Big Hierarchies, Little Hierarchies, and Flat Directions

in the SUSY LandscapeMatt Reece

Harvard UniversityJuly 3, 2017

P. Agrawal, J. Fan, MR to appear this summer Disclaimer: Preliminary! Numerical results are still in flux as we improve calculations.

Related discussions / work in progress with: M. Amin, K. Lozanov, P. Draper, D. Pinner

A Fine-Tuned Higgs?

Is our universe precariously balanced between no EWSB and extreme EWSB? If so, why?

I want to suggest some possible ingredients for an answer to this question. I’ll review the usual story of how EWSB relates to SUSY, then suggest a different one.

Where We’re Going

“Standard SUSY” “Flat-Higgs SUSY”

The typical SUSY naturalness story is that as we vary input UV parameters, the Higgs VEV should change by O(1) amounts. But in theories where the Higgs has a (badly broken!) shift symmetry, we will find exponential sensitivity of the VEV to UV parameters.

0 50 100 150 200

0.4

0.5

0.6

0.7

0.8

μ [TeV]

y t

105 GeV

108 GeV

1011 GeV1014 GeV

Hu = Hd = s = 10 TeV

Det[ℳ2( s )] > 0

dmD 2

dt< 0

b = Hu2+μ2

1×106 5×106 1×107 5×107 1×1080.4

0.5

0.6

0.7

0.8

0.9

1.0

b[GeV2 ]

y t

5 TeV

10 TeV

15 TeV

20 TeV Hu = Hd = s = 10 TeVμ = 1 TeV

An Observation

h h

t

+h h

t

h h

t

Consider the diagrams in Fig. 1. We’ve already observed that the one at left is problematic: it’s arenormalization of an external line, so we don’t want to include it when we compute a loop amplitude. Inshamplitude calculations, it shows up as unpleasant 1

s12...(n�1)⇥ ⇤ factors in the amplitudes we’re trying

to build the shamplitude out of, which we are currently removing by hand.The other kind of bubble diagram with one gluon connected at one end is shown on the right in Fig. 1.

It has a two-particle vertex at the other end. As a result, it has the structure:

�d4⇤

(2⇥)4�1µ (2⇤µ + kµ1 ) J(k2, . . . kj) · J(kj+1, . . . kn)

(⇤2 �m2)((⇤+ k1)2 �m2). (1)

Notice that this always contributes 0 to the loop integral: �1 · k1 = 0, and the bubble integral, linear in ⇤µ,can only be proportional to kµ1 , because all dependence on the other momenta factors out of the integrand.

So, we can in fact drop every diagram with only one gluon connected on one side of a bubble. It’s temptingto try to inductively turn this into a procedure for generating shamplitudes only from other shamplitudes,not from amplitudes, but the argument doesn’t work. It would be nice to do something more systematicthan dropping terms by hand. Is there a nice procedure that makes use of this fact?

At least for the 4-point shamplitude, it means computing it directly from Feynman diagrams only involvessumming up nine diagrams (Fig. 2). We can eliminate four of these with a convenient gauge choice.

Four-point loops from Feynman diagrams

If we want to compute the + + ++ amplitude, we can make �i · �j = 0 simply by taking �i =µ�i

hµ ii for all i.

In the + + +� case, we can make �i · �j = 0 by taking �i =�4�ih4 ii for i = 1, 2, 3 and �4 = �4�1

[4 1] . Thus, we can

discard all Feynman diagrams with 4-point (2-scalar 2-gluon) vertices. The remaining diagrams are boxes,triangles, and the bubble with two particles on each side attached at 3-gluon vertices.

1

An Observation

h h

t

Consider the diagrams in Fig. 1. We’ve already observed that the one at left is problematic: it’s arenormalization of an external line, so we don’t want to include it when we compute a loop amplitude. Inshamplitude calculations, it shows up as unpleasant 1

s12...(n�1)⇥ ⇤ factors in the amplitudes we’re trying

to build the shamplitude out of, which we are currently removing by hand.The other kind of bubble diagram with one gluon connected at one end is shown on the right in Fig. 1.

