What Can We Learn from the Higgs

Maxim Perelstein, Cornell Cornell Physics Colloquium, September 8 2014

Brief History of Weak Interactions

• 1896: Becquerel discovers radioactivity

• 1930: Pauli realizes that an invisible particle (later named “neutrino”) is involved

• 1934: Fermi proposes a successful theoretical model, based on 4-particle interaction in quantum field theory (relativity+QM)

1950s: Problem with the Fermi Model

• Experiment: accelerate electrons and protons to high energies and collide them!

• Weak interactions: inverse beta decay,

!

!

!

• Unitarity bound (quantum mechanics):

• Violate unitarity predict probability > 1!

• Unitarity bound violated at

• Fermi model agrees with experiment at “low” energies, but MUST break down at or below this scale!

e + p → n + ν

e−

p (u)

ν

n (d)

σ ∼ G2

F E2

c.m.

σ <π

E2c.m.

Ec.m.

≈ 300 GeV

GF = 1.166 × 10−5

GeV−2

• Energy: electron-volt (eV) =

• “Energy frontier” physics (Large Hadron Collider): protons with energies ~TeV =

• I will quote particle masses in GeV throughout this talk; what I mean is rest energy, or

• Proton , so e.g. “a 100 GeV particle” means 100 x proton mass, or

• Distance: relativistic collision with c.m. energy probes physics at length scale

• Length scales probed by the LHC (smallest ever):

• So, we will be discussing nano-nano-world in this talk

Aside: Units and Orders of Magnitude

cf: Quantum Electrodynamics!

• EM interactions (QED): elastic scattering,

• In QED, there is no direct 4-particle interaction: this scattering occurs via two 3-particle interactions, involving an exchange of a “virtual” photon

!

!

!

!

!

!

!

• No growth with E, no unitarity violation - could be valid at arbitrarily high energies!

e + p → e + p

σ ∼

α

E2c.m.

α =e2

4π

≈

1

137

“Feynman diagram”

1960s: From Fermi to Standard Model• Idea: “Massive Vector Boson” for weak interactions (Schwinger,

Glashow, Weinberg)

!

!

!

• Successes of the Fermi model are preserved

• Unitarity problem is avoided if

• QFT based on this idea has been called the “standard model of weak interactions” (SM)

• MVBs (W and Z) discovered at CERN in 1983, with masses slightly below 100 GeV

• Properties of W and Z predicted by the SM with just 3 parameters quantitative tests are possible

e−

p (u)

ν

n (d)

σ ∼ G2

F E2

c.m., Ec.m. ≪ MW

σ ∼αw

E2c.m.

, Ec.m. ≫ MW

MW < 300 GeV

1990s: Precision Electroweak Era!

• Experiments at CERN, SLAC, Fermilab and elsewhere have measured W and Z properties with high precision

!

!

!

!

!

!

!

!

• Result: complete agreement with SM predictions, at the level of ~0.1% for some observables!

Measurement Fit |Omeas<Ofit|/mmeas

0 1 2 3

0 1 2 3

6_had(mZ)6_(5) 0.02750 ± 0.00033 0.02759mZ [GeV]mZ [GeV] 91.1875 ± 0.0021 91.1874KZ [GeV]KZ [GeV] 2.4952 ± 0.0023 2.4959mhad [nb]m0 41.540 ± 0.037 41.478RlRl 20.767 ± 0.025 20.742AfbA0,l 0.01714 ± 0.00095 0.01645Al(Po)Al(Po) 0.1465 ± 0.0032 0.1481RbRb 0.21629 ± 0.00066 0.21579RcRc 0.1721 ± 0.0030 0.1723AfbA0,b 0.0992 ± 0.0016 0.1038AfbA0,c 0.0707 ± 0.0035 0.0742AbAb 0.923 ± 0.020 0.935AcAc 0.670 ± 0.027 0.668Al(SLD)Al(SLD) 0.1513 ± 0.0021 0.1481sin2eeffsin2elept(Qfb) 0.2324 ± 0.0012 0.2314mW [GeV]mW [GeV] 80.385 ± 0.015 80.377KW [GeV]KW [GeV] 2.085 ± 0.042 2.092mt [GeV]mt [GeV] 173.20 ± 0.90 173.26

March 2012

Electroweak Unification• SM predicts: EM and weak interactions have not just same

structure at high energies, but also same strength

!

!

!

!

!

!

!

