[PPT] - Gauge invariant Barr-Zee type contributions to fermionic EDMs in the PowerPoint Presentation

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北原鉄平 (Teppei Kitahara) University of Tokyo

Collaborators : 阿部智広 (Tomohiro Abe) [KEK], 久野純治 (Junji Hisano) [Nagoya Univ.],

飛岡幸作 (Kohsaku Tobioka) [Kavli IPMU] Based on T. Abe, J. Hisano, T.K and K. Tobioka, JHEP 1401 (2014) 106, arXiv:1311.4704

BURI 2014 February 13, 2014, Toyama University

Gauge invariant Barr-Zee type contributions to fermionic EDMs in the two-Higgs doublet models

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BURI 2014

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BURI 2014

Basis of the Universe

with ary Ideas 2014 ー

2 http://seriable.com/nbc-network/revolution-nbc-network/

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Our Revolutionary Idea

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We derive improved Barr-Zee type contributions → first make BZ contributions gauge invariant

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We become able to evaluate theoretical value of EDM more correctly conceptually and numerically

Our Revolutionary Idea

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Introduction

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Electric Dipole Moment (EDM)

What is EDM?

L = − eQ` 4m` a`¯ `µ⌫`F µ⌫ − i 2d`¯ `µ⌫5`F µ⌫

Non-relativistic Hamiltonian

H = −µ` ~ B · ˆ ~ s − d` ~ E · ˆ ~ s

Magnetic dipole moment term Electric dipole moment term Spin-Electric field int. Spin-Magnetic field int. C-even P-even T-even C-even P-odd T-odd

EDM : dl ✔ Non-zero EDM violates T and CP symmetry

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Why does one focus on EDM?

✔ Observation of EDM implies NEW CP violation source ✔ Experiments of EDM are very precise. We can seek TeV scale physics indirectly.

Since our Universe experiences Baryogenesis, new CP violation source is needed in somewhere. New Physics!! SM EDM is too small to observe...

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Electric Dipole Moment (EDM)

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The Model

✓ 2HDMs with softly-broken Z2 symmetry ✓ Higgs potential

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Z2 transformation relative phase of VEVS

Im m2

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λ5e2iφ

✔ CP violation phase

✔redefine Higgs field ✔stationary condition

λ5e2iφ

ne CP violation

physical phase

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The Yukawa interaction in this model is classified into the 4 types

SM + extra Doublet SUSY type Lepton Specific

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Flipped type Z2 transformation

One can avoid the dangerous FCNC problem

The Model

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EDM in 2HDM

In 2HDMs, leading EDM contributions come from not one-loop

but two-loop diagrams

[S. M. Barr and A. Zee, Phys. Rev. Lett. 65, 21(1990)]

αα2 Yukawa2

Barr-Zee diagram

Because electron Yukawa is too small,

φ = H1, H2, H3 φ = H1, H2, H3

e− e− e− e−

∼ αα2 (4π)2 me v2 ∼ 10−27 [e cm]

∼ 1 (4π)2 m3

e

v4 ∼ 10−36 [cm]

< 10−40 [e cm]

cf. SM (4 loop)

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Gauge invariance of Barr-Zee diagram

Actually, The Barr-Zee contribution to EDM is not gauge invariant

[R. G. Leigh, S. Paban, R. M. Xu, Nucl. Phys. B 352, 45(1991)]

Barr-Zee diagram

φ = H1, H2, H3

e− e−

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Gauge invariance of Barr-Zee diagram

Actually, The Barr-Zee contribution to EDM is not gauge invariant

[R. G. Leigh, S. Paban, R. M. Xu, Nucl. Phys. B 352, 45(1991)]

Barr-Zee diagram

φ = H1, H2, H3

e− e−

pµ

1Mµ 6= 0

pµ

1

The previous works did not care about the gauge invariance of BZ contributions

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Gauge invariance of Barr-Zee diagram

Actually, The Barr-Zee contribution to EDM is not gauge invariant

[R. G. Leigh, S. Paban, R. M. Xu, Nucl. Phys. B 352, 45(1991)]

Barr-Zee diagram

φ = H1, H2, H3

The previous works did not care about the gauge invariance of BZ contributions e− e−

We improved BZ contribution to be gauge invariant one

pµ

1Mµ 6= 0

pµ

1

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Pinch Technique

Pinch technique is general method which can decompose

gauge fixing parameter ξ independent subamplitudes

Diagrammatic representation

T1(t, ξ) = ˆ T1(t) − f(t, ξ)

T2(t, mi, ξ) = ˆ T2(t, mi) + f(t, ξ) − h(t, mi, ξ)

T3(t, s, mi, ξ) = ˆ T3(t, s, mi) + h(t, mi, ξ)

[John M. Cornwall, Phys. Rev. D26, 6(1982)]

T2(t, mi, ξ)

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Pinch Technique

Pinch technique is general method which can decompose

gauge fixing parameter ξ independent subamplitudes

Diagrammatic representation

T1(t, ξ) = ˆ T1(t) − f(t, ξ)

T2(t, mi, ξ) = ˆ T2(t, mi) + f(t, ξ) − h(t, mi, ξ)

T3(t, s, mi, ξ) = ˆ T3(t, s, mi) + h(t, mi, ξ)

[John M. Cornwall, Phys. Rev. D26, 6(1982)]

T2(t, mi, ξ)

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pinch!

