bremsstrahlung function, leading lüscher correction at

Bremsstrahlung function, leading Luschercorrection at weak coupling and localization

Michelangelo Preti

Universita degli studi di ParmaDipartimento di Fisica e Scienze della Terra

INFN Gruppo Collegato di Parma

Based on arXiv:1511.05016M.Bonini, L.Griguolo,M.P. and D.Seminara

Napoli, November 19th 2015

Introduction The Bremsstrahlung function An alternative route Leading Luscher correction Conclusions

Outline

Introduction

The Bremsstrahlung function

An alternative route

Leading Luscher correction

Conclusions

Michelangelo Preti Napoli, November 19th 2015

2/29


From N=4 SYM in d = 4 to pure YM in d = 2 (I)

If we restrict the DGRT contour on a two sphere the situation ismore interesting

W =1

NtrP exp

∮dτ(iAµx

µ + εµνρxµxνΦρ)

Indeed at leading order the expectation value ofthese WL on S2 is

〈W 〉 = 1 + g24dN

A1A2

2A2+O(g4

4d)

The VEV resembles a similar result in the 0 instant sector of 2dpure Yang-Mills on S2.


3/29


From N=4 SYM in d = 4 to pure YM in d = 2 (II)The pure YM theory in 2d is almost topological (invariant underarea preserving diffeomorphism) and exactly soluble. An exactcompact expression for the VEV of Wilson Loop on S2 is known

〈W 〉 =1

NL1N−1

(g2

2d

A1A2

2A

)exp

[− g2

2d

A1A2

4A

][A.Bassetto and L.Griguolo ’98]

DGRT conjecture (g22d = −2g2

4d/A)

VEV of Wilson loopon S2 in N = 4 SYM

in 4d

⇐⇒ VEV of Wilson loop on S2

in pure YM in 2d

〈W〉 =1

Z

∫DX

1

NTr(e iX)

exp

(− A

g22dA1A2

Tr(X 2)

)[N.Drukker, S.Giombi, R.Ricci, D.Trancanelli ’07]


4/29


Localization

Obtained by deforming the action of the theory by Q-exact term

SYM → S(t) = SYM + tQV

with V = (Ψ,QΨ) and setting t to infinity.

I Q squares to a symmetry of the theory, and the action andthe Wilson loop observable must be Q-closed;

I At the limit t →∞ we shall integrate in the path integralover the configurations solving QΨ = 0;

I The partition function and the expectation value ofobservables do not depend on the t-deformation (compactspace of fields);

I Localize usually means computing a classical action(fluctuation around classical solution) → Matrix model.


5/29


The matrix model

N = 4 SYM on S2 localization−−−−−−→ Zero instanton sector of pure 2d YMon S2 → Gaussian multi-matrix model [S.Giombi and V. Pestun ’09]

〈WR1 [C1]WR2 [C2]...OJ1(x1)OJ2(x2)...〉4d

=1

Z

∫[dX ][dY ]TrR1e

X1TrR2eX2 ...TrY J1

1 TrY J22 ...eS[X ,Y ]

I S [X ,Y ] is a quadratic form in Xi , Yi ;

I Coefficients depend on the areas singled out by the WLs;

I Checked in several configurations [M.Bonini, L.Griguolo, M.P. ’14]

I WL with operator insertions localize in the same way?

I What is the matrix model?


6/29


Outline

Introduction




Conclusions


7/29


Cusp anomalous dimension ΓCUSP

W ∼ e− log( Lε )ΓCUSP(λ,ϕ)

with L and ε the IR and UV cut-off

Supersymmetric configuration: ϕ = 0

I Universal cusp anomaly ΓCUSP(λ, ϕ)ϕ→iϕ−−−→ϕ→∞

ϕΓ∞CUSP(λ)

I QQ-potential ΓCUSP(λ, ϕ)δ→0−−−→ V (λ)

δ

I Bremsstrahlung function ΓCUSP(λ, ϕ)ϕ→0−−−→ −ϕ2B(λ)


