SUSY MEETS ITS TWIN Andrey Katz

based on work w/ A. Mariotti, S. Pokorski, D. Redigolo and R. Ziegler

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“Is SUSY Alive and Doing Well”, IFT Workshop September, 30, Madrid, Spain

Little About the Main Question.

Is SUSY alive and doing well??

Why Supersymmetry? Hierarchy problem

Gauge coupling unification

WIMP dark matter

For 30 years, experiments testing these suggestive reasons were “right around

the corner”, and SUSY became the dominant BSM paradigm.

The experiments are now here.

HP: Theory

GUT: data DM: D

ATA

3

What kind of SUSY are we talking about?

Original idea of 80’s and 90’s?

The LHC is finally here. What do we know about the FT in garden-variety SUSY

UDD

⇤

2ùñ

UDDU :D:D:

m

⇤

4(64)

G´1F

9 pmh

q

20 `

3y2t

4⇡2⇤

2` . . . “ 174 GeV (65)

L “ |H|

2O (66)

�pt¯tq “ 0.82 nb at

?

s “ 13 TeV (67)

�pt¯tq “ 0.97 nb at

?

s “ 14 TeV (68)

�pt¯tq « 32 nb at

?

s “ 100 TeV (69)

BRpt Ñ e b {ET

q “ 13.3% ˘ 0.4% ˘ 0.4% (70)

BRpt Ñ µ b {ET

q “ 13.4% ˘ 0.3% ˘ 0.5% (71)

BRpt Ñ ⌧h

b {ET

q “ 7.0% ˘ 0.3% ˘ 0.5% (72)

⌦h29

1

x�vy

(73)

T⌫

“

ˆ10.75

75

˙1{3T⌫

(74)

�Neff

« 0.075 (75)

�phidden Ñ SMq9

´mhidden

TeV

¯n

(76)

⌦

DM

« 5 ⌦

Baryon

(77)

⌦

DM

„

mDM

mp

⌦

B

(78)

m „ �T with � " 1 . (79)

↵ ° ↵8 (80)

↵ ”

�V

⇢rad

(81)

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

5

Bounds and Fine Tunings

[GeV]t~m200 400 600 800 1000 1200

[GeV

]10 χ∼

m

0

100

200

300

400

500

600

700

800

900

CMS Preliminary

10χ∼ t→ t~, t~t~ →pp ICHEP 2016

13 TeVExpectedObserved

-1), 12.9 fbmissTSUS-16-014, 0-lep (H

-1), 12.9 fbT2SUS-16-015, 0-lep (M-1), 12.9 fbTαSUS-16-016, 0-lep (-1SUS-16-029, 0-lep stop, 12.9 fb

-1SUS-16-030, 0-lep (top tag), 12.9 fb-1SUS-16-028, 1-lep stop, 12.9 fb

-1Combination 0-lep and 1-lep stop, 12.9 fb

01χ∼

+ m

t

= m

t~m

[GeV]g~m800 1000 1200 1400 1600 1800 2000

[GeV

]10 χ∼

m

0

200

400

600

800

1000

1200

1400

1600

1800

2000 CMS Preliminary

10χ∼t t→ g~, g~g~ →pp ICHEP 2016

(13 TeV)-112.9 fbExpectedObserved

)missTSUS-16-014, 0-lep (H

)T2SUS-16-015, 0-lep (M)TαSUS-16-016, 0-lep ()φ∆SUS-16-019, 1-lep (

2-lep (SS)≥SUS-16-020, 3-lep≥SUS-16-022,

SUS-16-030, 0-lep (top tag)

Generic bound on stops ~900 GeV

UDD

⇤

2ùñ

UDDU :D:D:

m

⇤

4(64)

G´1F

9 pmh

q

20 `

3y2t

4⇡2⇤

2` . . . “ 174 GeV (65)

L “ |H|

2O (66)

�pt¯tq “ 0.82 nb at

?

s “ 13 TeV (67)

�pt¯tq “ 0.97 nb at

?

s “ 14 TeV (68)

�pt¯tq « 32 nb at

?

s “ 100 TeV (69)

BRpt Ñ e b {ET

q “ 13.3% ˘ 0.4% ˘ 0.4% (70)

BRpt Ñ µ b {ET

q “ 13.4% ˘ 0.3% ˘ 0.5% (71)

