new physics / resonances / 15 in vector boson scattering...

J.R.Reuter New Physics in VBS @ LHC LHCP 2019, Puebla, 24.5.19

/ 15

Jürgen R. Reuter, DESY

New Physics / Resonances in Vector Boson Scattering

at the LHC

w. EPJC [1807.02512] PRD93(16),3. 036004 [1511.00022], PRD91(15) 096007 [1408.6207] JHEP 0811.010 [0806.4145]

based on work with A. Alboteanu, S. Brass, C. Fleper, W. Kilian, T. Ohl, M. Sekulla

1


/ 15Vector Boson Scattering after the Higgs Discovery

• 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 “the Higgs” 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)

7. Look for deviations in intricate cancellations of VBS amplitudes

2


/ 15

P1

P2

x1P1

x2P2

D(x1, Q2)

D(x2, Q2)

QCD

QCD

QGC

pp ! WWjj ! ``⌫⌫jj

3

VV Vjj VVjj VVV

Anatomy of Vector Boson Scattering (VBS)

Discovery for W+W+jj (electroweak production) ATLAS PRL 113(2014)14, 141803 [1405.6241] & 1611.02428; CMS PRL 114(2015), 051801 [1410.6315]

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

4α

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

5α

-0.6

-0.4

-0.2

0

0.2

0.4

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

68% CL95% CLexpected 95% CLStandard Model

confidence intervals


/ 15 4Anatomy of Vector Boson Scattering (VBS)

shown by Kenneth Long, Seoul, ICHEP 2018


/ 15 5More channels came/coming up …

1812.09740

pp → WZjj →lvlljj

pp → ZZjj →lllljj

PLB 774(2017) 682

PRL 120, 081801(2018)

pp → W+W+jj →lvlvjj

pp → WZjj →lljj + X

1901.04060

1905.07714

Philip Chang,plenary; Usama Hussain, parallel

24.5.

pp → WZjj →(l/v)(l/v)jjjj


/ 15 6Deviations as EFT — Dim 8 Operators

Mixed operators

Transversal operators

Longitudinal operators

Motivated by SMEFT: L = LSM +X

i

"c(5)i

⇤O

(5)i +

c(6)i

⇤2O

(6)i +

c(7)i

⇤3O

(7)i +

c(8)i

⇤4O

(8)i + · · ·

#

S. Weinberg, 1979 Buchmüller/Wyler, 1986

Grzadkowski/Iskrzysnki/Misiak/Rosiek, 2010

Eboli/Gonzalez-Garcia/Mizukoshi, 2006

Alboteanu/Kilian/JRR, 2008 Kilian/Ohl/JRR/Sekulla, 2014


/ 15 6Deviations as EFT — Dim 8 Operators

Mixed operators

Transversal operators

Longitudinal operators

Motivated by SMEFT: L = LSM +X

i

"c(5)i

⇤O

(5)i +

c(6)i

⇤2O

(6)i +

c(7)i

⇤3O

(7)i +

c(8)i

⇤4O

(8)i + · · ·

#

S. Weinberg, 1979 Buchmüller/Wyler, 1986

Grzadkowski/Iskrzysnki/Misiak/Rosiek, 2010

Eboli/Gonzalez-Garcia/Mizukoshi, 2006


Energy rise of operators lead to unitarity violation

Unitarity violation cancels between operators in UV-complete Theory


/ 15Procedures 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”)

Effect on BDT training not clear

✓(⇤2C � s)

7



0 1 2 3 40.0

0.5

1.0

s êH4 p vL

†AHsL§





✓(⇤2C � s)

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

7



0 1 2 3 40.0

0.5

1.0

s êH4 p vL

†AHsL§





✓(⇤2C � s)

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

7

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 amplitude [projection to unitarity circle], also for complex ampl., no additional parameters


��a� aK2

�� = aK2 ) a = 1

Re⇣

1a0

⌘�i


/ 15VBS 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

8



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

8



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

8

Much more leeway for new physics in

transversal gauge bosons and di-Higgs


/ 15 9(In)Validity of (In)Effective Field Theories

Resonances in direct reach (not clear: strongly interacting models [e.g. 𝜎 resonance]

