Symmetry and Geometry of Generalized Higgs Sectorsppp.ws/PPP2018/slides/nagai.pdfOutline...

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Ryo Nagai Tohoku University PPP2018 @ YITP, Aug. 6-10, 2018 in collaboration with M. Tanabashi (Nagoya U.), Y. Uchida (Nagoya U.), and K. Tsumura (Kyoto U.) Symmetry and Geometry of Generalized Higgs Sectors

Transcript of Symmetry and Geometry of Generalized Higgs Sectorsppp.ws/PPP2018/slides/nagai.pdfOutline...

Ryo Nagai Tohoku University

PPP2018 @ YITP, Aug. 6-10, 2018

in collaboration with M. Tanabashi (Nagoya U.), Y. Uchida (Nagoya U.),

and K. Tsumura (Kyoto U.)

Symmetry and Geometryof Generalized Higgs Sectors

Outline

Introduction

Higgs Effective Field Theory

Summary and Outlook

Perturbative Unitarity vs. EWPTs

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

We found a h(125) !

Symmetry and Geometry of Generalized Higgs Sectors

• The 125GeV Higgs boson was discovered at the LHC.

Ryo Nagai (Tohoku U)

• Higgs coupling measurements

V

V

κVκF

h(125)

Symmetry and Geometry of Generalized Higgs Sectors

h(125) = hSM ?

Ryo Nagai (Tohoku U)

~10% deviation from κV, F = 1 is still allowed.

ATLAS-CONF-2017-047

Vκ0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0.5

1

1.5

2

2.5 PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ZZ*→H and γγ→H

= 125.09 GeVHm

SM predictionBest fitCombined 68% CLCombined 95% CL

68% CLγγ→H 68% CL4l→ZZ*→H

Vκ0.5 1 1.5 2

0.5

1

1.5

2

bb→H ττ→H

ZZ→H γγ→H

WW→H Combined

Best fit SM expected

regionσ1 regionσ2

(13 TeV)-135.9 fbCMS Preliminary

CMS-PAS-HIG-17-031

κV κV

κFκF

• Higgs coupling measurements

h(125) = hSM ?

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

M.Peskin (2014)

5%

4%

3%

2%

1%

W

CMS-1 CMS-2

ILC

CMS 250 500 500up 1000up1000

hWW hZZ

LHC

5%

4%

3%

2%

1%

Z

CMS-1 CMS-2

ILC

CMS 250 500 500up 1000up1000

LHC

LHC

~1% precise measurement is expected to be realized.

• Higgs coupling measurements

h(125) = hSM ?

Symmetry and Geometry of Generalized Higgs Sectors

LHC

Ryo Nagai (Tohoku U)

WLWL → WLWL

What if κV ≠ 1 ?

Longitudinal

Longitudinal Longitudinal

Longitudinal

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

WLWL → WLWL

What if κV ≠ 1 ?

Energy

Energy Energy

Energy

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

Unitarity is violated ?

MWLWL!WLWL ' sv2

�1 � k2

V�

h

WLWL → WLWL

MWLWL!WLWL ⇠ E2

M2 (1 � k2V)

What if κV ≠ 1 ?

Gauge boson Higgs boson

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

MWLWL!WLWL ' sv2

�1 � k2

V�

h

WLWL → WLWL

MWLWL!WLWL ⇠ E2

M2 (1 � k2V)

What if κV ≠ 1 ?

• The SM higgs boson ( κV =1 ) keeps the theory perturbatively unitary.

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

-���� -���� -���� -���� �����

��

Δκ�

Λ[���

]

• If κV ≠ 1, perturbative unitarity in WLWL scattering seems to be violated at certain energy scale.

What if κV ≠ 1 ?

Perturbativeunitarity violation

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

ff → ff

What if κV ≠ 1 ?

S ' 112

�1 � k2

V�

lnL2

M2h

h

• Electroweak Precision Tests (EWPTs) at LEP :

S = 0.03 ± 0.10 Global fitting by Gfitter Group

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

-���� -���� -���� -���� �����

��

Δκ�

Λ[���

]

What if κV ≠ 1 ?

Perturbativeunitarity violation

• κV ≠ 1 causes not only perturbative unitarity violation but also inconsistency with EWPTs.

EWPTs

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

What if κV ≠ 1 ?

