tree-level vac cum stability in thdm r. santos next

September 18 2008 Uppsala

Tree-level Vaccum

stability in THDM

R. SantosR. Santos

NExT

with A. Barroso, P. Ferreira and N. Sá


Can the SM potential have a Normal anda CB minima simultaneously?

Local minimum –NORMAL (VN)

Global minimum – CHARGE BREAKING (VCB)

0m

! 0m

Charge breaking in the SM


0

vSM

43

21

viv

vivnewSM

What you see in books!

What you don’t see in books!


'2

2

newSM

aa

newSM BYg

iWg

iD

4

'

422

4400

20

40

200

4

22224

23

22

2132414231

24

23

22

21

22

3241

22

4231

22

Yvgv v vv

g g'Yv v vv

g g'Yv + vvv

g g'Y

v v vvg g'Yvg

v v vvg g'Yvg

v + vvvg g'Yvg

)()( newSM

newSM DD

00

00

01

2

1 3

vYQ SMSMSM


0

'4

4

24

2222

23

2222

21

m

Yggv

m

vgmm

24

23

22

21 vvvvv

U(1) always survives in the SM!

Gauge boson masses in the most

general case

Electric charge is conserved!Look, it’s the PHOTON!

The Weinberg angle is the same!



Charge breaking in the THDM

Conclusion: No Charge Breaking in the SM!

8 possible vevs

43

21

kk

kk

kviv

vivNow we have two

doublets 21

What about CB in THDM?

0

1

iev

3

2

v

v 2

d

c

id

ic

ev

ev

1

b

a

ib

ia

ev

ev

SU(2) U(1)



two ways to recover the photon

SM

alignment02 v

01 vLook It’s the…

PHOTON?

222

21

2222224222224

222

21

2222224222223

2222

21

168

1

168

14

Y v vg' gYg'gvYg'gvm

Y v vg' gYg'gvYg'gvm

vgmm

23

22

21 vvvv

mass eigenvalues

most simple configuration

0

1

iev

3

2

v

v

or else…



Are we in DANGER?!...Suppose we live in a THDM!

CB is possible in THDM!

22

11 '

e '

0

vv

► CHARGE BREAKING

2

21

1 '

0 e

'

0

viv ► CP BREAKING

22

11

0 e

0

vv► NORMAL

Three minimum field configurations


2

2 2 21 2 2 1 12' '

20

HCB N

MV V v v v v v

v

2

22H

M

v

Calculated at the Normal stationary point.

Some algebra

VN is a MINIMUM2

20

2H

M

v

N CBV V

Bonus VCB is a Saddle Point

Are we in DANGER?!...

No! VN is below VCB and tunnelling will

not happen!

For a safer THDM

2

22H

M

v

2

22H

M

v


2

22H

M

v

VN is a MINIMUM


2

22H

M

v

2

20

2H

M

v

VN is a MINIMUM


2

22H

M

v

N CBV V2

20

2H

M

v

VN is a MINIMUM


2

22H

M

v

Bonus

N CBV V2

20

2H

M

v

VN is a MINIMUM


2

22H

M

v

Bonus

N CBV V2

20

2H

M

v

VN is a MINIMUM


2

22H

M

v

VCB is a Saddle PointBonus

N CBV V2

20

2H

M

v

VN is a MINIMUM


2

22H

M

v


And for Cp

2

2 2 21 2 2 1 12

'' ''2

0

ACP N

MV V v v v v v

v

VCP is a Saddle PointBonus

N CPV V2

20

2AM

vVN is a MINIMUM

Calculated at the Normal stationary point.2

22AM

v


Too safe?

If VCP is a MINIMUM

Any competing N SP is a saddle point above it

However

Any competing CB SP is a saddle point above it

The THDM build on VCP

is also safe!


21

2212212

2

''2

vvvvvv

MVV H

NCB

21

2212212

2

''''2

vvvvvv

MVV A

NCP

For charge breaking

For CP breaking

21

2212212

2

2

2

''''2

1

21

12vvvvv

v

M

v

MVV

N

H

N

HNN

For 2 competing normal minima

No comments!

