High-precision calculations of Higgsproperties

Giampiero Passarino

Dipartimento di Fisica Teorica, Universita di Torino, ItalyINFN, Sezione di Torino, Italy

Precision Observables and Radiative Corrections relevantfor future e+e− colliders, 13−14 July 2015,CERN

This talk is about why NLO SMEFT + POs1, it is not

7 how NLO2

7 what NLO3however, see backup material go now

7 why POs4

fuel for discussion . . . nothing more

1What can be said at all should be said clearly and whereof one cannot speak thereof one must be silent2Covered in “ATLAS Higgs (N)NLO MC and Tools Workshop for LHC RUN-2”,

https://indico.cern.ch/event/345455/, see also https://indico.desy.de/conferenceDisplay.py?confId=4763same as above4Covered in “Pseudo-observables: from LEP to LHC”, https://indico.cern.ch/event/373667/

2/36

Part I

Warming up: mostly Lep

3/36

POs at Lep, the role of the ZZZ -pole

FromV µ

e+e−γV µ

f fγ

s+

V µ

e+e−Z V µ

f fZ

s−M2Z

+BoxesV µ

e+e−γV µ

f fγ

s+

V µ

e+e−Z V µ

f fZ

s−M2Z

+BoxesV µ

e+e−γV µ

f fγ

s+

V µ

e+e−Z V µ

f fZ

s−M2Z

+Boxes

To σpeakfσpeakfσpeakf = 12π

ΓeΓf

M2Z Γ2

Z

12πΓeΓf

M2Z Γ2

Z

12πΓeΓf

M2Z Γ2

Z

Caution: unstable particles present

From on-shell mass MZMZMZ To complex pole sZsZsZ

4/36

e+ f

γ

e− f

+

e+ f

Z

e− f

Figure 1: The process e+e−→ (Z, γ)→ f f in the Born approximation.

e+ f

(Z, γ)

e− f

+

e+ f

(Z, γ)

e− f

Figure 2: The process e+e−→ (Z, γ)→ f f ; final fermion vertex and its counter-terms.

1

e+ f

(Z, γ)

e− f

+

e+ f

(Z, γ)

e− f

Figure 3: Process e+e−→ (Z, γ)→ f f ; electron vertex and its counter-terms

e+ f

(Z, γ) (Z, γ)

e− f

+

e+ f

(Z, γ) (Z, γ)

e− f

Figure 4: Process e+e−→ (Z, γ)→ f f ; self-energies and kinetic counter-terms

2

DIAGRAMMATICA

at Lep1

role of theory:delivering boxes and crosseswith maniacal care for gauge invariance

5/36

The complete amplitude for the four-fermion process should bepresented in all schemes and all gauges with a generalstructure,

A ∼ 1s

α

fer(s)γµ ⊗ γµ + χ (s)[F

efQQ(s, t)γ

µ ⊗ γµ +FefLL(s, t)γ

µγ+ ⊗ γµ γ+

+FefQL(s, t)γ

µ ⊗ γµ γ+ +FefLQ(s, t)γ

µγ+ ⊗ γµ

]χ (s) = s χZ(s)

Again the raison d'etre of any renormalization scheme is deeplyconnected to the possibility of defining the form factors in agauge-invariant manner.

6/36

Where are the PO’s?

dσf

dΩ

dσf

dΩ

dσf

dΩ=

α2

4sNc

f βf

[(1+c2

)F1(s)

α2

4sNc

f βf

[(1+c2

)F1(s)

α2

4sNc

f βf

[(1+c2

)F1(s)

+++ 4 µ2f (1−c2)F2(s)+2βf c F3(s)

]4 µ

2f (1−c2)F2(s)+2βf c F3(s)

]4 µ

2f (1−c2)F2(s)+2βf c F3(s)

]where c = cosθ is the cosine of the scattering angle and

β 2f = 1−4 µ2

f with µ2f = m2

f /s.

