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

a > 0

a < 0

a = 0 f = 0

f = ∞

f =?

Review of broken CFT’ s Reducing the cosmological constant New 5D Dual of a Spontaneously Broken CFT

• 0.002
• 0.001

Relevant operators cause problems

With a consistent effective theory up to scale µ, without tuning coefficients go like:

µ4 µ2 ln µ 1 µn g2 16π2 ✓ ◆ , , ,

• bserved values:

O(1) < 1 (10 TeV)n (126 GeV)2 (10−12 GeV)4 L = Λ + m2H†H + Ld=4 + Ld=4+n

The Plan

LHC understand the weak scale string theory unique vacuum ?

The Plan

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

The Plan

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

Plan B

LHC we don’ t understand the weak scale string theory

?

unusual vacuum

• ne out of e500

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

Anthropic Speculations:

Approximate Symmetries can lead to large suppressions

terms can be forbidden to leading order Pseudo-Nambu-Goldstone bosons can have suppressed masses

Symmetry checklist

supersymmetry

m2H†H Λ

little Higgs

X

extra dim. gauge

X

conformal

X X X X

we’ll explore spontaneously broken CFTs

Effective theory for broken scale invariance

Goldstone boson

σ(x) → σ(eαx) + αf hOi = f n f → f χ ≡ f eσ/f Leff = X

n,m>0

an,m (4π)2(n−1) f 2(n−2) ∂2nχm χ2n+m−4 = −a0,0 (4π)2f 4χ4 + f 2 2 (∂µχ)2 + a2,4 (4π)2 (∂χ)4 χ4 + . . .

non-linear realization a la Callan, Coleman, Wess, and Zumino

Runaway Goldstone Boson

quartic coupling

dilaton potential

a > 0

a < 0

a = 0 f = 0

f = ∞

f =? for exact conformal symmetry need a flat direction to have spontaneous breaking

Leff = −a0,0 (4π)2f 4χ4 + f 2 2 (∂µχ)2 + . . .

Perturbation by an almost marginal operator

still need approximate flat direction QCD-like theories don’ t have light dilatons, Phys.Lett. B200 (1988) 338

wh

f δL = λ(µ)O V = a f 4 → V = a(λ(f)) f 4 V 0 = f 3 [4a(λ(f)) + βa0(λ(f))] = 0 ' 4f 2βa0(λ(f)) = 16f 2a(λ(f)) m2

dil = f 2β [βa00 + 4a0 + β0a0]

= O(16π2)f 2

Assuming an approximate flat direction

() = d d ln µ = ✏ + b1 4⇡ 2 + O3 a(λ) = (4π)2 " c0 + X

n

cn ✓ λ 4π ◆n# f ≈ µ0 ✓−16πc0 λ(µ0)c1 ◆1/✏ V 0 = f 3 [4a + βa0] ≈ (4π)2 f 3  4c0 + c1 4π λ0 ✓ f µ0 ◆✏

need a marginal operator, small function, and a tuned flat direction

β

Are there non-SUSY theories with approximate flat directions?

Turn to the AdS/CFT correspondence

h e

R d4x φ0(x)O(x)iCFT ⇡ e−S5Dgrav[φ(x,z)|z=0=φ0(x)]

O ⊂ CFT ↔ φ AdS5 field, φ0(x) is boundary value ds2 = R2 z2

• dx2 − dz2

With bulk mass

source

∆ = 2 + r 4 + m2 k2 φ = φ0 z4−∆ + c z∆

condensate

O has dimension ∆

CFT operator

m2 φ2

how do we spontaneously break the CFT?

Randall-Sundrum is dual to a spontaneously broken CFT

why is there a flat direction without SUSY?

IR UV W,Z e t

Tuning a flat direction in RS

brane tensions

Veff = V0 + V1 ✓ R R0 ◆4 + Λ(5)R 1 − ✓ R R0 ◆4!

5D cosmo. constant UV cosmo. constant quartic coupling

Veff = V0 + Λ(5)R + f 4χ4 V1R4 − Λ(5)R5

this solution is not stable to perturbations

Csaki, Graesser, Kolda, JT hep-ph/9906513, hep-ph/9911406

Goldberger-Wise stabilized RS

brane potentials

m2 = −4 ✏ k2 ∆ ≈ 4 − ✏ φ = φ0 e✏ky V0 = λ0(φ − v0)2 V1 = λ1(φ − v1)2 λ0, λ1 → ∞ Veff = V0 + Λ(5)R + f 4χ4 V1R4 − Λ(5)R5