Estimate of operator coefficients (difficult for strongly coupled models)

Partial wave unitarity: gives guidance on maximally possible event numbers

Positivity constraints on operator coefficients

Size of coefficients: dichotomy between

validity and detectability

EFT better/best[?] suited in intensity

frontier [example: HEFT @ 𝓞(100 GeV)]

EFT borderline in VBS/energy frontier physics

M. Szłeper Operator coefficient

Cut

off s

cale

ASM⇥Adim-6 & |Adim-6|2 ASM⇥Adim-8 & |Adim-8|2 ASM⇥Adim-6 & ASM⇥Adim-8<latexit sha1_base64="E/w7tFHdPngAgCyFMjDQmu9Ogtc=">AAADlnicpVLdatRAFJ5N1Nb4063eFLwZXAURuiR7UfdKqlL0RljRbQvNukwmJ9mhM5N05kS6pHkOn80n8C3EyXaRdlsE64GBj+/8fOecOUkphcUw/NHx/Fu376yt3w3u3X/wcKO7+WjfFpXhMOaFLMxhwixIoWGMAiUclgaYSiQcJMfvWv/BNzBWFPoLzkuYKJZrkQnO0FHTzc73OIFc6BpOqgX1sgkojRXDGWeyftNM6xjhFOvPH5uGxigU2OvcqVDbO21Ejk5MUdpWkZDh2V+jjchnePZ1QOMTp58GN1Qe/pPy8JLyH+n/n/rGNVxHQQw6vfAL024v7IcLo1dBtAQ9srTRtPszTgteKdDIJbP2KApLnNTMoOASXP3KQsn4McvhyEHNXD+TenFBDX3umJRmhXFPI12wFzNqpqydq8RFthPYVV9LXutLBcuT4nRFHrPhpBa6rBA0P1fPKkmxoO2N0lQY4CjnDjBuhBuA8hkzjKO75MBtJlrdw1WwP+hHYT/6NOjtvl3uaJ08IU/JCxKRV2SXfCAjMia888t75m17fX/Lf+3v+e/PQ73OMucxuWT+6DeYdC2x</latexit><latexit sha1_base64="E/w7tFHdPngAgCyFMjDQmu9Ogtc=">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</latexit><latexit sha1_base64="E/w7tFHdPngAgCyFMjDQmu9Ogtc=">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</latexit><latexit sha1_base64="E/w7tFHdPngAgCyFMjDQmu9Ogtc=">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</latexit>


/ 15 10Bumps vs. Tails


/ 15 Simplified signal models: generic resonancesRise of amplitude: is 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 (partially) different physics (mixing with W/Z)

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

. . . . . . . . .

11

Translation into Wilson coefficients below resonance

Courtesy: Jorge de Blas


/ 15Comparison: 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

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

12

Brass/Fleper/Kilian/JRR/Sekulla: w. EPJC [1807.02512]




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





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


12





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





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


12



/ 15Triple [multiple] Vector Boson Production ?

Relate to ?

Yes, same Feynman rule as in VBS, but …

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

Unitarization: work in progress (needs 2 ➝ 3 unitarizations, inelastic

channels) [Kilian/Kreher/JRR, w.i.p.]

Different Wilson coefficients dominate (particularly for resonances)

Important physics (partially) independent from VBS (“different fiducial vol.”)

13

➝ Philip Chang, plenary session, 24.05. Usama Hussain, parallel session, 24.05.

Markus Cristianziani, parallel/poster, 21./24.05.Evidence at ATLAS at 4𝜎 level: smallest SM cross section

1903.10415


/ 15Conclusions / Summary✦ Vector boson scattering a flagship measurement of Runs II/III (and FCC-hh !)