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Pinch Technique

Pinch technique is general method which can decompose

gauge fixing parameter ξ independent subamplitudes

Diagrammatic representation

T1(t, ξ) = ˆ T1(t) − f(t, ξ)

T2(t, mi, ξ) = ˆ T2(t, mi) + f(t, ξ) − h(t, mi, ξ)

T3(t, s, mi, ξ) = ˆ T3(t, s, mi) + h(t, mi, ξ)

pinch!

[John M. Cornwall, Phys. Rev. D26, 6(1982)]

f(t, ξ) T2(t, mi, ξ)

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Pinch :

q k p p-q k+q k+p

kµγµ = (6 k+ 6 p m) (6 p m) ⊃

gauge 3 point vertex

r gauge propagator

gauge-fermion-fermion vertex

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Pinch Technique

[John M. Cornwall, Phys. Rev. D26, 6(1982)]

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Pinch :

q k p p-q k+q k+p

kµγµ = (6 k+ 6 p m) (6 p m) ⊃

gauge 3 point vertex

r gauge propagator

gauge-fermion-fermion vertex

= i ✓ i 6 k+ 6 p m ◆−1 (6 p m)

second term cancel out

¯ u 6 p = ¯ um

Equation of mortion

Pinch Technique

[John M. Cornwall, Phys. Rev. D26, 6(1982)] 12

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Pinch :

q k p p-q k+q k+p

kµγµ = (6 k+ 6 p m) (6 p m) ⊃

gauge 3 point vertex

r gauge propagator

gauge-fermion-fermion vertex

= i ✓ i 6 k+ 6 p m ◆−1 (6 p m)

second term cancel out

¯ u 6 p = ¯ um

Equation of mortion

First term “pinch” fermion propagator

pinch!

Pinch Technique

[John M. Cornwall, Phys. Rev. D26, 6(1982)] 12

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Pinch :

Pinch

In order to obtain gauge invariant BZ type contribution, we

should sum these diagrams

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Pinch Technique

pinch!

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calculate all gauge invariant Barr-Zee

contributions

check the gauge invariance analytically
get analytical formula of improved BZ

contributions All charged particle run

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O(100) two-loop diagrams!

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Results of electron EDM

@YbF molecule @ThO molecule Recently ACME Collaboration got new upper bound by ThO molecule! current bound

15 [ACME Collaboration, Science 17 Vol.343 no.6168 (2014)]

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Parameter as a benchmark

tan β = O(10), λ1 = λ3 = λ4 = λ5 sin 2φ = 0.5, λ2 = 0.25

✔ Vacuum stability is safe ✔ EW precisions are safe @ M > 200GeV ✔ Lightest neutral scalar mass ∼ 126GeV ✔ not unnatural parameter region

tan β = v2 v1

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tanβ = 10

Gauge-inv. Barr-Zee vs. Ordinary Barr-Zee

We found that the difference between gauge-invariant and

rdinary (= not gauge invariant) Barr-Zee contribution to

electron EDM is about 5 - 8 %

Type II

Difference between de(gauge-inv. BZ) and de(ordinary BZ)

17 [T.Abe, J.Hisano, T.K, K.Tobioka, (2013)]

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tanβ = 10

We found that the difference between gauge-invariant and

rdinary (= not gauge invariant) Barr-Zee contribution to

electron EDM is about 5 - 8 %

Type II

Difference between de(gauge-inv. BZ) and de(ordinary BZ)

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This is not so big improvement from the numerical point of

view. However, we would like to emphasize that our result

is now gauge invariant, which must be satisfied when we discuss observables.