BRpt Ñ ⌧h

b {ET

q “ 7.0% ˘ 0.3% ˘ 0.5% (72)

⌦h29

1

x�vy

(73)

T⌫

“

ˆ10.75

75

˙1{3T⌫

(74)

�Neff

« 0.075 (75)

�phidden Ñ SMq9

´mhidden

TeV

¯n

(76)

⌦

DM

« 5 ⌦

Baryon

(77)

⌦

DM

„

mDM

mp

⌦

B

(78)

m „ �T with � " 1 . (79)

↵ ° ↵8 (80)

↵ ”

�V

⇢rad

(81)

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

5

Comparable result from gluino bounds

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

6

FT ~ 2%

Ways Out?

Pick the two:

Keep naturalness and Unification — RPV (also cornered , but in better shape). Give up on the DM. Choose a mechanism to bridge over the little hierarchy (NN ?). Usually screws unification. Mini-split — perfectly addresses unification + DM. OK w. 125 GeV higgs. Just give up on the little (?) hierarchy

Or one can keep believing in miracles

Outline

• Twin Higgs: Basic Idea and Fine Tuning

• Why marrying the Twin Higgs with SUSY?

• SUSY Twin Higgs: Natural, Non-Minimal

• Comments on Phenomenology — there is still a (weak) hope for the LHC

• Conclusions and Outlook

Logic: we can neither confirm nor rule out perturbative unification at the LHC. DM is also a long shot.

Let us see if we can bridge over the little hierarchy problem.

Twin Higgs (Still Alive and Well!!)

The idea fits into the paradigm of neutral naturalness (NN) solutions to the little hierarchy problem.

Why neutral?

Expect to get a natural model without light

colored particles — mild

constraints from the LHC

The Twin Higgs

In the Twin Higgs the lightness of the EW scale is explained by the fact, that the SM like higgs is a pGB of an approximate SU(4)

[enhanced to SO(8)] symmetry of the Lagrangian. This symmetry is not exact, but holds up to good approximation due to

a mirror symmetry of the Lagrangian

Chacko, Goh, Harnik; 2005

Z2 yt yt

−ytf

ytf

h h h h

t

t

Cancellation of the leading divergencies:

At the leading order only Z2 is needed

The Twin Higgs Potential

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

V “ m2p|A|

2` |B|

2q ` �p|A|

2` |B|

2q

2(86)

6

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

V “ m2p|A|

2` |B|

2q ` �p|A|

2` |B|

2q

2(86)

`p|A|

4` |B|

4q (87)

`�f 2|A|

2` ⇢|A|

4(88)

6

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

V “ m2p|A|

2` |B|

2q ` �p|A|

2` |B|

2q

2(86)

`p|A|

4` |B|

4q (87)

`�f 2|A|

2` ⇢|A|

4(88)

6

Standard SU(4) conserving and mirror symmetric term — massless higgs

Mirror symmetric, but breaks SU(4). We expect the SM higgs mass to come from

this term

Need to break the mirror symmetry, otherwise the SM higgs would be an equal mixture of our higgs and the twin higgs. The correct

alignment can be achieved via σ or ρ

Break SU(4) at a scale f ≫ v

Mirror Symmetry Breaking by the Higgs Potential

Most of the works until now considered soft mirror symmetry breaking — relative radiative safety.

Why bothering with the hard mirror symmetry?

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

V “ m2p|A|

2` |B|

2q ` �p|A|

2` |B|

2q

2(86)

`p|A|

4` |B|

4q (87)

`�f 2|A|

2` ⇢|A|

4(88)

v2

f 2«

1

2

2 ´ �

2(89)

m2h

« 8v2 (90)

6

The experimental bound is ~0.4, we have to fine tune σ against to make it small

In fact, in the soft-broken mirror symmetry scenario is not a free parameter:

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

V “ m2p|A|

2` |B|

2q ` �p|A|

2` |B|

2q

2(86)

`p|A|

4` |B|

4q (87)

`�f 2|A|

2` ⇢|A|

4(88)

v2

f 2«

1

2

2 ´ �

2(89)

m2h

« 8v2 (90)

6

It is simply a SM quartic!

leads to inevitable FT ≈ v2/f2

On the FT, More Carefully

What is our real gain in the FT due to the little Higgs?

multiple talks by Riccardo Rattazzi; 2015-2016

Sources: 1. IR effective theory — v/f FT 2. Is the scale f is stable??

