Gauge invariant Barr-Zee type contributions to fermionic EDMs in the two-Higgs doublet models 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
BURI 2014 Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 2 /21
B asis of the U niverse with ary I deas 2014 ー http://seriable.com/nbc-network/revolution-nbc-network/ BURI 2014 Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 2 /21
Our Revolutionary Idea Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 3 /21
We derive improved Barr-Zee type contributions → first make BZ contributions gauge invariant + We become able to evaluate theoretical value of EDM more correctly conceptually and numerically Our Revolutionary Idea Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 3 /21
Introduction Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 4 /21
Spin-Electric field int. T-odd Electric dipole moment term Magnetic dipole moment term C-even P-even T-even C-even P-odd Non-relativistic Hamiltonian What is EDM? Spin-Magnetic field int. Electric Dipole Moment (EDM) L = − eQ ` `� µ ⌫ ` F µ ⌫ − i a ` ¯ 2 d ` ¯ `� µ ⌫ � 5 ` F µ ⌫ 4 m ` EDM : d l H = − µ ` ~ s − d ` ~ B · ˆ E · ˆ ~ ~ s ✔ Non-zero EDM violates T and CP symmetry Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 5 /21
Since our Universe experiences Baryogenesis, Why does one focus on EDM? new CP violation source is needed in somewhere. New Physics!! SM EDM is too small to observe... Electric Dipole Moment (EDM) ✔ Observation of EDM implies NEW CP violation source ✔ Experiments of EDM are very precise. We can seek TeV scale physics indirectly . Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 6 /21
physical phase one CP violation Z2 transformation relative phase of VEVS The Model ✓ 2HDMs with softly-broken Z2 symmetry ✓ Higgs potential ✔ CP violation phase ✔ redefine Higgs field ✔ stationary condition λ 5 e 2 i φ λ 5 e 2 i φ Im m 2 3 Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 7 /21
• The Yukawa interaction in this model is classified into the 4 types SM + extra Doublet SUSY type Lepton Specific Flipped type Z2 transformation One can avoid the dangerous FCNC problem The Model Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 8 /21
cf. SM (4 loop) • 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)] Barr-Zee diagram Because electron Yukawa is too small, EDM in 2HDM φ = H 1 , H 2 , H 3 m 3 1 v 4 ∼ 10 − 36 [cm] e ∼ (4 π ) 2 e − e − v 2 ∼ 10 − 27 [e cm] ∼ αα 2 m e (4 π ) 2 φ = H 1 , H 2 , H 3 e − e − αα 2 � Yukawa 2 < 10 − 40 [e cm] Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 9 /21
• 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 Gauge invariance of Barr-Zee diagram φ = H 1 , H 2 , H 3 e − e − Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 10 /21
gauge invariance of BZ contributions • 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 The previous works did not care about the Gauge invariance of Barr-Zee diagram p µ 1 p µ 1 M µ 6 = 0 φ = H 1 , H 2 , H 3 e − e − Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 10 /21
gauge invariance of BZ contributions Barr-Zee diagram to be gauge invariant one We improved BZ contribution The previous works did not care about the [R. G. Leigh, S. Paban, R. M. Xu, Nucl. Phys. B 352, 45(1991)] Actually, The Barr-Zee contribution to EDM is not gauge invariant • Gauge invariance of Barr-Zee diagram p µ 1 p µ 1 M µ 6 = 0 φ = H 1 , H 2 , H 3 e − e − Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 10 /21
