PRIN meeting - Pisa, 17/5/2013

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A Survey in ABJM: Scattering Amplitudes

and Wilson loops Silvia Penati

University of Milano-Bicocca and INFN

A Survey in ABJM: Scattering Amplitudes

and Wilson loops Silvia Penati

University of Milano-Bicocca and INFN

In collaboration with Marco Bianchi, Gaston Giribet, Matias Leoni, Andrea Mauri, Alberto Santambrogio

The N=6 SCFT (ABJM) The N=6 SCFT (ABJM)

PRIN meeting - Pisa, 17/5/2013

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SU(2) x SU(2) global symmetry A B

The theory is at its superconformal point

Aharony, Bergman, Jafferis, Maldacena 0806.1218Aharony, Bergman, Jafferis 0807.4924

AdS4/CFT3 CORRESPONDENCE AdS4/CFT3 CORRESPONDENCE

PRIN meeting - Pisa, 17/5/2013

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For N << k type IIA on AdS x CP

5

4 3

N >> k

(Bagger, Lambert; Gustavsson; Aharony, Bergman, Jafferis, Maldacena)

Why AdS4/CFT3 correspondence? Why AdS4/CFT3 correspondence? The ultimate goal is to go deeper inside the nature of the AdS/CFT

correspondence by disclosing model independent features.(Not simple replication of the 4d case)

Why scattering amplitudes ?Why scattering amplitudes ? The main focus is on conceivable integrability structures at both sides of the

correspondence. Tracks in the on-shell sector of the theories

Why Wilson loops? Why Wilson loops? Light-like Wilson loops are strictly connected to the amplitudes, via AdS/CFT correspondence. BPS WL computed using localization. PRIN meeting - Pisa, 17/5/2013

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Plan of the talkPlan of the talk

1) Amplitudes in N=6 Chern-Simons-matter models (ABJM/ABJ models): Duality with light-like 4pt polygon WL, Dual superconformal invariance, BDS-like ansatz for 4pt

2) ½ BPS Wilson loops for ABJM and comparison with results from localization

PRIN meeting - Pisa, 17/5/2013

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PRIN meeting - Pisa, 17/5/2013

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Dualities Dualities Symmetries Symmetries

IntegrabilityIntegrability IntegrabilityIntegrability

of the planar on-shell sector of the theory

AdS/CFT perturbative investigation

In N=4 SYM the planar on-shell sector exhibits powerful symmetries and dualities that rely on an integrable structure at both sides of AdS/CFT correspondence

DUALITIESDUALITIES

PRIN meeting - Pisa, 17/5/2013

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CFT contains other non-trivial objects like Wilson loops and correlation functions

- Amplitudes are functions of p , p = 0 - Polygon WL and correlators are functions of x

ii2

Dual variables

ij

What happens in three dimensions ? • String side: Integrability is still an open question, although classically there

are strong evidences. • CFT side: All loop Bethe Ansatz for the dilatation operator. But effects on

the on-shell sector still to be deeply investigated at both sides of the correspondence.

PRIN meeting - Pisa, 17/5/2013

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1) Planar scattering amplitudes 1) Planar scattering amplitudes

• Only matter particles are dynamical. Therefore only scattering of scalars and fermions in the bifundamental repr.

• Gauge invariance, Lorentz and SU(2)xSU(2) invariances force the number of particles to be even and configurations with equal number of bifundamental and anti-bifundamental fields

PRIN meeting - Pisa, 17/5/2013

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N MHVn/2-2

Tree level results Tree level results

• General constraints from superconformal invariance, once solved, allow to determine tree level amplitudes in terms of a restrict number of functions (Bargheer, Loebbert, Meneghelli).

• The dual superconformal symmetry has been proved at tree level (Huang, Lipstein).

• The Yangian symmetry is realized at tree level. Proposed generating functional for tree amplitudes is manifestly Yangian invariant (Lee).

PRIN meeting - Pisa, 17/5/2013

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We move at loop level… Two loopsWe move at loop level… Two loops

PRIN meeting - Pisa, 17/5/2013

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JHEP 1201 (2012) 056, JHEP 1112 (2011) 073

(1) Amplitude/WL duality(1) Amplitude/WL duality

Duality with a light-like 4-polygon Wilson loop

It is zero at one-loop, but….

PRIN meeting - Pisa, 17/5/2013

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(Henn, Plefka, Wiegandt; Bianchi, Leoni, Mauri, SP, Santambrogio; Bianchi, Giribet, Leoni, SP, 1304.6085)

x

x x

x

1

2 3

4

(2) Dual conformal invariance(2) Dual conformal invariance

PRIN meeting - Pisa, 17/5/2013

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Chen-Huang (JHEP 11(2011) 057) have computed the 2 loop correction by assuming dual conformal invariance + generalized unitarity cuts

They find exactly the same result !!

(3) Another duality ??(3) Another duality ??

Identification:

PRIN meeting - Pisa, 17/5/2013

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At all orders in epsilon :

Bianchi, Leoni, SP, arXiv:1112.3649

(4) An all loop 4pt amplitude(4) An all loop 4pt amplitude

N=4 SYM case the BDS ansatz

Analogy between the integrable structures (Gromov, Vieira)

Identification of 4D 4pt at 1 loop and 3D 4pt at 2 loops

PRIN meeting - Pisa, 17/5/2013

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Therefore…..

PRIN meeting - Pisa, 17/5/2013

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Confirmed by our two-loop result

Confirmed at strong coupling

4pt amplitude at four loops needs be computed !!

PRIN meeting - Pisa, 17/5/2013

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2) 1/2 BPS vs 1/6 Wilson loops2) 1/2 BPS vs 1/6 Wilson loops

Superconnection

We consider U(N) x U(M) ABJ(M) theory

(Drukker-Trancanelli 0912.3006)

C = circle

PRIN meeting - Pisa, 17/5/2013

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Prediction from localizationPrediction from localizationPestun 0712.2824; Kapustin-Willett-Yaakov 0909.4559; Drukker-Marino-Putrov 1007.3837

Drukker-Trancanelli

PRIN meeting - Pisa, 17/5/2013

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Change of framing Change of framing

Guadagnini-Martellini-Mintchev NPB330 (1990) 575

Framing gives an overall phase

1loop

PRIN meeting - Pisa, 17/5/2013

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The two-loop calculationThe two-loop calculation (DRED scheme)

Bianchi, Giribet, Leoni, SP, 1303.6939; in preparation

Conclusions and outlookConclusions and outlook

Amplitudes: already remarkable results but still…..many open problems

• 4pt amplitude at four loops to confirm the BDS-like ansatz and amplitude/WL duality. • Are scattering amplitudes with n>4 dual to some Wilson loop? Which one?• Better comprehension of the holomorphic anomaly arising for n>4. • Correlators? • We have generalized the calculation to the case of ABJ theory. At the order we have

worked things do not seem to change much. What about higher orders?• Fermionic T-duality not completely understood at strong coupling

Wilson Loops:

• We have found perfect agreement with the two-loop result from localization• The calculation has required a consistent effort in evaluating loop integrals • We could exploit the techniques developed for evaluating more general BPS Wilson

loops (Cardinali, Griguolo, Martelloni, Seminara, 1209.4032)

PRIN meeting - Pisa, 17/5/2013

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PRIN meeting - Pisa, 17/5/2013

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