New Physics / Resonances in Vector Boson Scattering at...

J.R.Reuter New Physics in VBS at the LHC BSM Workshop, NTU Singapore, 04.03.2016

Jürgen R. Reuter, DESY

New Physics / Resonances in Vector Boson Scattering

at the LHC

PRD93(16),3. 036004 [1511.00022], PRD91(15),096007 [1408.6207], US Snowmass Summer Study 1310.6708,1309.7890,1307.8180,JHEP 0811.010 [0806.4145], EPJC48(06)353 [hep-ph/0604048]

based on work with A. Alboteanu, W. Kilian, T. Ohl, M. Sekulla


Seen signals, unseen signals, seen un-signals (?)• Discovery of a light Higgs boson leaves still open questions:

1. Nature of Electroweak Symmetry Breaking

2. Higgs boson potential, all the way like the Standard Model!?

3. Does it fulfill the US-fermion/Europe-boson rule?

4. Is the 125 GeV state the only resonance in the system of EW vector bosons?

5. How do EW vector bosons scatter? (true heart of weak interactions)

6. Is there something related to the Little Hierarchy problem (strong or weak)


Seen signals, unseen signals, seen un-signals (?)• Discovery of a light Higgs boson leaves still open questions:

1. Nature of Electroweak Symmetry Breaking

2. Higgs boson potential, all the way like the Standard Model!?

3. Does it fulfill the US-fermion/Europe-boson rule?

4. Is the 125 GeV state the only resonance in the system of EW vector bosons?

5. How do EW vector bosons scatter? (true heart of weak interactions)

6. Is there something related to the Little Hierarchy problem (strong or weak)

ATLAS-CONF-2015-081 CMS-PAS-EXO-15-004


Seen signals, unseen signals, seen un-signals (?) Evidence for W+W+jj (electroweak production) Talk by Brigitte Vachon ATLAS PRL 113(2014)14, 141803 [1405.6241]; CMS PRL 114(2015), 051801 [1410.6315]

First limits on New Physics in pure electroweak gauge/Goldstone sector




[GeV]jjm

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210 Data 2012 Syst. Uncertainty

jj Electroweak±W± Wjj Strong±W± W

Prompt Conversions Other non-prompt

ATLAS = 8 TeVs, -120.3 fb

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

5α

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0

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0.6ATLAS 20.3 fb-1, s = 8 TeV pp → W± W± jj K-matrix unitarization

68% CL95% CLexpected 95% CLStandard Model

confidence intervals




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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

5α

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0

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0.6ATLAS 20.3 fb-1, s = 8 TeV pp → W± W± jj K-matrix unitarization

68% CL95% CLexpected 95% CLStandard Model

confidence intervals

F. Gianotti, 01/2014


Anatomy of Vector Boson Scattering (VBS)

P1

P2

x1P1

x2P2

D(x1, Q2)

D(x2, Q2)

QCD

QCD

QGC

pp ! WWjj ! ``⌫⌫jj

Backgrounds [+ VTVT bkgd.]:• tt ➝ WbWb • W + jets • single top, misreconstructed jet• WWjj QCD production • ll + X + Emiss (“prompt”)

• lljj tag• mjj > 500 GeV (“jet recoil”)• yj,1 ⋅ yj,2 < 0 (“collinear beams”)• |Δyjj | > 2.4 (“rapidity distance”)• Cuts on Ej , pT

j

• No mini jet vetoes

Fiducial phase space volume:


EFTs: Higher-dimensional operators✦ Must include all dim 6 operators from SM fields Buchmüller/Wyler, 1986

✦ Redundancy of operators ⟹ minimal set of operators (in principle)

1. Equations of motion: 2. Gauge symmetry:3. Integration by parts:

DµWµ⌫ = �†(D⌫�)� (D⌫�)†�+ . . .

[Dµ, D⌫ ]� / Wµ⌫��†�

�2��†�

��! @µ

��†�

�@µ

��†�

�

✦ Further reduction by use of discrete / horizontal symmetries

✦ Assuming B and L conservation: number of operators (without )

1. B and L conservation (excludes 5 operators per generation)2. Flavor symmetries (assumption: Minimal Flavor Violation)3. CP symmetry

⌫R


EFTs: Higher-dimensional operators

• 1 dim-2 operator + 15 dim-4 operators• 59 dim-6 operators for 1 generation• 2499 dim-6 operators for 3 generations Alonso/Jenkins/Manohar/Trott, 2013

✦ Must include all dim 6 operators from SM fields Buchmüller/Wyler, 1986



DµWµ⌫ = �†(D⌫�)� (D⌫�)†�+ . . .

