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Holography and the dynamical breaking of Supersymmetry

Riccardo Argurio

Université Libre de Bruxelles

GGI Firenze, 8 April 2015

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 1 / 28

Outline & Credits

Outline

I Motivation I New domain wall SUGRA solutions I Holographic correlators

• massless modes
• massive resonances

I Applications to SUSY breaking hidden sectors I Outlook

Based on:

Collaboration with Bertolini, Di Pietro, Musso, Porri and Redigolo

I arXiv:1411.2658 [hep-th] AMR I arXiv:1412.6499 [hep-th] ABMPR I see also arXiv:1205.4709 [hep-th], arXiv:1208.3615 [hep-th],

arXiv:1310.6897 [hep-th] ABDPR

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 2 / 28

Motivation

Motivation

Symmetry breaking by strongly coupled dynamics is often crucial in physical theories Some examples:

I Chiral symmetry in QCD I Dynamical SUSY breaking $ hierarchy problem I High Tc superconductors I Higher spin symmetry $ string theory

In strongly coupled theories, both the fields responsible for symmetry breaking and the (pseudo)-Goldstone particles that result from it are typically composite, i.e. not elementary. Here we will focus on SUSY breaking at strong coupling.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 3 / 28

Motivation - Holography

Our approach to strong coupling is through holography. In a nutshell:

I Strongly coupled, large N 4d field theories from 5d near-AdS

gravitational theories.

I Vacua of the gauge theories correspond to specific solutions

(backgrounds) in the gravity theory.

I Gauge invariant (composite) operators correspond to classical

fields in the bulk.

I Quantum correlators are computed from fluctuations over the

background and a (classical) renormalization procedure.

I RG-flow from UV to IR in the quantum field theory is mapped to

radial evolution from the boundary to the deep bulk in the gravity theory. Motivated by the celebrated example of the correspondence between N = 4 SYM and IIB string theory on AdS5 ⇥ S5.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 4 / 28

Motivation - Holography

Models of SUSY breaking in holography: Roughly, in order to recover situations where SUSY breaking is spontaneous (or soft), we need non-SUSY solutions that asymptote to SUSY ones near the boundary/UV. Some examples:

I non SUSY versions of Klebanov-Strassler: anti-D3 branes (KPV),

Kuperstein et al., . . .

I d 6= 4 examples: Maldacena-Nastase, Massai et al.,. . . I T 6= 0: many! (but not our focus here–T = 0 from now on) I bottom-up: N = 2 SUGRA in 5d

The complexity of the bottom-up model (and its reach towards top-down models) depends on the hyper- and vector-multiplet content. ) it turns out N = 2 SUGRA coupled to a universal hypermultiplet is already rich enough!

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 5 / 28

Motivation - Holography

Given that we define the SUSY breaking strongly coupled theory through its holographic dual, how are we going to probe/characterize its SUSY breaking features? Our aim will be to compute two-point correlators through holographic renormalization This will give us information on several physical properties:

I Presence/absence of massless modes such as:

• dilaton
• R-axion
• Goldstino
• ’t Hooft fermions

I Violation of SUSY Ward identities I Spectrum of resonances I Stability: no tachyonic resonances I Can be directly useful for applications

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 6 / 28

Solutions

New RG-flow SUGRA solutions

We restrict to solutions with at most two backreacting scalars. We start form the action (choice of gauging!) S = Z d5x p G  1 2R + @M⌘@M⌘ + 1 4 cosh2 ⌘@M@M +3 4

• cosh2 2⌘ 4 cosh 2⌘ 5
• together with the 4d Poincaré invariant ansatz

ds2 = 1 z2

• dz2 + F(z)⌘µνdxµdxν

⌘ = ⌘(z) = (z)

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 7 / 28

Solutions

The system of equations we have to solve boils down to 12 ✓ 1 zF0 2F ◆2 + 3 2

• cosh2 2⌘ 4 cosh 2⌘ 5
• = 2z2⌘02 + 1

2z2 cosh2 ⌘02 z5 F2 @z ✓F2 z3 ⌘0 ◆ = 1 8z2 sinh 2⌘02 + 3 2 sinh 2⌘(cosh 2⌘ 2) @z ✓F2 z3 cosh2 ⌘0 ◆ = 0 [Note: we do not use any superpotential (true or fake)] 3-parameter worth of domain wall solutions, all are AdS near z = 0: = ˜ 4z4 + . . . ⌘ = ⌘0z + ˜ ⌘2z3 + . . . Most solutions are singular in the IR.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 8 / 28

Solutions

For some specific values of the parameters, we recover known solutions.

