constraining the higgs potential at e+ e-...

Constraining the Higgs potential at e+e− colliders

Xiaoran Zhao1

In collaboration with

Fabio Maltoni(CP3), Davide Pagani(TUM)

Based on JHEP 1807 (2018) 087

1Centre for Cosmology, Particle Physics and Phenomenology (CP3)Universite catholique de Louvain, Belgium

July 24, 2018

Xiaoran Zhao (CP3 UCL) Constraining the Higgs potential at e+e− colliders July 24, 2018 1 / 14

https://doi.org/10.1007/JHEP07(2018)087

The Higgs boson

Discovered at LHC, 4th July, 2012.

Properties measured at LHC.

Next step: measuring Higgs self-couplings to probe the Higgspotential.

e+e− colliders


The Higgs potential in SM

The SM Higgs potential is fixed by gauge symmetry and renormalisability:

V SM(Φ) = −µ2(Φ†Φ) + λ(Φ†Φ)2 (1)

After electroweak symmetry breaking:

V SM(Φ) =1

2m2

HH2 + λSM3 H3 +

λSM4

4H4 (2)

where λSM3 = λSM4 = λ =m2

H2v2 , fully determined in SM!

Various new physics models lead to non-SM values of λ3 and/or λ4.Measuring λ3 and λ4: the main targets of the current and future colliders!


Direct measurement

Directly measuring λ3: double Higgs production. Loose bound atLHC, good precision at future e+e− and hadron colliders.

g

g

H

H H

H

HH

g

g

HH

HH

Directly measuring λ4: triple Higgs production. Only loose bounds at100TeV pp collider, e.g. λ4/λ

SM4 ∈ [−20, 30] [Chen et.al. ’15]

O(1)ab at e+e−, too tiny for measurements, but sensitive on λ4!


http://arxiv.org/abs/1510.04013

Indirectly probe

λ3: single Higgs at one (more) loopAttract lots of attention recently!McCullough ’14 Gorbahn and Haisch ’16 Degrassi et.al. ’16 Bizon et.al. ’16 Di Vita et.al. ’17 Maltoni et.al. ’17

g

gHH

H W±H

W±H

H

Two-loop effects in electroweak precision measurement:Degrassi et.al. ’17 Kribs et.al. ’17

How about λ4? Double Higgs production at one-loop!










Higgs self-couplings at e+e− colliders

Previously only λ3 at tree-level of double Higgs and one-loop level of singleHiggs are studied.

λ3 λ4

Single Higgs one-loop (two-loop)Double Higgs tree-level one-loopTriple Higgs tree-level tree-level

We extend to one-loop level of double Higgs, which is sensitive on λ4. Wealso examine triple Higgs, where λ3 and λ4 appear at tree-level.

e−

e+

Z

Z

He

−

H

e+

Z

Z

H

H e−

νe

νe

W

e+

W

H

H

e−

νe

νe

W

e+

W

H

H

H

H


EFT parametrization and Renormalization

λ3

λ3

λ3 λ3λ4 λ5

UV Divergent⇒ EFT for renormalization.

V (Φ) = −µ2(Φ†Φ) + λ(Φ†Φ)2 +∑n=3

c2n

Λ2n−4(Φ†Φ− 1

2v2)2 (3)

To vary λ3 and λ4 independently, we need both c6 and c8.

κ3 =λ3

λSM3

= 1 +c6v

2

Λ2= 1 + c6 (4)

κ4 =λ4

λSM4

= 1 +6c6v

2

Λ2+

c8v4

Λ4= 1 + 6c6 + c8 (5)


Bounding the Higgs potential: setup

Two production mechanisms

Z boson associated production: ZHn

W boson fusion(WBF).

Colliders:

Circular colliders: CEPC and FCC-ee. Only single Higgs is available.

Linear colliders: ILC and CLIC. All processes are available.

Scenarios:

Scenario 1: c8 effects are negligible.

Scenario 2: c8 effects are non-negligible.


Scenario 1: single and double Higgs

c8 effects are negligible, we only have c6.

−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 −4 −3 −2 −1 0 1 2 3 4 5

c6

ctrue6

WBF H at CLIC-1400

ZH at CEPC-250−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 −4 −3 −2 −1 0 1 2 3 4 5

c6

ctrue6

ZHH at ILC-500

WBF HH at CLIC-1400

Left:Single Higgs Right:Double Higgs

“X”-shape band, different center point⇒ complementary.The band for double Higgs is thinner than single Higgs.