It has a two-particle vertex at the other end. As a result, it has the structure:

�d4⇤

(2⇥)4�1µ (2⇤µ + kµ1 ) J(k2, . . . kj) · J(kj+1, . . . kn)

(⇤2 �m2)((⇤+ k1)2 �m2). (1)

Notice that this always contributes 0 to the loop integral: �1 · k1 = 0, and the bubble integral, linear in ⇤µ,can only be proportional to kµ1 , because all dependence on the other momenta factors out of the integrand.

So, we can in fact drop every diagram with only one gluon connected on one side of a bubble. It’s temptingto try to inductively turn this into a procedure for generating shamplitudes only from other shamplitudes,not from amplitudes, but the argument doesn’t work. It would be nice to do something more systematicthan dropping terms by hand. Is there a nice procedure that makes use of this fact?

At least for the 4-point shamplitude, it means computing it directly from Feynman diagrams only involvessumming up nine diagrams (Fig. 2). We can eliminate four of these with a convenient gauge choice.

Four-point loops from Feynman diagrams

If we want to compute the + + ++ amplitude, we can make �i · �j = 0 simply by taking �i =µ�i

hµ ii for all i.

In the + + +� case, we can make �i · �j = 0 by taking �i =�4�ih4 ii for i = 1, 2, 3 and �4 = �4�1

[4 1] . Thus, we can

discard all Feynman diagrams with 4-point (2-scalar 2-gluon) vertices. The remaining diagrams are boxes,triangles, and the bubble with two particles on each side attached at 3-gluon vertices.

The box diagram is:

16

�d4⇤

(2⇥)4�1 · ⇤ �2 · (⇤+ k1) �3 · (⇤� k4) �4 · ⇤

(⇤2 �m2)((⇤+ k1)2 �m2)((⇤+ k1 + k2)2 �m2)((⇤� k4)2 �m2). (2)

1

Top Partners in Supersymmetry

�m2Hu

⇡ � 3y2t8⇡2

�m2

Q3+m2

u3+ |At|2

�log

Mmess

mt

Different-spin pieces combine to cancel large corrections.

“Stop” or “Scalar Top”: cancels the biggest correction. ~10% tuned if mass ~ 700 GeV.

Radiative Breaking in the MSSM

All of the scalars in the MSSM have a characteristic mass scale Msoft

At some mediation scale, we have m2

Hu⇠ M2

soft

d

dtm2

Hu=

1

16⇡2

⇥6|yt|2

�m2

Hu+m2

Q3+m2

u3

�+ 6|at|2 � 6g22 |M2|2 + . . .

⇤

Renormalization group running, driven by the large Yukawa coupling, tends to drive the up-type Higgs negative:

Let’s review the most familiar story for EWSB in SUSY.

Compare the stops, which run more slowly:d

dtm2

Q3=

1

16⇡2

⇥2|yt|2

�m2

Hu+m2

Q3+m2

u3

�+ 2|at|2 � 32/3g23 |M3|2 + . . .

⇤

d

dtm2

u3=

1

16⇡2

⇥4|yt|2

�m2

Hu+m2

Q3+m2

u3

�+ 4|at|2 � 32/3g23 |M3|2 + . . .

⇤

This naturally favors electroweak breaking with hhi ⇠ Msoft

The Higgs quartic coupling

A SUSY-breaking contribution to the Higgs quartic comes from loops of stops:

The tree-level MSSM has a Higgs quartic coupling from D-terms, completely fixed by the Higgs’ electroweak representations:

V = (|µ|2 +m2Hu

)|H0u|2 + (|µ|2 +m2

Hd)|H0

d |2 � (bH0uH

0d + c.c.)

+1

8(g2 + g02)(|H0

u|2 � |H0d |2)2

Notice the D-flat direction: |H0u| = |H0

d |

V1�loop

⇡ 3y4t16⇡2

(H†uHu)

2

"log

m2

˜t

m2

t

+

X2

t

m2

˜t

1� 1

12

X2

t

m2

˜t

!#

Two routes to tachyon: m2Hu

< 0 or b > |µ|2 +m2H

Natural SUSY: the expectation in 1984Slide shown by Lawrence Hall at Savasfest in 2012:

Natural SUSY: the status in 2017

The squarks certainly don’t seem to be below the W mass. In fact, they don’t seem to be below 20 times the W mass.

