The Planck Scale from Top Condensation

Yang Bai

Theoretical Physics Department, Fermilab

with Marcela Carena and Eduardo Pontón

arXiv:0809.1658

May 22, 2009@Collider Physics 2009

1 The hierarchy problem

In the standard model, the weak scale is not stable against radiative corrections: v EW 10−16 ??? MPlanck ≈

Proposed possible solutions:

Technicolor model (or its Ads/CFT version: Higgsless model)

Supersymmetry

Warped Extra Dimension

2 Randall-Sundrum model

2 −2ky µ ν 2 The 5-dimensional geometry is: ds = e ηµν dx dx dy −

c=1.4 c=-0.4 Gluon 1'st KK

UV IR

Need a Goldberger-Wise field φ propagating in the bulk to stabilize the distance between the IR and UV branes

The large hierarchy is given:

2 TeV − 4 k vIR ∼ = e m2 MPlanck vUV

3 Randall-Sundrum model

A list of unsatisfied points in the RS model:

It does not address how the electroweak symmetry is broken

There is no real dynamical linkage between the MPlanck and H = v = 175 GeV h i EW

There exists a little hierarchy between the KK-scale MKK and vEW . MKK > 2 TeV [Carena, Pontón, Santiago, Wagner] : Little hierarchy problem

4 How the electroweak symmetry is broken?

In the standard model a Higgs doublet with a potential

λ V (H) = m2 HH† + (HH†)2 m2 < 0 2

2 m h T vEW = − 174 GeV H = (0, vEW + ) r λ ≈ √2 the Higgs boson an elementary scalar ↔ Questions: 2 m < 0? v MPlanck ? EW ≪

5 Strong Dynamical Symmetry Breaking

Technicolor Model [Weinberg and Susskind] 3 Q¯ LQR Λ breaks the chiral symmetry Λ = O(100) GeV h i≈ TC TC QL weak doublet and QR singlet vEW Λ ∼ TC vEW /MPlanck is due to “dimensional transmutation” No fundamental scalar

Difficulties (for QCD-like dynamics) satisfying electroweak precision observables like the S parameter generate the top quark mass large enough Possible Solution:

Walking Technicolor (not QCD-like dynamics) [Appelquist, Shrock, Sannio ··· ]

6 Top Condensation [Nambu; Miransky, Tanabashi and Yamawaki]

Top quark is peculiar: the heaviest quark in the SM

the top quark mass mt = 172.4 GeV is not far from vEW 174 GeV ≈ the RG running equation of the top Yukawa coupling shows an infrared fixed point for the top quark mass [Pendleton and Ross; Hill]

Therefore

the electroweak breaking may be from ¯tL tR h i = Top Condensation ⇒

7 Nambu Jona Lasinio (NJL) model

An effective lagrangian with 4-fermion interactions

g2 = iψ Dψ + iψ Dψ + (ψ ψ )(ψ ψ ) L L 6 L R 6 R M2 L R R L Using an auxiliary “Higgs” field, H, rewrite

(M)= iψ Dψ + iψ Dψ M2H†H + (gHψ ψ + h.c.) L L 6 L R 6 R − L R Renormalization group running to a lower scale

(µ) = L iψ Dψ + R iψ Dψ + ( g gHψ ψ + h.c.) L Z L 6 L Z R 6 R Z L R † µ 2 † λ † 2 + ∂µH ∂ H m H H (H H) ZH − H − 2 2 2 Nc g M with ln H = 2 2 Z 16 π µ !

H is a composite particle

8 NJL model (cont.)

g2N µ2 g4N M2 m2 M2 1 c 1 c ln H 2 2 λ 2 2 ≈ " − 8π − M !# ≈ 8π µ !

2 2 8π2 H acquires a non-vanishing VEV if g > Gc (valid at large Nc limit) ≡ Nc The Higgs potential is

λ¯ m2 λ V H m2 H†H H†H 2 m2 H ¯ ( ) = H + ( ) with H = λ = 2 2 H Z ZH 2 ¯ The electroweak scale vEW = mH /λ q− 2 Origin of the strong coupling (g 1) or the UV theory? Topcolor[Hill] ≫ 2 Fine-tuning problem? For vEW M, need to adjust g very close to the 2 ≪ critical value Gc . So far, no solution

9 Wave-function

Consider an SU(Nc ) gauge theory in the AdS5 background (0 y L) ≤ ≤ 2 −2ky µ ν 2 ds = e ηµν dx dx dy −

( 1 −c)ky 1 1 f (y)= √ρ e 2 UV(IR) localized for c > (c < ) c c 2 2

c=1.4 c=-0.4 Gluon 1'st KK

UV IR otherfermions topquark

10 Four-fermion Operator

2 g5 k The 4-d SU(Nc) gaugecouplingis 4 π αs = k L

Integrating the 1’st KK mode leads to 4-fermion operators of the form

gc gc gc gc − 1 2 (ψ T Aγµψ )(ψ T Aγµψ ) = 1 2 (ψ ψ )(ψ ψ ) + O(1/N ) 2 1L 1L 2R 2R 2 1L 2R 2R 1L c MKK MKK

−kL The 1’st KK gluon mass is MKK = x1ke , where x1 ≈ 2.45.