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

10

10 3 10 4

HERA I high Q2 e-p

H1 e-p CCZEUS e-p CC 98-99SM e-p CC (CTEQ5D)

H1 e-p NCZEUS e-p NC 98-99SM e-p NC (CTEQ5D)

y < 0.9

Q2 (GeV2)

dm/d

Q2 (p

b/G

eV2 ) EM and Weak forces become

equally strong at energies above ~100 GeV

(or distances below )10−15

cm

Weak force is short-range, with range about 10

−15cm

Vweak ∝

e−r/r0

r

[blue=EM, red=weak] r0 ∼ M−1

W

1960s: Unitarity Strikes Back

• Experiment: accelerate W/Z bosons to high energies and collide them! e.g.

• The simple MVB theory suggested by Schwinger predicts

!

!

!

• Unitarity bound violated at

• Fix it: Introduce a new, spin-0 particle, with specific pattern of couplings to the MVBs (Higgs; Brout, Englert, 1964)

!

M ∝ E2

+ M ∝ E0!

• SM: simplest possible model with 1 Higgs boson (chosen by Occam’s razor)

• Parameters: Higgs mass and 16 “couplings” [Coupling = numerical constant which controls the strength of interaction between particles: e.g. electric charge of the electron = electron-photon coupling]

• Higgs couplings:

• Unitarity restoration requires

• Couplings to massless particles, e.g. photon, can still arise through virtual (loop) processes:

!

!

• Once all masses are measured, SM predicts all Higgs couplings unambiguously: only free parameter is !

More About the SM Higgs

1990s: Higgs Echoes in PEW• Virtual Higgs bosons lead to subtle modifications of the W/Z

boson properties: in QFT this is described by “loop” diagrams

• Precision electroweak measurements have sufficient precision to be sensitive to such effects

• SM predictions would not agree with data without including Higgs loops in the calculation; fit provides information on the Higgs mass

Large Hadron Collider (LHC) at CERN (Geneva, Switzerland) turned on in 2008

The LHC: 4 TeV protons colliding head-on, 17 miles long, 2 detectors - CMS and ATLAS

2012: Finally, the Higgs!

CERN Seminar, July 4, 2012

38 years after the theoretical prediction...

Physics Nobel Prize, 2013

Immediate Economic Impact

Before discovery After discovery

First Coupling Measurements1.2 Coupling Measurements 13

Parameter value-1 0 1

ATLAS

-1Ldt = 4.6-4.8 fb∫ = 7 TeV s-1Ldt = 20.7 fb∫ = 8 TeV s

= 125.5 GeVHm

, ZZ*, WW*γγ →Combined H

Vκ

σ1

σ2

Fκ

σ1

σ2

Model:Fκ, Vκ

FVλ

σ1

σ2 Model:VVκ, FVλ

WZλ

σ1

σ2 Model:,Zγλ, WZλ

ZZκ, FZλ

gκ

σ1

σ2

γκ

σ1

σ2

Model:γκ, gκ

Total uncertaintyσ 1± σ 2±

parameter value0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

BSMBR

γκ

gκ

tκ

τκ

bκ

Vκ

= 0.78 SM

p

= 0.88 SM

p

68% CL95% CL

-1 19.6 fb≤ = 8 TeV, L s -1 5.1 fb≤ = 7 TeV, L s

CMS Preliminary

1 ]≤ Vκ[

68% CL95% CL

Figure 1-1. Left: summary of the ATLAS coupling scale factor measurements for di↵erent models. Thesolid vertical lines are the best-fit values while the dark- and light-shaded band represent the total ±1� and±2� uncertainties. The curves are distributions of the likelihood ratios. Right: summary of the CMS fitsfor deviations in the coupling for the generic six-parameter model including e↵ective loop couplings. Theresult of the fit when extending the model to allow for beyond-SM decays while restricting the coupling tovector bosons to not exceed unity (V 1.0) is also shown.

the result in the table. These projections are based on the analysis of 7 and 8 TeV data, not all final stateshave been explored. They are expected to improve once more final states are included. CMS has consideredtwo scenarios of systematic uncertainties:

• Scenario 1: all systematic uncertainties are left unchanged (note that uncertainty reductions fromincreased statistics in data control regions are nevertheless taken into account);

• Scenario 2: the theoretical uncertainties are scaled by a factor of 1/2, while other systematic uncer-tainties are scaled by the square root of the integrated luminosity, i.e., 1/

pL.

