Perturbative spectra in gauge theories with gravity duals Christoph - - PowerPoint PPT Presentation

Perturbative spectra in gauge theories with gravity duals

Christoph Sieg

Niels Bohr International Academy Niels Bohr Institute

22.02.10, Nordic String Meeting, Hannover C.S., A. Torrielli: 0505071

• F. Fiamberti, A. Santambrogio, C.S., D. Zanon:

0712.3522 0806.2095 0806.2103 0811.4594

• F. Fiamberti, A. Santambrogio, C.S.:

0908.0234

• J. Minahan, O. Ohlsson Sax, C.S.:

0908.2463 0912.3460

Outline

Introduction and overview Perturbative calculations Conclusions and outlook

AdS/CFT correspondence

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OL ∝ tr φi1 . . . φiL integrable systems (asymptot.) Bethe ansätze

[Beisert, Dippel, Staudacher] [Arutyunov, Frolov, Staudacher]

string gauge

dressing phase

[Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher] [Minahan, Zarembo] [Bena, Polchinski, Roiban] [Kazakov, Marshakov, Minahan, Zarembo]

λ ≫ 1 λ ≪ 1

AdS/CFT correspondence

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OS ∝ tr φ DS φ integrable systems (asymptot.) Bethe ansätze

[Beisert, Dippel, Staudacher] [Arutyunov, Frolov, Staudacher]

string gauge

dressing phase

[Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher]

dilatation operator

[Beisert, Kristjansen, Staudacher]

D OL = γ OL Feynman graph computations in the flavour SU(2) subsector: 1-loop:

[Berenstein, Maldacena, Nastase]

2-loops: [Gross, Mikhailov, Roiban] checks at higher loops:

[Gross, Mikhailov, Roiban] [Beisert, McLoughlin, Roiban] [Fiamberti, Santambrogio, CS, Zanon] [Fiamberti, Santambrogio, CS] [Minahan, Zarembo] [Bena, Polchinski, Roiban] [Kazakov, Marshakov, Minahan, Zarembo]

λ ≫ 1 λ ≪ 1 [D,Qi] = 0

AdS/CFT correspondence

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OS ∝ tr φ DS φ integrable systems (asymptot.) Bethe ansätze

[Beisert, Dippel, Staudacher] [Arutyunov, Frolov, Staudacher]

string gauge

dressing phase

[Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher]

integral eq. for f(λ)

[Eden, Staudacher]

√ λ π − 3 π ln 2 λ 2π2 − ζ(2) λ2 16π4

[Benna, Benvenuti, Klebanov, Sardicchio] [Casteill, Kristjansen] [Alday, Arutyunov, Benna, Eden, Klebanov] [Basso, Korchemsky, Kotanski] [Minahan, Zarembo] [Bena, Polchinski, Roiban] [Kazakov, Marshakov, Minahan, Zarembo]

λ ≫ 1 λ ≪ 1 S ≫ 1 λ ≫ 1 λ ≪ 1

[Gubser, Klebanov, Polyakov] [Frolov, Tseytlin] [Kruczenski] [Kotikov, Lipatov, Velizhanin [Makeenko]

AdS/CFT correspondence

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OS ∝ tr φ DS φ integrable systems (asymptot.) Bethe ansätze

[Beisert, Dippel, Staudacher] [Arutyunov, Frolov, Staudacher]

string gauge

dressing phase

[Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher]

Konishi multiplet tr φiφi tr φ D2 φ tr φ [φ,Z] Z

• L=4

γasy =

• k<L

λkγk

[Minahan, Zarembo] [Bena, Polchinski, Roiban] [Kazakov, Marshakov, Minahan, Zarembo]

λ ≫ 1 λ ≪ 1 S = 2

AdS/CFT correspondence

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OS ∝ tr φ DS φ integrable systems (asymptot.) Bethe ansätze

[Beisert, Dippel, Staudacher] [Arutyunov, Frolov, Staudacher]

string gauge

dressing phase

[Beisert, Hernandez, Lopez] [Beisert, Eden, Staudacher]

Konishi multiplet tr φiφi tr φ D2 φ tr φ [φ,Z] Z

• L=4

γasy =

• k<L

λkγk finite size effects wrapping interactions

[Ambjørn, Janik, Kristjansen] [Schafer-Nameki] [Janik, Lukowski] [C.S.,Torrielli]

γ = γasy + λ4γ4 + . . .