✦ EFT provides well-defined (and very limited) framework for SM deviations

✦ There is not really a true model-independent parameterization! ✦ Unitarization for theoretically sane description (allows reliable BDT analysis)✦ T-matrix unitarization universal scheme for EFT and resonances✦ Simplified models: generic electroweak resonances

✦ Limits from LHC still quite limited: 𝜦new physics ∼ 700–800 GeV

14

Fby M. Szłeper

by M.-A. Pleier


/ 15 15

7th Workshop on Multi-Boson Interactions

1. TU Dresden

2. BNL (Brookhaven Ntl. Lab)

3. DESY

4. U. of Wisconsin — Madison

5. KIT Karlsruhe

6. U. of Michigan — Ann Arbor

7. Aristotle U. Thessaloniki

August 26-28, 2019Aristotle U., Thessaloniki


/ 15

BACKUP SLIDES

16


/ 15

P1

P2

x1P1

x2P2

D(x1, Q2)

D(x2, Q2)

QCD

QCD

QGC

pp ! WWjj ! ``⌫⌫jj Backgrounds:

• tt ➝ WbWb • W + jets • single top, misreconstructed jet• WWjj QCD production • ll + X + Emiss (“prompt”)

• lljj tag• mjj > 500 GeV (“jet recoil”)• |Δyjj | > 2.4 (“rapidity distance”)• Cuts on Ej , pTj

• No mini jet vetoes

Fiducial phase space volume:

17

VV Vjj VVjj VVV

Anatomy of Vector Boson Scattering (VBS)


/ 15Differential spectra in VBSpp ! e+µ+⌫e⌫µjj

ps = 14TeV L = 1ab�1

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

18

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

18






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

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

18






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

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

18



/ 15How to get EFTs from New Physics✦ Consider effects from heavy states by using (known) low-energy d.o.f.s

H. Georgi, 1993

19



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+M2)��'2��. . .+J�+j'

⌘�

19



H. Georgi, 1993



ZD[�]D['] exp

i

Zdx

⇣12 (@')

2� 12�(2+M2)��'2��. . .+J�+j'

⌘�

Completing the square (Gaussian integration)

�0 = �+�

M2

✓1 +

@2

M2

◆�1

'2 =)

19



H. Georgi, 1993



ZD[�]D['] exp

i

Zdx

⇣12 (@')

2� 12�(2+M2)��'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

19


/ 15

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

20

Im [aℓ]

Re [aℓ]

12

12

Im [aℓ]

Re [aℓ]

12

12

Im [aℓ]

Re [aℓ]

12

12

Im [aℓ]

Re [aℓ]

12

12

Im [aℓ]

Re [aℓ]

12

12

Scenarios for New Physics in VBS


/ 15Unitarity in vector boson scattering 21


/ 15Unitarity in vector boson scattering

Lee/Quigg/Thacker, 1973

21


/ 15Different 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)

22



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

22



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

22



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]

22



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

22


/ 15Complete 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

23


/ 15 24Remark on alternative unitarizations

• Independent Amplitude Method (IAM) [Truong, 1988; Dobado/Herrero/Truong, 1990]

• Padé Method [Padé, 1890; Basdevant/Lee, 1970]

• N/D method [Chew/Mandelstam, 1960]

• Focus on correct descriptions of certain explicit (known) resonance channels • Tied to chiral perturbation theory and QCD ⇒ more model-dependence

• Unitarization is not a tool to predict resonances


/ 15 24Remark on alternative unitarizations

• Independent Amplitude Method (IAM) [Truong, 1988; Dobado/Herrero/Truong, 1990]

• Padé Method [Padé, 1890; Basdevant/Lee, 1970]

• N/D method [Chew/Mandelstam, 1960]

• Focus on correct descriptions of certain explicit (known) resonance channels • Tied to chiral perturbation theory and QCD ⇒ more model-dependence

• Unitarization is not a tool to predict resonances

Unitarization of operators

Evaluate modified Feynman rules off-shell

Scale that is used for the diboson system in s-channel setups: psV V

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Clebsch-Gordan decomposition into spin–isospin eigenamplitudes

Amplitudes should be modified only in s–channel configurations


/ 15 25

ij

0 1 2 �

-6 -2 � 52 0 0 0 0 0 0 + + + +

0 0 0 0 0 0 0 � 25 � 4

5 � 12 + � + �

0 0 0 0 0 0 � 25 � 4

5 � 12 + � � +

� 223 � 14

3 � 116 0 0 0 � 2

15 � 415 � 1

30 + + � �0 0 0 0 0 0 0 0 0 + + + +

1 0 0 0 0 0 0 25 � 1

5 0 + � + �0 0 0 0 0 0 � 2

515 0 + � � +

0 0 0 23 � 1

316 0 0 0 + + � �

0 -2 -1 0 0 0 0 0 0 + + + +2 0 0 0 0 0 0 � 2

5 � 15 � 1

5 + � + �0 0 0 0 0 0 � 2

5 � 15 � 1

5 + � � +� 4

3 � 83 � 1

3 0 0 0 � 215 � 1

15 � 130 + + � �

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Use spin-isospin eigenamplitudes exclusive in helicities:

Can be obtained by using Wigner’s d-functions [Wigner, 1931]

AIJ(s;�) =

Z 0

�s

dt

sAI(s, t, u;�) · dJ�,�0

arccos

✓1 + 2

t

s

◆�

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� = (�1,�2,�3,�4)<latexit sha1_base64="fU5COVy6N2CutGBEnZJTIZtZkU8=">AAACKnicbZDLSsNAFIYn9VbrLerSzWArVJCSVEE3QtWNywr2Ak0Ik8mkHTq5MDMRSujzuPFV3HShFLc+iJM2Fq0eGObjP+dn5vxuzKiQhjHVCiura+sbxc3S1vbO7p6+f9AWUcIxaeGIRbzrIkEYDUlLUslIN+YEBS4jHXd4l/U7T4QLGoWPchQTO0D9kPoUI6kkR7+pWG7EPDEK1AUtppwegtfVnBzz7JvqCzpf0MVppeToZaNmzAr+BTOHMsir6egTy4twEpBQYoaE6JlGLO0UcUkxI+OSlQgSIzxEfdJTGKKACDudrTqGJ0rxoB9xdUIJZ+pPR4oCkS2jJgMkB2K5l4n/9XqJ9K/slIZxIkmI5w/5CYMygllu0KOcYMlGChDmVP0V4gHiCEuVbhaCubzyX2jXa6ZRMx/q5cZtHkcRHIFjUAUmuAQNcA+aoAUweAav4A28ay/aRJtqH/PRgpZ7DsGv0j6/AHqvpWE=</latexit><latexit sha1_base64="fU5COVy6N2CutGBEnZJTIZtZkU8=">AAACKnicbZDLSsNAFIYn9VbrLerSzWArVJCSVEE3QtWNywr2Ak0Ik8mkHTq5MDMRSujzuPFV3HShFLc+iJM2Fq0eGObjP+dn5vxuzKiQhjHVCiura+sbxc3S1vbO7p6+f9AWUcIxaeGIRbzrIkEYDUlLUslIN+YEBS4jHXd4l/U7T4QLGoWPchQTO0D9kPoUI6kkR7+pWG7EPDEK1AUtppwegtfVnBzz7JvqCzpf0MVppeToZaNmzAr+BTOHMsir6egTy4twEpBQYoaE6JlGLO0UcUkxI+OSlQgSIzxEfdJTGKKACDudrTqGJ0rxoB9xdUIJZ+pPR4oCkS2jJgMkB2K5l4n/9XqJ9K/slIZxIkmI5w/5CYMygllu0KOcYMlGChDmVP0V4gHiCEuVbhaCubzyX2jXa6ZRMx/q5cZtHkcRHIFjUAUmuAQNcA+aoAUweAav4A28ay/aRJtqH/PRgpZ7DsGv0j6/AHqvpWE=</latexit><latexit sha1_base64="fU5COVy6N2CutGBEnZJTIZtZkU8=">AAACKnicbZDLSsNAFIYn9VbrLerSzWArVJCSVEE3QtWNywr2Ak0Ik8mkHTq5MDMRSujzuPFV3HShFLc+iJM2Fq0eGObjP+dn5vxuzKiQhjHVCiura+sbxc3S1vbO7p6+f9AWUcIxaeGIRbzrIkEYDUlLUslIN+YEBS4jHXd4l/U7T4QLGoWPchQTO0D9kPoUI6kkR7+pWG7EPDEK1AUtppwegtfVnBzz7JvqCzpf0MVppeToZaNmzAr+BTOHMsir6egTy4twEpBQYoaE6JlGLO0UcUkxI+OSlQgSIzxEfdJTGKKACDudrTqGJ0rxoB9xdUIJZ+pPR4oCkS2jJgMkB2K5l4n/9XqJ9K/slIZxIkmI5w/5CYMygllu0KOcYMlGChDmVP0V4gHiCEuVbhaCubzyX2jXa6ZRMx/q5cZtHkcRHIFjUAUmuAQNcA+aoAUweAav4A28ay/aRJtqH/PRgpZ7DsGv0j6/AHqvpWE=</latexit><latexit sha1_base64="fU5COVy6N2CutGBEnZJTIZtZkU8=">AAACKnicbZDLSsNAFIYn9VbrLerSzWArVJCSVEE3QtWNywr2Ak0Ik8mkHTq5MDMRSujzuPFV3HShFLc+iJM2Fq0eGObjP+dn5vxuzKiQhjHVCiura+sbxc3S1vbO7p6+f9AWUcIxaeGIRbzrIkEYDUlLUslIN+YEBS4jHXd4l/U7T4QLGoWPchQTO0D9kPoUI6kkR7+pWG7EPDEK1AUtppwegtfVnBzz7JvqCzpf0MVppeToZaNmzAr+BTOHMsir6egTy4twEpBQYoaE6JlGLO0UcUkxI+OSlQgSIzxEfdJTGKKACDudrTqGJ0rxoB9xdUIJZ+pPR4oCkS2jJgMkB2K5l4n/9XqJ9K/slIZxIkmI5w/5CYMygllu0KOcYMlGChDmVP0V4gHiCEuVbhaCubzyX2jXa6ZRMx/q5cZtHkcRHIFjUAUmuAQNcA+aoAUweAav4A28ay/aRJtqH/PRgpZ7DsGv0j6/AHqvpWE=</latexit>