2 2 f2(c1, c2) gc1 gc2 gc1 gc2 ≈ g5 kx1 f1(c1, c2) − ≈ 30 (for c1 = c2 = −0.5) » kL – 4 π αs Randall-Sundrum Model Top Condensation:

The coupling of the KK gluon to fermions can be much larger than the zero mode.

The 4-fermion operator generated by integrating out the first KK gluon can be over the critical coupling in the NJL model and induce top condensation, if the top quark is localized very close to the IR brane.

11 Radion and Higgs Potential

Choosing the interval L between UV and IR branes as a free parameter

Referring back to the NJL analysis, we have a radion and Higgs coupled potential

λ¯(L) V (H, L) = m2 (L) H†H + (H†H)2 H 2 2 λ¯(L) m2 (L) m4 (L) = H†H + H H 2 " λ¯(L) # − 2λ¯(L)

2 4 g Nc µ2 g Nc M2 m2 M2 1 ψ 1 ψ ln KK H KK 2 2 λ 2 2 ≈ " − 8π − MKK !# ≈ 8π µ !

2 m2 λ Nc g M2 m2 H ¯ ψ ln KK M x ke−k L H = λ = 2 H = 2 2 KK 1 H Z 16 π µ ≡ Z ZH !

12 Radion and Higgs Potential

2 2 2 f2(c1, c2) g gc gc g k x f (c , c ) ψ ≡ 1 2 ≈ 5 1 1 1 2 − kL » – 2 2 2 Choosing c1 and c2 to require g5 k x1 f1(c1, c2) > Gc

There is a critical value, Lc , defined by

f (c , c ) g2k x2 f (c , c ) 2 1 2 = G2 5 1 1 1 2 − kL c » c –

such that when L > Lc, H = 0 and L < Lc , H = 0. h i 6 h i 2 M4 1 1 4 KK G2 g2 mH (L) c − ψ Veff(L) = θ(L Lc) „ « θ(L Lc) ¯ M2 − 2λ(L) − ≈− Nc log KK − 4π2 µ2

−k L L < Lc, Veff = 0; L , MKK x ke goes to zero and Veff 0 →∞ ≡ 1 →

There is a minimum for Veff(L)

13 Radion Potential

Lc 0 L L H eff V

L

The fermion condensation can stabilize the relative distance of two branes

A novel way to realize the Goldberger-Wise mechanism to stabilize the radion Top Condensation Randall-Sundrum Model

14 Scales: Leading Order Analysis

The minimum of the radion potential is

¯ f2 1 ¯ 2 2 kLmin + fi g k x fi (c1, c2) ≈ ¯f G2 2 ≡ 5 1 1 − c

¯ 2 2 2 2 (f1−Gc ) The 4-fermion coupling gψ Gc + ¯ ≈ 2f2 The electoweak scale is

¯f G2 1 ¯f G2 1 v M 1 − c = x ke−kLmin 1 − c EW KK 2 ¯ 1 2 ¯ ≈ Gc ! s 4 f2 Gc ! s 4 f2

g2 ¯ 2 ψ −3 MKK If f1 is 10% close to Gc , we have 2 1 10 and vEW = O( 100 ) Gc − ≈

vEW is two orders of magnitude below MKK (Little Hierarchy) A consequence of the radion stabilization using the strong dynamics

15 Scales: Leading Order Analysis

70 c1 = c2 c1 = c2 60 400 g5 k = 5.08 g5 k = 5.08 k = 2‰1017 GeV k = 2‰1017 GeV 50 300 EW v min 40  L KK k

30 XH\ = vEW M 200 XH\ = vEW

20 100

10

1.05 1.10 1.15 1.20 1.25 1.30 10 20 30 40 50 60 70 2 f 1Gc k Lmin

MKK is predicted to be around 35 TeV

16 Scales: Improved Analysis

2 d g¯ψ 9 2 2 9 2 17 2 16π = g¯ψ g¯ 8 g¯ g¯ g¯ d ln µ 2 ψ − 3 − 4 2 − 12 Y » – d λ¯ 1 3 1 1 3 16π2 = 12 λ¯2 + (g¯2 g¯2 g¯2) λ¯ + g¯4 + g¯2 g¯2 + g¯4 g¯4 d ln µ ψ − 4 Y − 4 2 16 Y 8 Y 2 16 2 − ψ » –