[T.Abe, J.Hisano, T.K, K.Tobioka, (2013)]

Gauge-inv. Barr-Zee vs. Ordinary Barr-Zee

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Electron EDM

Contour Plot of

eEDM [e・cm] Current exp. bound (90% CL) Future Prospects

YbF, WN Fr ThO [T.Abe, J.Hisano, T.K, K.Tobioka, (2013)]

Heavy Higgs mass

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Electron EDM

eEDM vs Heavy Higgs scale
Type X ~ Type II, Type Y ~ Type I

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Current exp. bound (90% CL) Future Prospects

YbF, WN Fr ThO

tanβ = 10

Future Prospects Exclude

tanβ = 10

[T.Abe, J.Hisano, T.K, K.Tobioka, (2013)]

Heavy Higgs mass Heavy Higgs mass

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eEDM vs Heavy Higgs scale
Type X ~ Type II, Type Y ~ Type I

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Current exp. bound (90% CL) Future Prospects

YbF, WN Fr ThO

tanβ = 10

Future Prospects Exclude

tanβ = 10

Heavy Higgs mass Heavy Higgs mass

~ 20 TeV ~ 2 TeV

Electron EDM

[T.Abe, J.Hisano, T.K, K.Tobioka, (2013)]

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Summary

2HDMs with Z2 symmetry have one new phase, and

are constrained by the electron/neutron EDM

An analytical formulae of full gauge-inv.

Barr Zee contribution is first derived by Using Pinch Technique

In Type II, X, Y 2HDMs, the future

expeRIments of electron/neutron EDM are expected to reach O(10) TeV new Heavy scalars

Type II tanβ = 10 ~ 20 TeV

200 400 600 800 1000 2 4 6 8 10 MH+ @GeVD D @%D

tanβ = 10 Type II

New New +

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Teppei Kitahara arXiv:1311.4704

Discussions

21 [G. C. Dorsch, S. J. Huber and J. M. No, JHEP 1310, 029 (2013)]

EWBG prefers low tanβ region

Is there a tension between EWBG and EDM?

EDM vs Electroweak Baryogenesis at the same

parameter region

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Backup

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neutron EDM

Future prospects can improve to 2 orders

f magnitude

current bound @Ultra cold neutron

The neutron EDM is obtained by QCD sum rule as follows

[J. Hisano, J. Y. Lee, N. Nagata, Phys. Rev. D85, 114044(2012)]

dn = 0.79dd − 0.20du + e(0.59dC

d + 0.30dC u )

(QCD sum rules)

assume PQ mechanism

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Neutron EDM

Contour Plot of

nEDM [e・cm] Current exp. bound (90% CL) Future Prospects Lightest Higgs Mass ~ 126 GeV Parameter Two Heavy Higgs Mass ~ MH+

λ5 sin 2φ = 0.5

[T.Abe, J.Hisano, T.K, K.Tobioka, (2013)]

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Neutron EDM

Lightest Higgs Mass = 126 GeV Parameter

In Type I and Type X, neutron EDM is too small
Type Y ~ Type II
nEDM vs Heavy Higgs scale

Two Heavy Higgs Mass ~ MH+

λ5 sin 2φ = 0.5

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tanβ = 10

Exclude Future Prospects Current exp. bound (90% CL)

[T.Abe, J.Hisano, T.K, K.Tobioka, (2013)]

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Neutron EDM

Lightest Higgs Mass = 126 GeV Parameter

In Type I and Type X, neutron EDM is too small
Type Y ~ Type II
nEDM vs Heavy Higgs scale

Two Heavy Higgs Mass ~ MH+

λ5 sin 2φ = 0.5

tanβ = 10

Exclude Future Prospects Current exp. bound (90% CL)

~ 5 TeV

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Barr-Zee diagram

In Barr-Zee diagrams calculation, we separated

them into “HVV effective couplings” part and

ther part

Generalize At first, we consider “HVV effective couplings”

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Tensor structure of effective coupling

ff-shell vector
n-shell photon

general tensor structure gauge symmetry (Ward-Takahashi identity) pµ

1Γµν = 0

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Tensor structure of effective coupling

ff-shell vector
n-shell photon

If HVV effective vertex is gauge invariant

pµ

1Mµ = 0

pµ

1Γµν = 0

Barr-Zee diagram is gauge invariant

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Effective coupling -W loop-

We explicitly calculate W loop contribution, and

result is as follows

ff-shell vector
n-shell photon

All set of W loop

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Effective coupling -W loop-

We explicitly calculate W loop contribution, and

result is as follows

ff-shell vector
n-shell photon

gauge invariant term gauge non-invariant term All set of W loop

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Effective coupling -W loop-

We explicitly calculate W loop contribution, and

result is as follows

ff-shell vector
n-shell photon

All set of W loop gauge non-invariant term

These terms drop when all external lines are on-shell. However, in this situation (external lines are off-shell), these term do not drop.