Let us now be very naive, assume that these two sources perfectly factorize and neglect other caveats:

2 3 4 5 6

1

2

3

4

5

fêv

Lt@TeV

Dfêv>2.3

Lt < lf

mhsoft êmh

mhhard êmh

Lt < lf

0.75

0.7

0.65

0.6

0.5

1

1.5

2

Figure 2. The blue contours indicates the irreducible contributions to the Higgs mass coming fromtop loops normalized to mh = 125 GeV, while the red dashed contours indicate the irreduciblecontributions to the Higgs mass in the pure hard breaking case (�

0

= 0) where we fix ⇤⇢ = 250 GeVand ✏ = +1 to minimize the soft Z

2

-breaking generated by radiative corrections (2.7). The greyshaded region at f < 2.3 v is excluded by Higgs coupling measurements and the grey region at thebottom of the plot is inconsistent due to ⇤ < �f .

2.3 Fine-Tuning in the Low Energy E↵ective Theory and Beyond

We are now ready to discuss the fine-tuning in the general Twin Higgs model quantitatively

and show that if hard Z2

-breaking is involved, one in fact expects to improve on the fine-

tuning compared to the soft breaking case. For illustration purposes let us start from the

pure soft case, where the IR fine-tuning is well-known to scale as ⇠ f2/v2. More precisely,

quantifying the fine tuning a la Barbieri-Giudice [31], one gets for the logarithmic derivative

of v2 with respect to �: �soft

v/f = f2�2v2

2v2. Of course, this is just a part of the total fine-tuning

that one estimates in the IR e↵ective theory. On top of that one should also consider a

fine-tuning of the scale f with respect to the top cuto↵ scale ⇤t. Strictly speaking this

fine-tuning should be computed in the full UV theory, but it turns out that one can get a

reasonable estimate by analyzing the threshold corrections to the scale f in the e↵ective

theory and further varying it with respect to the top Yukawa. These radiative corrections

are

�f2 =1

32⇡2

✓3y2t�

⇤2

t � 5⇤2

�

◆. (2.8)

We will see in the next section how they reproduce the dominant RGE e↵ects of the UVI see that there is

an overall factor of

2 mismatch with my

notes. Did you

check it and we dis-

agree? I think it is

just the normaliza-

tion of the CW po-

tential.

theory once the cut-o↵ scales are identified with physical mass thresholds. A-priori both

the sensitivity to the thresholds ⇤t and ⇤� are equally dangerous and, unlike in the SM,

it is not clear that the dominant sensitivity comes from the tops rather than the higgses

– 9 –

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

V “ m2p|A|

2` |B|

2q ` �p|A|

2` |B|

2q

2(86)

`p|A|

4` |B|

4q (87)

`�f 2|A|

2` ⇢|A|

4(88)

v2

f 2«

1

2

2 ´ �

2(89)

m2h

« 8v2 (90)

FT

twin`SUSY

FT

SUSY

„

�SM

�(91)

6

This coupling is measured to be ~0.06, the effect cannot be

moderate at best

Up to the denominator it is the

SUSY FT

Can be cancelled by

higgsinos, mild LHC bounds

What is the Full Parameter Space?

-0.10-0.05 0.00 0.05 0.10 0.15 0.20-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

mé2ê f 2

r

0.25

0.2

0.15

0.1

0.05

0

fêv > 2.3mh=130 GeVmh=125 GeVmh=120 GeV

k<0

soft Z2-breaking

k

pure hard breaking

pure soft breaking

With the given measurement of the higgs mass we are always constrained to a narrow strip

Always need small κ. Radiative corrections?

On the FT, Hard Breaking

Can we do a better job if we allow the hard symmetry breaking?

Simple answer: YES

vacuum, and only one of the Higgses, either HA or HB, gets a VEV. This possibility

usually does not lead to a viable phenomenology and we will not further consider it

here (see however [? ]). On the other hand, if is bigger than zero, the Z2

symmetry

is preserved by the vacuum, and HA and HB get equal VEVs vA = vB = f/p2.