Using the “compositeness conditions” g¯ = and λ¯ = at the MKK scale [Bardeen, Hill, Lindner] ψ ∞ ∞

20 000 20 000

10 000 10 000

5000 5000 D D 2 ΛHmhL vEW = mh

GeV g m v = m Ψ I ΨM EW Ψ GeV @ 1000 @ 1000 h Ψ m m 500 400 300 200

100 100 100 200 400 1000 5000 10 000 20 000 100 300 500 1000 5000 10 000 20 000 Μ @GeVD Μ @GeVD

m = 375 25GeV and m = 475 25 GeV ψ ± h ± 17 Caveats

The NJL analysis is only trustable in the large Nc approximation.

The Casimir energy generated by gauge boson loops can dominate the potential from top condensation and shift the VEV of radion. This can be suppressed by introducing “inert" fermions to match the d.o.f of bosons.

The non-calculable contributions to the radion potential through the IR tension, which are sensitive to the UV completion of the 5D theory, may be linked to the cosmological constant problem and suppressed by unknown 5D UV physics.

How to understand the interplay of Higgs and radion (dilaton) fields in terms of a 4d effective field theory is unknown now.

18 The realistic model

In the “top sector”, charged as 3 2 , and as 3 1 ΨQL ( , )1/3 Ψ1 Ψ2 ( , )4/3

choosing mixed boundary conditions for Ψ1 and Ψ2

one linear combination cos θ Ψ1R + sin θ Ψ2R has right-handed zero mode tR

another linear combination has ultralight Dirac KK mode: χL and χR, with a mass md (c1, c2,θ)

the low-energy fermions are QL, tR , χL and χR

In a new basis: t′ cos α t + sin αχ and χ′ sin α t + cos αχ R ≡ R R R ≡− R R g 0 t′ ¯′ ′ c1 A A µ R (t χ¯ ) Gµ T γ ′ R R 0 gc χ „ 2 « „ R «

−1 α = tan ( ρc /ρc tan θ) − 1 2 q

19 Top quark mass and EWSB

The top sector mass matrix is (“Top Seesaw” [Dobrescu and Hill])

sin α g¯Qχ H cos α g¯Qχ H − R h i R h i tR (t L χL) 0 1 χR 0 md „ « @ A In the limit that the Dirac mass is large, m g¯ H , the physical top quark mass is d ≫ QχR h i

m2 sin2α (g¯ H )2 t ≈ QχR h i m2 m2 + cos2 α (g¯ H )2 χ ≈ d QχR h i Using the RG improved value of g¯ H 375 GeV, we need sin 0 46 to fit the top QχR = α . quark mass. h i ≈

We have 5 model parameters: g5, cQ, c1, c2 and θ(α) to fit three observed quantities: αs, vEW and mt . We are left with two free parameters in the model: c1 and c2.

20 Top quark mass and EWSB

1 2 ′ ′ 2 ′ ′ 2 ′ ′ 2 ′ ′ g (χ χ )(χ χL)+ g (χ t )(t R χL)+ g (QLχ )(χ QL) + g (QLt )(t R QL) M2 χLχR L R R χLt L R QχR R R Qt R KK h i 2 2 2 2 2 Choosing cQ c2 < c1 < 1/2, g , g < 0 < g < Gc < g [χL is UV localized] ≤ − χLt χLχR Qt QχR the condensation and radion stabilization happen in the channel Q¯ χ′ h L Ri

34 700

-0.62

34 600 -0.64 D

-0.66 GeV

Q 34 500 @ c

-0.68 KK M 34 400 -0.70

-0.72 34 300 -0.60 -0.58 -0.56 -0.54 -0.52 -0.50 -0.60 -0.58 -0.56 -0.54 -0.52 -0.50 c2 c2

21 Radion

The radion-dependent terms arise from the 5D Einstein-Hilbert action

3 2 M5 4 φ 1 4 µ −1 S = d x g 1 + d x g ∂µφ∂ φ V H, k ln φ/F k − − F 2 R4 2 − − − Z ! Z p p n “ ”o where φ(x) Fe−k T (x) with T = L and F = 12M3/k. ≡ h i 5 q φ = F˜ = Fe−kLmin h i 4D Planck mass is M2 M3/k (2 1018 GeV)2, we have F 2√3M 6.9 1018 GeV P ≈ 5 ∼ × ≈ P ≈ × The mass of the radion takes the approximate form