Symmetry and Geometry of Generalized Higgs Sectors

Higgs Sector

h125

Ryo Nagai (Tohoku U)

What if κV ≠ 1 ?

Symmetry and Geometry of Generalized Higgs Sectors

Higgs Sector

π125

Ryo Nagai (Tohoku U)

Composite ?

What if κV ≠ 1 ?

Symmetry and Geometry of Generalized Higgs Sectors

Higgs Sector

h125

Elementary. But mixed state ?

Ryo Nagai (Tohoku U)

a H±….H

Outline

Introduction

Higgs Effective Field Theory

Summary and Outlook

Perturbative Unitarity vs. EWPTs

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

Ryo Nagai (Tohoku U)

Higgs EFT

• SU(2)W doublet filed : 4 = 3(NGBs) + 1(higgs)

Q. How can we formulate the Higgs sector ?

F =1p2

✓i(f1 + if2)

f0 � if3

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Symmetry and Geometry of Generalized Higgs Sectors

Higgs EFT

Q. How can we formulate the Higgs sector ?

125GeV higgsNGBs

F =1p2

exp✓

iv

pata◆✓

0v + h

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Ryo Nagai (Tohoku U) Roles of PU in extended Higgs scenarios

• SU(2)W doublet filed : 4 = 3(NGBs) + 1(higgs)

Higgs EFT

Q. How can we formulate the Higgs sector ?

F =1p2

exp✓

iv

pata◆✓

0v + h

<latexit sha1_base64="+AbJcqlzz61C9LJQkJ1u2NbL6zw=">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</latexit><latexit sha1_base64="+AbJcqlzz61C9LJQkJ1u2NbL6zw=">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</latexit><latexit sha1_base64="+AbJcqlzz61C9LJQkJ1u2NbL6zw=">AAAGo3icjVTNbtNAEJ4UCMVQ2oIESFwqoqAiINpUBSokpAouSFz630rdNnKcjWPVcdy1HZqu9sCBCy/AgRNIHBCPwYUX4NAnQIhjkbhwYHacRG1aB2x5dzwz3zc/O3Y19L0oZuwgN3Lm7Ln8+dEL1sVLY5fHJyavrEWtRDpi1Wn5LblRtSPhe4FYjb3YFxuhFHaz6ov16s4zY19vCxl5rWAl7oRiq2m7gVf3HDtGVWXiFV9oeE94XdqOKmvFo10ZqxmtLS72Qu5PpxZPqzYaXSlEoHjobdta89hOtm0u71jcF/V42uJV4XqBsqW0O1o52mKcW+27ilf9RKgGIjiyBrWuh8Wl5zbiO5WJAisxuqZOCuWuUIDutdCaHHkNFt4catACBxJogoAAYpR9sCHCexPKeIeo2wKFOomSR3YBmrAJegn0sFG7g6uLbwqZbPRskH4f2TUUoY67YZ/KxG12tQG+m1wiiuogzsdHIssUFNk39okdsq/sM/vB/gzJwXCYGjq4V1OsCCvjb24s//4nqpd/HzUE4aAU4Cowy+yemNpi7MEc1eQhIiSNICzFae+/PVx+vFRUt9kH9hPrfM8O2BesNGj/cj4uiqV3Q7Ko49pBriZadP9cBeo5Vi9gF31SyaXoinQBnanEtYMofQwl+iiTY+0/EAHxmt2cdEJ9F3RuR70kWhNiLOBsGf+wP1k29rBC+ixMkoHaPhUljkQxvd6j6e7Nk+p6aJRStB7om4/xxFCGnk8Wh0sxTAezWTbxcZHJTAaDe/iU4AHtWaxmdiSu/tDsjvOWYJY4U/65TO6Y6uJ0ei1kVKQZ9PJRVyO/Bs5jOo1mDu8T+0PKRaM1QnQ6qZK+EEVYw2FYPcyvQR6SzvglZXVyIkw2CfGdFm2uG20ww6OxBidD0dqLr/HvWR78V54U1mZKZVYqL84W5p92/6OjcBNuwTT28RHMw3NYgFXM7XtuLHctdz1fzL/IL+VXUteRXBdzFY5d+a2/jTVwXw==</latexit><latexit sha1_base64="+AbJcqlzz61C9LJQkJ1u2NbL6zw=">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</latexit>

