary CPT for Y/T-fn.
[Ito, Hatsuda, Sakai, YS, ’11; Ito, Hatsuda, YS, ’11, ’12]●

AdS5 n = 6 ⇐ Z4 integrable model[cf. Afree part, numerics: Ito, Hatsuda, Sakai, YS, ’10]

but, how to incorporate μinto boundary CPT ?
●AdS5 n ⩾ 7: corresponding integrable model ?

UV expansion from massless TBA●

AdS3, AdS4 case : remainder fn. are expanded around m=0
by ・ massive TBA of homogeneous sine-Gordon model・ conformal perturbation theory (CPT) for Afree・ boundary CPT for Y/T-fn.
[Ito, Hatsuda, Sakai, YS, ’11; Ito, Hatsuda, YS, ’11, ’12]●

AdS5 n = 6 ⇐ Z4 integrable model[cf. Afree part, numerics: Ito, Hatsuda, Sakai, YS, ’10]

but, how to incorporate μinto boundary CPT ?
●⇒AdS5 n ⩾ 7: corresponding integrable model ?
We derive UV expansion of Y-fn. and 6-pt remainder by・ “massless” TBA
・ quantum Wronskian relation

CFT/chiral limit and quantum Wronskian●T-system for 6pt case: obtained from XXZ (6-vertex) model2
1( =)= cos =・ anisotropy param.
23・ twist by D = diag(e /2 , e /2 ) , µ =: e3i /2

・ T-fn = transfer matrices
T1 ( ) =Tr0 D R0,2N ( 2i3 + i )R0,2N●[Λ: inhomogeneity]
2i(1 3First take scaling limit : N,2ii ) · · · R0,2 ( 2i+i)R(0,1 33; 4N e32i )= m : fixed[2N: # of sites]●log(m/2) , m

Further take CFT/chiral limit :TaCFT ( 2 ) :=log(m/2) =lim Tam02p[=e23]

p=0・ driving term : lim m coshm●(p,0) 4pta 2 30log(m/2)

0e“massless TBA”

One then has quantum Wronskian relations[Bazhanov,�Lukyanov, Zamolodchikov; ’94, ’96]

2i sin=e2· TaCFT ei(a 1)

i2 (a+1)

A+ ( q(a+1)22)A ( q(a+1)2)

eA± ∝ limits of Q-fn. Q± ( ) = e

λj: Bethe roots , q = eii
2 (a+1)±3A+ ( q2 sinhj(a+1)2)A ( qj (±)(a+1)2)

●

A± are expanded aslog A± =●a±n()

2(2n,))an ( ) = a+n()n=1

Substituting expansions into quant. Wronskian, e.g. , for a = 2(0,0)
t2= 2 cos

2

●

(083

1(2,0)t2= (2 cos
2Relation amongsin(n+ 1,2+)an±an(1,0)t2= (2 cos2+ 1) 2a+a+(a)1 11..
.

+ 1) a1 + a+122a2 + (a+)
12a+2CFT[T= 1]are given by Wronskian for a=00sin(n +R1 ( ) = 0 ,2432)an = Rn ( )sin 2+22(a+)+(a)+2cos2aR2 ( ) =1
11 a12..
.

●Assuming

・ analyticity: a+n (an ) analytic for φ> -γ	 (φ< γ)[consistent w/ BAE]・ asymptotics for φ→ ∞ :
a+n

1n24n
3n =,

( n3 ) ( 2n341212)n!

12( 14 )( 34 )

4n3[deduces from BAE/NLIE, data for AdS4 case]±a⇒ determine n⇒e.g. )13 + 223 + 2a+1( )=1CFT
CFT(T,Y)t(p,0)aaa

2 (2 +31+32 )
a2 ( ) =8 3 ( 31 + 2 )dx sinh x2
2 x+i

1ix
1( +) (3 23ix 3)2

UV expansion of 6pt remainder functionstrong● Substituting expansion of Y/T-fn. into R
6

Afree by (bulk) CPT[Ito, Hatsuda, Sakai, YS, ’10]

⇒ expansion around m = 0(k)r6 (strong
R6=, )m

[UV/regular polygonal limit]

43k[ µ =: e3i /2 ]k=0(0)
r6 =(2)
r6 =6343 2432(2 )231n =
223+ Li2 1
41u0

8 3(191n+2u0 1 = 4 cos2 ( /2) ,1,(1)r6 = 0u0 ) + log u0 · B
1

3n+2

12(2/n)(1/n)(z) = (z)/ (122nn+2z)

1
1+,
3 2 32・

dependence : allowed only for k = 3n・ for k=2,(1,0)
ta[ Z6-symm. ]is enough・ for k ⩾ 3, need t(p,q=0)a(massive corrections)

Comparison w/ perturbative results●

6pt remainder fn. : evaluated analytically at L=2,3,4 loops
(L)
(L 1)
R6 /R6: relatively constant●for large range of param.

●

useful to introduce rescaled remainder fn.
R6 R6 (m = 0)R̄6 =R6 (m = 0) R6 (m =+â+****
âââ+++*â+
â*+*â+*â+
â*+*â+
â24
6810*+
â
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â*+
â
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â
-0.2*+
ââ*+â*+â
*+-0.4â*+â* +â* +â
* +â
-0.6
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* ++
* +-0.8
* +
* *+
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* ++
* * ++* * +++* * * ++++****
-1.0(2)R̄6 (+) ,(3)R̄6 (),

R̄6strong (),R̄6strong;Exp ()mâ
+*-0.01-0.02)
â+
*+
*â
0.2

â
+
*0.4
â
+
*0.6
â
+
*

â
+
*0.8

â
+
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â+
*-0.03

â+*-0.04[ µ = e1/5 ,=/20 ]Summary●“Massless” TBA, quantum Wronskian give
UV expansion of chiral part of Y/T-fn.

⇒ UV expansion of 6-pt remainder fn.
●New formalism to computestrong-coupling remainder fn. around UV (regular-polygonal) lim.

・ allowed us to deal w/ AdS5 (μ≠ 0) case・ part of higher order terms ( t(p,0)) : derived systematicallya●

As in other known cases,(rescaled) remainder fn. are similar to each other