The ranges of the projections in the table represent the cases with and without theoretical uncertainties forATLAS and two scenarios of systematic uncertainties for CMS.

The estimates from ATLAS and CMS are similar for most of the final states with a few notable exceptions.ATLAS has no estimate for H ! bb̄ at this time, it’s estimate for H ! ⌧⌧ was based on an old study andsignificant improvement is expected. The large di↵erence between the two H ! Z� estimates needs to beunderstood. For these reasons, CMS projections are taken as the expected LHC per-experiment precisionsbelow.

Community Planning Study: Snowmass 2013

Current status: ~15-20% precision on some couplings (W/Z, g, gamma) No evidence of deviations from the SM so far…

Unreasonable Kindness of Nature

OURS IS HERE! Rich pattern of decays, many

measurable couplings

Higgs: Where Are We Headed

“Today’s Discovery is Tomorrow’s Background”

Higgs: Where Are We Headed“Today’s discovery is tomorrow’s precision tool

is the day after tomorrow’s background”20 Higgs working group report

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Figure 1-3. Measurement precision on W , Z , � , and g at di↵erent facilities.

A number of studies have presented results combining measurements from di↵erent facilities [88, 89]. Ageneral observation is that the precision in the measurement of many Higgs coupling at a new facility arereasonably or significantly improved, and these quickly dominate the combined results and overall knowledgeof the relevant coupling parameters. Exceptions are the measurements of the branching fractions of raredecays such as H ! �� and H ! µ+µ� where results from new lepton colliders would not significantlyimprove the coupling precisions driving these decays. However, precision measurements of the ratio of Z/�

at hadron colliders combined with the high-precision and model-independent measurements of Z at a leptoncollider would substantially increase the precision on � .

Community Planning Study: Snowmass 2013

From Snowmass-13 Higgs Report

2020s: Higgs Precision Era?• LHC upgrades will measure several couplings down to a few % level,

limited by composite nature of the proton

• Further improvements require a next-generation electron-positron collider - a “Higgs factory”, allowing 0.1-1% precision

• Currently the world-wide particle physics community is actively discussing design options for such a facility

• The most advanced proposal is “International Linear Collider” (ILC)

• Higgs discovery completes the Standard Model story: all particles predicted by the SM have now been verified

• But, strong theoretical arguments suggest that the SM story is incomplete: expect new particles, beyond the SM (BSM), with masses not too much higher than the Higgs mass

• These particles can have “virtual” effects on the Higgs, modifying its properties (e.g. couplings)

• I will outline two physical questions that can be probed by precise Higgs coupling measurements: Naturalness, and the dynamics of Electroweak Phase Transition

What Can We Learn?

Higgs and NaturalnessMarco Farina, MP, Nicolas Rey-Le Lorier, PRD 90 (2014) 015014

• Quantum Field Theory is the marriage of relativity and quantum mechanics

• Introduce a field for each kind of elementary particle: photon EM field, electron Electron field, Higgs boson Higgs field, etc.

• “Mattress Theory”: Each field is equivalent to a very large set of independent harmonic oscillators: its Fourier modes (or waves)

!

• Quantize oscillators particles: 1 quantum of energy in the mode = 1 particle with momentum

A Little Bit of QFT

• Consider a region of space filled with uniform Higgs field:

!

!

• SM postulates Higgs potential:

!

!

• Higgs field seeks to minimize its energy. The lowest-energy state has a non-zero Higgs field - “vacuum expectation value”

• Higgs boson = quantum of a small excitation around this vev

Higgs Field Energy (“Potential”)

• Imagine a thought experiment: vary inside the box

!

!

• In the SM, masses of all particles (e.g. electron) depend on the value of the Higgs field

• Electron field oscillators have zero-point energies:

!

• This is formally infinite, but unobservable since only differences in energy can be measured: for example

Higgs Energy: Quantum Correction

• Total energy = classical + quantum:

• Quantum part = Coleman-Weinberg potential:

!

• Higgs mass:

• Naturalness Problem: SM predicts

• Incredible level of “fine-tuning” between the 2 terms is required to fit the data!

Naturalness Problem

partner analysis remain valid. We discuss our findings and conclude in Section 5.

