A naturally light & bent dilaton

Transcription

1 A naturally light & bent dilaton Javi Serra with B.Bellazzini, C.Csaki, J.Hubisz, J.Terning arxiv: arxiv:14xx.xxxx SUSY 2014 Manchester July 22,

2 Motivation. DILATON =Goldstone Boson of Spontaneous Breaking of Scale Invariance L = Φ ) Φ)+m 2 Φ Φ λφ Φ) 2 Broken phase ξ Spontaneous internal Symmetry Breaking: G -> H approximately) Massless Goldstone mode appear = phase Φ = e iξ Φ + σ) V ξ) =0 2

3 Motivation. DILATON =Goldstone Boson of Spontaneous Breaking of Scale Invariance L = Φ ) Φ)+m 2 Φ Φ λφ Φ) 2 Broken phase ξ Spontaneous space-time Symmetry Breaking: G -> H The dilaton behaves more like the Amplitude mode Φ = e iξ Φ + σ) V σ) 0 3

4 What is the Dilaton? explicit breaking Λ UV ) Scale conformal) invariant sector x e α x, Φx) e dφα Φe α x) S CFT = d 4 x O i, d i =4 i. µ Λ UV f f spontaneous breaking d>4: Irrelevant invariant terms are unimportant at low energies. d < 4: No large relevant invariant terms can be present. 4

5 What is the Dilaton? explicit breaking Λ UV ) Scale conformal) invariant sector x e α x, Φx) e dφα Φe α x) S CFT = O i, d i =4 d 4 x i. µ Λ UV f f spontaneous breaking d>4: Irrelevant invariant terms are unimportant at low energies. d < 4: No large relevant invariant terms can be present. Spontaneous breaking of scale invariance Φx) = f d Φ SO4, 2)/SO3, 1) σx) σe α x)+αf χ fe σ/f e α χ 5

6 Effective theory of CFT spontaneous breaking. L eff = 1 2 χ)2 a 0 χ 4 + Quartic potential allowed Is there really a light scalar when scale symmetry spontaneously breaks? a 0 > 0 a 0 =0 a 0 < 0 V V V χ χ χ χ 0 χ = f =? χ CFT/AdS4 CFT/Poincare4 CFT/dS4 runaway flat direction runaway Fubini 76 6

7 Effective theory of CFT spontaneous breaking. L eff = 1 2 χ)2 a 0 χ 4 + Quartic potential allowed Is there really a light scalar when scale symmetry spontaneously breaks? a 0 > 0 a 0 =0 a 0 < 0 V V V χ χ χ χ 0 χ = f =? χ CFT/AdS4 CFT/Poincare4 CFT/dS4 runaway flat direction runaway TUNING Fubini 76 a 0 χ 4 = a 0 f 4 4πf) 2 f 2 vacuum energy 6

8 Effective theory of CFT spontaneous breaking. L eff = 1 2 χ)2 a 0 χ 4 + Quartic potential allowed Is there really a light scalar when scale symmetry spontaneously breaks? a 0 > 0 a 0 =0 a 0 < 0 V V V χ χ χ χ 0 χ = f =? χ CFT/AdS4 CFT/Poincare4 CFT/dS4 runaway flat direction runaway Fubini 76 a 0 χ 4 = a 0 f 4 4πf) 2 f 2 vacuum energy 6

9 Effective theory of CFT spontaneous breaking. We need a perturbation, aka explicit breaking L = L CFT + λo [O] =4 β/λ dλµ) d log µ = βλ) λ 0 spurion: µ χ V χ) =χ 4 F λχ)) F λχ)) = a 0 + n a n λ n χ) Quartic gets dependence on running coupling. V χ Coleman, Weinberg 73 The dilaton effectively scans the landscape of quartics. 7

10 Effective theory of CFT spontaneous breaking. Minimum and dilaton mass χ = f V = f 3 [4F λf)) + βf λf))] = 0 m 2 d 4f 2 βf λf)) = 16f 2 F λf)) = 16V f)/f 2 i) Dilaton mass prop. to explicit breaking at condensate scale, β = βλf)) ii) Potential at minimum, aka vacuum energy, also prop. to explicit breaking iii) Hierarchy between UV and IR scales fixed by dimensional transmutation, dependent on the explicit breaking along the whole running 8

