The Linear Dilaton: from the Clockwork Mechanism to Its Supergravity Embedding

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The linear dilaton: from the clockwork mechanism to its supergravity embedding

Chrysoula Markou

August 30, 2019 SQS ’19, Yerevan

Chrysoula Markou The linear dilaton 1 / 19 Overview

1 The clockwork mechanism The clockwork scalar Motivation and basic references The clockwork graviton Continuum

2 Little String Theory Deﬁnition Geometry Phenomenology

3 Supergravity embedding Structure of N = 2, D = 5 Supergravity Eﬀective supergravity

Chrysoula Markou The linear dilaton 2 / 19 The clockwork mechanism The clockwork scalar Setup

1 massless scalar π(x), symmetry U(1)

N+1 2 N+1 copies: πj(x), j = 0, 1,...,N , symmetry U(1) (at least)

3 explicit breaking U(1)N+1 → U(1) via a mass–mixing that involves near–neighbours only

N N−1 1 X 1 X L = − ∂ π ∂µπ − m2 (π − qπ )2 + ... 2 µ j j 2 j j+1 j=0 j=0

q treats the site j and the site j + 1 asymmetrically

1 1/q 1/q2 1/qN j = 0 j = 1 j = 2 j = N

Chrysoula Markou The linear dilaton 3 / 19 The clockwork mechanism The clockwork scalar Eigenmodes π = Oa

1 N0 massless Goldstone a0, Oj0 = qj

h (j+1)kπ i 2 jkπ massive Clockwork gears ak, Ojk = Nk q sin N+1 − sin N+1

now couple eg. topological term to the N–th site only

( N ) 1 µν 1 X k 1 µν πN GµνGe = a0 − (−) ak GµνGe f f0 fk k=1

N 1 f0/f ∼ q : exponential enhancement

3/2 2 fk/f ∼ N /k: mild N dependance

Chrysoula Markou The linear dilaton 4 / 19 The clockwork mechanism Motivation and basic references

the clockwork as a renormalizable scalar ﬁeld theory: Choi and Im ’15 Kaplan and Rattazzi ’15 (see also natural inﬂation and relaxion models)

the clockwork as an eﬀective low–energy ﬁeld theory: Giudice and McCullough ’16 Recipe: N + 1 of particle P whose masslessness is protected by symmetry S & explicit breaking via near–neighbour interactions

Known examples for the following spins: 0, 1/2, 1, 2

growing pheno literature (LHC signatures, inﬂation, . . . )

group theory of the clockwork: Craig, Garcia Garcia, Sutherland ’17 Giudice and McCullough ’17

Chrysoula Markou The linear dilaton 5 / 19 The clockwork mechanism The clockwork graviton

1 @ the linear level: use Fierz–Pauli 1939 mass term

N−1 " # 1 X 2 L = − m2 (hµν − qhµν )2 − η (hµν − qhµν ) int 2 j j+1 µν j j+1 j=0

Giudice and McCullough ’16

2 @ the nonlinear level: use Hinterbichler and Rosen ’12 interaction Z X a b c d SV = − Tijkl εabcd (Ei) ∧ (Ej) ∧ (Ek) ∧ (El) i,j,k,l

Niedermann, Padilla, Saﬃn ’18 µ 1 µ in both cases: asymmetric gear shifts ξi = qi ξ

Chrysoula Markou The linear dilaton 6 / 19 The clockwork mechanism Continuum Setup

1 discrete and ﬁnite lattice dimension → continuous y on S /Z2 Giudice and McCullough ’16

1 Consider real scalar φ in 5D with metric Ansatz: ds2 = X(|y|)dx2 + Y (|y|)dy2

2 Discretize: y → yj = ja , φ, X, Y → φj,Xj,Yj

N N−1 1/2 1/4 !2 1 X 1 X Xj Xj Yj L = − ∂ φ ∂µφ − φ − φ µ j j 2 j 1/2 1/4 j+1 2 2 a Yj j=0 j=0 Xj+1Yj+1

− 4 kπR j ka 3 Identify: Xj ∼ Yj ∼ e 3 N with q = e

Chrysoula Markou The linear dilaton 7 / 19 The clockwork mechanism Continuum Eigenmodes