2 General Argument: Top Partners, Naturalness, and

the Higgs Couplings

The starting point of our analysis is a single Higgs doublet H with the SM tree-level potential

V (H) = �µ2|H|2 + �|H|4. (1)

This hypothesis is the simplest interpretation of the LHC discovery consistent with all other

experimental data. In particular, there is no evidence in the data of H mixing with other scalar

fields, and the constraints on such mixing are now quite stringent. In the SM, the measurements

of the Higgs vacuum expectation value (vev) and mass provide precise values for the parameters

in the potential:

µ = 90 GeV, � = 0.13. (2)

How natural are these parameters? To address this question, we need to consider quantum

corrections to the potential (1). At the one-loop order, these corrections are conveniently given

by the Coleman-Weinberg (CW) formula

VCW(h) =1

2

X

k

gk

(�1)Fk

Zd4`

(2⇡)4log

�`2 + m2

k

(h)�

, (3)

where the sum runs over all particles in the model, and gk

and Fk

is the multiplicity and fermion

number of each particle, respectively. For example, for a gauge-singlet complex scalar, g = 2 and

F = 0; for a gauge-singlet Dirac fermion, g = 4 and F = 1. Here h/p

2 is the real part of the

U(1)em-neutral component of H; in the SM vacuum, hhi = 246 GeV. The one-loop correction to

the Higgs mass parameter is given by

�µ2 ⌘ �2VCW

�h2|h=0. (4)

In the SM, the largest contribution to the CW potential comes from the top quark, since the top

Yukawa is the strongest coupling of the Higgs:

�µ2 = �3y2t

8⇡2⇤2 + . . . , (5)

where ⇤ is the scale at which all loop integrals in VCW are cut o↵. Since we expect ⇤ � MEW, the

quantum correction to µ from the top loop is unreasonably large, and would require fine-tuning

if no new physics is present. If the theory is weakly coupled at the TeV scale, the only way to

2

• To restore naturalness: introduce new “BSM” particles, such that

• For example, “top partners” (top loops are the largest SM contribution):

• Sum rule: fix

• This can be enforced by symmetry: supersymmetry, global symmetry in Little Higgs models, etc.

• Residual fine-tuning is roughly : the heavier the top partner, the less precise the cancellation in becomes

• Target for experiment: search for top partner, if no fine-tuning expect to discover it with mass below ~TeV

Beyond the SM

• Direct searches at the LHC already place significant bounds on various models of top partners

• But there are many ways in which top partners can “hide” from direct searches, even in run-2 and HL-LHC

• Basic problem: in many cases top partner signatures look indistinguishable from much more frequent SM processes (“backgrounds”)

Direct Top Partner Searches

• Alternative route: indirect search for top partner effects in Higgs couplings (a la Higgs “discovery” in PEW)

• Top partner contribution to couplings to photons and gluons is given by “low-energy Higgs theorems”: e.g. 1 spin-0 TP

!

• Since is fixed, only depends on , which also controls fine-tuning

• Main result: robust, model-independent inverse correlation of deviations in and , and fine-tuning. If no FT, deviations should be easily observable in the precision Higgs program

Connection with Higgs Couplings

1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14

1000

500

2000

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Rg

Mass@G

eVD

Spin-0 Top Partner

0.95 0.96 0.97 0.98 0.99 1.00

1000

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Mass@G

eVD

Spin-0 Top Partner

A 1% measurement of would probe the top partner mass of

~1.2 TeV...

20 Higgs working group report

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Figure 1-3. Measurement precision on W , Z , � , and g at di↵erent facilities.

A number of studies have presented results combining measurements from di↵erent facilities [88, 89]. Ageneral observation is that the precision in the measurement of many Higgs coupling at a new facility arereasonably or significantly improved, and these quickly dominate the combined results and overall knowledgeof the relevant coupling parameters. Exceptions are the measurements of the branching fractions of raredecays such as H ! �� and H ! µ+µ� where results from new lepton colliders would not significantlyimprove the coupling precisions driving these decays. However, precision measurements of the ratio of Z/�

at hadron colliders combined with the high-precision and model-independent measurements of Z at a leptoncollider would substantially increase the precision on � .