11 Effective theory of CFT spontaneous breaking. Minimum and dilaton mass χ = f V = f 3 [4F λf)) + βf λf))] = 0 m 2 d 4f 2 βf λf)) = 16f 2 F λf)) = 16V f)/f 2 i) Dilaton mass prop. to explicit breaking at condensate scale, β = βλf)) ii) Potential at minimum, aka vacuum energy, also prop. to explicit breaking iii) Hierarchy between UV and IR scales fixed by dimensional transmutation, dependent on the explicit breaking along the whole running Dynamical theory space) requirement βλ) =ɛ bλ), ɛ 1, bλ) =O1) Large hierarchy, light dilaton, small cosmological constant 8

12 An Extra-D Computable Example Contino, Pomarol, Rattazzi, 10 Bellazzini, Csaki, Hubisz, Terning, JS, 13 Coradeschi, Lodone, Pappadopulo, Rattazzi, Vitale, 13 Megias, Pujolas 14 9

13 AdS/CFT phenomenological correspondence. UV brane ds 2 = e 2ky dx 2 dy 2 Randall & Sundrum setup IR brane y 0 y 1 S CFT = UV cutoff explicit breaking d 4 x i O i, Randall, Sundrum 99 y µ IR CFT scale spontaneous breaking AdS 5 e ky e ky 0 e ky 1 radion CFT 4 µ Λ UV χ dilaton χ = f The brane separation hierarchy of scales, is fixed by. 10

14 AdS/CFT phenomenological correspondence. Adding explicit breaking perturbation in AdS/CFT AdS 5 φ CFT 4 O V φ) =dv/dφ βλ) =dλ/d log µ V φ) =Λ 5) exactly marginal φ) y=y0 =0 φ y=y0 λ UV 4D gravity is readily included g MN graviton + dilaton 4D effective Lagrangian L = g [ L CFT + λo + M 2 P R] 11

15 Generalized Randall-Sundrum. S = Z d 5 x p g The general & stabilized & bent RS 1 2apple R N V ) Bellazzini, Csaki, Hubisz, Terning, JS, 13 Z d 4 x p g 0 V 0 ) Z d 4 x p g 1 V 1 ). UV brane ds 2 = e 2Ay) dt 2 e 2Ht d~x 2 dy 2 IR brane y We solve for the scalar & the warp factor profiles & Hubble bulk E.O.M. s A 2 + H 2 e 2A = κ2 φ 2 12 κ2 6 V φ) φ = 4A φ + V φ boundary conditions 2A 0 y=y0,y 1 = ± apple2 3 V 0,1 ) y=y0,y y=y0,y 1 = y=y0,y 1, 12

16 Generalized Randall-Sundrum. [ V IR = e 4Ay 1) [ General 4D effective potential V eff = V IR + V R ] V 1 φy 1 )) 6 κ 2 A y 1 ) V R = 3H 2 y 0,y 1 )M 2 Pl y 0,y 1 ) M 2 Pl = 2 apple 2 Z y1 y 0 dy e 2Ay) We obtain just what we expected Modified quartic dilaton potential from two sources of explicit breaking running UV perturbation V λ = χ 4 F λ λχ)) UV Hubble constant V H = χ 4 F H Hχ)) 13

17 Generalized Randall-Sundrum V φ) =Λ 5) + m 2 φ 2. m 2 = 2ɛk 2 ɛ 1 d O 4 ɛ F λ χ) Λ 1 Λ ) 5) + k F H χ) Λ 0 + Λ 5) k expansion in back-reaction [ ΛUV λ IR λ UV χ ) ) Λ 2 UV χ 2 1+ Λ2 UV χ 2 ) ɛ ] 2 14

18 Generalized Randall-Sundrum V φ) =Λ 5) + m 2 φ 2. m 2 = 2ɛk 2 ɛ 1 d O 4 ɛ F λ χ) Λ 1 Λ ) 5) + k F H χ) Λ 0 + Λ 5) k expansion in back-reaction [ ΛUV λ IR λ UV χ ) ) Λ 2 UV χ 2 H 2 UV/Λ 2 UV 1+ Λ2 UV χ 2 ) ɛ ] 2 β/λ 14

19 Generalized Randall-Sundrum V φ) =Λ 5) + m 2 φ 2. m 2 = 2ɛk 2 ɛ 1 d O 4 ɛ F λ χ) Λ 1 Λ ) 5) + k F H χ) Λ 0 + Λ 5) k expansion in back-reaction [ ΛUV λ IR λ UV χ ) ) Λ 2 UV χ 2 H 2 UV/Λ 2 UV 1+ Λ2 UV χ 2 ) ɛ ] 2 β/λ And eliminating cut-off effects UV cosmological constant tuning), V λ χ 4 { Λ 1 Λ } 5) k cosh [λ IR λ UV Λ UV /χ) ɛ ] H UV =0 14