[Dµ, D⌫ ]� / Wµ⌫��†�

�2��†�

��! @µ

��†�

�@µ

��†�

�




⌫R


EFTs: Higher-dimensional operators

• 1 dim-2 operator + 15 dim-4 operators• 59 dim-6 operators for 1 generation• 2499 dim-6 operators for 3 generations Alonso/Jenkins/Manohar/Trott, 2013

✦ Must include all dim 6 operators from SM fields Buchmüller/Wyler, 1986



DµWµ⌫ = �†(D⌫�)� (D⌫�)†�+ . . .

[Dµ, D⌫ ]� / Wµ⌫��†�

�2��†�

��! @µ

��†�

�@µ

��†�

�




⌫R

✦ No unique basis exists (more in a second)

✦ Well-known in B physics: different experimental measurements constrain different operators


Effective Field Theories: Operator BasesNo unique basis exists

‣ “HISZ” basis: no fermionic operators Hagiwara/Ishihara/Szalapski/Zeppenfeld, 1993

‣ “GIMR” basis: first minimal complete basis Grzadkowski/Iskrzyński/Misiak/Rosiek, 2010

‣ “SILH” basis: complete basis Giudice/Grojean/Pomarol/Ratazzi, 2007; Elias-Miró et al, 2013

‣ Dim. 8 operators: Eboli et al., 2006; Kilian/JRR/Ohl/Sekulla, 2014+2015







Grzadkowski et al.







Giudice et al. / Contino at al.


Operators and Multi(EW)-boson Physics (I)


Operators and Multi(EW)-boson Physics (I)Dimension-6 operators for Multiboson physics (CP-conserving)

OWWW = Tr[Wµ⌫W⌫⇢Wµ

⇢ ]

OW = (Dµ�)†Wµ⌫(D⌫�)

OB = (Dµ�)†Bµ⌫(D⌫�)

O@� = @µ⇣�†�

⌘@µ

⇣�†�

⌘

O�W =⇣�†�

⌘Tr[Wµ⌫Wµ⌫ ]

O�B =⇣�†�

⌘Bµ⌫Bµ⌫




⇢ ]

OW = (Dµ�)†Wµ⌫(D⌫�)

OB = (Dµ�)†Bµ⌫(D⌫�)

O@� = @µ⇣�†�

⌘@µ

⇣�†�

⌘

O�W =⇣�†�


O�B =⇣�†�

⌘Bµ⌫Bµ⌫

Dimension-6 operators for Multiboson physics (CP-violating)

OfWW= �†fWµ⌫W

µ⌫�

O eBB = �† eBµ⌫Bµ⌫�

OfWWW= Tr[fWµ⌫W

⌫⇢Wµ⇢ ]

OfW = (Dµ�)†fWµ⌫

(D⌫�)




⇢ ]

OW = (Dµ�)†Wµ⌫(D⌫�)

OB = (Dµ�)†Bµ⌫(D⌫�)

O@� = @µ⇣�†�

⌘@µ

⇣�†�

⌘

O�W =⇣�†�


O�B =⇣�†�

⌘Bµ⌫Bµ⌫



µ⌫�


OfWWW= Tr[fWµ⌫W

⌫⇢Wµ⇢ ]


(D⌫�)

Affect the following electroweak couplings:

ZWW AWW HWW HZZ HZA HAA WWWW ZZWW ZAWW AAWWOWWW X X X X X XOW X X X X X X X XOB X X X XO�d X XO�W X X X XO�B X X XOW̃WW X X X X X XOW̃ X X X X XOW̃W X X X XOB̃B X X X




⇢ ]

OW = (Dµ�)†Wµ⌫(D⌫�)

OB = (Dµ�)†Bµ⌫(D⌫�)

O@� = @µ⇣�†�

⌘@µ

⇣�†�

⌘

O�W =⇣�†�


O�B =⇣�†�

⌘Bµ⌫Bµ⌫



µ⌫�


OfWWW= Tr[fWµ⌫W

⌫⇢Wµ⇢ ]


(D⌫�)

Affect the following electroweak couplings:

ZWW AWW HWW HZZ HZA HAA WWWW ZZWW ZAWW AAWWOWWW X X X X X XOW X X X X X X X XOB X X X XO�d X XO�W X X X XO�B X X XOW̃WW X X X X X XOW̃ X X X X XOW̃W X X X XOB̃B X X X

connected to Higgs physics


Operators and Multi(EW)-boson Physics (II)