The energy dependence is confined in the F -functions

7/36

F1(s) = Q2e Q2

f +2QeQfgeVgf

V Re χ(s)

+[(

geV

)2 +(ge

A

)2] [(

gfV

)2+(

gfA

)2−4 µ

2f

]∣∣∣χ(s)∣∣∣2,

F2(s) = Q2e Q2

f +2QeQfgeVgf

V Re χ(s)

+[(

geV

)2 +(ge

A

)2] (

gfV

)2∣∣∣χ(s)∣∣∣2,

F3(s) = 2QeQfgeAgf

A Re χ(s)+4geVgf

VgeAgf

A

∣∣∣χ(s)∣∣∣2

χ is the reduced γ/Z propagator ratio. The form factors Finclude weak loop corrections but, in their construction, wehave completely ignored a few ingredients:

♣ QED radiation,

♦ weak boxes and

♠ all the imaginary parts8/36

Usually, 25 POs were introduced and discussed

the mass of the WWW (MWMWMW)

the hadronic peak cross-section (σhσhσh)

the partial leptonic and hadronic widths(Γf, f = ν,e,µ,τ,u,d,c,s,bΓf, f = ν,e,µ,τ,u,d,c,s,bΓf, f = ν,e,µ,τ,u,d,c,s,b)

the total width (ΓZΓZΓZ)

the total hadronic width (ΓhΓhΓh) Queen of new POs σZHσZHσZH

the total invisible width (ΓinvΓinvΓinv)

various ratios (Rl, Rb, RcRl, Rb, RcRl, Rb, Rc)

the asymmetries and polarization(Aµ

FB,AeLR,Ab

FB,AcFB,Pτ,PbAµ

FB,AeLR,Ab

FB,AcFB,Pτ,PbAµ

FB,AeLR,Ab

FB,AcFB,Pτ,Pb)

effective sines (sin2θe,sin2

θbsin2θe,sin2

θbsin2θe,sin2

θb)

9/36

Part II

Learning from LHC: mostly SMEFT

10/36

In the next few slides I will show you beauty in a handful of κs

m Start with EFT at a given order (here NLO)

m write any amplitude as a sum of κ -deformed SMsub-amplitudes

m add another sum of κ -deformed non-SM amplitudes

m show that κs are linear combinations of Wilson coefficients

m discover correlations among the κs

11/36

Rationale for this course of action

m Physics is symmetry plus dynamics

m Symmetry is quintessential (gauge invariance etc.)

m Symmetry without dynamics don’t bring you this far

¬ At Lep dynamics was SM, unknowns were MH (αs(MZ), . . .)MH (αs(MZ), . . .)MH (αs(MZ), . . .)

­ At LHC (post SM) unknowns are SM-deviations,dynamics?

* BSM is a choice. Something more model independent?

Ê An unknown form factor?

Ë A decomposition where dynamics is controlled by dim = 4amplitudes (with known analytical properties) and deviations(with a direct link to UV completions) are Wilson coefficients?

m It is for posterity to judge (for me deviations need a SM basis)

12/36

No NP yet?A study of SM-deviations: here the reference process is

H→ γγH→ γγH→ γγ

3 κκκ -approach: write the amplitude as

AAA = ∑i=t,b,W

κi Ai +κc∑

i=t,b,W

κi Ai +κc∑

i=t,b,W

κi Ai +κc

A tA tA t being the SM t -loop etc. The contact term (which is the LOSMEFT) is given by κcκcκc . Furthermore

κi = 1+∆κiκi = 1+∆κiκi = 1+∆κi i 6= ci 6= ci 6= c

13/36

3 For the sake of simplicity assume

κb = κW = 1κb = κW = 1κb = κW = 1(

κexpW = 0.95+0.14

−0.13ATLAS 0.96+35−16CMS

)and compute

. κγκγκγ 7→7→7→ R = Γ(

κt , κc)/ΓSM−1 [%]R = Γ

(κt , κc

)/ΓSM−1 [%]R = Γ

(κt , κc

)/ΓSM−1 [%]

In LO SMEFT κcκcκc is non-zero and κt = 1κt = 1κt = 1 5. You measure adeviation and you get a value for κcκcκc . However, at NLO ∆κt∆κt∆κt isnon zero and you get a degeneracy. The interpretation in termsof κ

LOcκLOcκLOc or in terms of κ

NLOcκNLOcκNLOc ,∆κ

NLOt κNLOt κNLOt could be rather different.