• thermodyn. Bethe ansatz

[Arutyunov, Frolov]

Y-system

[Gromov, Kazakov, Vieira]

(gen.) Lüscher formula

[Bajnok,Janik]

L = 4 loop Feynman graphs

[Fiamberti,Santambrogio,C.S.,Zanon] [Minahan, Zarembo] [Bena, Polchinski, Roiban] [Kazakov, Marshakov, Minahan, Zarembo]

λ ≫ 1 λ ≪ 1 S = 2

AdS4/CFT3 (ABJM) correspondence

[Aharony, Bergman, Jafferis, Maldacena]

Type II A ST AdS4 × CP3 ⇔ 3-dim. N = 6 CS theory

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OJ ∝ tr φn1 . . . φnJ integrable systems (asymptot.) Bethe ansätze

[Gromov, Vieira] [Ahn, Nepomechie]

magnon dispersion relation E =

• Q2 + 4h(λ)2 sin2 p

2 − Q

[Beisert, Dippel, Staudacher] [Beisert] [Arutyunov, Frolov, Zamaklar]

h(λ)2 = ? h(λ)2 =

λ 4π2

unexpectedly simple in AdS5/CFT4

[Minahan, Zarembo] [Bak, Rey] [Arutyunov, Frolov] [Stefanski] [Gromov, Vieira]

λ ≫ 1 λ ≪ 1

AdS4/CFT3 (ABJM) correspondence

[Aharony, Bergman, Jafferis, Maldacena]

Type II A ST AdS4 × CP3 ⇔ 3-dim. N = 6 CS theory

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OJ ∝ tr φn1 . . . φnJ integrable systems (asymptot.) Bethe ansätze

[Gromov, Vieira] [Ahn, Nepomechie]

magnon dispersion relation E =

• Q2 + 4h(λ)2 sin2 p

2 − Q

[Beisert, Dippel, Staudacher] [Beisert] [Arutyunov, Frolov, Zamaklar]

BMN limit giant magnons

[Nishioka, Takayanagi] [Gaiotto, Giombi, Yin] [Grignani, Harmark, Orselli]

quantum corr.

[McLoughlin, Roiban, Tseytlin]

two loops

[Nishioka, Takayanagi] [Minahan, Zarembo] [Bak, Rey]

λ 2 −

• λ

2 ln 2 π

λ2 h(λ)2 = ? h(λ)2 =

λ 4π2

unexpectedly simple in AdS5/CFT4

[Minahan, Zarembo] [Bak, Rey] [Arutyunov, Frolov] [Stefanski] [Gromov, Vieira]

λ ≫ 1 λ ≪ 1

AdS4/CFT3 (ABJM) correspondence

[Aharony, Bergman, Jafferis, Maldacena]

Type II A ST AdS4 × CP3 ⇔ 3-dim. N = 6 CS theory

energy E (semicl.) strings

• anom. dim. γ
• comp. operators

OJ ∝ tr φn1 . . . φnJ integrable systems (asymptot.) Bethe ansätze

[Gromov, Vieira] [Ahn, Nepomechie]

magnon dispersion relation E =

• Q2 + 4h(λ)2 sin2 p

2 − Q

[Beisert, Dippel, Staudacher] [Beisert] [Arutyunov, Frolov, Zamaklar]

BMN limit giant magnons

[Nishioka, Takayanagi] [Gaiotto, Giombi, Yin] [Grignani, Harmark, Orselli]

quantum corr.

[McLoughlin, Roiban, Tseytlin]

two loops

[Nishioka, Takayanagi] [Minahan, Zarembo] [Bak, Rey]

λ 2 −

• λ

2 ln 2 π

λ2 + λ4(−16 + 4ζ(2)) h(λ)2 = ? h(λ)2 =

λ 4π2

unexpectedly simple in AdS5/CFT4 four loops

[Minahan, Ohlsson Sax, C.S.] [Minahan, Zarembo] [Bak, Rey] [Arutyunov, Frolov] [Stefanski] [Gromov, Vieira]

λ ≫ 1 λ ≪ 1

Renormalization of composite operators

composite operator OL = L

• f length L (with L scalar fields)

two-point functions of composite operators: tree level

• OA

L (x), 1 , OB L (y)

• =

x y = δAB (x − y)2∆ , ∆ = L

Renormalization of composite operators

composite operator OL = L

• f length L (with L scalar fields)

two-point functions of composite operators: with loop corrections

• OA

L (x), V2L, OB L (y)

• =

x

V

y = δAB (x − y)2∆ , ∆ = L + γ + . . .