For charge breakingFor charge breaking


Pause! The global picture

1. THDM bounded from below have no maxima.

2. THDM have at most two minima.

3. Minima of different nature never coexist.

4. CB and CP minima are uniquely determined.

5. If a THDM has only one normal minimum than this is the absolute minimum – all other if they exist are saddle points and above it.

But how many normal minima?

Ivan Ivanov has helped to complete the picture! Proved 2. and the part of 3. we didn’t prove - that when the CP SP was above the Normal minimum it was always a saddle point – we only proved the reverse statement which is obvious from the mass expressions


Normal minima

The potential with the largest charged Higgs mass wins!

21

2212212

2

2

2

''''2

1

21

12vvvvv

v

M

v

MVV

N

H

N

HNN

Two competing normal minima


Two competing N minima?

Can we have 2 Normal minima with the same depth, with equal charged Higgs masses but with some of the other

masses different?

So, if

12 NN VV 21

2

2

2

2

N

H

N

H

v

M

v

M

No!

21

22NhNh MM Equal but

different?


The potential has 8 parameters:

It has a softly broken Z2 symmetry

It can break CP

It can break charge

The potential we love

But once we are at a Normal minimum, CB and CP will not occur. What about two normal minima for

the same parameter set?


80.4 GeVWm Is there a Normal minimum

below?

Bad news?

Local minimum –

v = 246 GeVv = 246 GeV

Global minimum – v = 329 GeV v = 329 GeV


N1 is “our” minimum

N2 is the other minimum

Well…

No pattern

No bounds

Note however that this potential is fine for CP-violating minima- in this case the minimum is uniquely determined!


Exact Z2 - the perfect THDM for Normal minima

2 22 2 1 3 5 21 2

3 5 1 2

2 22 1 2 3 5 12 2

3 5 1 2

2 ( )

( ) 4

2 ( )

( ) 4

m mv

m mv

212 0m

Using the condition for the potential to be bounded from below

we show that it is always above and it is a saddle point

Just one!

Normal

minimum

21

22 22

2

0

2

v

mv

2 22 1 2 3 5 1

1

2 ( )

2h

m mm

2 0hm


Force the CP minima not to exist (minimum equations have no solution).

2 easy (“natural”) ways of avoiding that the CP minimum equations have a solution:

• m12=0 and λ5 ≠ 0 (Φ1 Φ1 ; Φ2 Φ2)

• λ5 =0 and m12 ≠0 (Φ1 eiθ Φ1 ; Φ2 Φ2) softy broken

(otherwise axion)

Two 7-parameters models

Extending the symmetry to the fermions

Implies flavour conservation at tree level.

Z2, CP and FCNC and U(1)

Z2 and global U(1)


Conclusions I

CP

CB

N

NM

The very quiet world of the (tree-level) two-Higgs

doublet model!

When it exists, the N minimum is always below.

The CB SP is a saddle point.

NCB VvsV

CPCB VvsV

NCP VvsV

When it exists, the N minimum is always below.

The CP SP is a saddle point.

When it exists, the CP minimum is always below. The CB SP is a saddle

point (and so is the N SP).


Conclusions II

The conclusions are valid for all renormalizable THDM at tree level.

The soft broken (Z2) potential is fine for competing minima of different nature and for CP violating minima (which are

unique) but shows “problems” for two competing Normal minima. Exact Z2 potential is doing great for normal minima

but it doesn’t allow for CP violation.

The stability of the Normal minimum is not guaranteed. At one-loop the CB minimum could be below the Normal

one.


Thank youThank you


BUPBUP


Many Higgs

2 †1

2CP N A i jijV V M s s Yes!

No!

2 †1(terms that may be negative)

2CB N i jH ijV V M c c


Based on

► PLB322 (1994) Tree Level Vacuum Stability in THDM

► PLB603 (2004) Stability of the tree level vacuum in THDM

against charge or CP spontaneous violation

► PLB632 (2006) Charge and CP breaking in THDM

► PRD74 (2006) Stability of the normal vacuum in MHDM

► PLB652 (2007) Charge and CP breaking in THDM

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