� = �1 � �2 �0 = �3 � �4<latexit sha1_base64="8v+/Gf0r3Vb1G2INMv7ubdPTmJU=">AAACMnicbVBLSwMxGMzWV62vVY9egq3gxbJbBb0IRS96q2Af0F2XbDZtQ7MPk6xQlv4mL/4SwYMeFPHqjzDbbou2DoQMM/ORfONGjAppGK9abmFxaXklv1pYW9/Y3NK3dxoijDkmdRyykLdcJAijAalLKhlpRZwg32Wk6fYvU7/5QLigYXArBxGxfdQNaIdiJJXk6Ncli6m0h86z2zGPJqxSgtZ9jDw4ydxZEac+mUaPp9GTUsHRi0bZGAHOEzMjRZCh5ujPlhfi2CeBxAwJ0TaNSNoJ4pJiRoYFKxYkQriPuqStaIB8IuxktPIQHijFg52QqxNIOFJ/TyTIF2LguyrpI9kTs14q/ue1Y9k5sxMaRLEkAR4/1IkZlCFM+4Me5QRLNlAEYU7VXyHuIY6wVC2nJZizK8+TRqVsGmXzplKsXmR15MEe2AeHwASnoAquQA3UAQaP4AW8gw/tSXvTPrWvcTSnZTO74A+07x8EaajD</latexit><latexit sha1_base64="8v+/Gf0r3Vb1G2INMv7ubdPTmJU=">AAACMnicbVBLSwMxGMzWV62vVY9egq3gxbJbBb0IRS96q2Af0F2XbDZtQ7MPk6xQlv4mL/4SwYMeFPHqjzDbbou2DoQMM/ORfONGjAppGK9abmFxaXklv1pYW9/Y3NK3dxoijDkmdRyykLdcJAijAalLKhlpRZwg32Wk6fYvU7/5QLigYXArBxGxfdQNaIdiJJXk6Ncli6m0h86z2zGPJqxSgtZ9jDw4ydxZEac+mUaPp9GTUsHRi0bZGAHOEzMjRZCh5ujPlhfi2CeBxAwJ0TaNSNoJ4pJiRoYFKxYkQriPuqStaIB8IuxktPIQHijFg52QqxNIOFJ/TyTIF2LguyrpI9kTs14q/ue1Y9k5sxMaRLEkAR4/1IkZlCFM+4Me5QRLNlAEYU7VXyHuIY6wVC2nJZizK8+TRqVsGmXzplKsXmR15MEe2AeHwASnoAquQA3UAQaP4AW8gw/tSXvTPrWvcTSnZTO74A+07x8EaajD</latexit><latexit sha1_base64="8v+/Gf0r3Vb1G2INMv7ubdPTmJU=">AAACMnicbVBLSwMxGMzWV62vVY9egq3gxbJbBb0IRS96q2Af0F2XbDZtQ7MPk6xQlv4mL/4SwYMeFPHqjzDbbou2DoQMM/ORfONGjAppGK9abmFxaXklv1pYW9/Y3NK3dxoijDkmdRyykLdcJAijAalLKhlpRZwg32Wk6fYvU7/5QLigYXArBxGxfdQNaIdiJJXk6Ncli6m0h86z2zGPJqxSgtZ9jDw4ydxZEac+mUaPp9GTUsHRi0bZGAHOEzMjRZCh5ujPlhfi2CeBxAwJ0TaNSNoJ4pJiRoYFKxYkQriPuqStaIB8IuxktPIQHijFg52QqxNIOFJ/TyTIF2LguyrpI9kTs14q/ue1Y9k5sxMaRLEkAR4/1IkZlCFM+4Me5QRLNlAEYU7VXyHuIY6wVC2nJZizK8+TRqVsGmXzplKsXmR15MEe2AeHwASnoAquQA3UAQaP4AW8gw/tSXvTPrWvcTSnZTO74A+07x8EaajD</latexit><latexit sha1_base64="8v+/Gf0r3Vb1G2INMv7ubdPTmJU=">AAACMnicbVBLSwMxGMzWV62vVY9egq3gxbJbBb0IRS96q2Af0F2XbDZtQ7MPk6xQlv4mL/4SwYMeFPHqjzDbbou2DoQMM/ORfONGjAppGK9abmFxaXklv1pYW9/Y3NK3dxoijDkmdRyykLdcJAijAalLKhlpRZwg32Wk6fYvU7/5QLigYXArBxGxfdQNaIdiJJXk6Ncli6m0h86z2zGPJqxSgtZ9jDw4ydxZEac+mUaPp9GTUsHRi0bZGAHOEzMjRZCh5ujPlhfi2CeBxAwJ0TaNSNoJ4pJiRoYFKxYkQriPuqStaIB8IuxktPIQHijFg52QqxNIOFJ/TyTIF2LguyrpI9kTs14q/ue1Y9k5sxMaRLEkAR4/1IkZlCFM+4Me5QRLNlAEYU7VXyHuIY6wVC2nJZizK8+TRqVsGmXzplKsXmR15MEe2AeHwASnoAquQA3UAQaP4AW8gw/tSXvTPrWvcTSnZTO74A+07x8EaajD</latexit>