Renormalization of composite operators

composite operator OL = L

• f length L (with L scalar fields)

two-point functions of composite operators: with loop corrections

• OA

L (x), V2L, OB L (y)

• =

x

V

y = δAB (x − y)2∆ , ∆ = L + γ + . . . renormalization of composite operators in a CFT in D = 4 − 2ε dimensions Oa

L,ren = Za bOb L,bare ,

D = µ d dµ ln Z(λµ2ε)

Renormalization of composite operators

composite operator OL = L

• f length L (with L scalar fields)

two-point functions of composite operators: with loop corrections

• OA

L (x), V2L, OB L (y)

• =

x

V

y = δAB (x − y)2∆ , ∆ = L + γ + . . . renormalization of composite operators in a CFT in D = 4 − 2ε dimensions Oa

L,ren = Za bOb L,bare ,

D = µ d dµ ln Z(λµ2ε) anomalous dimensions: eigenvalues of the dilatation operator D =

• k≥1

λkDk D OL = γ OL

Bethe ansatz in the flavour SU(2) subsector

complex fields: φ =

1 √ 2(φ1 + iφ2), ψ = 1 √ 2(φ3 + iφ4), Z = 1 √ 2(φ5 + iφ6)

ψ only as internal flavour in Feynman diagrams

Bethe ansatz in the flavour SU(2) subsector

complex fields: φ =

1 √ 2(φ1 + iφ2), ψ = 1 √ 2(φ3 + iφ4), Z = 1 √ 2(φ5 + iφ6)

ψ only as internal flavour in Feynman diagrams map to integrable spin chain of length L OL = tr(φ . . . φ

M

ZZZ . . . Z

• L−M

) ↔ BPS operator tr(Z . . . Z) ↔ ferromagnetic vaccum impurities φ ↔ spin excitations (magnons) dilatation operator D ↔ Hamiltonian H anomalous dimensions γ ↔ energies E

Bethe ansatz in the flavour SU(2) subsector

complex fields: φ =

1 √ 2(φ1 + iφ2), ψ = 1 √ 2(φ3 + iφ4), Z = 1 √ 2(φ5 + iφ6)

ψ only as internal flavour in Feynman diagrams map to integrable spin chain of length L OL = tr(φ . . . φ

M

ZZZ . . . Z

• L−M

) ↔ BPS operator tr(Z . . . Z) ↔ ferromagnetic vaccum impurities φ ↔ spin excitations (magnons) dilatation operator D ↔ Hamiltonian H = HXXX1/2 + . . . anomalous dimensions γ ↔ energies E

• perator mixing problem solved by the asymptotic Bethe ansatz

M

• j=1

pj = 0 , eipjL =

M

• k=j

ˆ S(uj, uk)e2iθ(uj, uk) , E =

M

• j=1
• 1 + λ

π2 sin2 pj 2 − 1

• momentum

conservation matrix part dressing phase single magnon dispersion relation two-particle S-matrix

One loop

i tr(ψ [Z,φ]) = i

• −
• ,

−i tr( ¯ ψ [¯ φ,¯ Z]) = −i

• b

−

b

• b

=

b

−

b

= =

One loop

i tr(ψ [Z,φ]) = i

• −
• ,

−i tr( ¯ ψ [¯ φ,¯ Z]) = −i

• b

−

b

• b

= −

λ (4π)2ε

    

b

−

b

     = finite = finite D1 = 2

λ (4π)2

• 1

−

L

• i=1

Pi i+1

Chiral functions

b

→ − + = − χ(1) + {} {1}

b b

→ − − + = χ(1, 2) {} − {1} − {1} + {1, 2}

b b b

{a1, . . . , an} =

L

• i=1

Pi+a1 i+a1+1 . . . Pi+an i+an+1 = χ(1, 2, 3) − {} + 3{1} − 2{1, 2} − {1, 3} + {1, 2, 3}

Two loops

◮ all diagrams (apart from reflections, one-loop wave function ren.) ◮ finiteness Fiamberti, Santambrogio, CS, Zanon ◮ generalized finiteness ⇒ absence of χ() to all orders CS, to appear

R = 1 R = 2 R = 3 χ() χ(1) − χ(1, 2) − −

Two loops

◮ all diagrams (apart from reflections, one-loop wave function ren.) ◮ finiteness Fiamberti, Santambrogio, CS, Zanon ◮ generalized finiteness ⇒ absence of χ() to all orders CS, to appear

R = 1 R = 2 R = 3 χ() χ(1) − χ(1, 2) − −

Two loops

◮ all diagrams (apart from reflections, one-loop wave function ren.) ◮ finiteness Fiamberti, Santambrogio, CS, Zanon ◮ generalized finiteness ⇒ absence of χ() to all orders CS, to appear