OT,0 = Tr⇥Wµ⌫W

µ⌫⇤ · TrhW↵�W

↵�i

OT,1 = TrhW↵⌫W

µ�i· Tr ⇥Wµ�W

↵⌫⇤

OT,2 = TrhW↵µW

µ�i· Tr ⇥W�⌫W

⌫↵⇤

OT,5 = Tr⇥Wµ⌫W

µ⌫⇤ · B↵�B↵�

OT,6 = TrhW↵⌫W

µ�i· Bµ�B

↵⌫

OT,7 = TrhW↵µW

µ�i· B�⌫B

⌫↵

OT,8 = Bµ⌫Bµ⌫B↵�B

↵�

OT,9 = B↵µBµ�B�⌫B

⌫↵

OM,0 = Tr⇥Wµ⌫W

µ⌫⇤ ·h(D��)† D��

i

OM,1 = TrhWµ⌫W

⌫�i·h(D��)† Dµ�

i

OM,2 =⇥Bµ⌫B

µ⌫⇤ ·h(D��)† D��

i

OM,3 =hBµ⌫B

⌫�i·h(D��)† Dµ�

i

OM,4 =h(Dµ�)† W�⌫D

µ�i· B�⌫

OM,5 =h(Dµ�)† W�⌫D

⌫�i· B�µ

OM,6 =h(Dµ�)† W�⌫W

�⌫Dµ�i

OM,7 =h(Dµ�)† W�⌫W

�µD⌫�i

Dimension-8 operators for Multiboson physics OS,0 =h(Dµ�)

† D⌫�i⇥

h(Dµ�)† D⌫�

i

OS,1 =h(Dµ�)

† Dµ�i⇥

h(D⌫�)

† D⌫�i



WWWW WWZZ ZZZZ WWAZ WWAA ZZZA ZZAA ZAAA AAAAOS,0/1 X X X

OM,0/1/6/7 X X X X X X XOM,2/3/4/5 X X X X X XOT,0/1/2 X X X X X X X X XOT,5/6/7 X X X X X X X XOT,8/9 X X X X X

OT,0 = Tr⇥Wµ⌫W


↵�i

OT,1 = TrhW↵⌫W


↵⌫⇤

OT,2 = TrhW↵µW


⌫↵⇤

OT,5 = Tr⇥Wµ⌫W

µ⌫⇤ · B↵�B↵�

OT,6 = TrhW↵⌫W

µ�i· Bµ�B

↵⌫

OT,7 = TrhW↵µW

µ�i· B�⌫B

⌫↵


↵�


⌫↵

OM,0 = Tr⇥Wµ⌫W

µ⌫⇤ ·h(D��)† D��

i

OM,1 = TrhWµ⌫W

⌫�i·h(D��)† Dµ�

i

OM,2 =⇥Bµ⌫B

µ⌫⇤ ·h(D��)† D��

i

OM,3 =hBµ⌫B

⌫�i·h(D��)† Dµ�

i

OM,4 =h(Dµ�)† W�⌫D

µ�i· B�⌫

OM,5 =h(Dµ�)† W�⌫D

⌫�i· B�µ

OM,6 =h(Dµ�)† W�⌫W

�⌫Dµ�i

OM,7 =h(Dµ�)† W�⌫W

�µD⌫�i


† D⌫�i⇥

h(Dµ�)† D⌫�

i

OS,1 =h(Dµ�)

† Dµ�i⇥

h(D⌫�)

† D⌫�i



WWWW WWZZ ZZZZ WWAZ WWAA ZZZA ZZAA ZAAA AAAAOS,0/1 X X X

OM,0/1/6/7 X X X X X X XOM,2/3/4/5 X X X X X XOT,0/1/2 X X X X X X X X XOT,5/6/7 X X X X X X X XOT,8/9 X X X X X

OT,0 = Tr⇥Wµ⌫W


↵�i

OT,1 = TrhW↵⌫W


↵⌫⇤

OT,2 = TrhW↵µW


⌫↵⇤

OT,5 = Tr⇥Wµ⌫W

µ⌫⇤ · B↵�B↵�

OT,6 = TrhW↵⌫W

µ�i· Bµ�B

↵⌫

OT,7 = TrhW↵µW

µ�i· B�⌫B

⌫↵


↵�


⌫↵

OM,0 = Tr⇥Wµ⌫W

µ⌫⇤ ·h(D��)† D��

i

OM,1 = TrhWµ⌫W

⌫�i·h(D��)† Dµ�

i

OM,2 =⇥Bµ⌫B

µ⌫⇤ ·h(D��)† D��

i

OM,3 =hBµ⌫B

⌫�i·h(D��)† Dµ�

i

OM,4 =h(Dµ�)† W�⌫D

µ�i· B�⌫

OM,5 =h(Dµ�)† W�⌫D

⌫�i· B�µ

OM,6 =h(Dµ�)† W�⌫W

�⌫Dµ�i

OM,7 =h(Dµ�)† W�⌫W

�µD⌫�i


† D⌫�i⇥

h(Dµ�)† D⌫�

i

OS,1 =h(Dµ�)

† Dµ�i⇥

h(D⌫�)