5Certainly true in the linear realization

14/36

−0.4 −0.2 0 +0.2 +0.4

−0.4

−0.2

0

+0.2

+0.4

κNLO

cκNLO

cκNLO

c

∆κN

LO

t∆

κNLO

t∆

κNLO

t

Γ/ΓSM − 1 = +5 %Γ/ΓSM − 1 = +5 %Γ/ΓSM − 1 = +5 %

Γ/ΓSM − 1 = −5 %Γ/ΓSM − 1 = −5 %Γ/ΓSM − 1 = −5 %

Γ/ΓSM − 1 = +10 %Γ/ΓSM − 1 = +10 %Γ/ΓSM − 1 = +10 %

Γ/ΓSM − 1 = −10 %Γ/ΓSM − 1 = −10 %Γ/ΓSM − 1 = −10 %

H → γγH → γγH → γγ κb = κW = 1κb = κW = 1κb = κW = 1

••• •••

κLO

cκLO

cκLO

c

κLO

cκLO

cκLO

c

••• •••

κLO

cκLO

cκLO

c

κLO

cκLO

cκLO

c

Γ/ΓSMΓ/ΓSMΓ/ΓSM iso-lines

Γ(∆κt , κc

)=(42.29−23.87∆κt−13.01κc

) GFα2

128√

2π3 M3HΓ

(∆κt , κc

)=(42.29−23.87∆κt−13.01κc

) GFα2

128√

2π3 M3HΓ

(∆κt , κc

)=(42.29−23.87∆κt−13.01κc

) GFα2

128√

2π3 M3H

15/36

Fitting is not interpreting

Of course, depending on what you measure, the corresponding interpretation couldtell us that the required kappas or Wilson coeicients are too large to allow for a

meaningful interpretation in terms of a weakly coupled UV completion6

Caveat: SMEFT interpretation should include LO SMEFT and (atleast) RGE modified predictions (arXiv:1301.2588); furthermore, full one-loop

SMEFT gives you (new) logarithmic and constant terms that are not smallcompared to the one from RGE, see arXiv:1505.02646, arXiv:1505.03706

For interpretations other than weakly coupled renormalizable, seearXiv:1305.0017

EFT purist: there is no model independent EFT statement on some operatorsbeing big and other small (arXiv:1305.0017)

6Simpler theories are preferable to more complex ones because they are better testable and falsifiable

16/36

Going interpretational

AAASMEFT =g2s2

θ

8π2

[∑

i=t,b,W

κi Ai +

g6

g2s2θ

M2H

M2W

8π2 aAA

]g2s2θ

8π2

[∑

i=t,b,W

κi Ai +

g6

g2s2θ

M2H

M2W

8π2 aAA

]g2s2θ

8π2

[∑

i=t,b,W

κi Ai +

g6

g2s2θ

M2H

M2W

8π2 aAA

]3 Assumption: use arXiv:1505.03706, work in the

Einhorn-Wudka PTG scenario (arXiv:1307.0478), adoptWarsaw basis (arXiv:1008.4884)

¬ LO SMEFT: κi = 1κi = 1κi = 1 and aAAaAAaAA is scaled by 1/16π2 being LG

­ NLO PTG-SMEFT: κi 6= 1κi 6= 1κi 6= 1 but only PTG operators insertedin loops (non-factorizable terms absent), aAAaAAaAA scaled asabove