In the following we will always consider 0 < ⌧ � such that the SU(4) breaking

term gives a sub-leading contribution to the mass of the radial mode. but generates

a small SM-like higgs massp2f . The hierarchy between the two quartics in the

potential is technically natural and so is the hierarchy between the radial mode (the

“Twin Higgs”) and the SM Higgs. The latter is a pseudo-Goldstone-boson (PGB) in

the limit ⌧ �. The main problem with the model at the present stage is that the

unbroken Z2

implies that the SM-like Higgs mass eigenstate is an equal superposition

of the visible and the mirror Higgs. This would mean that SM Higgs couplings to

(visible) gauge bosons and fermions would be reduced by a factor 1/p2, a possibility

which is excluded both by LEP EWPM and the LHC Higgs coupling measurements.

Therefore we must include also explicit Z2

-breaking terms.

3. Z2

-breaking terms. In general one can break the mirror symmetry at the renormal-

izable level in two ways: with a relevant operator proportional to µ and/or with a

marginal operator proportional to ⇢. We will conveniently define µ2 ⌘ �f2 and work

with the dimensionless parameters � and ⇢, which are responsible for the Higgs VEV

misalignment v ⌘ vA 6= vB and a mixing between the SM and the twin Higgs. As

we have made clear before, maximal mixing and misalignment are already excluded

and the largest possible misalignment allowed by data translates into the bound

f/v & 2.3.

2.2 Electroweak Symmetry Breaking and Radiative Corrections

In order to analyze EWSB and the Higgs mass in the Twin Higgs it is instructive to

integrate out the heavy radial mode and switch to an e↵ective Higgs theory, where we can

write the Higgs fields in a non-linear realization as

HA = f sin�p2f

, HB = f cos�p2f

. (2.2)

Hereafter we identify � with the SM-like Higgs. It is straightforward to plug these expres-

sions into Eq. (??) and obtain the e↵ective SM Higgs potential in the low energy e↵ective

theory. Minimization of this potential with respect to � yields the following expressions

for the VEV and the mass of the SM-like Higgs:6

2v2

f2

=

✓2� �

2+ ⇢

◆, (2.3)

m2

h = 4v2 (2+ ⇢)

✓1� v2

f2

◆= 2f2 (2� �)

✓1� v2

f2

◆. (2.4)

6The same expressions can alternatively be obtained at the level of the linear sigma model (??) by

solving the EWSB conditions and expanding them at leading order in ,� ⌧ �. We refer to appendix ??

for a discussion of the subleading corrections in this expansion.

– 5 –

There is a-priori no need to fine-tune σ against κ. But…

ρ is a hard breaking term, we expect σ to be quadratically sensitive to it

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

V “ m2p|A|

2` |B|

2q ` �p|A|

2` |B|

2q

2(86)

`p|A|

4` |B|

4q (87)

`�f 2|A|

2` ⇢|A|

4(88)

6

And w/o UV completion we even do not know the sign of this term

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

6

FT, Hard Breaking

Let us assume for simplicity that we have only hard mirror symmetry breaking at the “cutoff ” scale. Of course in the real

theory we will have soft + hard.

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0-2-1012345

fêveLr@TeV

D

0.1

0.2

0.3

0.40.5

0.60.7

0.8

FHLr , fLe=+1 e=-1

Figure 3. Contours of F (v, f ;⇤⇢). The two di↵erent colors correspond to to the two di↵erent signsof ✏. maybe we can shaded in gray the region where ⇤⇢ < �f? - DR

IR e↵ective theory is given by the logarithmic variation of v2 with respect to the parameter

⇢ and reads

�hard

v/f =f2 � 2v2

2v2⇥ F (v, f ;⇤⇢) , (2.11)

where we traded ⇢ for the EW scale using the EWSB condition (2.3) and we have dropped

the log⇤⇢ contributions to � because they are largely subdominant to the quadratic con-

tributions in most portions of the parameter space (but not all!). Here we have introduced

for further convenience

F (v, f ;⇤⇢) ⌘3✏⇤2

⇢ + 32⇡2v2

3✏⇤2

⇢ + 16⇡2f2

. (2.12)