† D⌫�i

• Dim. 8 generate aQGCs independently• generate neutral quartic couplings


Unitarity in vector boson scattering


Unitarity in vector boson scattering

Lee/Quigg/Thacker, 1973


Scenarios for New Physics in VBS

1. SM or weakly coupled physics (e.g. 2HDM): amplitude remains close to origin

2. Rising amplitude (at least one dim-8 operator): rise beyond unitarity circle [unphys.], strongly interacting regime

3. Inelastic channel opens (form-factor description): new channels open out, multi-boson final states

4. Saturation of amplitude: maximal amplitude, strongly interacting continuum, K-/T-matrix unitarization

5. New resonance: amplitude turns over


Procedures to treat unitarity violations

0 1 2 3 40.0

0.5

1.0

s êH4 p vL

†AHsL§unitarity bound (0th partial wave) at ⇤C

no continuous transition beyond

Cut-off (a.k.a. “Event clipping”) ✓(⇤2C � s)



0 1 2 3 40.0

0.5

1.0

s êH4 p vL




Form factor

0 1 2 3 40.0

0.5

1.0

s êH4 p vL

†AHsL§

Unitarity bound

Unitarity broken

Form factor

Bare1⇣1+ s

⇤FF

⌘n

Applicable for arbitrary operators, tuning in 2 parameters: n damps unitarity violation, ΛFF highest value to satisfy 0th partial wave



0 1 2 3 40.0

0.5

1.0

s êH4 p vL




Form factor

0 1 2 3 40.0

0.5

1.0

s êH4 p vL

†AHsL§

Unitarity bound

Unitarity broken

Form factor

Bare1⇣1+ s

⇤FF

⌘n

Applicable for arbitrary operators, tuning in 2 parameters: n damps unitarity violation, ΛFF highest value to satisfy 0th partial wave

0 1 2 3 40.0

0.5

1.0

1.5

2.0

s êH4 p vL†AHsL§

Saturation

K-Matrix

Bare

K-/T-matrix saturation

saturates the amplitude, usable for complex amplitudes, no additional parameters


Different unitarity projections

i2

i

a

aK

K-matrix: Cayley transform of S-matrix Heitler, 1941; Schwinger, 1949; Gupta, 1950

Stereographic projection to Argand circle

S = 1+iK/21�iK/2 aK(s) = a(s)

1�ia(s)



i2

i

a

aK



S = 1+iK/21�iK/2 aK(s) = a(s)

1�ia(s)

Stereographic projection to Argand circle Formalism does a partial resummation of perturbative series need to construct (orig.) K-matrix as self-adjoint intermediate operator

Problems, if S-matrix non-diagonal, presence of non-perturbative contrib.

i2

i

a

aK

a0



i2

i

a

aKaK2

T-matrix: Thales circle construction Kilian/Ohl/JRR/Sekulla, 2014

Defined via ��a� aK

2

�� = aK2 ) a = 1

Re⇣

1a0

⌘�i

i2

i

a

aK



S = 1+iK/21�iK/2 aK(s) = a(s)

1�ia(s)



i2

i

a

aK

a0



i2

i

a

aKaK2



2

�� = aK2 ) a = 1

Re⇣

1a0

⌘�i

i2

i

a

aK



S = 1+iK/21�iK/2 aK(s) = a(s)

1�ia(s)



i2

i

a

aK

a0

Identical to K matrix for real amplitudesPoints on Argand circle left invariantDoes not rely on perturbation theoryApplicable for amplitudes with imaginary parts (models with resonances)

12

1

aT = aS

a0a02

Im [a]

Re [a]



i2

i

a

aKaK2



2

�� = aK2 ) a = 1

Re⇣

1a0

⌘�i

i2

i

a

aK



S = 1+iK/21�iK/2 aK(s) = a(s)

1�ia(s)



i2

i

a

aK

a0

Identical to K matrix for real amplitudesPoints on Argand circle left invariantDoes not rely on perturbation theoryApplicable for amplitudes with imaginary parts (models with resonances)

12

1

aT = aS

a0a02

Im [a]

Re [a]

i2

i

aS

a0

aT


VBS diboson spectra

200 400 600 800 1000 1200 1400 1600 1800 2000M(W+W+)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! W+W+jj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM

General cuts: Mjj > 500GeV; �⌘jj > 2.4; pjT > 20GeV; |�⌘j | < 4.5


VBS diboson spectra

200 400 600 800 1000 1200 1400 1600 1800 2000M(W+W+)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! W+W+jj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM


200 400 600 800 1000 1200 1400 1600 1800 2000M(W+W+)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! W+W+jj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM


VBS diboson spectra

200 400 600 800 1000 1200 1400 1600 1800 2000M(W+W+)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! W+W+jj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM


200 400 600 800 1000 1200 1400 1600 1800 2000M(W+W+)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! W+W+jj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM

200 400 600 800 1000 1200 1400 1600 1800 2000M(W+W�)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! W+W�jj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM

200 400 600 800 1000 1200 1400 1600 1800 2000M(W+Z)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! WZjj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM

200 400 600 800 1000 1200 1400 1600 1800 2000M(ZZ)[GeV]

10�4

10�3

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! ZZjj

FS,0 = 480 TeV�4

FS,1 = 480 TeV�4

FHD = 30 TeV�2

SM


Differential spectra in VBSpp ! e+µ+⌫e⌫µjj

ps = 14TeV L = 1ab�1

Simulations with WHIZARD [http://whizard.hepforge.org, Kilian/Ohl/JRR]

Mjj > 500GeV; �⌘jj > 2.4; pjT > 20GeV; |�⌘j | < 4.5; p`T > 20GeV

http://whizard.hepforge.org






0.5 1.0 1.5 2.0 2.5 3.0��eµ

0

50

100

150

200

250

300

350

N

bare

unit

SM

500 1000 1500 2000Pl=e,µ |pT(l)|

100

101

102

103

104

N

bare

unit

SM

FHD = 30 TeV�2







0.5 1.0 1.5 2.0 2.5 3.0��eµ

0

50

100

150

200

250

300

350

N

bare

unit

SM

500 1000 1500 2000Pl=e,µ |pT(l)|

100

101

102

103

104

Nbare

unit

SM

FS,0 = 480 TeV�4







0.5 1.0 1.5 2.0 2.5 3.0��eµ

0

50

100

150

200

250

300

350

N

bare

unit

SM

500 1000 1500 2000Pl=e,µ |pT(l)|

100

101

102

103

104

N

bare

unit

SM

FS,1 = 480 TeV�4



Resonances: Quantum numbers & simplified modelsRise of amplitude / anomalous coupling: Taylor expansion below a resonanceResonances might be in direct reach of LHC EFT framework EW-restored regime: SU(2)L ⨉ SU(2)R , SU(2)L ⨉ U(1)Y gauged

Include EFT operators in addition (more resonances, continuum contribution)Apply T-matrix unitarization beyond resonance (“UV-incomplete” model)




Spins 0, 2 considered, Spin 1 has different physics (mixing with W/Z)





SU(2)L ⇥ SU(2)R ! SU(2)C(0, 0) ! 0(1, 1) ! 2 + 1 + 0





SU(2)L ⇥ SU(2)R ! SU(2)C(0, 0) ! 0(1, 1) ! 2 + 1 + 0

isoscalar isotensor

scalar �0��t ,��

t ,�0t ,�

+t ,�

++t

��v ,�

0v,�

+v

�0s

tensor f0

0

@X��

t , X�t , X0

t , X+t , X++

t

X�v , X0

v , X+v

X0s

1

A

. . . . . . . . .





SU(2)L ⇥ SU(2)R ! SU(2)C(0, 0) ! 0(1, 1) ! 2 + 1 + 0

isoscalar isotensor

scalar �0��t ,��

t ,�0t ,�

+t ,�

++t

��v ,�

0v,�

+v

�0s

tensor f0

0

@X��

t , X�t , X0

t , X+t , X++

t

X�v , X0

v , X+v

X0s

1

A

. . . . . . . . .

Tensor resonances

• Symmetric tensor

• On-shell conditions: 10 ➝ 5 components

• Tracelessness:

• Transversality:

fµ⌫

fµµ = 0

@µfµ⌫ = 0

How to deal with off-shell tensor in realistic processes?


Tensor resonances: Fierz-Pauli vs. StückelbergStart with Fierz-Pauli Lagrangian for symmetric tensor

LFP =1

2@↵fµ⌫@

↵fµ⌫ � 1

2m2fµ⌫f

µ⌫ � 1

2@↵f

µµ@

↵f⌫⌫ +

1

2m2fµ

µf⌫⌫

� @↵f↵µ@�f�µ � f↵

↵@µ@⌫fµ⌫ + fµ⌫J

µ⌫f



LFP =1

2@↵fµ⌫@

↵fµ⌫ � 1

2m2fµ⌫f

µ⌫ � 1

2@↵f

µµ@

↵f⌫⌫ +

1

2m2fµ

µf⌫⌫

� @↵f↵µ@�f�µ � f↵


µ⌫f

Fierz-Pauli conditions not valid off-shell Fierz-Pauli propagator has bad high-energy behavior Use Stückelberg formalism to make off-shell high-energy behavior explicit Introduce compensator fields ⇒ no propagators with momentum factors

Crucial for MCs



LFP =1

2@↵fµ⌫@

↵fµ⌫ � 1

2m2fµ⌫f

µ⌫ � 1

2@↵f

µµ@

↵f⌫⌫ +

1

2m2fµ

µf⌫⌫

� @↵f↵µ@�f�µ � f↵


µ⌫f

Fierz-Pauli conditions not valid off-shell Fierz-Pauli propagator has bad high-energy behavior Use Stückelberg formalism to make off-shell high-energy behavior explicit Introduce compensator fields ⇒ no propagators with momentum factors

