Effective Field Theories in the Quest for BSM Physics

Effective Field Theories

Effective Field Theories in the Quest for BSM Physics

Raquel Gomez AmbrosioUniversita & INFN @Torino & CMS @CERN

First Annual Meeting of ITN HiggsTools

April 17, 2015

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Outline

Introduction:The Search of BSM physics

Effective Field theoryWhat is Effective Field TheoryWhy is Effective Field Theory useful

Hands on EFT: HowToSM EFTFrom the UV theory to the EFTThe Covariant Derivative Expansion

Summary & Open Questions

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Introduction:

The Search of BSM physics

Introduction

The Standard Model Today

I (Higgs) particle with JCP = 0++, MH = 125.09± 0.24 GeV found in 2012.

I No new physics found in the experiment (deviations, if any, are very tiny)

To-Do List

I Neutrino masses

I Dark Matter

I Graviton

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Introduction:


Search for BSM physics: Shifting the couplings

I Search deviations in the Higgs couplings, aka “kappa framework”

I Problem 1: All measurements seem consistent with no deviations:

1. Need NNLO and beyond (signal + background )

2. Need more precise experiments (more resolution)

I Problem 2: The κ’s don’t have a direct physical interpretation

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Introduction:








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Introduction:








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Introduction:








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Effective Field theory

What is Effective Field Theory


Definition:

An Effective field theory (EFT) is a field theory, designed to reproduce the

behavour of some underlying (in general, unknown) physical theory in some

limited regime. It focuses on the degrees of freedom relevant to that regime,

simplifying the problem though letting aside some important physics.

Don’t confuse it with “Emotional Freedom Technique” . . .

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What is EFT. Some examples

I Landau-Ginzburg theory for superconductivity, the (non)-linear sigma

model for (anti)-ferromagnetism, Fermi theory for β-decay . . .

I Example: Interactions between nucleons and pions. Promote the chiral

transformation for the nucleon field to a non linear one:

ψ(x) → e2iγ5−→τ ·−→φ (x)ψ(x)

ψ(x) Nucleon field

φ(x) Pion field.

I Caveat: The same effective phenomena can be achieved through many

different original scenarios. You need educated guesses, good luck, and

reliable experimental results in order to find the “real” one.

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Why is Effective Field Theory useful

Why EFT?

I Large scale physics, as we know it, is made of effective field theories: fluid

dynamics, solid state physics, condensed matter physics . . . Why shouldn’t

particle physics be one too?

I Newton’s theory of gravity is an effective low-energy theory of general rela-

tivity, which is itself some low-energy effective theory of a quantum theory

of gravity.

I As a hint: it doesn’t seem reasonable to use the renormalization-group flow

ad infinitum to reach arbitrarily short distance scales (Wilsonian interpreta-

tion of the SM)

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Why EFT?






of gravity.



tion of the SM)

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Why EFT?






of gravity.



tion of the SM)

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Why EFT(s), today?

I There is some hope that the SM might be an effective low-energy theory of

a higher unified theory. Maybe string theory.

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Why EFT(s), today?

I There is some hope that the SM might be an effective low-energy theory of

a higher unified theory. Maybe string theory.

I Maybe not . . .

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Hands on EFT: HowTo

Two basic things the EFT apprentice has to know:

1) Top-down Vs. Bottom-up approach

I In the Top-down approach,

I Start from a complete high energy theory

I Study its behaviour in its infrared regime

I Assuming heavy modes decouple, calculations become simpler.

I In the Bottom-up approach,

I Start from an low-energy known theory (the SM).

I Study its high energy behaviour.

I Add operators consistent with the symmetries.

I Match the unknown with experimental observations.

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Hands on EFT: HowTo

Two basic things the EFT apprentice has to know:

2) Relevant, Marginal and Irrelevant operators

I Introduced by K. Wilson, in the context of renormalization

I Check mass dimension Vs. spacetime dimensions

[O] < d relevant

[O] = d marginal

[O] > d irrelevant

I Relevant and marginal operators, like the ones in LSM , become increasingly largeat higher energies

I Irrelevant operators, decrease at higher energies, and are not “influential”

L = LSM + Ldim=5 + Ldim=6 + · · · =

= (∼ Λ0) + (∼ Λ−1) + (∼ Λ−2) + . . .

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Hands on EFT: HowTo

SM EFT

SM EFT (bottom-up approach)

Leff = LSM︸︷︷︸dim 4

+∑

i

ci Oi

Λ2︸︷︷︸dim 6

+ . . .︸︷︷︸higher dim. operators

I ci are Wilson coefficients

I One can build 80 dim-6 operators (compatible with SU(2)× SU(3)× U(1)

and lepton/baryon conservation)

I Eqs. of motion, reduce this set to a 59-operator basis

(for one generation of particles! for three → 2499 operators)

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Hands on EFT: HowTo

SM EFT

Hands on EFT: HowTo

From the UV theory to the EFT (Top-down approach)

I Integrate out the heavy fields of the UV theory

e iSeff [φ](µ) =

∫DΦ e iSUV [φ,Φ](µ)

I Where µ is the scale where the UV theory matches the EFT (µ ∼ m),don’t mistake it for ΛUV ,ΛPlanck .