At NLO, ∆κ = g6 ρ∆κ = g6 ρ∆κ = g6 ρ and aAA = s2θ

aφ W +c2θ

aφ B +sθ cθ aφ WBaAA = s2θ

aφ W +c2θ

aφ B +sθ cθ aφ WBaAA = s2θ

aφ W +c2θ

aφ B +sθ cθ aφ WB

AAA SMEFT = ∑i=t,b,W

(1+g6 ρi) A i +gc aAA∑i=t,b,W

(1+g6 ρi) A i +gc aAA∑i=t,b,W

(1+g6 ρi) A i +gc aAA

17/36

Warsaw basis

g−16

g−16

g−16

=√

2GF Λ2

√2GF Λ

2√

2GF Λ2

gcgcgc =12

g6

g2s2θ

M2H

M2W

12

g6

g2s2θ

M2H

M2W

12

g6

g2s2θ

M2H

M2W

ρtρtρt = −12

[aφ D −2s2

θ (at φ +2aφ2)] 1

s2θ

−12

[aφ D −2s2

θ (at φ +2aφ2)] 1

s2θ

−12

[aφ D −2s2

θ (at φ +2aφ2)] 1

s2θ

ρbρbρb = −12

[aφ D +2s2

θ (ab φ−2aφ2)] 1

s2θ

−12

[aφ D +2s2

θ (ab φ−2aφ2)] 1

s2θ

−12

[aφ D +2s2

θ (ab φ−2aφ2)] 1

s2θ

ρWρWρW = −12

[aφ D −4s2

θ aφ2

] 1s2

θ

−12

[aφ D −4s2

θ aφ2

] 1s2

θ

−12

[aφ D −4s2

θ aφ2

] 1s2

θ

ΓSMEFTΓSMEFTΓSMEFT =α2 GF M3

H

32√

2π3

M4W

M4H|ASMEFT |2

α2 GF M3H

32√

2π3

M4W

M4H|ASMEFT |2

α2 GF M3H

32√

2π3

M4W

M4H|ASMEFT |2 ΓSMΓSMΓSM = ΓSMEFT

∣∣∣∆κi=0 ,κc=0

ΓSMEFT

∣∣∣∆κi=0 ,κc=0

ΓSMEFT

∣∣∣∆κi=0 ,κc=0

Relaxing the PTG assumption introduces non-factorizable sub-amplitudes proportional to

at W ,at B ,ab W ,ab B ,aφW ,aφB ,aφWBat W ,at B ,ab W ,ab B ,aφW ,aφB ,aφWBat W ,at B ,ab W ,ab B ,aφW ,aφB ,aφWB with a mixing among aφW ,aφB ,aφWBaφW ,aφB ,aφWBaφW ,aφB ,aφWB. Meanwhile, renormalization has made

one-loop SMEFT finite, e.g. in the GFGFGF -scheme, with a residual µRµRµR -dependence

18/36

Appendix C. Dimension-Six Basis Operators for the SM22.

X3 (LG) ϕ6 and ϕ4D2 (PTG) ψ2ϕ3 (PTG)

QG fABCGAνµ GBρ

ν GCµρ Qϕ (ϕ†ϕ)3 Qeϕ (ϕ†ϕ)(lperϕ)

QG fABCGAνµ GBρ

ν GCµρ Qϕ2 (ϕ†ϕ)2(ϕ†ϕ) Quϕ (ϕ†ϕ)(qpurϕ)

QW εIJKW Iνµ W Jρ

ν WKµρ QϕD

(ϕ†Dµϕ

)⋆ (ϕ†Dµϕ

)Qdϕ (ϕ†ϕ)(qpdrϕ)

QW

εIJKW Iνµ W Jρ

ν WKµρ

X2ϕ2 (LG) ψ2Xϕ (LG) ψ2ϕ2D (PTG)

QϕG ϕ†ϕGAµνG

Aµν QeW (lpσµνer)τ

IϕW Iµν Q

(1)ϕl (ϕ†i

↔

Dµ ϕ)(lpγµlr)

QϕG ϕ†ϕ GAµνG

Aµν QeB (lpσµνer)ϕBµν Q

(3)ϕl (ϕ†i

↔

D Iµ ϕ)(lpτ

Iγµlr)

QϕW ϕ†ϕW IµνW

Iµν QuG (qpσµνTAur)ϕ G

Aµν Qϕe (ϕ†i

↔

Dµ ϕ)(epγµer)

QϕW

ϕ†ϕ W IµνW

Iµν QuW (qpσµνur)τ

I ϕW Iµν Q

(1)ϕq (ϕ†i

↔

Dµ ϕ)(qpγµqr)

QϕB ϕ†ϕBµνBµν QuB (qpσ

µνur)ϕ Bµν Q(3)ϕq (ϕ†i

↔

D Iµ ϕ)(qpτ

Iγµqr)

QϕB

ϕ†ϕ BµνBµν QdG (qpσ

µνTAdr)ϕGAµν Qϕu (ϕ†i

↔

Dµ ϕ)(upγµur)

QϕWB ϕ†τ IϕW IµνB

µν QdW (qpσµνdr)τ

IϕW Iµν Qϕd (ϕ†i

↔

Dµ ϕ)(dpγµdr)

QϕWB

ϕ†τ Iϕ W IµνB

µν QdB (qpσµνdr)ϕBµν Qϕud i(ϕ†Dµϕ)(upγ

µdr)

Table C.1: Dimension-six operators other than the four-fermion ones.