A0(s, t, u;�)<latexit sha1_base64="JZkwW7BaOpRepr7seSFzUeTiv5g=">AAACFXicbVBNS8MwGE79nPOr6tFLcBMmjNHuouBl6sXjBPcBaxlpmm5haVqSVBilf8KLf8WLB0W8Ct78N6ZbD7r5QsjD87xv8j6PFzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlVEiMOngiEWi7yFJGOWko6hipB8LgkKPkZ43ucn13gMRkkb8Xk1j4oZoxGlAMVKaGpr1qhMiNcaIpVfZ0KrJuqonl44XMV9OQ32lDtOv+Sg7q5aHZsVqWLOCy8AuQAUU1R6aX44f4SQkXGGGpBzYVqzcFAlFMSNZ2UkkiRGeoBEZaMhRSKSbzlxl8FQzPgwioQ9XcMb+nkhRKPMddWduQS5qOfmfNkhUcOGmlMeJIhzPPwoSBlUE84igTwXBik01QFhQvSvEYyQQVjrIPAR70fIy6DYbttWw75qV1nURRwkcgxNQAzY4By1wC9qgAzB4BM/gFbwZT8aL8W58zFtXjGLmCPwp4/MHQmCeOQ==</latexit><latexit sha1_base64="JZkwW7BaOpRepr7seSFzUeTiv5g=">AAACFXicbVBNS8MwGE79nPOr6tFLcBMmjNHuouBl6sXjBPcBaxlpmm5haVqSVBilf8KLf8WLB0W8Ct78N6ZbD7r5QsjD87xv8j6PFzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlVEiMOngiEWi7yFJGOWko6hipB8LgkKPkZ43ucn13gMRkkb8Xk1j4oZoxGlAMVKaGpr1qhMiNcaIpVfZ0KrJuqonl44XMV9OQ32lDtOv+Sg7q5aHZsVqWLOCy8AuQAUU1R6aX44f4SQkXGGGpBzYVqzcFAlFMSNZ2UkkiRGeoBEZaMhRSKSbzlxl8FQzPgwioQ9XcMb+nkhRKPMddWduQS5qOfmfNkhUcOGmlMeJIhzPPwoSBlUE84igTwXBik01QFhQvSvEYyQQVjrIPAR70fIy6DYbttWw75qV1nURRwkcgxNQAzY4By1wC9qgAzB4BM/gFbwZT8aL8W58zFtXjGLmCPwp4/MHQmCeOQ==</latexit><latexit sha1_base64="JZkwW7BaOpRepr7seSFzUeTiv5g=">AAACFXicbVBNS8MwGE79nPOr6tFLcBMmjNHuouBl6sXjBPcBaxlpmm5haVqSVBilf8KLf8WLB0W8Ct78N6ZbD7r5QsjD87xv8j6PFzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlVEiMOngiEWi7yFJGOWko6hipB8LgkKPkZ43ucn13gMRkkb8Xk1j4oZoxGlAMVKaGpr1qhMiNcaIpVfZ0KrJuqonl44XMV9OQ32lDtOv+Sg7q5aHZsVqWLOCy8AuQAUU1R6aX44f4SQkXGGGpBzYVqzcFAlFMSNZ2UkkiRGeoBEZaMhRSKSbzlxl8FQzPgwioQ9XcMb+nkhRKPMddWduQS5qOfmfNkhUcOGmlMeJIhzPPwoSBlUE84igTwXBik01QFhQvSvEYyQQVjrIPAR70fIy6DYbttWw75qV1nURRwkcgxNQAzY4By1wC9qgAzB4BM/gFbwZT8aL8W58zFtXjGLmCPwp4/MHQmCeOQ==</latexit><latexit