Community Planning Study: Snowmass 2013

3 Benchmark Model: a Single Top Partner

The simplest possibility is that there is a single top partner, in fundamental rep of SU(3) and

with electric charge 2/3, just like the SM top.3 This simple model can be used as a benchmark

for evaluating the potential of precision Higgs couplings to probe naturalness. In this case, c1 is

fixed by the sum rule (8), and m0,1 is the only free parameter in the predictions:

C�

= Cg

= 1 +1

4

y2t

v2

m20,1 + y2

t

v2(spin 0 partner);

C�

= Cg

= 1� y2t

v2

2m20,1 � y2

t

v2(spin 1/2 partner). (14)

The correlation between the Higgs coupling deviations from the SM and the mass of the top

partner is shown in Fig. 1. For reference, we also show constraints obtained from a fit to the

current LHC-7, LHC-8 [12, 13] and Tevatron [14] data, assuming that top-partner loops are the

only non-SM contribution to Higgs couplings. A broad range of top partner masses in the region

motivated by naturalness are currently allowed by data: the 95% c.l. limit on the top partner

mass is about 320 GeV for a spin-0 top partner, and 400 GeV for a spin-1/2 partner. However,

future precise measurements of the Higgs coupling at the LHC-14 and a future e+e� facility

would probe much of the interesting parameter space. For example, a 1% measurement of the

gluon coupling will probe the top partner masses in excess of 1 TeV, for both spin-0 and spin-1/2

top partners.

Since the one-partner model has only one free parameter, the deviations in gluon and pho-

ton couplings are correlated. This is shown in Fig. 2, along with the current and future LHC

constraints on the two couplings. (We used the information provided in Ref. [1] to estimate the

LHC-14 contours.) It is clear that the constraints are strongly dominated by the gluon coupling

measurement, due to both the slope of the trajectory and the stronger experimental bound on

Rg

. If a deviation from the SM is observed, it would be straightforward to check whether it can

be interpreted within the one-partner framework by simply checking whether the trajectories

shown here intersect with the experimentally determined region. If the answer is positive, these

measurements will also allow to unambiguously determine the top partner spin.

The connection between Higgs couplings and fine-tuning is illustrated more directly in Figs. 3

and 4. Since the top partners only cut o↵ the quadratic divergence in the top loop, leaving the

logarithmic divergence uncanceled, the value of the fine-tuning measure � depends logarithmi-

cally on the scale ⇤ where the logarithmic divergence is cut o↵. The value of ⇤ is very model-

3The single partner model of this section is applicable to models with multiple top partners if they have the

same m0,i parameters: for example, the MSSM with two degenerate stops.

6

* assuming that a ~1% direct measurement of coupling is

available! This needs e+e- @ 550 GeV

5

10

25

50

100

1.00 1.02 1.04 1.06 1.08 1.10 1.12

4

6

8

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12

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Rg

LogHLL

Spin-0 Partner, Fine Tuning

2

5

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25

50

1000.96 0.97 0.98 0.99 1.00

4

6

8

10

12

14

16

RgLogHLL

Spin-0 Partner, Fine Tuning

... and imply fine-tuning of at least 1/25 if no deviation from the SM is seen

Higgs and Electroweak Phase TransitionAndrey Katz, MP, JHEP 1408 (2014) 022

• Let’s fill our H-box with plasma of electrons/photons in thermal equilibrium

!

!

• Electron field oscillators have thermal energies:

!

!

• This gives an extra “thermal” contribution to the Higgs potential

Higgs Energy: Thermal Correction

• At sufficiently high T, the thermal correction drastically changes the nature of the Higgs potential - is preferred!

!

!

• Since all SM masses , at high T they disappear - a new phase (“unbroken electroweak symmetry”)

• Right after the Big Bang, Universe was filled with a high-T plasma, was in the unbroken-EW phase

• Expanding and cooling, the Universe undergoes the “electroweak phase transition” to the current, broken-symmetry phase, about after the Big Bang

Electroweak Phase Transition

H

V

H

V

T < Tc T > Tc

kT ∼ 100 − 1000 GeV, T ∼ 1015

K

10−10

sec

• Phase transition may occur through tunneling of the Higgs field - 1st order transition. Nucleation and merging of broken-EW bubbles (“boiling”). Non-equilibrium, entropy is produced.

!

!

• Or, it may occur gradually, adiabatically, with the Higgs field smoothly transitioning from 0 to v - 2nd order transition or crossover

!

Phase Transition Dynamics

• It is believed that the observed matter-antimatter asymmetry arose dynamically, very soon (<1 sec) after the Big Bang

• Many mechanisms proposed; one of the most theoretically attractive is “Electroweak Baryogenesis”, in which the asymmetry is generated during the EWPT

• It only works if transition is 1-st order (out-of-equlibrium)

EWPT and “Baryogenesis”

• In the SM, electroweak phase transition is 2nd order no EW baryogenesis!