• Examples :

LSM = (DµF)†(DµF) + · · ·<latexit sha1_base64="BaqWMYTtaud7SxZ3U26r/wT87gk=">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</latexit><latexit sha1_base64="BaqWMYTtaud7SxZ3U26r/wT87gk=">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</latexit><latexit sha1_base64="BaqWMYTtaud7SxZ3U26r/wT87gk=">AAAGWnicjVRLaxRBEK6sZs1DTaIeBC/BJRJRl16JughC0Bw8KORhHpAxy+xsZ3fIvJxHdDPM0Yt/wIMgKHgQf4YevOjNQ8A/IB4jePHg1zWzS7LJrE4z3TXV9X316Jque5YZhELsDhSOHR8snhgaHhk9eer02PjEmZXAjXxDLhuu5fprdT2QlunI5dAMLbnm+VK365ZcrW/dU/ur29IPTNd5FLY9+djWm465aRp6CFVtvKrZetgydCt+kNRizbfjpYdJcmd6rqbZkTbfMi9vaA29OT230fm+ohkNNwxq4yVRFvxMHhYqmVCi7Jl3JwrPaQRDowa5ZFBENklyKIRskU4BxjpVMDzoHlMMnQ/J5H1JCWMjWElY6NBuYW7iKwaTDssW63fAntAUbWJV7JO5uPVM6+BbxRKwVwM4C68PlkmaEt/Ee7EnPosP4of40ycGxaFyaGOtp1jp1cZenF/6/U9UJ/4uqg/CgORglogyvyYqtxA1qHJOJhAeayRj2c/2zsu9pduLU/El8Vb8RJ5vxK74iEyd7V/GuwW5+KpPFJuY2+CysZN0z1VCryF7SU9gk0pN9h6zzuEz9TG3gUoOoGQXpWJs/AfCYV61qpOOuO6Sz22/lY/diBlL6C1l73U7S0cNa6zPw0Q5qI0jUXKfF1XrZ9zdnX6KM4sEUopOeupmwZ/sy9CxyeNosg9VwXyWdbxNMKnOEHQVb5lu8JrHqnrHx2z1je4gb5lmmDPlr+Zyh5yXxqfngjFmTa+VBV2D7Vrox7QbVR9eY/abHEuC3QDotFN9/kNixioOxWoivhZb+HzGTzmqwx2hoomY7yhv1cxbb4T7ffV2Rsxzx3+C27PSe1ceFlaulyuiXFmYKc3eze7RIbpAF2kadbxFs3Sf5mkZsb2mT/SFvg5+LxaKw8XR1LQwkGHO0oGneO4v4QBU5w==</latexit><latexit sha1_base64="BaqWMYTtaud7SxZ3U26r/wT87gk=">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</latexit>

• What if e.g. Composite Higgs models h(125)�2 F ?<latexit sha1_base64="UCTKuNteCXb9dJWg3kwQlQcwZ4Y=">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</latexit><latexit sha1_base64="UCTKuNteCXb9dJWg3kwQlQcwZ4Y=">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</latexit><latexit sha1_base64="UCTKuNteCXb9dJWg3kwQlQcwZ4Y=">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</latexit><latexit sha1_base64="UCTKuNteCXb9dJWg3kwQlQcwZ4Y=">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</latexit>

Ryo Nagai (Tohoku U) Roles of PU in extended Higgs scenarios

• SU(2)W doublet filed : 4 = 3(NGBs) + 1(higgs)

LSMEFT = (DµF)†(DµF) +C

L2 F†F (DµF)†(DµF) + · · ·<latexit 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Symmetry and Geometry of Generalized Higgs Sectors

CCWZ method

• Systematic way for constructing model with G → H

Coleman-Wess-Zumino (1969) Coleman-Callen-Wess-Zumino (1969)

• General Lagrangian for NGBs arising through EWSB :

DµU = ∂µU � igUWµ + igYBµU

• NGB fields :

• Gauge bosons :

U = eiv pata

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Lp =v2

4Tr[DµU†DµU]

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* We here assume “custodial symmetry” for simplicity.