Interestingly the first piece in (2.11) is precisely the fine-tuning in the pure soft breaking

scenario, but in the hard breaking case it is multiplied by a function F (v, f ;⇤⇢). We

illustrate the behavior of this function on Fig. 3. This function is always smaller than 1

and in the limit 3⇤2

⇢ ⌧ 32⇡2v2 reduces to 2v2/f2, while in the limit 3⇤2

⇢ � 16⇡2f2 this

function simply becomes 1. If we assume that the fine-tuning of the scale f2 with respect to

the top cuto↵ scale is not di↵erent from the soft case and the factorization works, this tells

us, at least naively that the reduction of the fine-tuning with respect to common SUSY

scenarios should now be �SM/� ⇥ F (v, f ;⇤⇢), which can be a significant improvement

compared to the Twin Higgs with softly broken Z2

.

Alternatively one can understand the parametric of the fine tuning measure in the

hard-breaking scenario fixing the cut-o↵ scale ⇤⇢ using the EWSB condition (2.3) so that

the fine tuning in (2.11) can be rewritten as

�hard

v/f =f2 � 2v2

2v2⇥ 2

2+ ⇢=

f2 � 2v2

2v2⇥ 8v2

m2

h

✓1� v2

f2

◆. (2.13)

– 11 –

Limits:

Λ goes to infinity ☛ the function F reduces to 1, no gain

Λ ≪ 4πf ☛ the function F ~ v2/f2

In the IR:

There is a true gain if the “cutoff ” is low enough 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0-2

-1

0

1

2

3

4

5

fêv

eLr@TeV

D

FHv, f ; LrL

Lr<l f

0.80.7

0.60.5

0.40.3

0.2

0.1

Caveats

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

1

2

3

4

5

fêv

Lt@TeV

D

0.001

0

-0.02

-0.01

0.1

0.2

0.3

0.4 e=+1, Lr=1 TeV______________________

k 0DhardêDsoft

k0<0 k0>0

Gains in the FT

Desired values of quartic at the “cutoff ” scale

The desired values of the quartic are of order 10-2, but…dimensionless parameters then read

� =3y4t16⇡2

log⇤2

t

m2

tB

+3�⇢

32⇡2

log

⇤2

⇢

m2

rad

+ log⇤2

⇢

m2

h

!, (2.5)

�⇢ =3y4t16⇡2

logf2

v2, (2.6)

�� =3⇢

16⇡2

✏⇤2

⇢

f2

+ 2� log⇤2

⇢

m2

h

!. (2.7)

Several clarifications are needed at this point. First, we have introduced in these expressions

two di↵erent mass thresholds: ⇤t and ⇤⇢. Usually one assumes that there is a single scale,

often loosely dubbed “the cut-o↵ of the IR e↵ective theory”. Here we will be slightly more

precise. Strictly speaking ⇤t and ⇤⇢ are the mass scales at which one finds the new states

of the UV complete, natural theory that cut the corresponding quantum contributions of

the low-energy theory. In strongly coupled UV completions of the Twin Higgs it is hard to

imagine the situation in which these scales are qualitatively separated. However in weakly

coupled UV completions, e.g. SUSY, one might expect appreciable di↵erences between

these mass scales, because of naturally small couplings that control these expressions.

Therefore we will keep track of these scales separately and we will later see that in the

concrete models that we analyze in the next section these scales are indeed di↵erent. Of

course, due to higher orders corrections these scales cannot be arbitrarily di↵erent from

one another.

Second, in the above expressions we have kept the Higgs dependence in the logarithms

frozen. Moreover we have expanded these expressions to the first order in and v/f .

Third, in Eq. (2.7) we have introduced a new parameter ✏. This parameter stand for

the sign of the UV mass threshold corrections and a-priori ✏ = ±1. Since one cannot

calculate the sign of these radiative corrections within the IR e↵ective theory, we will be

agnostic about the sign of ✏ and consider both positive and negative threshold corrections.

As will see in the next section, within a full UV complete theory this sign is determined

unambiguously.

Having at hand all the radiative contributions to ⇢, and �, we can estimate how

big are the contributions of the radiative corrections to the Higgs mass. Because it would

be di�cult to illustrate this point for generic Z2

-breaking, we concentrate on two extreme

cases: pure soft breaking, defined as ⇢0

= 0, and pure hard breaking, defined as �0

= 0.

