Effective Theories for Precision Higgs and Flavor Physics · Effective Theories for Precision Higgs and Flavor Physics Aneesh Manohar University of California, San Diego 15 Apr 2019

Effective Theories for Precision Higgs and FlavorPhysics

Aneesh Manohar

University of California, San Diego

15 Apr 2019HEFT 2019

Aneesh Manohar 15.04.2019 1 / 23

Experimental Input

An SU(2)×U(1) gauge theory spontaneously broken to U(1)em ata scale v = 246 GeV.

A particle has been seen with a mass Mh ∼ 126 GeV;0+ quantum numbers favored

No evidence for any new particles in the few hundred GeV range.

Many theoretical ideas on how the electroweak symmetry isspontaneously broken. Need precision tests of the EW breakingmechanism and whether it is due to a single scalar doublet.

Precision measurements in flavor physics place strong constraints onnew physics. Have to resort to minimal flavor violation for BSM modelsto survive.


Standard Model – A fundamental scalar doublet

A field H which transforms as 21/2

H =

[φ+

φ0

]=

1√2

[iφ1 + φ2φ4 − iφ3

]Expanding about the minimum,

H =1√2

[iφ1 + φ2

v + h − iφ3

]h is the neutral scalar with mass

m2h = 2λv2

of order the electroweak scale. φ1,2,3 are the eaten Goldstone bosonsthat give mass to the W and Z .


φ = (φ1, φ2, φ3, φ4), 〈φ〉 = (0,0,0, v)

L =12∂µφ · ∂µφ−

λ

4

(φ · φ− v2

)2

v

φ0

Experiments can only probe tiny deviations from 〈φ〉. Can’t study thefull structure of V (φ).


Scalar ManifoldMThe SM scalar manifold is flat, with fields (φ1, φ2, φ3, φ4) ∈ R4. Thisfollows from the kinetic term

L =12

gij(φ) ∂µφi∂µφj gij = δij

The potential V (φ) gives mass to radial oscillations. The angularoscillations are massless Goldstone bosons.

Can have generalizations (e.g. dynamical symmetry breaking) wherethe radial and angular variables are not related as in the SM.

The angular degrees arise from breaking SU(2)× U(1)→ U(1), e.g.in technicolor theories. The h has to be added in somehow, since ithas been observed.

In such theories, (φ1, φ2, φ3) and h not related as in the SM. Thescalars can live on a curved manifold.


SM in Polar CoordinatesKinetic energy

L =12

(∂µh)2 +12

(v + h)2(∂µn)2,

where

φ = (v + h) n n · n = 1

Just like spherical polar coordinates in R3

ds2 = dr2 + r2(dθ2 + sin2 θdφ2) = dr2 + r2dΩ2

But if we modify the radial depedence: r is not related to θ, φ as in R3

ds2 = dr2 + f (r)2r2dΩ2

then space is no longer flat. Similarly,

L =12

(∂µh)2 +12

v2F (h)2(∂µn)2, F (h)SM= 1 +

hv,


Extensions of the SM

Some dynamics at a scale f v .

Rather than worry about details about the high energy theory, try andcharacterize what aspects of the theory can be measured at low (i.e.LHC and below) energies.

SMEFT and HEFT

SMEFT is a special case of HEFT where we assume the existence ofH.


Testing EW symmetry breaking mechanism

Experiments can only measure the properties near 〈φ〉.

The S-matrix is invariant under field redefinitions.The local properties of the Lagrangian which are coordinateinvariant are the curvature and the eigenvalues of the massmatrix.Mass eigenvalues are known to be mh ∼ 125 GeV and 0,0,0(Higgs mechanism).

Invariant objects are:

r4 = Rππππ(φ0) r2h = Rπhπh(φ0)

R is the Riemann curvature tensor. [Note that there are three π(Goldstone boson) directions.]Alonso, Jenkins, AM


Experimental Consequences

Observables depend on r4,2h

analogous to S,T ,U for the Higgs sector.In composite Higgs models

r ∼ 1f 2

f is the scale of new physics.

Aim is to determine whether r = 0 (i.e. SM) or not.

S parameter (Λ = 4πf ):

∆S =1

12πr4 log

(Λ2

M2Z

).