Crucial for MCs L =1

2ffµ⌫

��@2 �m2

f

�fµ⌫f +

1

2fµf µ

✓�1

2

��@2 �m2

f

�◆f⌫f ⌫

+1

2Afµ

�@2 +m2

f

�Aµ

f +1

2�f

��@2 �m2

f

��f

+

✓fµ⌫ � 1p

6�fgµ⌫

◆Jµ⌫f

�

1p2mf

(Afµ@⌫ +Af⌫@µ)�p2p

3m2f

�f@µ@⌫

!Jµ⌫f

• fµ⌫: on-shell fµ⌫

• �: @µ@⌫fµ⌫

• Aµ: @⌫fµ⌫

• �: fµµ

Gauge fixing: � = ��


Comparison: Simplified Models & EFT

Mjj > 500GeV; �⌘jj > 2.4; pjT > 20GeV; |�⌘j | < 4.5

Black dashed line: saturation of A22(W+W+)/A00(ZZ)

400 600 800 1000 1200 1400 1600 1800 2000M(ZZ)[GeV]

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! ZZjj

FS,1 = 12.3 TeV�4

F� = 4.0 TeV�1

SMlimit of A00

M� = 800GeV�� = 80GeV

• EFT fails at resonance

• aQGC describe rise of resonance

• Unitarization applied

• Tensor resonances better visible than scalars

ATLAS PRL 113(2014)14, 141803 [1405.6241]

Kilian/Ohl/JRR/Sekulla: PRD93(16),3. 036004 [1511.00022]


Comparison: Simplified Models & EFT

Mjj > 500GeV; �⌘jj > 2.4; pjT > 20GeV; |�⌘j | < 4.5

Black dashed line: saturation of A22(W+W+)/A00(ZZ)

400 600 800 1000 1200 1400 1600 1800 2000M(ZZ)[GeV]

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! ZZjj

FS,1 = 12.3 TeV�4

F� = 4.0 TeV�1

SMlimit of A00

M� = 800GeV�� = 80GeV

• EFT fails at resonance

• aQGC describe rise of resonance

• Unitarization applied

• Tensor resonances better visible than scalars

400 600 800 1000 1200 1400 1600 1800 2000M(W+W+)[GeV]

10�2

10�1

100

101

@�

@M

⇥fb

100G

eV

⇤

pp ! W+W+jj

FS,0 = 19.2 TeV�4FS,1 = -134.1 TeV�4

FX = 38.6 TeV�1

SMlimit of A22

MX = 1800GeV�X = 720GeV

ATLAS PRL 113(2014)14, 141803 [1405.6241]

Kilian/Ohl/JRR/Sekulla: PRD93(16),3. 036004 [1511.00022]


Complete LHC process at 14 TeV

0 500 1000 1500 2000

M (e+, e�, µ+, µ�) [GeV]

0

50

100

150

200

250

N

pp ! e+e�µ+µ�jj at 3 ab�1

Ff = 17.4 TeV�1

SM

Mf = 1.0TeV


Triple [multiple] Vector Boson Production ?

Relate to ?

Yes, same Feynman rule as in VBS, but …

one external W/Z/γ always far off-shell

Unitarization formalism not available (would need 2 ➝ 3 unitarizations)

Different Wilson coefficients dominate (particularly for resonances)

Important physics (partially) independent from VBS


Conclusions / Summary

✦ Vector boson scattering one of flagship measurements of Runs II/III

✦ EFT provides a (!) [not the] consistent framework for SM deviations

✦ Very well-defined (and limited) range of applicability

✦ Accounts for access to New Physics in VBS and Di-/Triboson channels

✦ Unitarization for theoretically sane description (allows to calculate ‘best limit’)✦ T-matrix unitarization universal scheme for EFT and resonances✦ Simplified models: generic electroweak resonances ✦ Limits from LHC still incredibly limited: M ∼ 200-300 GeV

✦ Make sure that actual limits are meaningful and comparable









Spin I = 0 I = 1 I = 2

0 1.55 � 1.951 � 2.49 �2 3.29 � 4.30

Spin I = 0 I = 1 I = 2

0 1.39 1.55 1.951 1.74 2.67 �2 3.00 3.01 5.84

ILC 1 TeV, 1/ab expectations: LHC 13/14 TeV, 0.3-3/ab









Spin I = 0 I = 1 I = 2

0 1.55 � 1.951 � 2.49 �2 3.29 � 4.30

Spin I = 0 I = 1 I = 2

0 1.39 1.55 1.951 1.74 2.67 �2 3.00 3.01 5.84

ILC 1 TeV, 1/ab expectations: LHC 13/14 TeV, 0.3-3/ab

light Higgs: als

o needs to be updated

ca. 1-3 TeV[very preliminary]


whatever approach ....