I Use saddle point approximation:

Seff ≈ S [ΦC ]︸︷︷︸tree-level

+i

2Tr log

− δ2S

δΦ2

∣∣∣∣∣ΦC

︸︷︷︸

one-loop

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Hands on EFT: HowTo

SM EFT

Diagrammatical interpretation

I Clear form the path integral point of view: Only Φ is “dynamical” (φ is fixed)

∫DΦ e iSUV [φ,Φ](µ) 6=

∫DΦDφ e iSUV [φ,Φ](µ)

I Recall the background field method: φ→ φ+ φ

L(φ)→ L(φi + φi ) = L(φi ) + L1︸︷︷︸=0

+1

2

δ2Lδφiφj

∣∣∣∣∣φ=φ

φiφj︸︷︷︸1-loop

+ . . .

Φ

Φ

ΦΦ

Φ

φ

φ

φ

φ

φ

φ

φ

φ

φ

φ

φ

φ

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Hands on EFT: HowTo

The Covariant Derivative Expansion

Covariant Derivative Expansion

I First proposed by M. Gaillard (1986), see also Murayama (1412.1837)

I Elegant method to evaluate the one loop effective action:

Seff ≈ S [ΦC ]︸︷︷︸tree-level

+i

2Tr log

− δ2S

δΦ2

∣∣∣∣∣ΦC

︸︷︷︸

one-loop

I Bonus: Respect gauge invariance at each step!

I Bonus 2: Covariant expansion of Seff leads directly to the Oi ’s

Underlying idea

I Dµ = ∂µ − iAµ → DµO = [D,O]

I Make a expansion on D2, instead of the usual (∂2 −m2)

I All terms in the expansion are proportional to D2 or [D,O] (i.e. gaugeinvariant/covariant)

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Hands on EFT: HowTo


CDE. A simple example: Scalar L

L(Φ, φ) =[Φ†B(φ) + B†(φ)Φ

]+ Φ†

((iD)2 −m2 − U(φ)

)Φ + . . .

I Find Φc : δS

δΦ

∣∣∣∣Φ=Φc

= 0 → Φc =−B(φ)

(iD)2 −m2 − U(φ)

I Expand:

Φc =

[1

1− 1m2 (−D2 − U)

]1

m2B ∼

1

m2B +

1

m2(−D2 − U)

1

m2B +

+1

m2(−D2 − U)

1

m2(−D2 − U)

1

m2B + . . .

I And replace:

Leff ,tree = B†1

m2B + B†

1

m2(−D2 − U)

1

m2︸︷︷︸dim-6 operator

B + higher dim. ops

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Hands on EFT: HowTo


Once you know Φc you can evaluate:

For the previous example:

i

2Tr log

δ2S

δφ2

∣∣∣∣∣ΦC

= (D)2 −m2 − U(φ)

Or, in the general case:

∆Seff ,1loop ∝ i cs Tr log[D2 + m2 + U(φc (x)

] cs = 1/2 Real scalar

cs = 1 Complex scalar

cs = −1/2 Fermion

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Hands on EFT: HowTo


And, after some simple algebra, also Leff ,1-loop . . .

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I We presented a method to write down an operative EFT from a given UV model

I Integrating out the heavy fields and expanding the result in a CDE we find the

Leff for such models

I After that, we can do physics with the effective model, and match its observables

with standard model ones by means of the RG flow equations (next time!)

Open Questions

I What is the range of validity of the effective theory?

I Is there a smart way to combine the bottom-up and top-down approaches of

EFT?

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Backup

Backup

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Backup

1-Loop Leff

∆Seff ,1loop ∝ i cs Tr log[D2 + m2 + U(φc (x)

]∆Seff ,1loop = i cs

∫d4x

∫d4q

(2π)4tr(

e iq·x log[D2 + m2 + U(φc (x))

]e−iq·x

)

∆Seff ,1loop = i cs

∫d4x

∫d4q

(2π)4tr

(e

iD· ∂∂q log

[−(iDµ − qµ)2 + m2 + U(φc (x))

]e−iD· ∂

∂q

)

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Backup

eiDµ

∂∂q (iDµ − qµ)e

−iDµ∂∂q =

∞∑n=0

1

n!

[P ·

∂

∂q

]n

(iDµ)−∞∑

n=0

1

n!

[P ·

∂

∂q

]n

qµ = . . .

= −(qµ + Gµν)

where,

Gµν =∞∑

n=0

n + 1

(n + 2)![iDα1 , [iDα2 , [. . . , [Pαn , [Dµ,Dν ]]]]]

∂n

∂qα1∂qα2 . . . ∂qαn

And,

eiDµ

∂∂q (U)e

−iDµ∂∂q = · · · = U

where,

U =∞∑

n=0

1

n![iDα1 , [iDα2 , [. . . , [Pαn ,U]]]]

∂n

∂qα1∂qα2 . . . ∂qαn

Science

Effective Field Theories in the Quest for BSM Physics