The Higgs looks anomalously light compared to other scalars, which are yet to be found.

A Typical Supersymmetric Scenario

Naturalness: visualize how EWSB changes as we vary UV parameters.

Higgs VEV ~ Soft masses ~ 10 TeV.

1×106 5×106 1×107 5×107 1×1080.4

0.5

0.6

0.7

0.8

0.9

1.0

b[GeV2 ]

y t

5 TeV

10 TeV

15 TeV

20 TeV Hu = Hd = s = 10 TeVμ = 1 TeV

Right-hand side: we have b-driven EWSB instead of radiative EWSB.

Here the VEV runs away to large values. More on this soon.

What is Fine Tuning? One Possible Answer:Barbieri and Giudice, Nucl.Phys. B306 (1988) 63-76

Fine tuning is the sensitivity of observable physics to high-scale parameters. Is this consistent with our intuitions about what fine tuning means?

��

�1O

�p=

����@ logO@ log p

����

O = m2Z , p = µ : ��1 / µ2

m2Z

if µ � mZex. 1:

ex. 2: O = ⇤QCD, p = g3 : �

�1 ⇡ 2 log

⇤UV

⇤QCD⇠ 100

What is Fine Tuning? Is QCD Really Tuned?Potter Stewart, Concurring opinion in Jacobellis v. Ohio 378 U.S. 184 (1964)

"I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description; and perhaps I could never succeed in intelligibly doing so. But

I know it when I see it, and [dimensional transmutation in QCD] is not that."

What I mean by tuning is really about typicality.

In standard SUSY with radiative EWSB, it is typically true that the Higgs VEV is near the soft scale. You must tune parameters to avoid this.

In QCD, it is typically true that the QCD scale is exponentially below the cutoff. The precise value is very sensitive to the parameters, but this is not tuning.

My working opinion for this talk:

V1�loop

⇡ 3y4t16⇡2

(H†uHu)

2

"log

m2

˜t

m2

t

+

X2

t

m2

˜t

1� 1

12

X2

t

m2

˜t

!#

EWSB Along the Flat DirectionSuppose there is a tachyonic direction pointing along the flat direction, that is, that we have

�1 1

�✓|µ|2 +m2Hu

�b�b |µ|2 +m2

Hd

◆✓11

◆= m2

Hu+m2

Hd+ 2|µ|2 � 2b < 0

How large will the Higgs VEV be? At first, you would expect to be stopped by the loop-level quartic coupling:

But importantly, the stop mass here is the geometric mean of the physical stop masses,

m2t ⇡ m2

Q3,u3+ y2t |H0

u|2

and as we move far out along the flat direction the stop and top become degenerate:

hH0

ui � Msoft

) m˜t ⇡ mt

Approximate SUSY suppresses the quartic by a factor of Msoft2/H2, allowing Higgs VEVs much larger than soft masses!

Flat directions should always be lifted at very large field values.

Zd4✓

X†X

⇤4

(H†uHu)

2 ! m2

soft

⇤2

(H†uHu)

2

Kähler corrections are compatible with VEVs of order the cutoff:

Zd2✓

✓µHu ·Hd +

1

M(Hu ·Hd)

2

◆Superpotential terms at first glance appear more dangerous.

µ†

M(H†

uHu)(Hu ·Hd) + . . . ) hhi ⇠pµM

gives rise to quartics:

but given that some spurion forbids the mu term we expect

1

M. µ

⇤2) hhi ⇠ ⇤

Higher-Dimension Operators Lifting the Flat Direction

A Shift-Symmetric Kähler Potential

A range of theories, including various forms of “Gauge-Higgs Unification,” predict that the tree-level Kähler potential has the form

f(X,X†)���Hu +H†

d

���2

which is invariant under the continuous shift symmetry

Hu ! Hu + c, Hd ! Hd � c†

This symmetry is explicitly broken—for instance, the top Yukawa coupling breaks it badly. Nonetheless, it can have a profound effect on the physics of the theory.