Ryo Nagai (Tohoku U)

• General Lagrangian for NGBs and singlet scalar(s)

Symmetry and Geometry of Generalized Higgs Sectors

CCWZ method

L =v2

4Tr[DµU†DµU]

1 + 2 Â

hkh

WWhv+ Â

h,h0khh0

WWhh0

v2 + · · ·!

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LSM =v2

4Tr[DµU†DµU]

✓1 + 2

hv+

h2

v2

• Examples:

LSM+S =v

2

4Tr[DµU

†D

µU]

✓1 + 2c

h

v+ 2s

H

v+ c

2 h2

v2 + s2 H

2

v2 + 2cshH

v2

Nagai-Tanabashi-Tsumura (2016)

Ryo Nagai (Tohoku U)

LMCSM =v2

4Tr[DµU†DµU]

✓1 + 2

✓1 � v2

8 f 2

◆hv+

✓1 � v2

f 2

◆h2

v2 + · · ·◆

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Symmetry and Geometry of Generalized Higgs Sectors

CCWZ method

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=12

∂µ(pa h h0 · · · )

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1

CCCA

<latexit sha1_base64="h4UMxh15t7lTn/70jWNjZpXMGog=">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</latexit><latexit sha1_base64="h4UMxh15t7lTn/70jWNjZpXMGog=">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</latexit><latexit sha1_base64="h4UMxh15t7lTn/70jWNjZpXMGog=">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</latexit><latexit sha1_base64="h4UMxh15t7lTn/70jWNjZpXMGog=">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</latexit>

=12

gij(f) ∂µfi ∂µfj<latexit sha1_base64="XD6I0wmgqfaUUJIrLAwhsJhngOc=">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</latexit><latexit sha1_base64="XD6I0wmgqfaUUJIrLAwhsJhngOc=">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</latexit><latexit sha1_base64="XD6I0wmgqfaUUJIrLAwhsJhngOc=">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</latexit><latexit sha1_base64="XD6I0wmgqfaUUJIrLAwhsJhngOc=">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</latexit>

• NGBs and Higgs field(s) define coordinates on a scalar manifold.

• General Lagrangian for NGBs and singlet scalar(s)

Ryo Nagai (Tohoku U)

Generalized Higgs sector

• The EW sector in general extended Higgs scenarios can be expressed by the following form:

fi = (pa, h, h0, · · · )<latexit 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NGBs Higgses

• The interaction between π (~ WL/ZL) and Higgs bosons is determined by the structure of gij (Φ) Alonso-Jenkins-Manohar (2016)

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

L =12

gij(f) ∂µfi ∂µfj<latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">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</latexit><latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">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</latexit><latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">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</latexit><latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">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</latexit>

Outline

Introduction

Higgs Effective Field Theory

Summary and Outlook

Perturbative Unitarity vs. EWPTs

Perturbative Unitarity

• The EW sector in general extended Higgs scenarios can be expressed by the following form:

fi = (pa, h, h0, · · · )<latexit 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NGBs Higgses

• The interaction between π (~ WL/ZL) and Higgs bosons is determined by the structure of gij (Φ) Alonso-Jenkins-Manohar (2016)

• Perturbative Unitarity / EWPTs → gij (Φ) ??