−k|y| 1 zero–mode ψ0 ∼ e ! n|y| n|y| 2 kR Kaluza–Klein excitations ψn ∼ n sin R + cos R

now couple eg. topological term to y = 0 only (ﬂipped k sign for pheno reasons)

Z ( N ) 1 µν 1 ˆ X 1 ˆ µν dy δ(y) φ GµνGe = φ0 + φn GµνGe f f f 0 n=1 n √ kπR 1 f0/f ∼ e / kπR: exponential enhancement

2 fn ∼ f: roughly equal

Chrysoula Markou The linear dilaton 8 / 19 The clockwork mechanism Continuum Origin and comparison

Continuum clockwork 2 4 k|y| 2 2 5D metric: ds = e 3 (dx + dy )

2 2 n 2 KK spectrum: mn = k + R ⇒ precisely those of the 5D toy model for the holographic dual of 6D Little String Theory as we will see now!

Discretize 5th dimension of other models: large extra dimensions: no warping and no “clockworking”

Randall–Sundrum: warping and “clockworking”

but @ simple one–parameter deformation from the clockwork to Randall–Sundrum

to explain eg. hierarchy, need only O(10) sites and q ∼ 4

Chrysoula Markou The linear dilaton 9 / 19 Little String Theory Deﬁnition there exist two inequivalent limits of string theory: 1 E << mS: low–energy limit

2 gS → 0: “little string theories” Berkooz, Rozali, Seiberg ’97, Seiberg ’97, . . .

simple example of LST: take gS → 0 for a stack of NNS5–branes in type IIB 1 mS and N are the only parameters

2 bulk dynamics decouple

Properties of LST 1 non–gravitational, since MP l → ∞ as gS → 0

2 non–local, eg. Hagedorn density of states at high energies, T–duality

3 1 2 interacting, with 2 = mS gN

Chrysoula Markou The linear dilaton 10 / 19 Little String Theory Geometry

5,1 3 “holographic” dual of 6D LST: string theory on IR × IRy × SN Aharony, Berkooz, Kutasov, Seiberg ’98 infinite throat connected to asymptotically flat spacetime linear dilaton Φ = − √mS y N compactify y on interval (therby treating strong coupling problem and obtaining a large but finite MP l in the weak coupling regime): cigar–like geometry, Antoniadis, Dimopoulos, Giveon ’01

Chrysoula Markou The linear dilaton 11 / 19 Little String Theory Phenomenology

5D graviton–dilaton toy model for cigar: Antoniadis, Arvanitaki, Dimopoulos, Giveon ’11

Φ − 3/2 −1 M 3 2 e LLST = e 5 M5 R + (∇Φ) − Λ + branes

impose linear dilaton background Φ = α|y|: 2 − 2 α|y| 2 2 background metric: ds = e 3 (dx + dy )

2 2 KK spectrum: m2 = α + nπ n 2 rc ⇒ same as continuum clockwork!

supergravity embedding Kehagias, Riotto ’17, Antoniadis, Delgado, CM, Pokorski ’17

Chrysoula Markou The linear dilaton 12 / 19 Supergravity embedding Structure of N = 2, D = 5 Supergravity N = 2, D = 5 Supergravity

m i 0 pure supergravity multiplet: (eM , ψM ,AM ) i 1 vector multiplet: (φ, λ ,AM ) m i I a x Maxwell–Einstein supergravity: (eM , ψM ,AM , λi , φ ) 1 {φx}: coordinates of M 2 a {λi }: tangent space group SO(n) Lagrangian:

−1 1 h i MNP i 1 I MNJ e LME = 2κ2 R(ω) − ψM Γ DN ψP i − 4 GIJ FMN F

1 ia / ab ab / x b 1 x M y − 2 λ Dδ + Ωx ∂φ λi − 2 gxy(∂M φ )(∂ φ )

i ia M N a x 1 a ia M ΛP I − 2 λ Γ Γ ψMi fx ∂N φ + 4 hI λ Γ Γ ψMi FΛP

ia −1 + iκ Φ λ ΓMN λb F I + κe√ C MNP ΣΛF I F J AK 4 Iab i MN 6 6 IJK MN P Σ Λ

i Mi − 3√i h ψ ΓMNP Σψ F + 2ψ ψN F + ... 8κ 6 I M Ni P Σ i MN G¨unaydin–Sierra–Townsend ’84

