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Electroweak Theory, SSB, and the Higgs: Lecture 3

• Theoretical aspects of spontaneous symmetry breaking

• Standard model Higgs scalar H (general discussion for arbitrary mass)

!1

!2!

!!

!!

!!

V (!)

!1

!2!

!!

!!

!!

!!

V (!)

φ =

(φ+

φ0

)−−−−−−−−→unitary gauge

1√2

(0

ν +H

)

- Gauge interactions: ZZH,ZZH2,W+W−H,W+W−H2

(Dµφ)†Dµφ =1

2(∂µH)2 +M2

WWµ+W−µ

(1 +

H

ν

)2

+1

2M2ZZ

µZµ

(1 +

H

ν

)2

(quartic and induced cubic interactions, ∝ (ν)M2/ν2)

ν =2MW

g=

2MZ cos θWg

∼ 246 GeV

2

W!!

W+µ

H2igµ!M2

W!

W+!

W!µ

H

H

2igµ!M2

W!2

f

f

H!imf!

Z!

Zµ

H2igµ!M2

Z!

Z!

Zµ

H

H

2igµ!M2

Z!2

H

H

H!3iM2

H!

H

H

H

H

!3iM2

H!2

– Typeset by FoilTEX – 1

- Higgs potential:

V (φ) = +µ2φ†φ+ λ(φ†φ)2

→ −µ4

4λ− µ2H2 + λνH3 +

λ

4H4

Fourth term: quartic self-interaction

Third: induced cubic self-interaction

Second: (tree level) H mass-squared, MH =√−2µ2 =

√2λν

First: constant (irrelevant until gravity added ⇒ cosmological

constant)

!

"

!

– Typeset by FoilTEX – 1

3

- Yukawa couplings of Higgs to fermions

−LY uk =∑i

mi ψ̄iψi

(1 +

H

ν

)

– Coupling mi/ν is flavor diagonal and small except t quark

– H → b̄b dominates for MH . 2MW (H →W+W−, ZZ, t̄t

dominate when allowed because of larger gauge/top couplings)

– Flavor diagonal: only one doublet couples to fermions ⇒fermion mass and Yukawa matrices proportional

– Often flavor changing couplings in extended models withtwo doublets coupling to same kind of fermion (not MSSM)

(Stringent limits, e.g., tree-level Higgs contribution to KL −KS

mixing (loop in standard model) ⇒hd̄s/MH < 10−6 GeV−1)

• Theoretical bounds on MH

M2H = 2λν2, λ =

g2M2H

8M2W

=GFM

2H√

2

- Tree-level: no a priori constraint on λ except vacuum stability

(λ > 0 ⇒ 0 < MH <∞)

- Loop-level: triviality, tree unitarity, (meta) stability

- MSSM: much of parameter space has standard-like Higgs with

MH < 130 GeV (λ related to gauge couplings)

(MH < 150 GeV in extensions)

- Running gauge couplings (vacuum polarization)

dg2id lnQ2

= big4i︸︷︷︸

1 loop

– Typeset by FoilTEX – 1

4

bgs = − 1

16π2

[11− 4F

3

]−−−−−−−→F=3,nH=1

1

16π2(−7)

bg = − 1

16π2

[22

3− 4F

3− nH

6

]−−−−−−−→F=3,nH=1

1

16π2

(−19

6

)bg′ = +

1

16π2

[+

20F

9+nH6

]−−−−−−−→F=3,nH=1

1

16π2

(+

41

6

)

0 2 4 6 8 10 12 14 16 18 20

log10

µ (GeV)

02

04

06

08

01

00

!i-1

!1

"1

!2

"1

!3

"1

!1

"1

!2

"1

!3

"1

0 2 4 6 8 10 12 14 16 18 20

log10

µ (GeV)

02

04

06

08

01

00

!i-1

Standard Model

Supersymmetric Standard Model

MSUSY

= MZ

(αi = g2i /4π, g1 =

√53g′ (GUT normalization))

5

- Quartic coupling λ and top-Yukawa ht also run

dλ(Q2)

d lnQ2=

1

32π2

[24λ2 + 24λh2t − 24h4t − 3λ

(3g2 + g′2

)+

3

8

(2g4 + (g2 + g′2)2

)]dht(Q

2)

d lnQ2=

1

32π2

[9h3t − ht

(8g2s +

9

4g2 +

17

12g′2)]

HH

H H

H H

t

t t

t

H H

H H

– Typeset by FoilTEX – 1

– λ(ν2) ≡ GFM2H√

2(> 1 for MH & 350 GeV)

– ht(ν2) ≡ mt/ν ∼ 0.7

• The triviality upper limit

- λ2 dominates for large MH ⇒

λ(Q2) =λ(ν2)

1− 3λ(ν2)4π2 ln Q2

ν2

- Diverges at Landau pole QLP = νe2π2/3λ(ν2)

- Require QLP > Λ = new physics scale (triviality limit) ⇒

MH <

(2√

2π2

3GF ln(Λ/ν)

)1/2

∼{O(140) GeV, Λ ∼MP

O(650) GeV, Λ ∼ 1500 GeV

(Planck scale: MP = G−1/2N ∼ 1.2× 1019 GeV)

- Lattice: MH < 650− 700 GeV

6

(Hambye and Riesselmann, hep-ph/9708416)

• Tree unitarity

- W+LW

−L →W+

LW−L scattering

(longitundinal εµ(~k, 3) ∼ kµ/MW,Z grows with k)

M = −i√

2GFM2H

(s

s−M2H

+t

t−M2H

)(for s,M2

H �M2W ; s ≡ E2

CM , t ≡ momentum transfer2)

W + W !

W + W !

Z, !, H

W + W !

W + W !

Z, !, H

W + W !

W + W !

– Typeset by FoilTEX – 1

- M → constant for MH →∞, violating tree-unitarity(equivalent to no Higgs)

- s�M2H : violate tree-unitarity unless

MH ≤

(4π√

2

3GF

)1/2

∼ 700 GeV

(or else strong coupling)

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• Lower limit from loop corrections to vacuum stability

- h4t dominates for small λ⇒

λ(Q2) ∼ λ(ν2)− 3h4t4π2

lnQ2

ν2

- λ(Q2) < 0 for Q2 > Q2−, with M2

H =3h4

t√2π2GF

ln Q−ν

- Vacuum unstable unless Q− > Λ ⇒ lower bound on MH

(or upper bound on Λ)

- MH = 125 GeV: λ→ 0 at ∼ 1011 GeV

- Stability for Λ = MP requires MH & 130 GeV

- Weaker constraint for metastable vacuum (with lifetime > 13.8

Gy): MH & 115 GeV for Λ = MP

- Stability limit doesn’t apply in MSSM (Λ ∼MSUSY )

• Complementary limits from standard model (SM) and MSSM:

MH & 130 (115) GeV (SM for large Λ), MH < 130 GeV (MSSM)

- 125 GeV does not distinguish and is challenging for both

• Extended Electroweak Symmetry Breaking Sectors

- Extended elementary Higgs sectors: (extra doublets (e.g., MSSM),

singlets, triplets) (Higgs production/decays; additional

charged/neutral spin-0; possible FCNC, violation of custodial

symmetry, CP ; possible electric charge violation)

- Strong dynamics

– Break SU(2)× U(1) directly or by BC (usually no Higgs-like

particle)

– Composite Higgs (by strong dynamics, e.g., pseudo-Goldstone

boson)

(new particles/interactions)