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

L =12

gij(f) ∂µfi ∂µfj<latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">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</latexit><latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">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</latexit><latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">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</latexit><latexit sha1_base64="biweoCQ52cFMMedpuAvcDjBN9DM=">AAAGdnicjVRNb9NAEJ0WGkr5aAsXJA5ERIEilWhTFShISBVcOHDoB6GVaho57iZx69jGH4XUcq9I/AEOnEDigPgB/AAu/AEOPXAFIY5FQkgceDt2QpvWAVveHb+d92Z2duyaa5l+IMTOwOCRo0O5Y8PHR06cPHV6dGz8zEPfCT1DVgzHcrzlmu5Ly7RlJTADSy67ntRbNUsu1TbuqvWlTen5pmM/CNqufNTSG7ZZNw09AFQdm9daetA0dCu6H2/f3tbqnm5E5TiairXJRjUy1+MJzW2aV7RJzdW9wNStqtYK8wpbNf+Cq11wvTpWECXBV/6gUU6NAqXXnDM++IxG0lujNXLIoJBaJMmmALZFOvm4V6iM2wX2iCJgHiyT1yXFzA3hJeGhA93A2MBbBCUdnk3Gt6AeU5HqmJV6PpO3kqI23lUuPkc1wLPweFDJU1F8Em/Frvgo3olv4nefHJSG2kMbcy3hSrc6+vzc4s9/sjr5d1l9GAYsG6NEltk1UXsLUIMZ3pMJhsuIZC7H2dx6sbt4a6EYXRKvxXfs85XYER+wU3vzh/FmXi687JNFHWMbWi2sxOmpSqAa9i7pMTwSq8GxI8ZsPlEPYxuc/SzZZakM1/6DYbOumtU5h1x1yae218vDasiKBXSW8ne7faWjglXGszhhBmv1UJbcE0VV+in3dqebotQjhpWwe+tmIZ7sq9DxydJocAxVwWyVFTwNKKm+EDSJp0TXeM5SVZ3jYbT6Zrdft0TTrJnoz2RqB7wvjU/PgWLESK+XBWyN/ZroxqQXVRdeZfXrnEuMVR/spE89/j4i5ioNpWoivyZ7eHzGTzirgx2hsglZ77BoM2m03gz3xurtjIjHTnys4g9a7v1fHjQqU6WbpfL8dGH2TvorHabzdJEmUMgbNEv3aI4qSO49faYv9HXoV+5Crpi7nLgODqScs7Tvyok/Ag5gAA==</latexit>

Perturbative Unitarity

• High-energy behavior of WL/ZL ~ π. (Equivalence Theorem).

• NGBs/Higgses scattering amplitudes can be expressed in terms of geometry of the scalar manifold.

VL VL

VLVL

π π

ππ

~

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

Mfifj!fkfl ⇠s3�

Riklj + Rilkj�+

t3�

Rijlk + Riljk�+

u3�

Rijkl + Rikjl�

Φi Φj → Φk Φl

• Tree-level scattering amplitude:

Riemann curvature tensor

Perturbative Unitarity

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

Mfifj!fkfl ⇠s3�

Riklj + Rilkj�+

t3�

Rijlk + Riljk�+

u3�

Rijkl + Rikjl�

Φi Φj → Φk Φl

• Tree-level scattering amplitude:

• Perturbative unitarity condition: Rijkl = 0

Perturbative Unitarity

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

EWPTs

• The EW sector in general extended Higgs scenarios can be expressed by the following form:

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NGBs Higgses

• The interaction between π (~ WL/ZL) and Higgs bosons is determined by the structure of gij (Φ) Alonso-Jenkins-Manohar (2016)

• Perturbative Unitarity / EWPTs → gij (Φ) ??

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

L =12

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EW corrections

• EW oblique corrections depend on not only geometry but also symmetry of the scalar manifold.

• The scalar manifold respects SU(2)W × U(1)Y gauge sym.

4 Killing vectorswi

a yi(a=1,2,3)

SU(2)W symmetry U(1)Y symmetry

For examples:wb

a =v2

dab �12

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yb = �v2

d3b �12

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Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

Slog µ (wj3);i(yi);j

EW corrections

S-parameter at one-loop

• UV divergence:

Alonso-Jenkins-Manohar (2016)

(wia);j =

∂fj wia + Gi

kjwka

(yi);j =∂

∂fj yi + Gikjy

k

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

EW corrections

S-parameter at one-loop

• UV divergence:

[wa, wb] = �eabcwc<latexit sha1_base64="kratI5Y/1lkgi4woC0SMGw38Lc4=">AAAGQXicjVRNb9NAEJ0WAqV8tIULEhwioiIObbRBbUmRkCq4cOwHoZWSYtnuNrXq2MYfCamVCxck/gAHTiAhgTjBX4ADf4BDfwJwQRSJAxx4O3aqNK0Dtrw7fjvvzezs2IZnW0EoxO7Q8LHjuRMnR06Nnj5z9tzY+MT5+4Eb+aasmK7t+muGHkjbcmQltEJbrnm+1BuGLVeN7TtqfbUp/cBynXth25PrDb3uWJuWqYeAtPHL1ZamT7U0Y/3WdE16gWUDjXXD7LQ0UxsviKLgK99jzIrS/FwpX0qRAqXXojsx/IRG07tGG+SSSRE1SJJDIWybdApwV6mE2wO2TjEwH5bF65I6zI3gJeGhA93GWMdbDCUdnluM70C9Q5O0iVmp5zN51RR18K5yCTiqCZ6Nx4dKnibFZ/FG7IlP4q34Iv4MyEFpqD20MRsJV3ra2NOLK7/+yermv88awDBhORglssyuidpbiBqUeU8WGB4jkrkcp7nzbG/l5vJkfFW8FN+wzxdiV3zATp3mT/PVklx+PiCLTYxtaDWw0klPVQKtYe+SHsIjseocO2bM4RP1MbbBOciS+yyV4cZ/MBzWVbM654irLvnUer18rEasWEBnKX9vv690VFBjPIsTZbAeHMmSPVFUpR9xb3e7KU49OrASdn/dbMSTAxW6PlkadY6hKpitUsVTh5LqC0FTeIo0y3OWquocH6M9MLuDukWaYc1Ev5ypHfK+anx6LhRjRvq9bGAb7LeFbkx6UXXhNKvPcS4drAZgJ33q8/cRM1dpKFUL+W2xh89n3OKsDneEyiZivaOildNo/Rn2xurvjJjHbnys4g/a/U3ms43K9eJ8sbQ0U1i4nf5KR+gSXaFrKOQNWqC7tEgVJPeYXtM7ep/7mPua+577kbgOD6WcC3Tgyv3+CzSJTeA=</latexit><latexit sha1_base64="kratI5Y/1lkgi4woC0SMGw38Lc4=">AAAGQXicjVRNb9NAEJ0WAqV8tIULEhwioiIObbRBbUmRkCq4cOwHoZWSYtnuNrXq2MYfCamVCxck/gAHTiAhgTjBX4ADf4BDfwJwQRSJAxx4O3aqNK0Dtrw7fjvvzezs2IZnW0EoxO7Q8LHjuRMnR06Nnj5z9tzY+MT5+4Eb+aasmK7t+muGHkjbcmQltEJbrnm+1BuGLVeN7TtqfbUp/cBynXth25PrDb3uWJuWqYeAtPHL1ZamT7U0Y/3WdE16gWUDjXXD7LQ0UxsviKLgK99jzIrS/FwpX0qRAqXXojsx/IRG07tGG+SSSRE1SJJDIWybdApwV6mE2wO2TjEwH5bF65I6zI3gJeGhA93GWMdbDCUdnluM70C9Q5O0iVmp5zN51RR18K5yCTiqCZ6Nx4dKnibFZ/FG7IlP4q34Iv4MyEFpqD20MRsJV3ra2NOLK7/+yermv88awDBhORglssyuidpbiBqUeU8WGB4jkrkcp7nzbG/l5vJkfFW8FN+wzxdiV3zATp3mT/PVklx+PiCLTYxtaDWw0klPVQKtYe+SHsIjseocO2bM4RP1MbbBOciS+yyV4cZ/MBzWVbM654irLvnUer18rEasWEBnKX9vv690VFBjPIsTZbAeHMmSPVFUpR9xb3e7KU49OrASdn/dbMSTAxW6PlkadY6hKpitUsVTh5LqC0FTeIo0y3OWquocH6M9MLuDukWaYc1Ev5ypHfK+anx6LhRjRvq9bGAb7LeFbkx6UXXhNKvPcS4drAZgJ33q8/cRM1dpKFUL+W2xh89n3OKsDneEyiZivaOildNo/Rn2xurvjJjHbnys4g/a/U3ms43K9eJ8sbQ0U1i4nf5KR+gSXaFrKOQNWqC7tEgVJPeYXtM7ep/7mPua+577kbgOD6WcC3Tgyv3+CzSJTeA=</latexit><latexit sha1_base64="kratI5Y/1lkgi4woC0SMGw38Lc4=">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</latexit><latexit sha1_base64="kratI5Y/1lkgi4woC0SMGw38Lc4=">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</latexit>

Alonso-Jenkins-Manohar (2016)

[wa, y] = 0<latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit><latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit><latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit><latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit>

vi;j;k = Rlkjivl

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(v = wa, y)<latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">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</latexit><latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">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</latexit><latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">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</latexit><latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">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</latexit>

SU(2)W×U(1)Y sym.