I For ˜

4 = 0 and ⌘0 = 0 we have the SUSY solution of GPPZ: F(z) =

• 1 z6˜

⌘2

2

1

3 ,

⌘(z) = 1 2 ln ✓1 + ˜ ⌘2z3 1 ˜ ⌘2z3 ◆ Our generic (numerical) solution is a non-SUSY generalization of it:

0.5 1 z 0.5 1 F@zD 0.5 1 z 2 4 Η@zD 0.5 1 z 2 4 Φ'@zD

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Solutions

I For ⌘ = 0, we have the dilaton domain wall of Gubser,

Kehagias-Sfetsos, Constable-Myers: F(z) = 1 ˜ 2

4z8

6 !1/2 , (z) = p 6 arctanh ˜ 4z4 p 6 ! Again, we have a two-dimensional subspace of solutions generalizing it, with a non-trivial but non-singular profile for ⌘:

0.5 1 z 0.5 1 F@zD 0.5 1 z 0.25 0.5 Η@zD 0.5 1 z 7 14 Φ'@zD

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 10 / 28

Solutions

I For = 0 and ˜

⌘2 a specific function of ⌘0, we also recover a family

• f numerical non-singular solutions, known as the Distler-Zamora

solutions. They are domain walls where ⌘ interpolates between the maximum and the minimum of the potential. We generalize this family by “almost getting there": walking solutions

5 15 25 z 0.5 1 F@zD 5 15 25 z 2 4 Η@zD

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 11 / 28

Holographic correlators

Holographic correlators

2-point correlators of gauge invariant operators hO(k)O(k)i contain lots of information on the theory and on the vacuum in which it finds itself.

I Euclidean momentum k2 > 0: UV and IR asymptotics I Minkowskian momentum k2 < 0: spectrum of theory (resonances) I When SUSY is unbroken, different 2-point functions are related. I When SUSY is broken the 2-point functions will differ at low

momenta/large distances. We will employ holographic renormalization to compute two-point func- tions in the gauge theories dual to our backgrounds.

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Holographic correlators

The prescription relating quantum correlators to fluctuations of bulk fields is the following (in the large N, strong coupling limit): ZQFT(0) ⌘ he

R φ0Oi = eSSUGRA(φ|z=0=φ0)

It follows that two-point correlators are obtained taking the second

• rder variation of SSUGRA with respect to the leading mode 0.

The on-shell action SSUGRA reduces to a 4d boundary integral, which however diverges due to the infinite volume of AdS. The procedure of holographic renormalization goes through regularization (introducing a z = ✏ surface), covariant counter-terms, and eventually leads to a renormalized action Sren.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 13 / 28

Holographic correlators

More specifically, if the fluctuation of the bulk field dual to the dimension ∆ operator O has a near-boundary expansion like = 0z4∆ + · · · + ˜ 2∆4z∆ + . . . then the renormalized action will be Sren / Z d4x(0 ˜ 2∆4 + local terms) Solving for the linear fluctuations of in the non-trivial background, including boundary conditions in the bulk, ˜ 2∆4 is typically a non-local but linear function of 0. ) Eventually, the two point function is given by hO(k)O(k)i = 2Sren 2 = ˜ 2∆4 + local terms Note that sometimes the local terms are crucial!

