Dilatons and Fine Tuning John Terning, UC Davis with Csaba Cski, - PowerPoint PPT Presentation

Dilatons and Fine Tuning John Terning, UC Davis with Csaba Csáki, Brando Bellazzini, Jay Hubisz, Javi Serra hep-ph/1209.3299 and 1305.3919

dilaton potential f = 0 a > 0 Review of broken CFT’ s f = ∞ a < 0 A ¢ k 5 f =? New 5D Dual of a 4 3 a = 0 Spontaneously Broken CFT 2 1 y yc 0.0 0.2 0.4 0.6 0.8 1.0 Reducing the 0.002 cosmological constant 0.001 16.0 16.5 17.0 17.5 18.0 - 0.001 - 0.002

Relevant operators cause problems L = Λ + m 2 H † H + L d =4 + L d =4+ n With a consistent effective theory up to scale µ, without tuning coefficients go like: 1 ◆ g 2 ✓ µ 2 µ 4 ln µ , µ n , , 16 π 2 observed values: 1 (10 − 12 GeV) 4 O (1) (126 GeV) 2 < (10 TeV) n

The Plan string theory unique vacuum ? understand the weak scale LHC

The Plan string theory unique vacuum ? understand the weak scale find a bunch of new particles LHC

The Plan string theory a miracle occurs unique vacuum ? understand the weak scale find a bunch of new particles LHC

Plan B string theory unusual vacuum one out of e 500 ? we don’ t understand the weak scale LHC

Anthropic Speculations: The Universe must be such that life can be advanced enough to contemplate the Universe and primitive enough to contemplate the anthropic principle.

Approximate Symmetries can lead to large suppressions terms can be forbidden to leading order Pseudo-Nambu-Goldstone bosons can have suppressed masses

Symmetry checklist m 2 H † H Λ supersymmetry X X little Higgs X extra dim. gauge X conformal X X we’ll explore spontaneously broken CFTs

Effective theory for broken scale invariance h O i = f n σ ( x ) → σ ( e α x ) + α f Goldstone boson f → f χ ≡ f e σ /f non-linear realization ∂ 2 n χ m a n,m X L eff = (4 π ) 2( n − 1) f 2( n − 2) χ 2 n + m − 4 n,m > 0 = − a 0 , 0 (4 π ) 2 f 4 χ 4 + f 2 ( ∂χ ) 4 2 ( ∂ µ χ ) 2 + a 2 , 4 + . . . (4 π ) 2 χ 4 a la Callan, Coleman, Wess, and Zumino

Runaway Goldstone Boson L e ff = − a 0 , 0 (4 π ) 2 f 4 χ 4 + f 2 dilaton potential 2 ( ∂ µ χ ) 2 + . . . f = 0 quartic coupling a > 0 f = ∞ a < 0 for exact conformal symmetry f =? need a flat direction to have spontaneous breaking a = 0

Perturbation by an almost marginal operator δ L = λ ( µ ) O f V = a f 4 → V = a ( λ ( f )) f 4 V 0 = f 3 [4 a ( λ ( f )) + β a 0 ( λ ( f ))] = 0 dil = f 2 β [ β a 00 + 4 a 0 + β 0 a 0 ] m 2 ' 4 f 2 β a 0 ( λ ( f )) = � 16 f 2 a ( λ ( f )) = O (16 π 2 ) f 2 still need approximate flat direction QCD-like theories don’ t have light dilatons, Phys.Lett. B200 (1988) 338 wh

Assuming an approximate flat direction d ln µ = ✏� + b 1 d � 4 ⇡ � 2 + O � 3 � ( � ) = ✓ λ " ◆ n # X a ( λ ) = (4 π ) 2 c 0 + c n 4 π n ✓ f ◆ ✏ �  4 c 0 + c 1 V 0 = f 3 [4 a + β a 0 ] ≈ (4 π ) 2 f 3 4 π λ 0 µ 0 ◆ 1 / ✏ ✓ − 16 π c 0 f ≈ µ 0 λ ( µ 0 ) c 1 need a marginal operator, small function, β and a tuned flat direction

Are there non-SUSY theories with approximate flat directions? Turn to the AdS/CFT correspondence d 4 x φ 0 ( x ) O ( x ) i CFT ⇡ e − S 5Dgrav [ φ ( x,z ) | z =0 = φ 0 ( x )] R h e ds 2 = R 2 dx 2 − dz 2 � � z 2 O ⊂ CFT ↔ φ AdS 5 field, φ 0 ( x ) is boundary value

m 2 φ 2 With bulk mass φ = φ 0 z 4 − ∆ + c z ∆ source condensate r 4 + m 2 ∆ = 2 + k 2 CFT operator O has dimension ∆ how do we spontaneously break the CFT?