Experimental Consequences

The scattering amplitude at high energies of longitudinal W -bosonsWL depends on the curvature:

A (WLWL →WLWL) = −4λ+ (s + t)r4 , K (Xπ,Yπ)

A (WLWL → hh) = 2λ− 2s r2h . K (Xπ,Yh)

The scale of new physics governing the mass of these resonances isΛ ∼ 4π/

√r

Unitarity is restored by these higher resonances.

In composite Higgs models, the sign is fixed; r4,2h ≥ 0

Aneesh Manohar 15.04.2019 10 / 23

SMEFTAn effective theory with the same field content as the SM — a field Hthat transforms as 21/2.

L = LD≤4 + LD=6 + . . .

D = 5 only has |∆L| = 2 terms.

Electroweak symmetry is broken by minimizing V , and expandingabout the non-trivial vacuum.

6 classes of operators B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek

X 3 : f ABCGAνµ GBρ

ν GCµρ H6 : (H†H)3

H4D2 :(

H†DµH)∗ (

H†DµH)

ψ2H3 : (H†H)(lper H)

X 2H2 : H†H GAµνGAµν ψ2XH : (lpσµνer )τ IHW I

µν

ψ2H2D : (H†i←→D µH)(epγ

µer ) ψ4 : (epγµer )(esγµet )

Aneesh Manohar 15.04.2019 11 / 23

L = LD≤4 + ciOi + . . . [ci ] =1

mass2 ci →ci

Λ2

Λ is determined from experiment after you see deviations from the SM.

53 CP-even and 23 CP-odd operators for one generation (total 76)1350 CP-even and 1149 CP-odd operators for three generations(total 2499)59 operators in the original table — some are real and somecomplex.

SMEFT gives a model independent way of testing for deviations fromthe SM.

Aneesh Manohar 15.04.2019 12 / 23

SMEFT

One-loop RGE computed Alonso, Jenkins, AM, Trott

Operator mixing: can test MFV. The SM respects MFV bydefinition.Holomorphy in the one-loop anomalous dimensions.Alonso, Jenkins, AM

Relation between the amplitude method and EFTs. Henning, Melia

Experimental obervables computed to one loop in SMEFTstarting with Hartmann and Trott for h→ γγ

Calculational packages developedSMEFTsim: Brivio, Jiang, Trott

DsixTools: Celis, Fuentes-Martin, Vicente, Virto

Aneesh Manohar 15.04.2019 13 / 23

Below the EW scale

Fermi theory, WEFT, LEFT, . . .

Integrate out t , h, W , Z to get a low-energy effective theory.Integrating out t breaks EW symmetry.Can do this all at once, since log mt/mW are not large.

Compared with SMEFT:Gauge group now SU(3)× U(1)em.H not a field in the Lagrangian.Q → uL,dL and L→ eL, νL.∆B = ∆L = 0 magnetic moment operators at dimension five, e.g.

Oeγ = eLpσµνeRr Fµν

Magnetic moment operators lead to g − 2 for e and µ, as well as EDM,b → sγ, etc.

Aneesh Manohar 15.04.2019 14 / 23

Below the EW scale

Classified operators and worked out one-loop RGE Jenkins, AM,

Stoffer.70 hermitian dimension 5 and 3631 hermitian dimension 6operators for ng = 3. Lots of flavor channels.Pieces of this have already been done in the context of flavorphysics, with anomalous dimensions known in some cases to 4loops. Czakon, Haisch, Misiak for b → sγ

Allows one to connect high energy LHC experiments withlow-energy precision measurements in a consistent frameworkLogs are not small

Aneesh Manohar 15.04.2019 15 / 23

Below the EW scale

SM→ LEFT : expansion inp

MW

SMEFT→ LEFT : expansion inpΛ,

vΛ,

pMW

HEFT→ LEFT : expansion inpΛ,

pMW

SMEFT puts constraints on LEFT parameters from SU(2)× U(1)invariance.

Aneesh Manohar 15.04.2019 16 / 23

RK ,K ∗

Humair: Moriond 2019

RK =Γ(B → Kµ+µ−)

Γ(B → Ke+e−)= 0.745+0.090

−0.074 (stat) ± 0.036 (syst) for 1.1 < q2 < 6.0 GeV2 ,

→ 0.846+0.060−0.054 (stat) +0.016

−0.14 (syst) for 1.1 < q2 < 6.0 GeV2 , new

RK∗ =Γ(B → K∗µ+µ−)

Γ(B → K∗e+e−)=

0.66 + 0.11

− 0.07 (stat)± 0.03 (syst) for 0.045 < q2 < 1.1 GeV2 ,

0.69 + 0.11− 0.07 (stat)± 0.05 (syst) for 1.1 < q2 < 6.0 GeV2 .