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 14 / 28

Massless modes

Massless modes

As a warm up, we consider a toy model for the holographic realization

• f Goldstone bosons: 5d model with a vector and an axion-like scalar

[See also Bianchi, Freedman, Skenderis 01]

S = Z d5x p G 1 4FMNFMN + 1 2m(z)2(@M↵ AM)(@M↵ AM)

• The bulk vector’s U(1) symmetry is broken by the profile

m(z) = m0z + ˜ m2z3 (∆ = 3 for simplicity) From the boundary FT point of view:

I m0 explicit breaking of global U(1) I ˜

m2 spontaneous breaking of global U(1)

[From the SUGRA point of view it is always spontaneous breaking.]

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 15 / 28

Massless modes

The correlator we will be probing is hJµ(k)Jν(k)i = (k2µν kµkν)C(k2) m2 kµkν k2 F(k2) In holographic renormalization, the transverse and longitudinal parts

• f Aµ are dual to the transverse and longitudinal parts of Jµ,

respectively. The longitudinal part of Aµ however mixes with ↵, which is dual to ImOm, i.e. the imaginary part of the operator that breaks the symmetry. Expectations:

I spontaneous breaking: F(k2) = 0, massless pole in C(k2),

Schwinger term in hJµ ImOmi

I explicit breaking: no massless poles, F(k2) 6= 0, operator identity

@µJµ = m0 ImOm

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 16 / 28

Massless modes

Holographic renormalization: Spontaneous: Sren = R d4k n at

0µ˜

at

2µ + ˜

m2

2↵0(˜

↵2 + al

0) + local

• I at

0µ˜

at

2µ term leads to C(k2). Massless pole appears because of

bulk boundary conditions.

I ˜

m2

2↵0al 0 term leads to the Schwinger term h@µJµ ImOmi = ˜

m2, which leads to the massless pole hJµ(k)Im Om(k)i = ˜ m2

kµ k2 .

Explicit: Sren = R d4k n at

0µ˜

at

2µ + m2 0(↵0 al 0)˜

↵2 + local

• I massless pole in C(k2) can be cancelled by local terms / m2

(cancels anyhow from full hJµJνi)

I Operator identity consequence of Sren depending only on ↵0 al

(bulk gauge symmetry): Sren al0 = Sren ↵0 , h@µJµi = m0hIm Omi

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Massless modes

In full N = 2 SUGRA models, these considerations apply to the R-axion and the dilaton. The correlators are hTµν(k) Tρσ(k)i = 1 8Xµνρσ C2(k2) 1 12 m2 k2 PµνPρσ F2(k2) hjR

µ(k) jR ν(k)i = Pµν C1R(k2) 1

3 m2 kµkν k2 F1(k2)

I F2 = 0 = F1 if conformal and R-symmetry are not explicitly broken. I SUSY Ward identities require C2 = C1R and F2 = F1 for a SUSY

preserving RG-flow.

I Tµν dual to hMN, the graviton, and jR µ dual to RM, the graviphoton,

• f the N = 2 gravity multiplet.

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Massless modes

Both C2 and C1R have a massless pole when ⌘0 = 0. SUSY is recovered when also ˜ 4 = 0. In this case we expect C2 = C1R.

1 2 3 4 k 2 4 6 8 10 12 14 C2- C1 R 3

0.0 0.5 1.0 1.5 2.0 2.5 3.0f é

4

0.0 0.1 0.2 0.3 0.4 Res@C2H0LD-Res@C1 RH0LD

The pole in C2 corresponds to the dilaton of broken conformal symmetry, and the pole in C1R to the R-axion of broken R-symmetry. In the SUSY limit, the R-axion and dilaton become superpartners.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 19 / 28

Massless modes

The massless poles disappear when ⌘0 is turned on:

10 20 30 40 k 0.0 0.5 1.0 1.5 2.0 2.5 3.0 C1 RHkL h0=0 h0=1 h0=3 h0=5

The R-axion acquires a mass proportional to ⌘0.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 20 / 28

Goldstino

Goldstino

When SUSY is spontaneously broken, a massless fermion, the Goldstino is expected. Problem: spontaneous SUSY breaking is allowed only when conformal symmetry is explicitly broken. This is because spontaneous SUSY breaking is heralded by hTµνi = F2⌘µν This relation is impossible in a conformal theory in which we have the

• perator constraint Tµ

µ = 0.