22These tables are taken from [5], by permission of the authors.

20

Einhorn, Wudka

Grz

adko

wsk

i,Is

krzy

nski

, Mis

iak,

Ros

iek

19/36

3 Demonstration strategy:

¬ Allow each Wilson coefficient to vary in the intervalI2 = [−2 , +2]I2 = [−2 , +2]I2 = [−2 , +2] (naturalness7; put Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV (conventionalpoint)

­ LO: generate points from I2I2I2 for aAAaAAaAA with uniform probabilityand calculate RLORLORLO

­ NLO: generate points from I52I52I52 for aφ D,aφ2,at φ,ab φ,aAAaφ D,aφ2,at φ,ab φ,aAAaφ D,aφ2,at φ,ab φ,aAA

with uniform probability and calculate RNLORNLORNLO

® Calculate the RRR pdf

N.B. | aAA |< 1| aAA |< 1| aAA |< 1 is equivalent to | gc aAA |< 8.610−2| gc aAA |< 8.610−2| gc aAA |< 8.610−2

7Disregarding TH bias for the sign (Sect. D of arXiv:0907.5413)

20/36

−10 −5 0 +5 +10

0

+0.05

+0.1

Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]

pdf[%

]

LO

NLO

H→ γγH→ γγH→ γγ

PTG scenario

Benc

hmar

ksc

enar

io

21/36

−10 −5 0 +5 +10

0

+0.05

+0.1

Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]

pdf[%

]

LO

NLO

H→ γγH→ γγH→ γγ

PTG scenario

Λ=

4Te

V

Λ=

4Te

V

Λ=

4Te

V

Cha

ngin

gth

ecu

toff

22/36

−10 −5 −2.5 0 +2.5 +5 +10

0

+0.05

+0.1

Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]

pdf[%

]

LO

NLO

H → γγH → γγH → γγ

PTG scenario

| ai | < 1| ai | < 1| ai | < 1

Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV

−10 −5 0 +5 +10

0

+0.05

+0.1

Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]

pdf[%

]

LO

NLO

H → γγH → γγH → γγ

PTG scenario

| ai | < 2| ai | < 2| ai | < 2

Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV

−10 −5 0 +5 +10

0

+0.05

+0.1

Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]Γ/ΓSM − 1[%]

pdf[%

]

LO

NLO

H → γγH → γγH → γγ

PTG scenario

| ai | < 3| ai | < 3| ai | < 3

Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV Changing the interval

23/36

−0.1 0 +0.1

0

+0.1

+0.2

∆κt∆κt∆κt

pdf[%

]

H→ γγH→ γγH→ γγ

PTG scenario

−0.1 0 +0.1

0

+0.1

+0.2

∆κW∆κW∆κW

pdf[%

]

H→ γγH→ γγH→ γγ

PTG scenario

From Wilson coefficients (aaa) to κκκ

Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV

−2≤ ai ≤+2−2≤ ai ≤+2−2≤ ai ≤+2

ATLAS κt = 1.28±0.35κt = 1.28±0.35κt = 1.28±0.35

CMS κt = 1.60+0.34

−0.32κt = 1.60+0.34

−0.32κt = 1.60+0.34

−0.32, 2σ2σ2σ?

other couplings < 10−2< 10−2< 10−2, MAGA notehttps://cds.cern.ch/record/2001958/files/LHCHXSWG-INT-2015−001−2.pdfIs κ

tκ

tκ

t the only window? Relax bounds compared to LO analysis (arXiv:1502.02570)?Correctly define kappas? κ

ttHt 6= κ

Hγ γ

tκttHt 6= κ

Hγ γ

tκttHt 6= κ

Hγ γ

t etc.

