Siva Swaminathan
October 20, 2015
71

# Can a pseudo-Nambu-Goldstone-Boson Higgs lead to symmetry non-restoration?

Talk given in the "Many-Body Theory" class.

October 20, 2015

## Transcript

1. Can a pNGB Higgs lead to symmetry
Can a pNGB Higgs lead to symmetry
non-restoration?
non-restoration?
(w/ Can Kilic)
(w/ Can Kilic)
Sivaramakrishnan Swaminathan
Sivaramakrishnan Swaminathan
arXiv:1508.05121
arXiv:1508.05121
20 October 2015 (Many-body theory)
20 October 2015 (Many-body theory)

2. The big picture
The big picture

3. Motivations
Motivations
What happens to electroweak symmetry at high temperatures?
Baryogenesis (How was matter created?)
Heuristic thermodynamic argument hints at symmetry restoration
Counter examples (Rochelle salts, �nite chemical potential, etc.)
F = E − TS

4. The effective potential
The effective potential

5. The idea (a la Landau-Ginzburg)
The idea (a la Landau-Ginzburg)
Let the order parameter be a �eld
Free-energy cost of homogeneous �eld con�gurations
Include quantum effects ("integrate out" all �uctuations)
Think "bubbles" ( )
Position of minimum corresponds to particle density in the condensate
F = −β log Z

6. Integrating out the �uctuations
Integrating out the �uctuations
Cost of zero-point energies of �eld �uctuations around a putative
background �eld value: harmonic oscillator for each mode
VCW
(H)
VCW
= STr ∫ [ ]
d3k⃗ 1
2
+
k2 m2
− −

= STr [ (H) + (log ( ) − )]
m2
Λ2
16π2
(H)
m4
64π2
(H)
m2
Λ2
3
2
At one-loop, the cost of one-�uctuation depends only on the background,
and is independent of other �uctuations
"Integrate out �uctuations" : up to what scale?

7. At �nite temperature
At �nite temperature
For each �uctuation: Modify spectral function by adding thermal
occupations
(ϕ, T) = (ϕ) + Δ (ϕ, T)
Veff, 1-loop VCW V T
1
Δ = ( )
V T
1

i
T 4
2π2
ni
Jb/f
m2
i
T 2
( )
Jb
xi
( )
Jf
xi
= dt log [1 − ]

0
t2 e− +
xi
t2

= dt log [1 + ]

0
t2 e− +
xi
t2

8. High temperature expansion
High temperature expansion
(x)
Jb
− (x)
Jf
=
=
− π4
45

7π4
360
+ x
π2
12
+ x
π2
24
− πx 3
2
6
− log ( )
x2
32
x
ab
+ log ( )
x2
32
x
af
+ …
+ …
Note the correspondence with the zero temperature contribution
bosons:
fermions: −
Λ2
16π2
Λ2
16π2
⟶ T 2
12
⟶ .
T 2
24
Both fermions and bosons contribute to with the same sign!
T 2

9. Particle physics today
Particle physics today

10. We
We �nally
�nally found the Higgs!
found the Higgs!
v ≈ 173GeV , ≈ 125GeV
mh

11. The Standard Model
The Standard Model

12. The hierarchy problem
The hierarchy problem
Mass term: Self energy of background �eld, at zero-external momentum
Mass term, and hence VEV depends sensitively on the UV-cutoff!
Supersymmetry models try to exploit the opposite sign between fermion
(quark) and scalar ("squark") corrections above.

13. The Twin Higgs
The Twin Higgs

14. SU(4)/SU(3)
SU(4)/SU(3) NLSM
NLSM
H = [ ] = − [− + λ ]
HA
HB
Lscalar
|∂H|2
m2|H|2 |H|4
Integrate out the radial mode; work with NLSM
symmetry prevents contributions to !
H = exp i
f

h∗
1
0
h∗
2
h∗
3
h1
h2
h3
h0

0
0
0
f

SU(4) Λ2 m2
h

15. SU(4)/SU(3)
SU(4)/SU(3) NLSM (2)
NLSM (2)
H [ ] = ≈ [ ]
− →

Exploit [ ]
SU(2)
A
SU(2)
B
|⟨ ⟩|
HA
|⟨ ⟩|
HB

f sin ( )
v
f
f cos ( )
v
f

v
f − v2
2f
What happens to ?
3 : eaten by gauge bosons
3 : eaten by gauge bosons
1 : "magnitude" of the SM Higgs
H
R SU(2)
A
R SU(2)
B
R
R

16. A caricature model
A caricature model
Twin image of SM (A B)
Forget the gauge bosons for now (Set )
Focus on the top quark sector (forget all other fermions)

= 0
gi
Top sector
Top sector
= y [ (ϵ ) + (ϵ ) ] + h.c.
Lup-type
yukawa
Q
¯
A
HA
UA
Q
¯
B
HB
UB
Yukawas break but have twin symmetry
SU(4) (A ↔ B)

17. Twin mechanism
Twin mechanism
Discrete symmetry causes quadratic terms to mimic symmetry
(which is otherwise broken)
Z2 SU(4)
− ( + ) = −
3y2Λ2
8π2
H†
A
HA
H†
B
HB
3y2
8π2
Λ2|H|2

18. High temperature behaviour
High temperature behaviour

19. Observations
Observations
Symmetry does get restored!
f = 450GeV, μ = 90GeV, = 1500GeV
Tmax

20. Insights
Insights
The leading dependence cancels away ( just like )
Subleading temperature dependence ( ) restores symmetry
Those corrections are unavoidable, since terms are what give the
Higgs a �nite mass ( ), in such models
T T 2 Λ2
log T
log Λ
125GeV

21. Thank you!
Thank you!