An incomplete list of references for such symmetries:

Lopes Cardoso, Luest, Mohaupt ’94; Antoniadis, Gava, Narain, Taylor ’94; Brignole, Ibáñez, Muñoz ’96; Burdman, Nomura ’02; Choi, Haba, Jeong, Okumura, Shimizu, Yamaguchi ’03; Hebecker, March-Russell, Ziegler ’08; Brümmer, Fichet, Hebecker, Kraml ’09; Hebecker, Knochel, Weigand ’12; Ibáñez, Valenzuela ’13

Consequences of the Tree-Level Shift SymmetryThe F-term VEVs of the various fields X (e.g. moduli) determine the Higgs soft masses, the mu term, and the B term. But the symmetry constrains the mass matrix to have the form:

✓|µ|2 +m2

Hu�b

�b |µ|2 +m2Hd

◆= b

✓1 �1�1 1

◆

At tree level, there is one massless eigenvalue which is oriented along the D-flat direction Hu = Hd.

We must compute loop corrections to understand the fate of this flat direction, but it raises the possibility that these realizations of SUSY prefer

hhi � Msoft

Obviously the real world doesn't obey this. But it alters the way we think about the hierarchy problem. Let’s explore further.

RG Running and Electroweak BreakingIn the UV this theory is on the edge of breaking EWSB. Let’s work in a limit where gaugino masses and A-terms are negligible, to start.

✓|µ|2 +m2

Hu�b

�b |µ|2 +m2Hd

◆The evolution of the mass matrix

depends mostly on wanting to run tachyonic and on , which gets either larger or smaller depending on how big the top Yukawa is relative to the gauge coupling:

m2Hu

µ

d

dtµ ⇡ 1

16⇡2µ

✓3|yt|2 � 3g22 �

3

5g21

◆

Let’s explore the toy landscape where we vary the UV values of

yt,µ2

m2Hu

,m2

Q3

m2Hu

=m2

u3

m2Hu

keeping the other parameters fixed and keeping the shift-symmetry conditions

m2Hu

= m2Hd

= b� |µ|2

SUSY Near a Cliff

0 50 100 150 200

0.4

0.5

0.6

0.7

0.8

μ [TeV]

y t

105 GeV

108 GeV

1011 GeV1014 GeV

Hu = Hd = s = 10 TeV

Det[ℳ2( s )] > 0

dmD 2

dt< 0

b = Hu2+μ2

hHi ⇠ ⇤ runaway along flat direction

hHi ⇠ 0 mu term stabilizes Higgs (QCD EWSB)

Exp. sensitivity to parameters.

SUSY Near a Cliff: RGE Running Behaviors

0 50 100 150 200

0.4

0.5

0.6

0.7

0.8

μ [TeV]

y t

105 GeV

108 GeV

1011 GeV1014 GeV

Hu = Hd = s = 10 TeV

Det[ℳ2( s )] > 0

dmD 2

dt< 0

b = Hu2+μ2

5 10 15 20 25 300

5

10

15

t

D2

S2

Det ℳ2

S4

5 10 15 20 25 30

-20

-15

-10

-5

0

t

D2

S2

Det ℳ2

S4

5 10 15 20 25 30-12

-10

-8

-6

-4

-2

0

t

D2

S2

Det ℳ2

S4

stabilize where mass2 crosses zero

mHu2 tachyonic running dominates

hHi ⇠ 0

hHi ⇠ ⇤ hHi ⇠ e�t⇤⇤

mu runs positive, dominates

mu running arcs

SUSY Near a Cliff: Dimensional Transmutation

Taking a slice through the plot:

Exponential variation in the Higgs VEV as we change input parameters like the top Yukawa coupling. 0.4 0.5 0.6 0.7 0.8 0.9 1.0

100

105

108

1011

1014

1017

yt

v[GeV

]

Hu = Hd = s = 10 TeVμ = 150 TeVb = Hu

2+μ20 50 100 150 200

0.4

0.5

0.6

0.7

0.8

μ [TeV]

y t

105 GeV

108 GeV

1011 GeV1014 GeV

Hu = Hd = s = 10 TeV

Det[ℳ2( s )] > 0

dmD 2

dt< 0

b = Hu2+μ2

The ratio v/Msoft is exponentially sensitive to the parameters!

Summary So Far:

Everything else I will say will be very preliminary and speculative—this is work in progress.