• However BSM particles can give new contributions to the thermal potential, change its shape: recall

!

!

• Significant modification in the thermal potential requires a fairly light particle (roughly <300 GeV), with significant coupling to the Higgs

• Virtual effects of such a particle necessarily affect the Higgs couplings to SM we can use precision measurements of Higgs couplings to determine the order of the transition!

Higgs Couplings and EWPT

Analytic Example• A special case can be studied analytically*:

• High-temperature expansion of the thermal potential:

!

• Location of the broken-symmetry minimum at finite T:

• Critical temperature:

• Solve together:

• Strongly 1-st order if

• Gluon-Higgs coupling:

3 EWPT/Higgs Coupling Connection: Analytic Treatment

Before presenting numerical results, let us consider a much-simplified treatment of the

problem which can be carried through analytically. Even though the approximations

made here are often not strictly valid in examples of real interest, this analysis never-

theless provides a qualitatively correct and useful illustration of the physics involved.

To drive a first-order EWPT, the BSM scalar � should provide the dominant loop

contribution to the Higgs thermal potential at T ⇠ Tc

. Let us therefore ignore the SM

contributions. If Tc

is significantly higher than all other mass scales in the problem,

a high-temperature expansion of the thermal potential can be used to analyze the

phase transition, and zero-temperature loop corrections to the e↵ective potential can

be ignored. For simplicity, we will also omit the resummed daisy graph contributions

to the thermal potential. In this approximation,

VT

('; T ) ⇡ g�m2�(')T 2

24� g�m3

�(')T

12⇡+ . . . (3.1)

The � mass in the presence of a background Higgs field is given by

m2�(') = m2

0 +

2'2. (3.2)

If m0 is su�ciently small, the second term in the thermal potential (3.1) is e↵ectively

cubic in '. Such a negative '3 term can result in a stable EWSB minimum of the

potential at high temperature, as required for first-order EWPT. Motivated by this, let

us consider the case m0 = 0, which allows for simple analytic treatment. The e↵ective

potential is

Ve↵('; T ) = V0(') + VT

('; T ) ⇡ 1

2

✓�µ2 +

g�T 2

24

◆'2 � g�3/2T

24p

2⇡'3 +

�

4'4. (3.3)

The unbroken symmetry point ' = 0 is a local minimum as long as

g�T 2

24� µ2 > 0. (3.4)

The location of the other minimum is given by the larger root, '+, of the quadratic

equation

�'2 � g�3/2T

8p

2⇡'� µ2 +

g�T 2

24= 0. (3.5)

The critical temperature Tc

for the first-order transition is determined by the condition

V (0; Tc

) = V ('+(Tc

); Tc

). (3.6)

– 11 –

Solving Eqs. (3.5), (3.6) yields

T 2c

=24µ2

g�⇣1� g�

2

24⇡

2�

⌘ , '+(Tc

) =g�3/2T

c

12p

2⇡�. (3.7)

Requiring that a first-order transition occurs, T 2c

> 0, and is strongly first-order,

'+(Tc

)/Tc

> 1, yields a range of acceptable values of :

5.5

g1/2�

> >3.6

g2/3�

. (3.8)

As an example, consider a color-triplet, weak-singlet � field, as in the “RH stop” or

“Exotic Triplet” benchmark models of Table 1. In this case, our estimate suggests that

a strongly first-order transition occurs for values of between 1.1 and 2.2. At the same

time, the � loop contribution to the Higgs-gluon coupling is

Rg

=1

8

v2

m20 + v2

. (3.9)

In the limit m20 ⌧ v2, which for ⇠ 1 corresponds to a broad range of m0, we obtain

Rg

⇡ 1/8, or a 12.5% enhancement in the hgg coupling compared to the SM. In fact,

even larger enhancements are possible for negative values of m20. Of course, the hgg

deviations from the SM become small when m20 � v2; however, in this regime, the �

mass is well above the weak scale, and it does not a↵ect the EWPT dynamics either.

Thus, models with first-order EWPT should produce a large e↵ect, of order 10% or

more, in the Higgs-gluon coupling. This conclusion will be confirmed by the numerical

analysis in the following section.

