This document discusses using effective field theory (EFT) to search for new physics beyond the Standard Model. It introduces the kappa framework, which parametrizes deviations from the Standard Model in a non-gauge invariant way. EFT provides an alternative approach that is compatible with quantum field theory. The document outlines how to build a basis of higher-dimensional operators in the Standard Model EFT and perform calculations to NLO while renormalizing ultraviolet divergences. Open questions remain about the validity range, possible high-energy completions, and combining bottom-up and top-down EFT approaches.

1. NLO Higgs Effective Field Theory and κ Framework
arXiv:1505.03706. M. Ghezzi, R.G. , G.Passarino, S.Uccirati
Raquel G´omez-Ambrosio
HiggsTools @Universit`a & INFN @Torino & CMS @CERN
Planck Conference 2015, Ionannina, GR
May 27, 2015

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Outline
Introduction:
The Search of BSM physics
The kappa framework
Effective Field theory
What is Effective Field Theory
Why choose Effective Field Theory
Hands on EFT:
HowTo
SM EFT
Summary & Open Questions

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Introduction:
The Search of BSM physics
Introduction: Why are we here?
The Standard Model Today:
Higgs-like particle with JCP = 0++,
MH = 125.09 ± 0.24 GeV found in 2012.
No new physics found yet:
Neutrino masses
Dark Matter
Graviton

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Introduction:
The Search of BSM physics
CMS results: “Constraints on the Higgs boson width . . . ” (1405.3455)
MH can be extracted from the peak, for ΓH we have to look at the off-shell region

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Introduction:
The kappa framework
Search for BSM physics: The kappa framework
First proposed by the LHC-HXSWG in 1209.0040
Idea: Introduce ad-hoc deviations for some SM observables
(Higgs’ σ’s and Γ’s)
Provide a series of benchmark parametrizations in order to
test deviations against experimental data

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Introduction:
The kappa framework
Search for BSM physics: The kappa framework
The simplest example:
Gamma-Gamma state originated from Gluon-Gluon fusion
(σ · BR)(gg→H→γγ) = (σggH )SM
· (BRHγγ)SM
·
κ2
g κ2
γ
κ2
H
κ2
g =
σggH
(σggH )SM
, κ2
γ =
Γγγ
(Γγγ)SM
, κ2
H =
ΓH
(ΓH )SM

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Introduction:
The kappa framework
Search for BSM physics: The kappa framework
Disadvantages . . .
Ad-hoc deviations are not compatible with QFT
(they break gauge invariance and unitarity)
The κ’s don’t have a direct physical interpretation
With the available amount of data and theoretical
predictions, no deviation has been found
Need to go to higher orders in perturbation theory
(NLO for signal AND background processes)
Need higher experimental accuracy
Or, maybe, need another approach . . .

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Introduction:
The kappa framework
Search for BSM physics: The kappa framework
Disadvantages . . .
Ad-hoc deviations are not compatible with QFT
(they break gauge invariance and unitarity)
The κ’s don’t have a direct physical interpretation
With the available amount of data and theoretical
predictions, no deviation has been found
Need to go to higher orders in perturbation theory
(NLO for signal AND background processes)
Need higher experimental accuracy
Or, maybe, need another approach . . .

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Introduction:
The kappa framework
Search for BSM physics: The kappa framework
Disadvantages . . .
Ad-hoc deviations are not compatible with QFT
(they break gauge invariance and unitarity)
The κ’s don’t have a direct physical interpretation
With the available amount of data and theoretical
predictions, no deviation has been found
Need to go to higher orders in perturbation theory
(NLO for signal AND background processes)
Need higher experimental accuracy
Or, maybe, need another approach . . .

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Introduction:
The kappa framework
Search for BSM physics: The kappa framework
Disadvantages . . .
Ad-hoc deviations are not compatible with QFT
(they break gauge invariance and unitarity)
The κ’s don’t have a direct physical interpretation
With the available amount of data and theoretical
predictions, no deviation has been found
Need to go to higher orders in perturbation theory
(NLO for signal AND background processes)
Need higher experimental accuracy
Or, maybe, need another approach . . .

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Effective Field theory
What is Effective Field Theory
Alternative strategy: Effective field theory
Definition:
An effective field theory (EFT) is a field theory, designed to reproduce the
behaviour of some underlying physical theory in some limited regime. It focuses
on the degrees of freedom relevant to that regime, simplifying the problem but
letting aside some physics.