R = 1 R = 2 R = 3 χ() χ(1) − χ(1, 2) − − D2 = 4χ(1) − 2[χ(1, 2) + χ(2, 1)]

Checks at higher loops

Feynman diagrams with χ(a1, . . . , an) at k loops are simplest if: purely chiral → n = k

• f maximum range → maxa1,...,an − mina1,...,an = k − 1

D4 = + 200χ(1) − 150[χ(1, 2) + χ(2, 1)] + 8(10 + ǫ3a)χ(1, 3) − 4χ(1, 4) + 60[χ(1, 2, 3) + χ(3, 2, 1)] + (8 + 2β + 4ǫ3a − 4iǫ3b + 2iǫ3c − 2iǫ3d)χ(1, 3, 2) + (8 + 2β + 4ǫ3a + 4iǫ3b − 2iǫ3c + 2iǫ3d)χ(2, 1, 3)] − (4 + 4iǫ3b + 2iǫ3c)[χ(1, 2, 4) + χ(1, 4, 3)] − (4 − 4iǫ3b − 2iǫ3c)[χ(1, 3, 4) + χ(2, 1, 4)] − (12 + 2β + 4ǫ3a)χ(2, 1, 3, 2) + (18 + 4ǫ3a)[χ(1, 3, 2, 4) + χ(2, 1, 4, 3)] − (8 + 2ǫ3a + 2iǫ3b)[χ(1, 2, 4, 3) + χ(1, 4, 3, 2)] − (8 + 2ǫ3a − 2iǫ3b)[χ(2, 1, 3, 4) + χ(3, 2, 1, 4)] − 10[χ(1, 2, 3, 4) + χ(4, 3, 2, 1)]

Checks at higher loops

Feynman diagrams with χ(a1, . . . , an) at k loops are simplest if: purely chiral → n = k

• f maximum range → maxa1,...,an − mina1,...,an = k − 1

D4 = + 200χ(1) − 150[χ(1, 2) + χ(2, 1)] + 8(10 + ǫ3a)χ(1, 3) − 4χ(1, 4) + 60[χ(1, 2, 3) + χ(3, 2, 1)] + (8 + 2β + 4ǫ3a − 4iǫ3b + 2iǫ3c − 2iǫ3d)χ(1, 3, 2) + (8 + 2β + 4ǫ3a + 4iǫ3b − 2iǫ3c + 2iǫ3d)χ(2, 1, 3)] − (4 + 4iǫ3b + 2iǫ3c)[χ(1, 2, 4) + χ(1, 4, 3)] − (4 − 4iǫ3b − 2iǫ3c)[χ(1, 3, 4) + χ(2, 1, 4)] − (12 + 2β + 4ǫ3a)χ(2, 1, 3, 2) + (18 + 4ǫ3a)[χ(1, 3, 2, 4) + χ(2, 1, 4, 3)] − (8 + 2ǫ3a + 2iǫ3b)[χ(1, 2, 4, 3) + χ(1, 4, 3, 2)] − (8 + 2ǫ3a − 2iǫ3b)[χ(2, 1, 3, 4) + χ(3, 2, 1, 4)] − 10[χ(1, 2, 3, 4) + χ(4, 3, 2, 1)]

β = 4ζ(3) is leading coeff. of the dressing phase

Beisert, McLoughlin, Roiban

Checks at higher loops

Feynman diagrams with χ(a1, . . . , an) at k loops are simplest if: purely chiral → n = k

• f maximum range → maxa1,...,an − mina1,...,an = k − 1

D4 = + 200χ(1) − 150[χ(1, 2) + χ(2, 1)] + 8(10 + ǫ3a)χ(1, 3) − 4χ(1, 4) + 60[χ(1, 2, 3) + χ(3, 2, 1)] + (8 + 2β + 4ǫ3a − 4iǫ3b + 2iǫ3c − 2iǫ3d)χ(1, 3, 2) + (8 + 2β + 4ǫ3a + 4iǫ3b − 2iǫ3c + 2iǫ3d)χ(2, 1, 3)] − (4 + 4iǫ3b + 2iǫ3c)[χ(1, 2, 4) + χ(1, 4, 3)] − (4 − 4iǫ3b − 2iǫ3c)[χ(1, 3, 4) + χ(2, 1, 4)] − (12 + 2β + 4ǫ3a)χ(2, 1, 3, 2) + (18 + 4ǫ3a)[χ(1, 3, 2, 4) + χ(2, 1, 4, 3)] − (8 + 2ǫ3a + 2iǫ3b)[χ(1, 2, 4, 3) + χ(1, 4, 3, 2)] − (8 + 2ǫ3a − 2iǫ3b)[χ(2, 1, 3, 4) + χ(3, 2, 1, 4)] − 10[χ(1, 2, 3, 4) + χ(4, 3, 2, 1)]