Randall-Sundrum is dual to a spontaneously broken CFT t e W,Z UV IR why is there a flat direction without SUSY?

Tuning a flat direction in RS ✓ R ✓ R ◆ 4 ◆ 4 ! + Λ (5) R V eff = V 0 + V 1 1 − R 0 R 0 brane tensions 5D cosmo. constant V eff = V 0 + Λ (5) R + f 4 χ 4 � V 1 R 4 − Λ (5) R 5 � UV cosmo. constant quartic coupling this solution is not stable to perturbations Csaki, Graesser, Kolda, JT hep-ph/9906513, hep-ph/9911406

Goldberger-Wise stabilized RS m 2 = − 4 ✏ k 2 ∆ ≈ 4 − ✏ ds 2 = e − 2 A ( y ) dx 2 − dy 2 φ = φ 0 e ✏ ky λ 0 ( φ − v 0 ) 2 = V 0 λ 0 , λ 1 → ∞ brane potentials λ 1 ( φ − v 1 ) 2 = V 1 two fine tunings V eff = V 0 + Λ (5) R + f 4 χ 4 � V 1 R 4 − Λ (5) R 5 � hep-ph/9907447

Goldberger-Wise stabilized RS λ 0 ( φ − v 0 ) 2 = V 0 brane potentials λ 1 ( φ − v 1 ) 2 = V 1 3 2.5 2 1.5 1 0.5 0 1 2 3 4 5 0 1 2 3 4 5

Goldberger-Wise stabilized RS λ 0 ( φ − v 0 ) 2 = V 0 brane potentials λ 1 ( φ − v 1 ) 2 = V 1 3 2.5 2 1.5 1 0.5 0 1 2 3 4 5 0 1 2 3 4 5

Goldberger-Wise stabilized RS λ 0 ( φ − v 0 ) 2 = V 0 brane potentials λ 1 ( φ − v 1 ) 2 = V 1 3 2.5 2 1.5 1 0.5 0 1 2 3 4 5 0 1 2 3 4 5 now we can parameterize a broken CFT

Reducing the Cosmological Constant Riccardo Rattazzi perturbing the CFT and allowing sufficient running allows the IR brane to sit at a location with a vanishing c.c. Planck conference 2010

Weinberg’ s No-Go Theorem You can’ t have your cake and eat it too! Exact conformal symmetry can remove the c.c. but to have a unique vacuum it must be broken. S. Weinberg Rev. Mod. Phys. 61, 1 (1989)

Back reaction in AdS 5 ds 2 = e − 2 A ( y ) dx 2 − dy 2 κ 2 φ 0 2 − κ 2 A 0 2 = 6 V ( φ ) 12 4 A 0 φ 0 + ∂ V φ 00 = ∂φ  � V 1 ( φ ( y 1 )) + 6 V eff ( y 1 ) = e � 4 A ( y 1 ) κ 2 A 0 ( y 1 ) assuming we have fine-tuned the UV contributions

UV behavior with a bulk mass � 00 − 4 A 0 � 0 + 4 ✏ k 2 � = 0 dominant balance φ 0 e ✏ ky φ = ky − κ 2 φ 2 0 e 2 ✏ ky − 1 � � A = 12 perturbative RG evolution

IR behavior with a bulk mass � 00 − 4 A 0 � 0 + 4 ✏ k 2 � = 0 dominant balance A 0 ( y ) = − k coth (4 k ( y − y c )) √ 3 c φ ( y ) = 2 κ log (tanh (2 k ( yc − y ))) κ − 8 singularity at y c 6 4 2 y 0.0 0.2 0.4 0.6 0.8 1.0 y_c Csaki, Erlich, Grojean, Hollowood hep-th/0004133