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Effective coupling -W loop-

ff-shell vector
n-shell photon

We can transform pinch term into effective coupling form

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Effective coupling -W loop-

ff-shell vector
n-shell photon

We can transform pinch term into effective coupling form

second term cancel out

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Effective coupling -W loop-

ff-shell vector
n-shell photon

We can transform pinch term into effective coupling form

last two terms does not contribute to dipole operator

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Effective coupling -W loop-

ff-shell vector
n-shell photon

We can transform pinch term into effective coupling form

We checked analytically and found that these sum become gauge invariant

last two terms does not contribute to dipole operator

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Some demands to parameter region

Vacuum Stability of Higgs potential
Electroweak Precessions
Higgs mass = 126 GeV
Before we calculate EDMs, we should consider the

following some demands to parameter region

demands to parameter region

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Vacuum stability

[N. G. Deshpande and E. Ma, Phys. Rev. D 18 (1978) 2574] [A. W. El Kaffas, W. Khater, O. M. Ogreid and P. Osland, Nucl. Phys. B 775, 45 (2007)]

The stability condition of the Higgs potential at tree-

level (one demand that EW vacuum becomes global minimum)

Tree-level stability condition

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Electroweak precision / ρ (T) parameter

The custodial SU(2) symmetry is broken in the CP

violation 2HDM at the tree level, and ρ(T) parameter might deviate from 1 at one-loop level

ρ ≡ m2

W

m2

Z cos2 θw

= 1 + ∆ρ ' 1 + αEMT

Theρ(T) parameter depends on the quadratic mass splitting among particles in the same isospin multiplet. If mass splitting is small, or heavy Higgs mass scale is large, theρ(T) parameter is small.

αEMT = ΠW W (0) m2

W

− ΠZZ(0) m2

Z

∝ m2

H± − m2 H

m2

W

[A. Pomarol and R. Vega, Nucl. Phys. B 413, 3 (1994)]

Texp. = 0.05 ± 0.12 (1 σ)

experimental bound @ not mH >> VEVs

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@BSM effect

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Higgs mass = 126 GeV

2HDM with CP violation have physical scalar

bosons as 3 neutral Higgs and 1 charged Higgs

3 neutral Higgs 1 charged Higgs 2 neutral Higgs and 1 charged Higgs are same mass scale M

where

mh1 ' v sin2 β p λ2

1 neutral Higgs is light

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λ2 Plot

demand h1 = 126.0 GeV

200 400 600 800 1000 0.245 0.250 0.255 0.260 0.265 0.270 M @GeVD Λ2

Higgs mass = 126 GeV

mh1 ' v sin2 β p λ2

We require the mass of lightest neutral scalar to be 126 GeV, then parameter λ2 is uniquely determined

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QCD correction from Four Fermi Operator

[J. Hisano, K. Tsumura, M. J. S. Yang, Phys. Lett. B713, 473(2012)]

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gg -> bb H/A, H/A -> tautau search Type II

[CMS PAS HIG 12 050]

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Flavor constraint

[F. Mahmoudi and O. Stal, Phys.Rev. D81 (2010) 035016]

Type I Type II Type Y Type X

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electron EDM tanB = 3

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electron EDM tanB = 50

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M_H+ = 150 GeV M_H+ = 380 GeV

p

2

p

4 p 4 p 2

4. ¥10-27
2. ¥10-27
2. ¥10-27
4. ¥10-27

f dêe @cmD

p

2

p

4 p 4 p 2

2. ¥10-27
1. ¥10-27
1. ¥10-27
2. ¥10-27

f dêe @cmD

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Higgs mass h1 h2 h3

200 400 600 800 1000 200 400 600 800 1000 MH+ @GeVD mh @GeVD

Neutral Higgs Mass

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Top couplings h1 h2 h3 h1 h2 h3

Vector coupling Axial coupling

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 MH+ @GeVD »gtthV»êSM

200 400 600 800 1000 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 MH+ @GeVD »gtthA»êSM

絶対値絶対値

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Bottom couplings h1 h2 h3 h1 h2 h3

Vector coupling Axial coupling

200 400 600 800 1000 1 2 3 4 5 6 7 MH+ @GeVD »gbbhV»êSM 200 400 600 800 1000 1 2 3 4 5 6 7 MH+ @GeVD »gbbhA»êSM

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WWh couplings h1 h2 h3

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 MH+ @GeVD »gWWh»êSM

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1.5
1.0
0.5

0.0 0.5 1.0 MH+ @GeVD geehAêSM * gtthVêSM 200 400 600 800 1000

0.4
0.2

0.0 0.2 0.4 MH+ @GeVD geehVêSM * gtthAêSM

Some couplings h1 h2 h3 h1 h2 h3

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Electroweak precision / ρ (T) parameter

Texp. = 0.05 ± 0.12 (1 σ)

experimental bound

0.02
0.04
0.04