Scenario 1: combined

c8 effects are negligible, we only have c6.

−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 −4 −3 −2 −1 0 1 2 3 4 5

c6

ctrue6

ILC-250ILC-500ILC-1000

−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 −4 −3 −2 −1 0 1 2 3 4 5

c6

ctrue6

CLIC-350CLIC-1400CLIC-3000

Left: ILC Right: CLIC

A good constraint over all region can be obtained at ILC.

CLIC cannot resolve degeneracy around c6 = 0.5.


Scenario 2: double Higgs

Non-negligible c8, assuming SM cross sections:

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHH at ILC-500

WBF HH at ILC-1000

SM

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5c8

c6

WBF HH at CLIC-1400

WBF HH at CLIC-3000

SM

Combining ZHH and WBF HH yields better bounds.


Scenario 2: triple Higgs

Tiny cross section⇒ zero events once SM is assumed.

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHHH at ILC-1000

WBF HHH at ILC-1000

SM

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

WBF HHH at CLIC-1400

WBF HHH at CLIC-3000

SM

Useless at ILC.

WBF HHH is important at CLIC, especially at 3TeV.


Scenario 2: combined

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ILC-500ILC-1000

CLIC-1400CLIC-3000

SM

ILC is better on c6

CLIC is better on c8


Conclusion

e+e− colliders

One-loop for single and double Higgs

tree-level for triple Higgs

(c6, c8): EFT description to cancel UV divergence

WBF HH and ZHH are complementary

Strong dependence on c8 for triple Higgs

c8, comparable or even better than 100TeV.


Backup slides


Counter term

δc6 =∆

16π2

[c6

(54λ− 9

m2Z + 2m2

W

v2+ 6

Ncm2t

v2

)+c8v

2

Λ2

(64λ− 6

m2Z + 2m2

W

v2+ 4

Ncm2t

v2

)+

45c26v

2

Λ2

+20c10λv

4

Λ4+

36c6c8v4

Λ4

]where ∆ = 1

ε − log(4π) + γE is the divergence in MS convention.


Results for single Higgs

σNLO =σLO + σ1−loop, σLO = σSMLO (6)

σ1−loop =σ0 + σ1c6 + σ2c26 (7)

σphenoNLO =σLO + σ1c6 + σ2c26 (8)

=σLO(1 + (κ3 − 1)C1 + (κ23 − 1)C2) (9)

C2 =δZSM,λH ≈ −0.00154 (10)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

250 350 500 1000 2000 3000

σLO

[fb]

√

s [GeV]

ZH

WBF H

−0.5

0

0.5

1

1.5

2

2.5

250 350 500 1000 2000 3000

C1[%

]

√

s [GeV]

ZH

WBF H


Results for double Higgs

σLO =σ0 + σ1c6 + σ2c26 (11)

σNLO =σLO + σ1−loop (12)

σ1−loop =σ00 + σ10c6 + σ20c26 + σ30c

36 + σ40c

46 (13)

+ c8(σ01 + σ11c6 + σ21c26 ) + c10(σ001 + σ101c6) (14)

c10 contribution can be written as a kinematically independent shift to c6:

c6 → c6 +5λc10

4π2(1− log

m2H

µ2r

) (15)

σphenoNLO =σLO + ∆c6 + ∆c8 (16)

∆c6 =σ30c36 + σ40c

46 (17)

∆c8 =c8(σ01 + σ11c6 + σ21c26 ) (18)



LO cross sections:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

350 500 1000 2000 3000

σLO

[fb]

√

s [GeV]

ZHH (c6 = −4)ZHH (c6 = −2)ZHH (c6 = 0)ZHH (c6 = 2)ZHH (c6 = 4)

0

5

10

15

20

25

30

35

40

350 500 1000 2000 3000

σLO

[fb]

√

s [GeV]

WBF HH (c6 = −4)WBF HH (c6 = −2)WBF HH (c6 = 0)WBF HH (c6 = 2)WBF HH (c6 = 4)

0.01

0.1

1

10

100

−5 −4 −3 −2 −1 0 1 2 3 4 5

σLO

[fb]

c6

ZHH: 500 GeVWBF HH: 1 TeV

WBF HH: 1.4 TeVWBF HH: 3 TeV



The one-loop contributions:

0.0001

0.001

0.01

0.1

1

10

−5 −4 −3 −2 −1 0 1 2 3 4 5

ZHH: 500 GeV

σ[fb]

c6

σLO∆σc6

∆σc8/c80.0001

0.001

0.01

0.1

1

10

100

−5 −4 −3 −2 −1 0 1 2 3 4 5

WBF HH: 1 TeV

σ[fb]

c6

σLO∆σc6

∆σc8/c80.0001

0.001

0.01

0.1

1

10

100

−5 −4 −3 −2 −1 0 1 2 3 4 5

WBF HH: 3 TeV

σ[fb]

c6

σLO∆σc6

∆σc8/c8

Sensitivity on c8 depends on value of c6


Results for Triple Higgs

σLO = σ00 +∑

1≤i+2j≤4

σij ci6c

j8 (19)

0

0.2

0.4

0.6

0.8

1

1.2

500 1000 1500 2000 2500 3000

σLO

[ab]

√s [GeV]

ZHHH (c6 = 0, c8 = 0)ZHHH (c6 = 0, c8 = −1)ZHHH (c6 = 0, c8 = 1)

100 × σ02(ZHHH)0

0.5

1

1.5

2

2.5

3

500 1000 1500 2000 2500 3000

σLO

[ab]

√

s [GeV]

WBF HHH (c6 = 0, c8 = 0)WBF HHH (c6 = 0, c8 = −1)WBF HHH (c6 = 0, c8 = 1)

σ02(WBF HHH)

Although σSM = σ00 is tiny, for WBF HHH the cross section stronglydepends on (c6, c8): large c8 ⇒ σLO ≈ c2

8σ02 ≈ c28σ00.


Bounding the Higgs potential: double Higgs

Scenario 2: non-negligible c8, BSM cases:

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHH at ILC-500

WBF HH at CLIC-1400

ctrue6

= −4

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHH at ILC-500

WBF HH at CLIC-1400

ctrue6

= −2

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHH at ILC-500

WBF HH at CLIC-1400

ctrue6

= −1

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHH at ILC-500

WBF HH at CLIC-1400

ctrue6

= 4

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHH at ILC-500

WBF HH at CLIC-1400

ctrue6

= 2

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ZHH at ILC-500

WBF HH at CLIC-1400

ctrue6

= 1


Bounding the Higgs potential: combined

Scenario 2: assuming BSM cases:

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ILC-500ILC-1000

CLIC-1400CLIC-3000

ctrue6

= −4

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ILC-500ILC-1000

CLIC-1400CLIC-3000

ctrue6

= −2

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ILC-500ILC-1000

CLIC-1400CLIC-3000

ctrue6

= −1

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ILC-500ILC-1000

CLIC-1400CLIC-3000

ctrue6

= 4

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ILC-500ILC-1000

CLIC-1400CLIC-3000

ctrue6

= 2

−30

−20

−10

0

10

20

30

−5 −4 −3 −2 −1 0 1 2 3 4 5

c8

c6

ILC-500ILC-1000

CLIC-1400CLIC-3000

ctrue6

= 1


Bounding the Higgs potential: scenarios

√s [GeV] P(e−, e+) Luminosity [ab−1] Relevant final states

CEPC 250 (0.0,0.0) 5.0 ZH, WBF H

FCC-ee240 (0.0,0.0) 10.0 ZH, WBF H350 (0.0,0.0) 2.6 ZH, WBF H

ILC250 (-0.8,0.3) 2.0 ZH, WBF H500 (-0.8,0.3) 4.0 ZHH, WBF H

1000 (-0.8,0.2) 2.0 ZHHH, WBF H(H(H))

CLIC350 (-0.8,0.0) 0.5 ZH, WBF H

1400 (-0.8,0.0) 1.5 ZHHH, WBF H(H(H))3000 (-0.8,0.0) 2.0 WBF H(H(H))

Scenario 1: c8 effects are negligible. Single, double and triple Higgs are considered.We allow both c6 = 0(SM-like) and c6 6= 0(BSM case)

Scenario 2: c8 effects are non-negligible. Here only double and triple Higgs areconsidered, since two-loop for single Higgs are not available. We explore how wellwe can constrain c8 and how much c8 can affect the measurement of c6.


constraining the higgs potential at e+ e-...

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