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Effective Field theory
Why choose Effective Field Theory
Why choose EFT?
Historically legitimated: Large scale physics, as we know it, is made of EFTs: fluid
dynamics, solid state and condensed matter physics.
Newton’s theory of gravity is an effective low-energy theory of general relativity,
which is itself some low-energy effective theory of a quantum theory of gravity.

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Hands on EFT:
HowTo
Things the apprentice has to know: Top-down Vs. Bottom-up approach
In the Top-down approach: (model dependent)
Start from a complete high energy theory.
Integrate out heavy fields: eiSeff [φ](µ) = DΦ eiSUV [φ,Φ](µ)
Use RG-flow to study the resulting theory in its low-energy regime
In the Bottom-up approach: (model independent)
Start from a low-energy known theory (the SM).
Add operators consistent with the symmetries
(recall Wilson: only dim > 4 makes sense)
Calculate (pseudo)-observables and compare with experiments

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Hands on EFT:
HowTo
SM EFT (bottom-up approach)
Leff = LSM
dim 4
+
i
ai Oi
Λ2
dim 6
+ . . .
higher dim. operators
ai can be Wilson coefficients or the κ’s introduced previously
For current experimental thresholds, dim 6 operators are enough.
Using eqs. of motion and gauge symmetries, one can build a 59-operator
basis (for one generation of particles! for three → 2499 operators)

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Hands on EFT:
SM EFT
SM EFT
Some Assumptions
There is one Higgs doublet with a linear representation
The EFT does not add new light degrees of freedom
The heavy degrees of freedom of the EFT decouple
The heavy degrees of freedom do not mix with the Higgs doublet
The UV completion is weakly coupled and renormalizable
Also: Restrict to dim 6 and NLO
As a consequence: 5 TeV < Λ < 7 TeV

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Hands on EFT:
SM EFT
Hands on: The Strategy to follow
1. Start from the SM L
2. Add all possible dim 6 operators
(“a basis”)
3. Redefine fields and parameters to
recover the wanted expression:
L = LSM + Ldim 6
4. Write down Feynman rules and
renormalize this L
5. Do your calculations!

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Hands on EFT:
SM EFT
Redefiniton and Renormalization
Include wave-function factors and counterterms
Φ = ZΦΦren Mi = Zi Mi,ren Zi = 1 +
g2
16π2
dZ
(4)
i + g6dZ
(6)
i
counterterms
Dyson-resum the propagators. For example, Higgs self energy:
SHH =
g2
16π2
ΣHH =
g2
16π2
Σ
(4)
HH + Σ
(6)
HH
Add counterterms. Remove UV divergencies.
Use Ward-Slavnov-Taylor identities to check consistency.
OBS: Counterterms remove O(4) UV divergencies, not the O(6):
Wilson coefficients mix!
Finite renormalization instead of RG flow: Connect with pseudo observables

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Hands on EFT:
SM EFT
The message
We want to do precision physics: We are looking for tiny deviations from the SM,
and the energy scale of our theory is relatively narrow
× Therefore cannot use the renormalization group equations
We want to isolate the O(6) contributions to the amplitudes from the O(4)
If you manage to do this, it should be easy to measure SM deviations in Higgs
production and decays (through the couplings)

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Hands on EFT:
SM EFT
Step 5. Calculations: Higgs decays and all that
Example: H → γγ
The amplitude for the process is:
Aµν
HAA = THAA
pµ
2 pν
1 − p1 · p2δµν
M2
H
were we find T to be,
THAA = i
g3
16π2
(T
(4)
HAA + g6 T
(6),b
HAA
UV divergent
) + igg6 T
(6),a
HAA
UV finite
(1)
Need to renormalize T
(6),b
HAA → mixing of Wilson coefficients ( ≡ κ)
Find a final expression for the amplitudes in terms of κ’s and subamplitudes!

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Hands on EFT:
SM EFT
NLO Higgs EFT
More details on the paper: 1505.03706

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Summary & Open Questions
Summary & Open Questions
We present an effective field theory approach to BSM physics
EFT is a very good choice,regarding model independence.
It also goes beyond LO:
Starting from the kappa-framework, propose an NLO extension for it.
We can identify the deviations inside the amplitudes
and therefore compare with LHC data
Open Questions
What is the range of validity of the effective theory?
Which kind of theory we find at (even) higher energies?
How to combine the bottom-up and top-down approaches of EFT?
Do SM deviations have a SM basis?
What happens to PDFs & theoretical uncertainties?

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Summary & Open Questions

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Backup
Backup

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Backup
List of relevant dim 6 operators

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Backup
Realtion with the Wilson coefficients