important for leading wrapping correction

Fiamberti, Santambrogio, CS, Zanon

Finite size effects / wrapping interactions

dilatation operator Dk = k + 1

• Dk
• k + 1 at order k, i.e. ∼ λk

composite operator OL = L

• f length L

action of the dilatation operator: k < L: DkOL =

Dk

+

Dk

+ . . . +

Dk

+ . . . k ≥ L: DkOL =

Dk

+

Dk

universal part finite size effects: wrapping interactions

Dk from integrability: only in the asymptotic limit k < L

Calculation of wrapping effects

Fiamberti, Santambrogio, C.S., Zanon: 0712.3522, 0806.2095, 0806.2103, 0811.4594 Fiamberti, Santambrogio, C.S.: 0908.0234

• 1. analyze the properties of the wrapping interactions

C.S., Torrielli: 0505071

• 2. use efficient formalism → N = 1 superfields
• 3. compute all k-loop diagrams? No!

use known asymptotic dilatation operator Dk

◮ correct it for the application to OL with k = L ◮ need appropriate basis for flavour permutations

→ chiral functions

◮ only have to compute subtraction and wrapping

• 4. compute the divergences of the loop integrals analytically

◮ we improved the Gegenbauer polynomial x-space technique

(correct treatment of traceless products in numerators)

◮ we introduced recursion chains for the radial integrations,

→ at k = 11 loops: 225 975 instead of 39 916 800 terms

Our results as tests of AdS/CFT

Fiamberti, Santambrogio, C.S., Zanon: 0712.3522, 0806.2095 ◮ four-loop result of O4 = tr(φ [φ,Z] Z)

γ4 = −2496 + 576ζ(3) − 1440ζ(5)

◮ matches result from the string integrable model [Bajnok, Janik]

now available also at five-loops

[Bajnok, Hegedus, Janik, Lukowski]

Our results as tests of AdS/CFT

Fiamberti, Santambrogio, C.S., Zanon: 0712.3522, 0806.2095 ◮ four-loop result of O4 = tr(φ [φ,Z] Z)

γ4 = −2496 + 576ζ(3) − 1440ζ(5)

◮ matches result from the string integrable model [Bajnok, Janik]

now available also at five-loops

[Bajnok, Hegedus, Janik, Lukowski] Fiamberti, Santambrogio, C.S.: 0908.0234 ◮ five-loop result of O5 = tr(φ [φ,Z] ZZ)

γ5 = 6664 + 1152ζ(3) + 3840ζ(5) − 2240ζ(7)

◮ matches result from the string integrable model [Beccaria, Forini, Lukowski, Zieme]

Our results as tests of AdS/CFT

Fiamberti, Santambrogio, C.S., Zanon: 0712.3522, 0806.2095 ◮ four-loop result of O4 = tr(φ [φ,Z] Z)

γ4 = −2496 + 576ζ(3) − 1440ζ(5)

◮ matches result from the string integrable model [Bajnok, Janik]

now available also at five-loops

[Bajnok, Hegedus, Janik, Lukowski] Fiamberti, Santambrogio, C.S.: 0908.0234 ◮ five-loop result of O5 = tr(φ [φ,Z] ZZ)

γ5 = 6664 + 1152ζ(3) + 3840ζ(5) − 2240ζ(7)

◮ matches result from the string integrable model [Beccaria, Forini, Lukowski, Zieme] Fiamberti, Santambrogio, C.S., Zanon: 0806.2103, 0811.4594 ◮ L ≤ 11 loop results of OL = tr(φZ . . . Z) in β-deformed N = 4 SYM ◮ match results from the string integrable model [Beccaria, De Angelis]

Conclusions and outlook

◮ perturbative computations important:

first computation of the 4-loop anomalous dimension of the Konishi

• perator

◮ N = 1 supergraphs is an efficient tool (finiteness theorems) ◮ refined tests at four and five loops ◮ wrapping increases transcendentality (degree of harmonic sums) ◮ all results confirm the duality and are in accord with the Y-system ◮ still non-trivial cancellations and simplifications

→ more efficient formalism: required for calculations at higher orders and beyond the critical wrapping orders Thank you!