From Eq. (2.4) we can infer that the radiative contributions to the Higgs mass squared,

up to O(v2/f2) corrections, are given by �m2

h = 4v2(2�+�⇢). We show the results in

Fig. 2. The first important conclusion that we draw from this figure is that while in the soft-

breaking radiative corrections contribute at least ⇠ 60% of the Higgs mass, the radiative

corrections of the hard Z2

-breaking typically overshoot for the Higgs mass, though not by

orders of magnitude. In the soft breaking case the contributions grow with ⇤t, showing

only mild dependence on the value of f . On the other hand, in the hard breaking case this

contributions grow significantly with the value of f . This fact will be very important for

the next subsection, where we will see that in the hard breaking case the fine tuning �v/f

can be ameliorated compared to the soft case at the price of adjusting the Higgs mass.

– 8 –

Should we FT the higgs mass?yes, but the extra FT is of order 1

Typically full models with hard breaking will slightly overshoot for the Higgs mass, and will lack perfect factorization, but these caveats

turn to be minor.

UV Completion. Why Do We Prefer SUSY?

• The scale f is unstable by itself and requires a natural UV completion. Known options: strong coupling, SUSY or “turtles all the way down”.

• Examples of the latest version are not know in the Twin Higgs context.

• Strongly coupled Twin Higgs has its own problems, e.g. precisions EW. In the strongly coupled models all the mass scales should come out similar.

• We would like to have some moderate separation between the top partners mass scale and the higgs partners mass scale — hard in a strongly coupled theory

• SUSY can naturally explain why various mass scales that we have introduced are slightly different from one another — technically natural, moderately small couplings.

SUSY Meets Its TwinFalkowski, Pokorski, Schmaltz; 2006;; Chang, Hall, Weiner; 2006

Z2

How do we get the necessary

couplings?

Getting SU(4) conserving quartic: NMSSM (well, almost)

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

� “ 0.9 � “ 1.4 (87)

tan � “ 1.2 tan � “ 2.0 (88)

W “ �SHu

Hd

(89)

6

full multiplets of the approximate SU(4)

assume to be order-1 The singlet should be integrated out non-

supersymmetrically (soft mass ≫ SUSY mass)

The Soft SUSY Model

Straightforward, but do not expect big gains in the FT!

κ — comes out from the D-terms. Given that the SM higgs mass ~ κ, it very similar to the MSSM (but factor of 2 helps!) Mirror symmetry breaking terms — simply assume asymmetric soft masses (definitely soft).

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

1

2

3

4

5

fêv

Ms

lS=1

2.7

33.3 3.6

Dsoft

100

200

300

400

500

600

-6-4-20246

Rattazzi’s logic clearly works — no FT better than 1% (in agreement with Craig

& Howe)

Higgs mass condition cannot be satisfied for any tan β.

The Hard SUSY Model — Wrong Way

Very naive way to break the mirror symmetry:

Just introduce a mirror symmetry breaking W:

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

� “ 0.9 � “ 1.4 (87)

tan � “ 1.2 tan � “ 2.0 (88)

W “ �SHu

Hd

(89)

W “ �1HA

u

HA

d

(90)

6

Perfectly maps on a ρ-term. Small problem:

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

6

Now we can really calculate the sign, and it turns out to be negative

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

� “ 0.9 � “ 1.4 (87)

tan � “ 1.2 tan � “ 2.0 (88)

W “ �SHu

Hd

(89)

W “ �1HA

u

HA

d

(90)

⇤

⇢

„ mSA (91)

6

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

1

2

3

4

5

fêvLt@TeV

D0

2

4

6

0.5

0.3

0.1

k0=0__________________

e Lr @TeVDFHv, f; LrLe=+1 e=-1

This negative threshold drags the soft mass negative, allows only low values of the stop masses.

Way out — negative κ.

The Bi-Doublets

Desiderata: negative mirror symmetry preserving quartic, hard mirror symmetry breaking terms. Any way to get this?