whatever approach .... always get the correct ellipses


BACKUP SLIDES


Effective Field Theory (EFT) for Weak Interactions✴ SppS: discovery of W, Z (on-shell) ✴ SLC/LEP: proof of non-Abelian weak structure, failure to find (very) light Higgs✴ Measurement of longitudinal Ws: ee ➝ WW (LEP), t ➝ Wb (Tevatron)✴ Using all known d.o.f., parameterizing all possible interactions

SM fermions weak bosons hypercharge boson longitudinal d.o.f.

Building blocks for EFT: , Wµ , Bµ , ⌃ = exp

�i

vw⌧

�





�i

vw⌧

�

Minimal Lagrangian describing measurements at SLC / LEP [II] / Tevatron

Lpre-LHC =X

(i /D) � 1

2g2tr [Wµ⌫W

µ⌫ ]� 1

2g0 2tr [Bµ⌫B

µ⌫ ]+v2

4tr [(Dµ⌃)(D

µ⌃)]

with the following useful definitions: Dµ = @µ +i

2g⌧ IW I

µ +i

2g0Bµ⌧

3

Wµ⌫ =i

2g⌧ I(@µW

I⌫ � @⌫W

Iµ + g✏IJKW J

µWK⌫ )

Bµ⌫ =i

2g0(@µB⌫ � @⌫Bµ)⌧

3





�i

vw⌧

�


Lpre-LHC =X

(i /D) � 1

2g2tr [Wµ⌫W

µ⌫ ]� 1

2g0 2tr [Bµ⌫B

µ⌫ ]+v2

4tr [(Dµ⌃)(D

µ⌃)]


2g⌧ IW I

µ +i

2g0Bµ⌧

3

Wµ⌫ =i

2g⌧ I(@µW

I⌫ � @⌫W

Iµ + g✏IJKW J

µWK⌫ )

Bµ⌫ =i

2g0(@µB⌫ � @⌫Bµ)⌧

3Electroweak Chiral Lagrangian





�i

vw⌧

�


Lpre-LHC =X

(i /D) � 1

2g2tr [Wµ⌫W

µ⌫ ]� 1

2g0 2tr [Bµ⌫B

µ⌫ ]+v2

4tr [(Dµ⌃)(D

µ⌃)]


2g⌧ IW I

µ +i

2g0Bµ⌧

3

Wµ⌫ =i

2g⌧ I(@µW

I⌫ � @⌫W

Iµ + g✏IJKW J

µWK⌫ )

Bµ⌫ =i

2g0(@µB⌫ � @⌫Bµ)⌧

3Electroweak Chiral Lagrangian

Ruled out by LHC data (Higgs discovery)


Parameterizing SM deviations★ Specific models (SUSY, Compositeness, Little Higgses, 2HDM, Modified Higgses, Xdim, ......)

• Could give strong signals in VBS (presumably Little Higgses, Compositeness, Xdim .... )• Could give faint signals in VBS (presumably SUSY, 2HDM [Higgs data!], ....)• Up to parametric uncertainties precise predictions from the models (new independent couplings)• Mostly even beyond tree level predictable • Analysis has to be repeated for each and every model, introduces new parameters




★ Anomalous Couplings

• Usually first “model-independent” proposal• At the moment applied by HXSWG (but under debate)• Only modifications of SM couplings or introduction of new (Lorentz) structures ? • Allows fits of coupling strengths• Possible deviations difficult to interpret in terms of quantum field theory, unitarity!!






★ Form factors

• Similar approach to anomalous couplings, partially resums perturbative series• (Almost) completely general and model-independent• Needs parameterizations, violates gauge invariance, unitarity ad-hoc curable (new parameters)






★ Form factors

• Similar approach to anomalous couplings, partially resums perturbative series• (Almost) completely general and model-independent• Needs parameterizations, violates gauge invariance, unitarity ad-hoc curable (new parameters)

★ Effective Field Theory

• (Almost) model-independent, consistent calculation of perturbative corrections (power counting !?)• Depends on (possibly) many free parameters• Requires decoupling of New Physics • Range of applicability strongly depends on couplings and scales (unitarity issue)


General Procedure using EFTs• Use all experimental observables ⟹ global fit to all Wilson coefficients

• Would-be optimal approach

• Too many independent variables ⟹ needs staged fitting

• Need for simplifying assumptions

• Try to find minimal “physically well motivated” operator basis

• Experimental bias: consider only LHC-accessible operators

• Cross check from low-energy physics (flavor / EDMs / EWPO etc. / Higgs!)