) We need a solution where conformal symmetry is broken by a SUSY source.

[This requires a different gauged SUGRA–no details here]

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 21 / 28

Goldstino

Q: Where should the Goldstino massless pole appear? A: in the correlator of the supercurrent! More precisely, in a Schwinger term in this correlator. h(@µSµα)(k)¯ Sν ˙

β(k)i = hα¯

Sν ˙

βi = 2iν α ˙ β h Tµνi 6= 0

implies hSµα(k) ¯ Sν ˙

β(k)i = · · · + F2(µ¯

ρν)α ˙

β

2kρ k2 ) boundary analysis in holographic renormalization is enough to find the Goldstino.

[See Matteo’s talk next week!]

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 22 / 28

Spectrum of resonances

Spectrum of resonances

The poles appearing in the correlators give the spectrum in the corresponding channel. If all the poles lie on the k2 < 0 axis, then no tachyonic resonance is present and the solution represents an RG-flow to a stable vacuum.

5 10 15 20 k

• 2

2 4 6 C1 RHikL

In the generic case the first pole corresponds to a mass of the order of the mass gap.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 23 / 28

Spectrum of resonances

We have checked that there are no tachyonic poles in any channel for a sample of our class of solutions. This does not mean that the solutions are stable also when enlarging the SUGRA model (for instance in string theory embeddings). However it is a check in principle of the stability of the solution. Note that tachyons in the QFT spectrum of resonances are not the same thing as scalars below the BF bound in the bulk!

I We look for tachyons that would occur in a channel associated to

an operator with real dimension.

I Scalars below the BF bound would lead to complex dimensions

! more drastic instability.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 24 / 28

SUSY breaking hidden sectors

Application to SUSY breaking hidden sectors

Q: How are the soft terms in the SSM generated? SUSY breaking must be communicated to the SSM through higher dimension operators. This leads to the mediation paradigm:

SSM DSB

I hidden sector with (dynamical) SUSY breaking I visible SSM sector I mediating fields in between: one option is SM gauge fields (and

superpartners) ) Gauge Mediation of SUSY breaking

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SUSY breaking hidden sectors

The hidden sector has to have a global symmetry, which is then gauged by the SM gauge fields. All data of interest from the hidden sector is contained in the 2-pt correlators of operators in the supermultiplet to which the conserved currents belongs to. This is the formalism of General Gauge Mediation.

[Meade, Seiberg, Shih 08]

An early top-down attempt was Holographic Gauge Mediation.

[Benini et al, McGuirk et al 09]

I In our holographic set up, we need to add to our supergravity

an N = 2 vector multiplet dual to the conserved current supermultiplet.

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 26 / 28

SUSY breaking hidden sectors

Our results:

I Set up for holographic computation of correlators relevant

for GGM.

I R-preserving cases ⌘ = 0: we find a composite ’t Hooft fermion,

leading to a Dirac mass for gauginos. A heavy gaugino leads to a gaugino mediation scenario.

I Turning on the R-breaking

scalar ⌘ produces Majorana masses, eventually covering all GGM parameter space.

I For dominant ⌘0 gaugino

masses are suppressed–‘screened’.

0.001 0.01 0.1 1 10 50 50 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 h0 f4 HMgêMsfL for N 1 & 2 1

Gaugino Mediation Gaugino Screening

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.0 50 10 1 0.1 0.01 0.001 50

˜ φ4

η0

Riccardo Argurio (ULB) Holography & SUSY-breaking GGI 08/04/15 27 / 28

Outlook

Outlook

I New classes of SUGRA solutions representing SUSY-breaking

RG-flows.

I Holographic realization of massless particles related to dynamical

symmetry breaking.

I These particles arise as part of the spectrum of resonances, and

also from Schwinger terms.

I More general and more realistic classes of solutions

! more hyper- and vector multiplets in the N = 2 SUGRA.

I Embedding in string theory/consistent truncations

! top down models.

I Finding the Goldstino can be a useful discriminant to determine

whether we have the gravity dual of spontaneous SUSY breaking.

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