4 Results

We developed a numerical code to analyze the dynamics of the electroweak phase

transition in each of the benchmark models listed in Table 1. Given the model and the

values of the free parameters, m0, and ⌘, the code computes the e↵ective potential

as a function of temperature, Eq. (2.5), and identifies the critical temperature Tc

. The

x integral in the finite-temperature potential (2.7) is performed numerically, with no

high-temperature approximation. This is important since the critical temperature in

our models is typically of order 100 GeV, which is at the same scale as the masses of

the particles involved. To identify the region in the parameter space of a given model

where a strongly first-order phase transition occurs, we compute Tc

and ⇠ on a dense

– 12 –

0.5

0.5

0.50.5

0.5

0.9

0.90.90.9

0.9

0.9

1.31.31.3

1.3

1.3

1.3 1.3

1.3

0.17

0.25

0.37

1.5 2.0 2.5 3.0150

200

250

300

350

k

mfHGe

VL

h=1, f~H3, 1L-4ê3, hgg

0.5

0.5

0.50.5

0.5

0.9

0.90.90.9

0.9

0.9

1.31.31.3

1.3

1.3

1.3 1.3

1.3

0.15

0.25

0.45

1.5 2.0 2.5 3.0150

200

250

300

350

k

mfHGe

VL

h=1, f~H3, 1L-4ê3, hgg

Figure 2. Same as Fig. 1, for the Exotic Triplet model (see Table 1).

0.6

0.60.6 0.6

0.90.90.9

0.9

1.31.3 1.3

1.3

1.31.3

1.3

0.7

1

1.5

2

1.5 2.0 2.5 3.0150

200

250

300

350

400

450

k

mfHGe

VL

h=1, f~H6, 1L1ê3, hgg0.6

0.60.6 0.6

0.90.90.9

0.9

1.31.3 1.3

1.3

1.31.3

1.3

0.015

0.025

0.045

1.5 2.0 2.5 3.0150

200

250

300

350

400

450

k

mfHGe

VL

h=1, f~H6, 1L1ê3, hgg

Figure 3. Same as Fig. 1, for the Sextet model (see Table 1).

in precision expected in future experiments will allow these models to be probed. In

both models, the minimal deviation in the hgg coupling compatible with a strongly

– 15 –

Example: BSM “Sextet”

0.5

0.5

0.5

0.50.5

0.9

0.90.90.9

0.9

0.9

1.3 1.31.3

1.3

1.3

1.3 1.3

1.3

0.17

0.25

0.37

1.5 2.0 2.5 3.0150

200

250

300

350

k

mfHGe

VL

h=1, f~H3, 1L2ê3, hgg

0.5

0.5

0.5

0.50.5

0.9

0.90.90.9

0.9

0.9

1.3 1.31.3

1.3

1.3

1.3 1.3

1.3

0.05

0.07

0.1

1.5 2.0 2.5 3.0150

200

250

300

350

k

mfHGe

VL

h=1, f~H3, 1L2ê3, hgg

Figure 1. The region of parameter space where a strongly first-order EWPT occurs in the“RH stop” benchmark model. Also shown are the fractional deviations of the hgg (left panel)and h�� (right panel) couplings from their SM values. Solid/black lines: contours of constantEWPT strength parameter ⇠ (see Eq. (2.9)). Dashed/orange lines: contours of constanthgg/h�� corrections. (For the case of h�� the correction is always negative, and the plotsshow its absolute value.) In the shaded region, phase transition into a color-breaking vacuumoccurs before the EWPT.

reported by the ATLAS collaboration [? ] are

Rg

= 1.08± 0.14,

R�

= 1.23+0.16�0.13. (4.2)

These results already have interesting implications for the possibility of a strongly first-

order EWPT. In particular, the Sextet model, where the deviations in the hgg coupling

in the region with first-order EWPT are predicted to be 70% or above, is completely

excluded.4 It is clear that models where � is in even larger representations of SU(3)c

,

e.g. an octet, are also ruled out. The RH Stop and Exotic Triplet models, on the other

hand, are still compatible with data at 68% CL. However, a dramatic improvement

4A potential loophole that should be kept in mind is that these bounds assume no sizable BSMcontributions to the Higgs width. If such a contribution is allowed, a 70% deviation in the hgg couplingis only excluded at a 2 sigma level, and thus the Sextet model remains marginally compatible withdata.