Trick: introduce vectorlike bidoublets:

The simplest setup to get a negative contribution to is to integrate out a bi-doublet

transforming both under SU(2)A and SU(2)B. Consider for example a vectorlike bidoublet

B +B with hypercharge ±1, with the superpotential and soft mass term

�W = �BBhAu hBu +MBDBB , V

soft

= m2

BD

�|B|2 + |B|2

�. (3.18)

Integrating out the bidoublet in the limit of mBD � MBD and matching gives

�V = s2As2

B|�B|2|HA|2|HB|2 , (3.19)

which in our general parametrization gives a negative contribution to (and positive to

�):

� = ��� = �1

2s2As

2

B|�B|2 . (3.20)

However, the presence of a state charged under both A and B gauge group would generically

generate a large mixing between the photon and the dark photon and makes it non-trivial

to achieve gauge coupling unification.

⇡ ��t��2

S

4s4�c

2

� , (3.21)

µ2 ⇡ �t�c2

�

⇥�m2

u �m2

d

�s2� � 2b c

2�

⇤(3.22)

⇢ ⇡ �t�2

s4�c

2

�

�2

S � g2ew

2

�. (3.23)

4 Higgs sector phenomenology

In the previous sections we have studied the (SUSY) Twin Higgs model in full generality

and identified the regions in the parameter space which exhibit the lowest amount of fine

tuning, allowing for generic soft as well as hard breaking Z2

terms. In this section we

discuss possible phenomenological signatures of the (SUSY) Twin Higgs. We focus on the

Higgs sector of the (SUSY) Twin Higgs, which features the most distinctive signatures of

the model that we expect to be accessible at the LHC. We have seen that, for interesting

viable scenarios, also the SUSY (visible) spectrum can be quite light, at the edge of LHC

reach. For such states the expected signals are the canonical ones of SUSY, possibly

augmented with extra missing energy when mixing with the dark sector is relevant (i.e. for

very small f/v).

The Higgs sector of the SUSY Twin Higgs presents new properties with respect to

the standard SUSY case, being e↵ectively a double copy of the MSSM. This includes new

scalar states that can be possibly probed directly at the LHC. In Table 1 we summarize

the complete scalar spectrum of the Higgs sector of the Twin SUSY model at leading order

in the expansion �2 � g2eff , f2 � v2, m2

A � �2f2.

The CP-even states exhibit a rich structure with possible sizeable mixing. The CP-odd

Higgs mixing is exact and depends only on vf , and we have indicated with AT the CP-odd

– 19 –

Automatically get negative quartic which can outweigh the D-terms 20

10

6

5

252.5 3.0 3.5 4.0 4.5 5.01.0

1.5

2.0

2.5

3.0

fêvMsusyHTeV

L

Min D

Contours of the FT (scan!!) We can improve the situation

qualitatively

Hopes for the LHC Pheno?

We have plenty of the Higgses in the story? Who is the next to the lightest higgs

Red — more radial mode-like

Blue — more MSSM heavy-

higgs—like

Green, black — contours of the other CP-even higgs masses.

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

� “ 0.9 � “ 1.4 (87)

6

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

� “ 0.9 � “ 1.4 (87)

6

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

� “ 0.9 � “ 1.4 (87)

tan � “ 1.2 tan � “ 2.0 (88)

6

↵8 “

30

24⇡2

ca

�m2a

p'q

g˚T 2˚(82)

m2h

9

3y2t

4⇡2M2

stops

log

ˆ⇤

2

M2stops

˙(83)

�

´1«

ˇˇ m2

h

2�m2h

ˇˇ

« 1.4% (84)

�m2h

«

y2t

↵s

⇡3M

g

log

2

ˆ⇤

2

Mg

˙(85)

�� „

3⇢

16⇡2

⇤

2⇢

f 2(86)

� “ 0.9 � “ 1.4 (87)

tan � “ 1.2 tan � “ 2.0 (88)

6

Searches for the New Higgses

What can we search for? Decays into di-Z and di-h — very promising, BRs are not independent

due to the Higgs equivalence theorem. Decays into tops — very challenging due to the interference with the

SM background.

Promising

coverage for the

HL LHC and

future b ⇾ sɣ

Conclusions

• Twin Higgs is still an alive mechanism to bridge over the looming little hierarchy problem

• Hard mirror symmetry breaking can reduce the FT, with certain caveats

• Twin SUSY is a natural candidate to UV complete the Twin Higgs + hard mirror symmetry breaking

• A model with bidoublets and generic terms in the superpotential can do the job

• LHC signals are possible, both in the searches for the extra higgses and in the precsion measurements of the higgs couplings

• Where are the stops??