Slog µ e3bcRijc3(w

jb);i + e3bcRi

jbc(yj);i

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Slog µ (wj3);i(yi);j

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

EW corrections

S-parameter at one-loop

• UV divergence:

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Slog µ e3bcRijc3(w

jb);i + e3bcRi

jbc(yj);i

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π π

ΦjΦi ΦjΦi

f f

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

Ulog µ (wj1);i(wi

1);j � (wj3);i(wi

3);j

EW corrections

U-parameter at one-loop

• UV divergence:

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[wa, y] = 0<latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit><latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit><latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit><latexit sha1_base64="t/aX0a7CEL94umVpoy6aV4RwNHg=">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</latexit>

vi;j;k = Rlkjivl

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(v = wa, y)<latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">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</latexit><latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">AAAGLnicjVTNa9RQEJ9WV2v9aKsXwUtwqVSoy0up7VYQil489sO1hd26ZLOv29BsEvOxdRv2LHgXD4Ki4EE8+Td48R8Q6UG8Kh4rePHg702yZbs2WxPy3mRmfr+ZNzNJzbOtIBRib2j4xMncqdMjZ0bPnjt/YWx84uKDwI18U5ZM13b99ZoRSNtyZCm0Qluue740mjVbrtW27yr7Wkv6geU698O2JzeaRsOxNi3TCKEqT7W029pO1ZhuX6+O50VBCKHruqYEfX5OQFhYKM7oRU1XJlx5Sq8ld2L4CY2md4Xq5JJJETVJkkMhZJsMCnCXScftQbdBMXQ+JIvtkjqMjeAl4WFAu421gbcYTAY8t1i/C/YOTdImdsWuZeLKqdbBu8ol4KgmcDYeHywaTYrP4p3YF5/Ee/FD/BmQg+JQZ2hjryVY6VXHnl5e/X0sqpv/AWoAwoTkYJXIMrsm6mwhalDkM1lAeKyRjOU4rd3n+6u3Vibja+KN+IlzvhZ74iNO6rR+mW+X5cqLAVlsYm2DqwlLJ+2qhLaCs0t6BI9EanDsmHUOd9TH2gbmMEoeoFSG9f9AOMyrdtXniKsuuWu9Xj6sETPmMVnK3zuYKwMVrLI+CxNloB4eiZI9UVSlH/Nsd6cpTj06kBJ0f91sxJMDGbo+WRwNjqEqmM1SxtMAk5oLQdN4CnST9yxWNTk+Vntgdod5CzTLnAl/MZM75HNVuHsuGGPW9HvZ0NXZbwvTmMyimsIbzD7HuXRgDYBO5tTn7yNmrOJQrBby22IPn3u8w1n9OxEqm4j5jopWTKP1Z9gbq38yYl678WHFH7T7m9SyhdJMYaGgL8/mF++kv9IRukJXaQqFnKdFukdLVOIWPKOX9Cr3Ifcl9zX3LXEdHkoxl+jQlfv+F6QdRXw=</latexit><latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">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</latexit><latexit sha1_base64="s/3yChOIGn9KvAZ0lFLmNXrbSow=">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</latexit>

SU(2)W×U(1)Y sym.

Ulog µ e1bcRijbc(w

j1);i � e3bcRi

jbc(wj3);i

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Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

PU vs. EWPTs

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Ulog ⇠ e1bcRijbc(w

j1);i

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*

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Perturbative Unitarity EW oblique corrections

L =12

gij(f)DµfiDµfj fi = (pa, h, h0, · · · )<latexit 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Φk Φl

ΦjΦi

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)

Outline

Introduction

Higgs Effective Field Theory

Summary and Outlook

Perturbative Unitarity vs. EWPTs

Summary

• We have discussed extended Higgs scenarios from the view pointof perturbative unitarity and consistency with EWPTs.

• Physics in extended Higgs scenarios can be understood in terms ofgeometry and symmetry of “the scalar manifold”.

• Consistency with EWPTs does not imply the complete flatness of thescalar manifold.

• Precise measurements of the 125GeV Higgs couplings reveal the structure of the scalar manifold.

Flat → Perturbative

Curved → Non-Perturbative

Symmetry and Geometry of Generalized Higgs SectorsRyo Nagai (Tohoku U)