– 14 –

ATLAS: already ruled out!*

[* usual caveat: SM total width assumed]

First-order EWPT implies

0.5

0.5

0.5

0.50.5

0.9

0.90.90.9

0.9

0.9

1.3 1.31.3

1.3

1.3

1.3 1.3

1.3

0.17

0.25

0.37

1.5 2.0 2.5 3.0150

200

250

300

350

k

mfHGe

VL

h=1, f~H3, 1L2ê3, hgg

0.5

0.5

0.5

0.50.5

0.9

0.90.90.9

0.9

0.9

1.3 1.31.3

1.3

1.3

1.3 1.3

1.3

0.05

0.07

0.1

1.5 2.0 2.5 3.0150

200

250

300

350

kmfHGe

VL

h=1, f~H3, 1L2ê3, hgg

Figure 1. The region of parameter space where a strongly first-order EWPT occurs in the“RH stop” benchmark model. Also shown are the fractional deviations of the hgg (left panel)and h�� (right panel) couplings from their SM values. Solid/black lines: contours of constantEWPT strength parameter ⇠ (see Eq. (2.9)). Dashed/orange lines: contours of constanthgg/h�� corrections. (For the case of h�� the correction is always negative, and the plotsshow its absolute value.) In the shaded region, phase transition into a color-breaking vacuumoccurs before the EWPT.

reported by the ATLAS collaboration [? ] are

Rg

= 1.08± 0.14,

R�

= 1.23+0.16�0.13. (4.2)

These results already have interesting implications for the possibility of a strongly first-

order EWPT. In particular, the Sextet model, where the deviations in the hgg coupling

in the region with first-order EWPT are predicted to be 70% or above, is completely

excluded.4 It is clear that models where � is in even larger representations of SU(3)c

,

e.g. an octet, are also ruled out. The RH Stop and Exotic Triplet models, on the other

hand, are still compatible with data at 68% CL. However, a dramatic improvement

4A potential loophole that should be kept in mind is that these bounds assume no sizable BSMcontributions to the Higgs width. If such a contribution is allowed, a 70% deviation in the hgg couplingis only excluded at a 2 sigma level, and thus the Sextet model remains marginally compatible withdata.

– 14 –

Another Example: “RH Stop”

This will be probed at 3-sigma level at LHC-14

First-order EWPT implies

Final Example: BSM “Singlet”

No deviation in couplings to photons/gluons No LHC signature

However a small deviation in coupling to Z, can be probed at the ILC/Higgs factory

0.70.7

0.9

0.9

-0.01

-0.007

-0.004

1.8 2.0 2.2 2.4 2.6 2.8 3.0

160

180

200

220

240

260

280

k

mfHGe

VL

h=2, Singlet, hZZ

0.7

0.7

0.9

0.9

0.1

0.15

0.2

1.8 2.0 2.2 2.4 2.6 2.8 3.0

160

180

200

220

240

260

280

k

mfHGe

VL

h=2, Singlet, h^3

Figure 6. The region of parameter space where a strongly first-order EWPT occurs in theSinglet benchmark model. Also shown are the fractional deviations of the e+e� ! hZ

cross section (left panel) and Higgs cubic self-coupling (right panel) from their SM val-ues. Solid/black lines: contours of constant EWPT strength parameter ⇠ (see Eq. (2.9)).Dashed/orange lines: contours of constant �

hZ

/�3 corrections. In the shaded region, phasetransition into a color-breaking vacuum occurs before the EWPT.

are in the 10 � 20% range, making them di�cult to test at the proposed facilities.

(The accuracy of the self-coupling measurement at an ILC-1T with luminosity upgrade

is estimated to be about 13% [? ], while at TLEP it can be measured with a preci-

sion of about 30% via its contribution to Higgsstrahlung [? ].) Thus, it appears that

the Higgsstrahlung cross section provides the most sensitive probe of this challenging

scenario.

5 Discussion

In this paper, we considered several toy models which can induce a first-order elec-

troweak phase transition in the early Universe. In all models, we found a strong cor-

relation between the strength of the phase transition and the deviations of the Higgs

couplings from the SM. This suggests that precise measurements of the Higgs couplings

have a potential to definitively determine the order of the electroweak phase transition.

Such a determination would be not only fascinating in its own right, but would also

– 18 –

• Higgs discovery completes the Standard Model story: all particles predicted by the SM have now been verified

• We’re entering the era of precision measurement of Higgs properties, such as its couplings to other SM particles

• Higgs couplings can be used as an indirect probe to search for physics beyond the SM

• Higgs couplings to photons and gluons provide a robust, model-independent signature of naturalness

• Precise measurements of Higgs couplings will allow to determine the order of electroweak phase transition, testing viability of electroweak baryogenesis scenario

Conclusions