20 Generalized Randall-Sundrum. The large hierarchy 25 ɛ = V χ = f f Λ UV = c=e -k y 1 ) 1/ɛ λ UV λ IR signɛ)arcsechλ 5) /kλ 1 ) Because of slow running for a long time. 15

21 Generalized Randall-Sundrum The light dilaton ɛ = V m 2 χ ɛ 48λ UV tanh c=e -k y 1 [ ] 1 2 λ IR λ UV Λ UV /f) ɛ ) Because of slow running at the minimum. Λ UV /f) ɛ f 2 16

22 Generalized Randall-Sundrum. The small cosmological constant ɛ = V [ ] V min IR = ɛ 3λ UV tanh c=e -k y 1 [ ] 1 2 λ IR λ UV Λ UV /f) ɛ ) Because of slow running at the minimum. Λ UV /f) ɛ f 4 17

23 Conclusions. Approximate Spontaneous Breaking of Scale Invariance offers a natural way to obtain a light scalar, the Dilaton, βλ) =ɛ bλ), ɛ 1, bλ) =O1) and to suppress the spontaneously generated Vacuum energy. Is this possibility realized in Nature? Inflaton as Dilaton Higgs as Dilaton Dilaton in Phase Transitions... arxiv: arxiv: arxiv:14xx.xxxx We just have to wait and see 18

24 NP: Themes 1. Necessity for new particles at TeV mass the questions of fine tuning and dark matter are still ope 2. Candidate TeV particles weakly coupled: SUSY, Dark Matter, Long-lived strongly coupled/composite: Randall-Sundrum, KK and Z resonances, long-lived particles evolution of robust search strategies Thank you for your attention 3. Connection to dark matter problem 4. Connection to flavor issues 19

25 Comments on the Cosmological Constant. 1) Small cosmological constant & light dilaton signal the approximate scale invariance at the IR scale: V φ) =dv/dφ βλ) =dλ/d log µ Change the bulk potential, change the running. Chacko, Mishra, Stolarski 13 20

26 Comments on the Cosmological Constant. 1) Small cosmological constant & light dilaton signal the approximate scale invariance at the IR scale: V φ) =dv/dφ βλ) =dλ/d log µ Change the bulk potential, change the running. 2) The suppression is parametrically better than in SUSY: Chacko, Mishra, Stolarski 13 SUSY Λ IR CC = cm4 b m4 f ) cm2 b + m2 f )g2 s F 2 s CFT Λ IR CC = c ɛ4π) 2 f 4 c ɛλ 2 IRf 2 20

27 Comments on the Cosmological Constant. 1) Small cosmological constant & light dilaton signal the approximate scale invariance at the IR scale: V φ) =dv/dφ βλ) =dλ/d log µ Change the bulk potential, change the running. 2) The suppression is parametrically better than in SUSY: Chacko, Mishra, Stolarski 13 SUSY Λ IR CC = cm4 b m4 f ) cm2 b + m2 f )g2 s F 2 s CFT Λ IR CC = c ɛ4π) 2 f 4 c ɛλ 2 IRf 2 3) Our result is consistent with Weinberg s no-go theorem: ɛ =0can remove the CC, but ɛ 0 is required for a unique vacuum A very light state must be in the spectrum. 20

28 Comments on the Cosmological Constant. 1) Small cosmological constant & light dilaton signal the approximate scale invariance at the IR scale: V φ) =dv/dφ βλ) =dλ/d log µ Change the bulk potential, change the running. 2) The suppression is parametrically better than in SUSY: Chacko, Mishra, Stolarski 13 SUSY Λ IR CC = cm4 b m4 f ) cm2 b + m2 f )g2 s F 2 s CFT Λ IR CC = c ɛ4π) 2 f 4 c ɛλ 2 IRf 2 3) Our result is consistent with Weinberg s no-go theorem: ɛ =0can remove the CC, but ɛ 0 is required for a unique vacuum A very light state must be in the spectrum. 4) UV contribution to the cosmological constant must be tuned away. 20

29 AdS/CFT phenomenological correspondence. perturbative) Example: bulk mass V φ) =Λ 5) + m 2 φ 2 Scaling dimension of operator d O =2+ 4+m 2 /k 2 Background scalar solution of E.O.M. φy) =φ 0 e ky4 d O) + φ 1 e kyd O { { running condensate φ 0 = lim Λ UV Λ4 d O UV λ UV φ 1 = O 2d O 4 dλ d log µ βλ) =4 d O)λ 21