Chrysoula Markou The linear dilaton 13 / 19 Supergravity embedding Structure of N = 2, D = 5 Supergravity Constraints

1 Algebraic:

I I I J hxhI = hIxh = 0 , hxhy GIJ = gxy x y x GIJ = hI hJ + hI hJ gyx = hI hJ + hI hJx I I y y h hI = 1 ,h xhI = δx

I I ⇒ h GIJ = hJ , hxGIJ = hJx

2 Diﬀerential:

q 2 I q 2 I hI,x = 3 hIx , h,x = − 3 hx

q 2 z I q 2 I Iz hIx;y = 3 gxyhI + TxyzhI , hx;y = − 3 gxyh + Txyzh

Chrysoula Markou The linear dilaton 14 / 19 Supergravity embedding Structure of N = 2, D = 5 Supergravity Embedding

E with coordinates XI = XI (φx, N )

hypersurfaces Mk deﬁned by ln N = k with normal vectors nI = ∂I ln N

orthonormality relations: 3 XI n = 0 , β2 XI X,y = δy ,x I 2 ,x I x

! I ! I ⇒ identify: hI = αnI , h = βX

can also show that:

3 I J K N = β CIJK X X X

3 I J K M: special case k = 0 ⇒ β CIJK X X X = 1

Chrysoula Markou The linear dilaton 15 / 19 Supergravity embedding Structure of N = 2, D = 5 Supergravity Gauging

I gauge U(1) subgroup of SU(2) w.r.t. AM = υI AM new Lagrangian terms: √ −1 0 g2 i 6 g i MN j e L = − κ4 P − 8 κ3 ψM Γ ψN δij P0 ia ia − √1 g λ ΓM ψj δ P + √i g λ λjbδ P 2 κ2 M ij a 2 6 κ ij ab new terms in the fermion transformations: δ0ψ = i√ gP Γ δjk Mi 2κ 6 0 M ji k δ0λa = 1√ gP a δjk i κ2 2 ji k algebraic and diﬀerential constraints

G¨unaydin–Sierra–Townsend ’84, ’85

Chrysoula Markou The linear dilaton 16 / 19 Supergravity embedding Eﬀective supergravity Embedding

use a U(1) gauging of N = 2, D = 5 Maxwell–Einstein supergravity with one vector multiplet

Ansatz: N = st2 + at3 ,X0 ≡ s , X1 = t Klemm, Lerche, Mayr ’95, Antoniadis, Ferrara, Taylor, ’95

calculate potential: A 1 P = −3A P t2 + B P 4 P t √ 3 2 2 redeﬁne 3 ln t = Φ, set 4 g AP = −Λ ,BP = 0 2 −1 1 1 M √ Φ ⇒ e L = R − (∂ Φ)(∂ Φ) − e 3 Λ LST 2 2 M

Chrysoula Markou The linear dilaton 17 / 19 Supergravity embedding Eﬀective supergravity Partial breaking

Φ = Cy, ﬁne–tuning condition: C = ± gA√ P 2

fermion transformations: δψ = √i Γ iCΓ5 + gA√ P ε δjk e µi 2 3 µ i 2 ji k

1 1 √ Cy gA δλ = − e 3 iCΓ5 + √ P ε δjk e i 2 i 2 ji k so that 5 δe λ1 − iΓ λ2 = 0

5 5 δe λ1 + iΓ λ2 ∼ 2 − iΓ 1 ⇒ partial supersymmetry breaking

three free parameters: g, υ0, υ1. Mass gap: ∼ g

Chrysoula Markou The linear dilaton 18 / 19 Supergravity embedding Eﬀective supergravity Summary

Gauged N = 2, D = 5 supergravity can accommodate the linear dilaton model (minimal case)

Partial breaking due to the linear dilaton bkg

5D supergravity embedding of other models Im, Nilles, Olechowski ’18

Branes compatible with the unbroken supersymmetry Antoniadis, Delgado, CM, Pokorski (ongoing work)

deeper link between LST and the clockwork?

Chrysoula Markou The linear dilaton 19 / 19