• Explore full structure of EW Higgs doublet:










O�f = (�†$Dµ�)

�f�µf

� O�W = (�†�)Wµ⌫Wµ⌫

W W W W

Effects in H ➝ Zff related to Z ➝ ff ⟹ constrained by SLC/LEP

visible in Higgs physics: H ➝ WW

redefines only SM parameters⟹ no experimental constraints










O�f = (�†$Dµ�)

�f�µf

� O�W = (�†�)Wµ⌫Wµ⌫

W W W W

Effects in H ➝ Zff related to Z ➝ ff ⟹ constrained by SLC/LEP

visible in Higgs physics: H ➝ WW

redefines only SM parameters⟹ no experimental constraints

• Often use vev-subtracted operators: O(�†�) �! O0(�†�� v2)

• EFT allows to systemically calculate higher-order corrections


How to get EFTs from New Physics✦ Consider effects from heavy states by using (known) low-energy d.o.f.s

H. Georgi, 1993



H. Georgi, 1993

✦ Integrating out heavy d.o.f.s marginalizes over details of short-distance interactions✦ Toy Example: two interacting scalar fields ',�

Path integralZ[j, J ] =

ZD[�]D['] exp

i

Zdx

⇣12 (@')

2� 12�(2+M

2)��'

2��. . .+J�+j'

⌘�



H. Georgi, 1993



ZD[�]D['] exp

i

Zdx

⇣12 (@')

2� 12�(2+M

2)��'

2��. . .+J�+j'

⌘�

Completing the square (Gaussian integration)

�0 = �+�

M2

✓1 +

@2

M2

◆�1

'2 =)



H. Georgi, 1993



ZD[�]D['] exp

i

Zdx

⇣12 (@')

2� 12�(2+M

2)��'

2��. . .+J�+j'

⌘�

Completing the square (Gaussian integration)

�0 = �+�

M2

✓1 +

@2

M2

◆�1

'2 =)

In the Lagrangian remove the high-scale d.o.f.s:

1

2(@�)2� 1

2M2�2��'2� = �1

2�0(M2 + @2)�0

| {z }+

�2

2M2'2

✓1 +

@2

M2

◆�1

'2

| {z }Irrelevant normalization

of the path integralTower of higher and

higher-dim. operators of light fields


Generation of Higher-dimensional Operators

O(I)JJ =

1

⇤2trhJ (I) · J (I)

i

O0�,1 = 1

⇤2

�(D�)†�

� · ��†(D�)�� v2

2 |D�|2

O0�� = 1

⇤2 (�†�� v2/2) (D�)† · (D�)

O0�,3 =

1

⇤2

1

3(�†�� v2/2)3

O0WW = � 1

⇤2

1

2(�†�� v2/2)tr [Wµ⌫W

µ⌫ ]

OB =1

⇤2

i

2(Dµ�)

†(D⌫�)Bµ⌫

O0BB = � 1

F 2

1

4(�†�� v2/2)Bµ⌫B

µ⌫

OV q =1

⇤2q�( /D�)q

Couplings of new states to the longitudinal / transversal diboson system

J = 0 J = 1 J = 2I = 0 �0 (Higgs singlet?) !0 (�0/Z 0 ?) f0 (Graviton ?)I = 1 ⇡±,⇡0

(2HDM ?) ⇢±, ⇢0 (W 0/Z 0 ?) a±, a0

I = 2 �±±,�±,�0 (Higgs triplet ?) — t±±, t±, t0


Generation of Higher-dimensional Operators

O(I)JJ =

1

⇤2trhJ (I) · J (I)

i

O0�,1 = 1

⇤2

�(D�)†�

� · ��†(D�)�� v2

2 |D�|2

O0�� = 1

⇤2 (�†�� v2/2) (D�)† · (D�)

O0�,3 =

1

⇤2

1

3(�†�� v2/2)3

O0WW = � 1

⇤2

1

2(�†�� v2/2)tr [Wµ⌫W

µ⌫ ]

OB =1

⇤2

i

2(Dµ�)

†(D⌫�)Bµ⌫

O0BB = � 1

F 2

1

4(�†�� v2/2)Bµ⌫B

µ⌫

OV q =1

⇤2q�( /D�)q

Couplings of new states to the longitudinal / transversal diboson system

J = 0 J = 1 J = 2I = 0 �0 (Higgs singlet?) !0 (�0/Z 0 ?) f0 (Graviton ?)I = 1 ⇡±,⇡0

(2HDM ?) ⇢±, ⇢0 (W 0/Z 0 ?) a±, a0

I = 2 �±±,�±,�0 (Higgs triplet ?) — t±±, t±, t0

Different power counting for weakly and strongly interacting theoriesci⇤

⇠ g

4⇡⇤vs.

ci⇤

⇠ g

⇤

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