sha1_base64="JZkwW7BaOpRepr7seSFzUeTiv5g=">AAACFXicbVBNS8MwGE79nPOr6tFLcBMmjNHuouBl6sXjBPcBaxlpmm5haVqSVBilf8KLf8WLB0W8Ct78N6ZbD7r5QsjD87xv8j6PFzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlVEiMOngiEWi7yFJGOWko6hipB8LgkKPkZ43ucn13gMRkkb8Xk1j4oZoxGlAMVKaGpr1qhMiNcaIpVfZ0KrJuqonl44XMV9OQ32lDtOv+Sg7q5aHZsVqWLOCy8AuQAUU1R6aX44f4SQkXGGGpBzYVqzcFAlFMSNZ2UkkiRGeoBEZaMhRSKSbzlxl8FQzPgwioQ9XcMb+nkhRKPMddWduQS5qOfmfNkhUcOGmlMeJIhzPPwoSBlUE84igTwXBik01QFhQvSvEYyQQVjrIPAR70fIy6DYbttWw75qV1nURRwkcgxNQAzY4By1wC9qgAzB4BM/gFbwZT8aL8W58zFtXjGLmCPwp4/MHQmCeOQ==</latexit>

Extract all partial waves:

Aij(s;�)/(g4s2) =

(c0FT0 + c1FT1 + c2FT2)<latexit sha1_base64="dPB1DegmIomJcGVmkY5S6eONCSU=">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</latexit><latexit sha1_base64="dPB1DegmIomJcGVmkY5S6eONCSU=">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</latexit><latexit sha1_base64="dPB1DegmIomJcGVmkY5S6eONCSU=">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</latexit><latexit sha1_base64="dPB1DegmIomJcGVmkY5S6eONCSU=">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</latexit>

Braß/Fleper/Kilian/JRR/Sekulla, 1807.02512

Unitarization of [transverse] operators

new physics / resonances / 15 in vector boson scattering...

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