We typically expect that in a supersymmetric theory, the soft scale and the Higgs VEV are closely related,

hHi ⇡ Msoft

If there is a good UV reason for a tachyon to point along the flat direction, this is replaced by something more dramatic:

usually hHi ⇡ ⇤ or hHi ⇡ 0

but there is a slice of parameter space where

with

which looks more like how we understand the origin of the QCD scale (i.e., dimensional transmutation). Also resembles Coleman-Weinberg.

This could change how we think about fine-tuning in SUSY.

hHi ⇡ 10xMsoft

�1 . x . 12

Realism

The minimal version of this story has viable vacua only with

mh < mZ

We can deform the story to get less minimal theories, but we want UV boundary conditions that stick close to a flat direction to have an interesting story.

Still exploring different options for doing this.

AnthropicsCan this help us find a nice anthropic explanation of the weak scale?

In the large swath of parameter space where , no large structures form in the universe.

In other parts of parameter space, it is less clear is there is a “simple” anthropic argument. Would like to not have to appeal to all the details of chemistry or nuclear physics.

hHi ⇠ ⇤

Cosmological Consequences of the Flat Direction

M2

soft

⇠ M2

soft,0 +�

fM2

soft,1

Suppose that we have a modulus field ɸ whose VEV determines the Higgs mass matrix:

and similarly for the b term. As ɸ oscillates, the shape of the Higgs potential changes. If

m� . Msoft

then the Higgs clearly has time to respond to these changes; it can fluctuate out to large field values when there is a tachyon along the flat direction.

Similar to “tachyonic preheating”: Dufaux, Felder, Kofman, Peloso, Podolsky, hep-ph/0602144.

Cosmological Consequences of the Flat Direction

Similar to “tachyonic preheating”: Dufaux, Felder, Kofman, Peloso, Podolsky, hep-ph/0602144.

What is this good for?

Remove unwanted cosmological moduli? New approaches to baryogenesis? Produce detectable gravitational wave signals? [*]

Work in progress on various possibilities.

[*] Discussions with Mustafa Amin, Kaloian Lozanov

Consequences for Hidden Sectors?In general, string compactifications predict the existence of hidden sectors: particles not charged under the Standard Model gauge group.

From cosmology, we have at least a mild “Missing Hidden Sector Problem.”

If there are ultraviolet reasons for many hidden sectors to lie near a flat direction, then large VEVs could decouple large numbers of these particles.

Could ameliorate the Missing Hidden Sector Problem. Massless dark photons and their superpartners—photini with masses near the SUSY breaking scale—could survive.

Photini phenomenology: Arvanitaki, Craig, Dimopoulos, Dubovsky, March-Russell ’09; Baryakhtar, Craig, Van Tilburg ’12; Acharya, Ellis, Kane, Nelson, Perry ’16

Ne↵ = 3.15± 0.23 (1�)Planck constraint on relativistic degrees of freedom (“effective neutrinos”):

Standard Model expectation:

NSMe↵ = 3.046

Idea of Graham, Kaplan, Rajendran arXiv:1504.07551:

Relaxion Prospects?

g�h†h+ V (h) +�

fGa

µ⌫Gaµ⌫

Suppose we realize a variation on this idea close to a SUSY flat direction. Not far from our vacuum, the Higgs VEV can be much bigger, raising the QCD scale—and hence the potential barrier height—significantly.

Can we use the trick of living near a flat direction to improve the relaxion scenario? (Smaller field range, less need for strong dissipation, fewer e-folds….)?

original relaxion near a cliff?

Outlook

A fairly mild constraint from UV physics—a badly broken shift symmetry (not at all like strong CP or large-field inflation or the relaxion)—can significantly alter the form the hierarchy problem takes in SUSY.

Unlifted flat directions for the Higgs may not be everywhere in the landscape, but they do arise.

There might be highly nontrivial cosmological dynamics associated with these corners of the landscape. There is much to explore here. The intersection of anthropics and SUSY could also take a different form.

Question for top-down string landscape studies: How often do approximate shift symmetries enforce alignment of tachyonic directions with D-flat directions? Is it plausible that this should play a role in how we think about the MSSM, or about the fate of hidden sector particles?