ATLAS: κγ = 0.90+0.16−0.14κγ = 0.90+0.16−0.14κγ = 0.90+0.16−0.14

CONFNOTES/ATLAS-CONF-2015-007/tab-08.pngCMS: κγ = 1.14+0.12

−0.13κγ = 1.14+0.12−0.13κγ = 1.14+0.12−0.13

http://arxiv.org/pdf/1412.8662.pdfthese unc. cannot be underestimated

24/36

−0.1 −0.05 0 +0.05 +0.1

0

+0.1

+0.2

∆κt∆κt∆κt

pdf[%

]

∆κZγt∆κZγt∆κZγt

∆κγγt∆κγγt∆κγγt

PTG scenario

κZγ t6=

κγγ

t

κZγ t6=

κγγ

t

κZγ t6=

κγγ

t

25/36

Conclusions:

¬ For the SMEFT, (almost) regardless of the κcκcκc , to have morethan 5%5%5% deviation (at Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV ) you have to go NLO, orunnatural8 (Wilson coefficients not O(1)O(1)O(1))

­ The LO, NLO pdfs are different, therefore interpretation isdifferent, how to reweight once your analysis was LOinterpreted? It all depends on the new central value for κ

expγκexpγκexpγ

presently ATLAS: aLOAA = +3.79+5.31

−6.06aLOAA = +3.79+5.31

−6.06aLOAA = +3.79+5.31

−6.06 CMS: aLOAA =−5.31+4.93

−4.55aLOAA =−5.31+4.93

−4.55aLOAA =−5.31+4.93

−4.55

naive dimensional estimate aAA ≈ 1aAA ≈ 1aAA ≈ 1

® Chi ha avuto, ha avuto, ha avuto . . . i ha dato, ha dato, ha dato . . .scurdammoce o ppassatoThose who’ve taken, taken, taken . . . Those who’ve given, given, given. . . Let’s forget about the past

8from the point of view of a weakly coupled UV completion

26/36

Other than Higgs (just one example): if we neglect LGoperators in loops, the following result holds for vacuum

polarization:

Π(dim=6)AA (0) =−8

c2θ

s2θ

aφ D Π(dim=4)AA (0)

One of the key ingredients in computing precision(pseudo-)observables is αQED at the mass of the Z. Define

α(MZ)

=α(0)

1−∆α(5)(MZ)−∆αt

(MZ)−∆α

ααst(MZ)

∆α(5) (MZ

)= ∆αl

(MZ)+∆α

(5)had

(MZ)

27/36

∆α(5)had

(MZ)

= 0.0280398104 × ∆αl

(MZ)

= 0.0314976104 × ∆αt

(MZ)

≈ [−0.62 ,−0.55]104 × ∆α

ααst(MZ)≈ [−0.114 ,−0.095]

The SMEFT effect is equivalent to replace

∆αl(MZ)+∆αt

(MZ)

→ (1−κα)[∆αl

(MZ)+∆αt

(MZ)]

κα = 8g6

c2θ

s2θ

aφ D = 0.188aφ D at Λ = 3 TeV

*| κα ∆αt | > ∆αl

| κα ∆αt | ≈ | ∆αααst |

28/36

Part III

Mostly POs

29/36

s2s2s2

s1s1s1

Z , γZ , γZ , γ

Z , γZ , γZ , γ

ExpansionADR(s1,s2;...)

(s1−sZ)(s2−sZ) = ADR(sZ ,sZ ;...)(s1−sZ)(s2−sZ) +

A(2)DR(sZ ,s2;...)

s1−sZ

. . . +A restDR (s1,s2; . . .)

ASR(s1;...)s1−sZ

= ASR(sZ ;...)s1−sZ

+A restSR (s1; . . .)

ANR(. . .)

+ (Z→ γ)

remember LEP

σpeakf = 12π

ΓeΓf

M2ZΓ2

Z

Γ(H → ZZ)Γ(H → ZZ)Γ(H → ZZ) etc.

Γ(H → f fγ) etc.

the difficult part (e.g. VBF)

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interpretation: POs a la LEPhttps://indico.cern.ch/event/373667/ arXiv:1504.04018

H→ γγ (γZ)H→ γγ (γZ)H→ γγ (γZ) 7→7→7→ ργγ(Z)H

p1 ·p2 gµν −pµ

2 pν

1MH

ργγ(Z)H

p1 ·p2 gµν −pµ

2 pν

1MH

ργγ(Z)H

p1 ·p2 gµν −pµ

2 pν

1MH

H→ VVH→ VVH→ VV 7→7→7→ ρVH

(MH gµν +

GVL

MHpµ

2 pν

1

)ρ

VH

(MH gµν +

GVL

MHpµ

2 pν

1

)ρ

VH

(MH gµν +

GVL

MHpµ

2 pν

1

)H→ bbH→ bbH→ bb 7→7→7→ ρ

bH uvρbH uvρbH uv

etc.