two fine tunings

ds2 = e−2A(y)dx2 − dy2

hep-ph/9907447

Goldberger-Wise stabilized RS

brane potentials

V0 = λ0(φ − v0)2 V1 = λ1(φ − v1)2

1 2 3 4 5 1 2 3 4 5 0.5 1 1.5 2 2.5 3

Goldberger-Wise stabilized RS

brane potentials

V0 = λ0(φ − v0)2 V1 = λ1(φ − v1)2

1 2 3 4 5 1 2 3 4 5 0.5 1 1.5 2 2.5 3

Goldberger-Wise stabilized RS

brane potentials

V0 = λ0(φ − v0)2 V1 = λ1(φ − v1)2

1 2 3 4 5 1 2 3 4 5 0.5 1 1.5 2 2.5 3

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 AdS5

assuming we have fine-tuned the UV contributions

ds2 = e−2A(y)dx2 − dy2 A02 = κ2φ02 12 − κ2 6 V (φ) φ00 = 4A0φ0 + ∂V ∂φ Veff(y1) = e4A(y1)  V1 (φ(y1)) + 6 κ2 A0(y1)

UV behavior with a bulk mass

φ = φ0 e✏ky A = ky − κ2φ2 12

• e2✏ky − 1
• 00 − 4A00 + 4✏k2 = 0

dominant balance

perturbative RG evolution

IR behavior with a bulk mass

00 − 4A00 + 4✏k2 = 0 dominant balance

singularity at yc

A0(y) = −k coth (4k(y − yc)) φ(y) = c κ − √ 3 2κ log (tanh (2k(yc − y)))

Csaki, Erlich, Grojean, Hollowood hep-th/0004133

Boundary Layer Matching

yc = 1 ✏k log c 0 − √ 3 20 log 1 − (0)4 1 + (0)4 ! φ(y) = φ0e✏ky − √ 3 2κ log [tanh (2k(yc − y))] − φ0e✏kyc + c κ

• cf. Chacko, Mishra, Stolarski hep-ph/1304.1795

5D Dual of Spontaneously Broken Scale Invariance

A0(y) = k coth (4k(yc − y)) φ(y) = φ0 e✏ky − √ 3 2κ log (tanh (2k(yc − y))) Veff = e4A(y1)  Λ1 + 6 κ2 A0(y1, yc)

• = e4A(y1)Vb

V 0

eff = e4A(y1)

 −4A0Vb + d dy1 Vb + dyc dy1 d dyc Vb

• by boundary condition

φ

potentially 24 orders of magnitude better than SUSY

V ∼ ✏ (TeV)4 m2

dil ∼ ✏ (TeV)2

A0(y) = k coth (4k(yc − y)) φ(y) = φ0 e✏ky − √ 3 2κ log (tanh (2k(yc − y)))

5D Dual of Spontaneously Broken Scale Invariance

Cosmological Constant in TeV4

16.0 16.5 17.0 17.5 18.0

• 0.02
• 0.01

0.01 0.02 16.0 16.5 17.0 17.5 18.0

• 0.002
• 0.001

0.001 0.002

✏ = 0.01 ✏ = 0.001 Λ1 = 2ΛRS

Weinberg’ s No-Go Theorem

You can’ t have your cake and eat it too! can remove the c.c. but to have a unique vacuum

✏ = 0 ✏ 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

is a massless Nambu-Goldstone boson

L = 1 2 ∂νφ ∂νφ hφi = v φ

SUSY 3-2 Model is a Broken CFT

SU(3) SU(2) U(1) U(1)R Q 1/3 1 L 1 1 3 U 1 4/3 8 D 1 2/3 4

W =

Λ7

3

det(QQ) + λ Q ¯

DL

⌧ V = | ∂W

∂Q |2 + | ∂W ∂U |2 + | ∂W ∂D |2 + | ∂W ∂L |2

⇡

Λ14

3

φ10 + λ Λ7

3

φ3 + λ2φ4

hφi = Λ3 λ1/7

SUSY 3-2 Model is a Broken CFT

SU(3) SU(2) U(1) U(1)R Q 1/3 1 L 1 1 3 U 1 4/3 8 D 1 2/3 4

mdil = λhφi = λ6/7Λ3 hφi = Λ3 λ1/7 Vmin ≈ λ2 < φ >4

Scale invariant action

invariance requires

x → x0 = eαx O(x) → O0(x) = eα∆O(eαx) S = X

i

Z d4x giOi(x) − → S0 = X

i

Z d4xeα(∆i4)giOi(x) ∆i = 4

5 spontaneously broken conformal generators give one Goldstone Boson

Sydney Coleman, “Aspects of Symmetry” see also Ian Low, hep-th/0110285

Mµν = −i(xµ∂ν − xν∂µ) Pµ = −i∂µ Kµ = −i(x2∂µ − 2xµxα∂α) D = ixα∂α