Boundary Layer Matching √ �!  1 − ( � 0 ) 4 y c = 1 3 c ✏ k log log − 1 + ( � 0 ) 4 2 � 0 � 0 √ 3 2 κ log [tanh (2 k ( y c − y ))] − φ 0 e ✏ ky c + c φ ( y ) = φ 0 e ✏ ky − κ 2.0 1.5 1.0 0.5 0 5 10 15 20 cf. Chacko, Mishra, Stolarski hep-ph/1304.1795

5D Dual of Spontaneously Broken Scale Invariance A 0 ( y ) = k coth (4 k ( y c − y )) √ 3 φ 0 e ✏ ky − φ ( y ) = 2 κ log (tanh (2 k ( y c − y )))  � Λ 1 + 6 V e ff = e � 4 A ( y 1 ) = e � 4 A ( y 1 ) V b κ 2 A 0 ( y 1 , y c )  � d V b + dy c d e ff = e � 4 A ( y 1 ) V 0 − 4 A 0 V b + V b dy 1 dy 1 dy c by boundary condition φ 0

5D Dual of Spontaneously Broken Scale Invariance A 0 ( y ) = k coth (4 k ( y c − y )) √ 3 φ 0 e ✏ ky − φ ( y ) = 2 κ log (tanh (2 k ( y c − y ))) V ∼ ✏ (TeV) 4 m 2 dil ∼ ✏ (TeV) 2 potentially 24 orders of magnitude better than SUSY

Cosmological Constant in TeV 4 0.02 Λ 1 = 2 Λ RS ✏ = 0 . 01 0.01 16.0 16.5 17.0 17.5 18.0 0.002 - 0.01 0.001 ✏ = 0 . 001 - 0.02 16.0 16.5 17.0 17.5 18.0 - 0.001 - 0.002

Weinberg’ s No-Go Theorem You can’ t have your cake and eat it too! can remove the c.c. ✏ = 0 but to have a unique vacuum ✏ 6 = 0 ✏ = 10 − 12 ?

Conclusions we have a new 5D dual of a spontaneously broken CFT conformal symmetry is better than SUSY for reducing the cosmological constant but not nearly a solution by itself

Backup

Soluble Conformal Field Theory L = 1 2 ∂ ν φ ∂ ν φ h φ i = v is a massless Nambu-Goldstone boson φ

SUSY 3-2 Model is a Broken CFT SU (3) SU (2) U (1) U (1) R Q 1 / 3 1 L � 1 � 3 1 U � 4 / 3 � 8 1 D 2 / 3 4 1 Λ 7 det( QQ ) + λ Q ¯ W = DL 3 ⌧ ∂ Q | 2 + | ∂ W ∂ U | 2 + | ∂ W ∂ D | 2 + | ∂ W | ∂ W ∂ L | 2 = V Λ 14 φ 10 + λ Λ 7 φ 3 + λ 2 φ 4 ⇡ 3 3 h φ i = Λ 3 λ 1 / 7

SUSY 3-2 Model is a Broken CFT SU (3) SU (2) U (1) U (1) R Q 1 / 3 1 L � 1 � 3 1 U � 4 / 3 � 8 1 D 2 / 3 4 1 h φ i = Λ 3 λ 1 / 7 m dil = λ h φ i = λ 6 / 7 Λ 3 V min ≈ λ 2 < φ > 4

Scale invariant action x → x 0 = e � α x O ( x ) → O 0 ( x ) = e α ∆ O ( e α x ) Z Z → S 0 = X X d 4 x g i O i ( x ) − d 4 xe α ( ∆ i � 4) g i O i ( x ) S = i i invariance requires ∆ i = 4

5 spontaneously broken conformal generators give one Goldstone Boson = − i ( x µ ∂ ν − x ν ∂ µ ) M µ ν = − i ∂ µ P µ − i ( x 2 ∂ µ − 2 x µ x α ∂ α ) = K µ = ix α ∂ α D Sydney Coleman, “Aspects of Symmetry” see also Ian Low, hep-th/0110285