a middle way languagewolf, goat, and cabbage

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POs (container) at LHC: summary table

¬ external layer (similar to LEP σpeakfσpeakfσpeakf )

ΓVV AZZFB N4 l

off etcΓVV AZZFB N4 l

off etcΓVV AZZFB N4 l

off etc

­ intermediate layer (similar to LEP geV AgeV AgeV A )

ρVH G

VL ρ

γγ

H , ργZH ρ

fHρ

VH G

VL ρ

γγ

H , ργZH ρ

fHρ

VH G

VL ρ

γγ

H , ργZH ρ

fH

® internal layer: the kappas

κγγ

f κγγ

W κγγ NF

i etcκγγ

f κγγ

W κγγ NF

i etcκγγ

f κγγ

W κγγ NF

i etc

¯ innermost layer: Wilson coeff. or non-SM parameters inBSM (e.g. α,β ,Msb etc. in THDMs)

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How to inlude EWPD? The case of the WWW mass

Working in the α -scheme we can predict MW. The solution is

M2W

M2Z

= c2θ +

α

πRe(

1− 12

g6 aφ D

)∆

(4)B (MW)

+ ∑gen

[(1+4g6 a(3)

φ l

)∆

(4)l (MW)+

(1+4g6a(3)

φq

)∆

(4)q (MW)

]+ g6

[∆

(6)B (MW)+ ∑

gen

(∆

(6)l (MW)+∆

(6)q (MW)

)]

The expansion can be improved when working within the SM(dim = 4). Any equation that gives dim = 6 corrections to the SMresult will always be understood as

O = OSM∣∣∣imp

+α

πg6 O(6)

in order to match the TOPAZ0/Zfitter SM results whe g6 → 0.33/36

THE Example

SWW =g2

16π2 ΣWW

SZZ =g2

16π2 c2θ

(Σ33−2s2

θ Σ3Q−s4θ ΠAA s

)ΣF = ΣWW(0)−Re Σ33(M2

Z)+Re Σ3Q(M2Z)

Define ρ−1 = 1+ GF2√

2π2 ΣFρ−1 = 1+ GF2√

2π2 ΣFρ−1 = 1+ GF2√

2π2 ΣF = 0.99490 , ∆ρ∆ρ∆ρ contains (PTG only):

aφD, aΦ2, aφ f, a(1,3)φ f , f = l, u, daφD, aΦ2, aφ f, a(1,3)φ f , f = l, u, daφD, aΦ2, aφ f, a(1,3)φ f , f = l, u, d

Leading term (don’t use it for precision) is

∆ρ = M2t

[κρ ∆ρ

(4) +g6 ∑i

Fi ai

]ai = aφD, aφ t, a(1,3)

φq

κρ = 1+g6

11

[76

aφD +28(a(1)φq +a(3)

φq )−20aφ t

]34/36

How to inlude EWPD?

¬ By reducing (a priori) the number of dim = 6dim = 6dim = 6 operators

­ By imposing penalty functions ω on the global fit, that isfunctions defining an ω -penalized LS estimator for a set ofglobal penalty parameters (perhaps using merit functionsand the homotopy method)

® Using a Bayesian approach, with a flat prior for theparameters. One κ at the time? Fit first to the EWPD andthen to H observables? Combination of both?

Of course, all EWPO must be rewritten in the κ -EFT approach. . .

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Thank you for your attention

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Backup Slides(moving backward)

Return

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How/what NLO?

3 Start with Warsaw basis, full set, write down Lagrangianand Feynman rules

3 Normalize the quadratic part of the Lagrangian and paydue attention to the FP ghost sector

3 Compute (all) self-energies (up to one Odim=6Odim=6Odim=6 insertion),write down counterterms, make self-energies UV finite

3 Compute the set of processes you like/want (don’t forgetnon-SM topologies), mix Wilson coefficients to make themUV finite, check closure under renormalization

3 Perform finite renormalization, selecting a scheme (betterthe GFGFGF -scheme), introduce wave-function factors, get theanswer

3 Start making approximations now (if you like), e.g.neglecting operators etc.

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How/what NLO? (cont.)

