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 "abcd (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(jyj)dx2 + Y (jyj)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 −kjyj 1 zero{mode 0 ∼ e ! njyj njyj 2 kR Kaluza{Klein excitations n ∼ 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 ^ X 1 ^ µν dy δ(y) φ GµνGe = φ0 + φn GµνGe f f f 0 n=1 n p 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 kjyj 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 ! 1 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 = − pmS 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 + (rΦ) − Λ + branes impose linear dilaton background Φ = αjyj: 2 − 2 αjyj 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 ; M ;AM ) i 1 vector multiplet: (φ; λ ;AM ) m i I a x Maxwell{Einstein supergravity: (eM ; M ;AM ; λi ; φ ) 1 fφxg: coordinates of M 2 a fλi g: tangent space group SO(n) Lagrangian: −1 1 h i MNP i 1 I MNJ e LME = 2κ2 R(!) − M Γ DN P 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 Mi fx @N φ + 4 hI λ Γ Γ Mi FΛP ia −1 + iκ Φ λ ΓMN λb F I + κep C MNP ΣΛF I F J AK 4 Iab i MN 6 6 IJK MN P Σ Λ i Mi − 3pi h ΓMNP Σ F + 2 N 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: p −1 0 g2 i 6 g i MN j e L = − κ4 P − 8 κ3 M Γ N δij P0 ia ia − p1 g λ ΓM j δ P + pi g λ λjbδ P 2 κ2 M ij a 2 6 κ ij ab new terms in the fermion transformations: δ0 = ip gP Γ δjk Mi 2κ 6 0 M ji k δ0λa = 1p 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 p 3 2 2 redefine 3 ln t = Φ, set 4 g AP = −Λ ;BP = 0 2 −1 1 1 M p 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 = ± gAp P 2 fermion transformations: δ = pi Γ iCΓ5 + gAp P " δjk e µi 2 3 µ i 2 ji k 1 1 p Cy gA δλ = − e 3 iCΓ5 + p P " 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.