¯ 3 x1 f2 k MKK k mϕ 1 (4 GeV) ≈ 3 2 x1 k MKK ≈ M 64 π log log 2 MP P MKK µ “ ” “ ”

22 Radion (cont.)

3.5 500

3.0

400 2.5 D D 2.0

TeV 300 GeV @ @ Φ 1.5 \ Φ m X 200 1.0

0.5 100 0.0 5. ´ 1017 1. ´ 1018 1.5 ´ 1018 2. ´ 1018 5. ´ 1017 1. ´ 1018 1.5 ´ 1018 2. ´ 1018 k @GeVD k @GeVD

−1 LEP imposes bounds on the couplings of a light scalar to Z gauge bosons: vEW/ φ < 10 h i

g e φ ¯ 2 µ 2 +µ β( s) aµν β( ) µν int = Fcψ mψψ ψ + MZ Z Zµ + 2MW W W−µ + G Gaµν + F Fµν L φ 2 2 gs 2 e 3 h i Xψ 4 5 −4 In our case: vEW/ φ = few 10 h i × [arXiv:0903.3410: Davoudiasl, Pontón] 23 Electroweak Precision Constraints

the T parameter [He, Hill and Tait] the 475 GeV Higgs boson contributes 0.2 to ∆T the χ fermion loop provides a positive and∼− non-negligible contribution Therefore, there is an upper bound on mχ

the S parameter the 475 GeV Higgs boson contributes 0.07 to ∆S while χ contributes 0.01 ∼ ∆S = 0.08 for a wide∼ range of mode parameter space

the Z b¯L bL coupling

2 4 2 2 e (g¯Qχ cos α H ) m m δgloop R h i 1 + 2 t log χ 1 bL ≈ 64π2s2 M2 m2 (g¯ cos α H )2 m2 − W W χ " QχR h i t !#

loop −3 δg /gb 2.4 10 for mχ = 2 TeV, g¯Qχ H = 375 GeV and α = 0.48 bL L ≈− × R h i 24 T and S parameters

T parameter S parameter

-0.520 -0.520

0.0 0.079 -0.525 -0.525

0.04 0.05 -0.530 -0.530 0.080 2 2 c c -0.535 0.10 -0.535

0.15 0.081 -0.540 -0.540

0.20 -0.545 -0.545 0.082 0.25 0.23

0.30 -0.550 -0.550 -0.54 -0.52 -0.50 -0.48 -0.46 -0.44 -0.42 -0.40 -0.54 -0.52 -0.50 -0.48 -0.46 -0.44 -0.42 -0.40 c1 c1

25 Constrains on the fermion χ mass

S=0.08 0.003

99% 0.002

95% 3500

0.001 3000 d g D 0.000 2500 GeV @

Χ 2000

-0.001 m

1500

-0.002 1000 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.05 0.10 0.15 0.20 T DT

1.6 TeV < mχ < 2.9 TeV

26 LHC Collider Phenomenology

The heavy Higgs boson with a mass around 475 GeV and SM-like a total width of Γtotal 56.6 GeV; h ≈ Br(h ¯tt) 19.5%, Br(h W +W −) 54.5% and Br(h ZZ ) 26.0%, → ≈ → ≈ → ≈ the decay h ZZ 4 ℓ is the “gold-plated” mode for discovery at the LHC → → The vector-like fermion χ with a mass around 2 TeV; The mixing angle of the χ and t is β 0.16; its total width is around 140 GeV L L L ≈ 2 2 1 cot α mt Γ(χ th) = Γ(χ tZ ) = Γ(χ bW ) = mχ → → 2 → 64 π v 2

−1 can be discovered up to mχ = 2.5 TeV with 300 fb at 5 σ in the decay chain χ bW ℓν b [G. Azuelos et al.] → → χ h t ZZWb , 3 W b and 3 W 3 b would provide interesting signal topologies → → for the presence of the heavy quark and Higgs boson

27 Conclusions

Randall-Sundrum Model ⇋ Top Condensation

From left to right : the flavor structure of the RS model naturally provides a 4-fermion operator with a large coefficient, which is necessary for top quarks to condense and to break the electroweak symmetry dynamically.

From right to left : the top condensation can stabilize the radion (the distance between the UV and IR branes). Therefore directly links the Planck and the eletroweak scale or the top condensation scale. All together

A complete story from the Planck scale to the weak scale is known.

The Kaluza-Klein scale ( 35 TeV) is predicted to be about two orders of magnitude ∼ above the electroweak scale.

In a realistic model constructed, three new particles below 10 TeV: heavy Composite Higgs ( 500 GeV), heavy top quark ( 2 TeV) and light radion ( 1 GeV). ∼ ∼ ∼

28