8/29


Deforming the observable

R-symmetry deformation ~n · ~n ′ = cos θ

W ∼ e− log( Lε )ΓCUSP(λ,θ,ϕ)

with ΓCUSP(λ, θ, ϕ) the generalized cusp anomalous dimension[N.Drukker, V.Forini ’11][D.Correa, J.Henn, J.Maldacena, A.Sever ’12]

Supersymmetric configuration: ϕ = ±θ

ΓCUSP(λ, θ, ϕ)ϕ→±θ−−−−→ −(ϕ2−θ2)H(λ, ϕ) +O(ϕ− θ)2


9/29


The generalized Bremsstrahlung function

The cusp on the Euclidean plane (for θ = ±ϕ) is conformally

equivalent to the 1/4 BPS wedge on S2 (λ = λ(

1− ϕ2

π2

))

〈Wwedge〉 =2√λI1(√λ)

The function H can be computed as logarithmic derivative

H(λ, ϕ) = −1

2∂ϕ log 〈Wwedge(ϕ)〉 = −1

2

∂ϕ〈Wwedge(ϕ)〉〈Wwedge(ϕ)〉

H(λ, ϕ) =2ϕ

1− ϕ2

π2

B(λ) ⇒ B(λ) =

√λ

4π2

I2(√λ)

I1(√λ)

[D.Correa, J.Henn, J.Maldacena, A.Sever ’12]


10/29


The Integrability way (I)

I Complex scalars Z not mix with the scalars coupled to lines⇒ the configuration θ = ±ϕ is still supersymmetric

I Local operator insertion on the cusp ⇒ Spin chain descriptionI The associated TBA describes the propagation of magnons

between two boundaries associated to the Wilson linesseparated by the ”mirror” time L

I One of the boundaries is rotated relative to the other by afunction of the angles θ and ϕ

[N.Drukker ’12] [D.Correa, J.Maldacena, A.Sever ’12]


11/29


The Integrability way (II)

The near BPS limit drastically simplify the TBA[N.Gromov, F.Levkovich-Maslyuk, G.Sizov ’13]

ΓL(λ, θ, ϕ) =ϕ− θ

4∂ϕ log

detM2L+1

detM2L−1

MN =

Iϕ1 Iϕ0 · · · Iϕ2−N Iϕ1−N

Iϕ2 Iϕ1 · · · Iϕ3−N Iϕ2−N...

.... . .

......

IϕN IϕN−1 · · · Iϕ1 Iϕ0IϕN+1 IϕN · · · Iϕ2 Iϕ1

Iϕn =1

2In(√

λ)[(√π + ϕ

π − ϕ

)n

−(−√π − ϕπ + ϕ

)n]


12/29


Wrapping corrections (Luscher term)Exchange of virtual magnons along the ”mirror” channel from oneboundary to the other

[N.Drukker ’12] [D.Correa, J.Maldacena, A.Sever ’12]

[N.Gromov, F.Levkovich-Maslyuk, G.Sizov ’13]

ΓL(λ, θ, ϕ) ' (ϕ− θ)(−1)LλL+1

4π(2L + 1)!B2L+1

(π − ϕ

2π

)I Leading correction to the energy of the ground state in the

spin-chain which happens at order L + 1I It corresponds to the leading order at weak coupling of ΓL

⇓Can we identify a class of Feynman diagrams

which produces the same expression?


13/29


Recap


14/29


Recap


15/29


Outline

Introduction




Conclusions


16/29


BPS wedge on S2 with scalar insertions in N = 4 SYM

Al =xµl Aµ − iΦ2

Ar =xµr Aµ − i sin δΦ1 + i cos δΦ2

OL(xN) =(Φ3 + iΦ4

)L ≡ ZL

OL(xS) =(−Φ3 + iΦ4

)L ≡ (−1)LZL

WL(δ) = Tr

[ZL Pexp

(∫Cl

Aldτ

)ZL Pexp

(∫Cr

Ardσ

)]

HL(λ, ϕ) =2ϕ

1− ϕ2

π2

BL(λ, ϕ) =1

2∂δ log 〈WL(δ)〉

∣∣∣∣δ=π−ϕ


17/29


The wedge on S2 with field strength insertions in YM2

light cone gauge Az = 0

OL(x) 7→(i ∗F (z)