LEFT:

L = c7sLσµνbR Fµν + c′7 sRσ

µνbL Fµν

+ cV ,LL` (lLγµlL)(sLγµbL) + cV ,RR

` (lRγµlR)(sRγµbR)

+ c′V ,LR` (lRγµlR)(sLγµbL) + cV ,LR

` (lLγµlL)(sRγµbR)

+ cS,RL` (lLlR)(sRbL) + c′S,RL

` (lR lL)(sLbR)

+ cS,RR` (lLlR)(sLbR) + c′S,RR∗

` (lR lL)(sRbL)

+ cT ,RR` (lLσµν lR)(sLσµνbR) + c′T ,RR∗

` (lRσµν lL)(sRσµνbL) ,

Aneesh Manohar 15.04.2019 17 / 23

Constraints

Assuming that all BSM physics is via the SMEFT, i.e. it respects theSM electroweak symmetry-breaking mechanism, leads to the relations

cS,RR` = cT ,RR

` = c′S,RR` = c′T ,RR

` = 0

Alonso, Grinstein, Camalich

Jenkins, AM, Stoffer:Worked out the tree-level matching to LEFT. There are

3631 + 70− 2499 = 1202

constraints which can be tested.

Aneesh Manohar 15.04.2019 18 / 23

Loop Corrections

To work to dimension six in LEFT, have to includeSingle insertions of dimension 6 operatorsUp to two insertions of dimension 5 operators

We can make field redefinitions to eliminate redundant operators — notthe same as using equations of motion if one works to second order.

Worked out the one-loop RGE including double insertions of dipoleoperators. Jenkins, AM, Stoffer

SMEFT holomorphy survives in a large part of the calculation, but isviolated by terms proportional to the fermion mass matrix in doubledipole insertions. cf. amplitude method

Aneesh Manohar 15.04.2019 19 / 23

Dipole Operators

Dipole operators in LEFT are dimension five — no dimension five∆B = ∆L = 0 operators in SMEFT.

Oeγ = eLpσµνeRr Fµν and quark analogs

eLσµνeR Fµν , µLσ

µνµR Fµν , eLσµνµR Fµν , sLσ

µνbR Fµν

Active experimental programElectron g − 2Muon g − 2Electron EDMµ→ eγNEDM (from quark operators)b → sγ

Aneesh Manohar 15.04.2019 20 / 23

Dipole Operators

In SMEFT, need H field for SU(2)× U(1) invariance

(lpσµνer )τ IHW Iµν , (lpσµνer )HBµν → v eLσ

µνeR Fµν

SMEFT dipole effects are v/Λ suppressed in LEFT, and act likedimension six,

1Λ→ v

Λ2

Enhaced dipole effects would indicate HEFT 6= SMEFT.

Aneesh Manohar 15.04.2019 21 / 23

EFT and light particles.Actually, EFTs can be used even if there are very weakly coupled lightparticles (e.g. axions).

− g2

2M2W

+1f 2a

q2

q2 −m2a

= − 2v2

[1 +

v2

2f 2a

q2/m2a

q2/m2a − 1

]v = 246 GeV, fa ∼ 2× 1012 GeV, v/fa ∼ 10−10, ma ∼ 2µeV. Need∣∣∣∣ q2

m2a− 1∣∣∣∣ ∼ v2

f 2a∼ 10−20 =⇒ ∆q ∼ 10−25 GeV

EFT valid unless one is doing very specialized experiments (i.e. axiondetection experiments) with precise control. For the LHC, the EFT canbe used.

Aneesh Manohar 15.04.2019 22 / 23

Conclusions

Can experimentally measure the curvature of the scalar fieldmanifold by looking at high energy WW scattering and testwhether it is flat (SM) or not.Flavor physics can be used to test SMEFT constraints onlow-energy parameters to determine if the LEFT arose via SMEFT.Determine whether EW symmetry breaking is due to a complexscalar doublet as in the Higgs mechanism.Model-independent way of testing deviations from the SM withoutassumptions other than unitarity, locality and causality.

Aneesh Manohar 15.04.2019 23 / 23

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Effective Theories for Precision Higgs and Flavor Physics · Effective Theories for Precision Higgs and Flavor Physics Aneesh Manohar University of California, San Diego 15 Apr 2019