3 Transform the answer in terms of κκκ -shifted SMsub-amplitudes and non SM factorizable sub-amplitudes

3 Derive κκκ -parameters in terms of Wilson coefficients

3 Write Pseudo-Observables in terms of κκκ -parameters

3 Decide about strategy for including EWPD

3 Claim you invented the whole procedure 222

NLO is like biking, you learn it when you are a kid

Fade Out Round House Fast Pace 222 Coked Pistol

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SMEFT evolution

LO A SMEFT = A SM +aiA SMEFT = A SM +aiA SMEFT = A SM +ai , where ai ∈ V6ai ∈ V6ai ∈ V6 and V6V6V6 is the set ofdim = 6dim = 6dim = 6 Wilson coefficients

RGE ai → Zij(L)ajai → Zij(L)ajai → Zij(L)aj , where L = ln(Λ/MH)L = ln(Λ/MH)L = ln(Λ/MH) and j ∈ H6 ⊂ V6j ∈ H6 ⊂ V6j ∈ H6 ⊂ V6

NLO A SMEFT = A SM +Ak (L , const) akA SMEFT = A SM +Ak (L , const) akA SMEFT = A SM +Ak (L , const) ak , where k ∈ S6k ∈ S6k ∈ S6 andH6 ⊂ S6 ⊂ V6H6 ⊂ S6 ⊂ V6H6 ⊂ S6 ⊂ V6

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How/what NLO? FAQ

3 Are there some pieces that contain the dominant NLOeffects

3 It depends on the TH bias:

À For EFT purists there is no model independent EFTstatement on some operators being big and other small

Á Remember, logarithms are not large, constants matter too

3 which could be easily incorporated in othercalculations/tools?

3 Well, Well, Well, its certainly a compelling provocativeexciting to think about idea

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How/what NLO? FAQ

3 NLO SMEFT availability? From arXiv:1505.03706 Return

¬ Counterterms (SM fields and parameters): all

­ Mixing: those entries related to H→ γγ,Zγ,ZZ,WWH→ γγ,Zγ,ZZ,WWH→ γγ,Zγ,ZZ,WW

® Self-energies, complete and at p2 = 0p2 = 0p2 = 0: all

¯ Amplitudes, sub-amplitudes (both SM andnon-factorizable, full PTG + LG scenario)

¶ H → γγH → γγH → γγ · H → ZγH → ZγH → Zγ ¸ H → ZZ,WWH → ZZ,WWH → ZZ,WW9 ¹ H → ffH → ffH → ff (thelatter available, although not public)

° EWPD, MWMWMW, TTT -parameter; Z→ ffZ→ ffZ→ ff available, although notpublic.

9Green’s functions in well-defined kinematic limit, i.e. residue of the poles after extracting the parts which are 1Preducible

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Backup Plots(the role of κWκWκW)

43/36

−3 0 +3

−3

0

+3

aAAaAAaAA

at

φa

tφ

at

φ

Γ/ΓSM − 1Γ/ΓSM − 1Γ/ΓSM − 1

[−10 , −9] %[−10 , −9] %[−10 , −9] %

[+10 , +9] %[+10 , +9] %[+10 , +9] %

Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV κW = κb = 0κW = κb = 0κW = κb = 0

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!

−3 0 +3

−3

0

+3

aAAaAAaAA

at

φa

tφ

at

φ

Γ/ΓSM − 1Γ/ΓSM − 1Γ/ΓSM − 1

[−10 , −9] %[−10 , −9] %[−10 , −9] %

[+10 , +9] %[+10 , +9] %[+10 , +9] %

Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV κW = 0.95 κb = 0κW = 0.95 κb = 0κW = 0.95 κb = 0

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−3 0 +3

−3

0

+3

aAAaAAaAA

at

φa

tφ

at

φ

Γ/ΓSM − 1Γ/ΓSM − 1Γ/ΓSM − 1

[−10 , −9] %[−10 , −9] %[−10 , −9] %

[+10 , +9] %[+10 , +9] %[+10 , +9] %

Λ = 3 TeVΛ = 3 TeVΛ = 3 TeV κW = 1.05 κb = 0κW = 1.05 κb = 0κW = 1.05 κb = 0

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