)L

W(2d)L (δ)=Tr

[(i ∗F (0)

)LPexp

(∫C2

w Awds

)(i ∗F (∞)

)LPexp

(∫C1

z Azdt

)]

HL(λ, ϕ) =2ϕ

1− ϕ2

π2

BL(λ, ϕ) =1

2∂δ log 〈W(2d)

L (δ)〉∣∣∣∣ δ=π−ϕg2

2d=−2g2/A


18/29


L = 1 test in N = 4 SYM

z

x2

x1

x

x

S

N

w

z x2

x1

x

x

S

N

w

z x2x1

x

x

S

N

z x2x1

x

x

S

N

Π4

Π2

3 Π4 Π

Φ

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

H11-loop


19/29


L = 1 test in YM2

H(1)1 (λ, ϕ) = − λ2

24πB3

(π − ϕ

2π

)


20/29


Outline

Introduction




Conclusions


21/29


The leading Luscher term from YM2 P.T. (I)

sss

s ss

1m+q

m+q−1

m+1 m

2ssq

t1

2

n t

t

ttt

tn+p

n+p−1

n+1

m

pn

. . .

. . . . . .

. . .

I (s,t)

(−1)q

(−1)p

M(1)L =

(−1)L+123L πL

g 2L2d NL

L∑n,m=0

∫ ∞0

dsdt (−1)n+mgn(t, ε)gL−n(s, ε)Iδ(s, t)gm(1/s, ε)gL−m(1/t, ε)

Iδ(s, t)=

(−g2

2d

2π

)1

(t2 + 1)(s2 + 1)

te iδ−st−e iδs

, gn(t, ε)=

(−g2

2d N

4π

)n∫ t

εdtn

∫ tn

εdtn−1 ...

∫ t2

εdt1

n∏i=1

∆(ti )


22/29


The leading Luscher term from YM2 P.T. (II)Recurrence relation for gn and its derivative from its definition:

gn(t, ε) =

(−g2

2d N

4π

)∫ t

εdtn ∆(tn) gn−1(tn, ε)

d

dtgn(t, ε) =

(−g2

2d N

4π

)∆(t) gn−1(t, ε)

We can combine these two equations into a single recurrence

gn(t, ε) = −n∑

k=1

(−α)k

k!logk

(t2

t2 + 1

)gn−k(t, ε)− (t → ε)

where α = −g22d N8π . The generating function is

gn(t, ε) =1

2πi

∮γ

dz

zn+1G (t, ε, z) ⇒ G (t, ε, z) =

(t2

t2 + 1

ε2 + 1

ε2

)αzfor a suitable closed curve γ around the origin.


23/29


The leading Luscher term from YM2 P.T. (III)

M(1)L =

(−1)L+1

(2πi)4

23L πL

g 2L2d NL

L∑n,m=0

∫ ∞0dsdt

∮G(t, ε, z)

zn+1

G−1(s, ε,w)

wL−n+1Iδ(s, t)

G(1/s, ε, v)

vm+1

G−1(1/t, ε, u)

uL−m+1

H(1)L =

1

2∂δM(1)

L =(−1)Lα2L

L!(L− 1)!

23L πL

g 2L2d NL

(−g 2

2d

2π

)β

(1

2, L

){sin δ

∫ 1

0

dρlog2L ρ

ρ2 − 2ρ cos δ + 1

}δ=π−ϕ

g22d

=− g2

2π

⇓

H(1)L = − (−1)LλL+1

4π(2L + 1)!B2L+1

(π − ϕ

2π

)Michelangelo Preti Napoli, November 19th 2015

24/29


Outline

Introduction




Conclusions


25/29


Conclusions and outlook

Conclusions

I We identify an operator and a class of Feynman diagrams onS2 in N = 4 SYM that reproduce the integrability results;

I Using the supersymmetric localization we move the system inYM2 where perturbation theory is under control;

I We reproduce the leading Luscher correction at weak couplingto the generalized cusp anomalous dimension usingperturbation theory in d=2;

Outlook

I Determine the complete matrix model for Wilson loop withlocal operator insertions and the related Bremsstrahlungfunction for any coupling.


26/29

bremsstrahlung function, leading lüscher correction at

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