• Pinch technique is general method which can decompose gauge fixing parameter ξ independent subamplitudes • Diagrammatic representation [John M. Cornwall, Phys. Rev. D26, 6(1982)] Pinch Technique T 1 ( t, ξ ) = ˆ T 1 ( t ) − f ( t, ξ ) T 2 ( t, m i , ξ ) = ˆ T 2 ( t, m i ) + f ( t, ξ ) − h ( t, m i , ξ ) T 3 ( t, s, m i , ξ ) = ˆ T 3 ( t, s, m i ) + h ( t, m i , ξ ) T 2 ( t, m i , ξ ) Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 11 /21
[John M. Cornwall, Phys. Rev. D26, 6(1982)] • Pinch technique is general method which can decompose gauge fixing parameter ξ independent subamplitudes • Diagrammatic representation Pinch Technique T 1 ( t, ξ ) = ˆ T 1 ( t ) − f ( t, ξ ) T 2 ( t, m i , ξ ) = ˆ T 2 ( t, m i ) + f ( t, ξ ) − h ( t, m i , ξ ) T 3 ( t, s, m i , ξ ) = ˆ T 3 ( t, s, m i ) + h ( t, m i , ξ ) T 2 ( t, m i , ξ ) pinch! Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 11 /21
[John M. Cornwall, Phys. Rev. D26, 6(1982)] • Pinch technique is general method which can decompose gauge fixing parameter ξ independent subamplitudes • Diagrammatic representation Pinch Technique T 1 ( t, ξ ) = ˆ T 1 ( t ) − f ( t, ξ ) T 2 ( t, m i , ξ ) = ˆ T 2 ( t, m i ) + f ( t, ξ ) − h ( t, m i , ξ ) T 3 ( t, s, m i , ξ ) = ˆ T 3 ( t, s, m i ) + h ( t, m i , ξ ) f ( t, ξ ) T 2 ( t, m i , ξ ) pinch! Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 11 /21
[John M. Cornwall, Phys. Rev. D26, 6(1982)] gauge 3 point vertex gauge-fermion-fermion vertex or gauge propagator Pinch Technique p k q k+p ● Pinch : k+q p-q k µ γ µ = ( 6 k + 6 p � m ) � ( 6 p � m ) ⊃ Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 12 /21
gauge-fermion-fermion vertex gauge 3 point vertex Equation of mortion second term cancel out [John M. Cornwall, Phys. Rev. D26, 6(1982)] or gauge propagator Pinch Technique p k q k+p ● Pinch : k+q p-q k µ γ µ = ( 6 k + 6 p � m ) � ( 6 p � m ) ⊃ ◆ − 1 ✓ i u 6 p = ¯ ¯ = i � ( 6 p � m ) um 6 k + 6 p � m Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 12 /21
second term cancel out gauge 3 point vertex Equation of mortion First term “pinch” fermion propagator [John M. Cornwall, Phys. Rev. D26, 6(1982)] gauge-fermion-fermion vertex or gauge propagator Pinch Technique p k q k+p ● Pinch : k+q p-q k µ γ µ = ( 6 k + 6 p � m ) � ( 6 p � m ) ⊃ ◆ − 1 ✓ i u 6 p = ¯ ¯ = i � ( 6 p � m ) um 6 k + 6 p � m pinch! Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 12 /21
should sum these diagrams Pinch Pinch Technique pinch! ● Pinch : ● In order to obtain gauge invariant BZ type contribution, we + Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 13 /21
• calculate all gauge invariant Barr-Zee contributions • check the gauge invariance analytically • get analytical formula of improved BZ contributions All charged particle run O(100) two-loop diagrams! Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 14 /21
@YbF molecule @ThO molecule Recently ACME Collaboration got new upper bound by ThO molecule! current bound [ACME Collaboration, Science 17 Vol.343 no.6168 (2014)] Results of electron EDM Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 15 /21
Parameter as a benchmark tan β = O (10) , λ 1 = λ 3 = λ 4 = λ 5 sin 2 φ = 0 . 5 , tan β = v 2 λ 2 = 0 . 25 v 1 ✔ Vacuum stability is safe ✔ EW precisions are safe @ M > 200GeV ✔ Lightest neutral scalar mass ∼ 126GeV ✔ not unnatural parameter region Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 16 /21
We found that the difference between gauge-invariant and ordinary (= not gauge invariant) Barr-Zee contribution to electron EDM is about 5 - 8 % [T.Abe, J.Hisano, T.K , K.Tobioka, (2013)] Gauge-inv. Barr-Zee vs. Ordinary Barr-Zee Di ff erence between d e (gauge-inv. BZ) and d e (ordinary BZ) tan β = 10 Type II Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 17 /21
We found that the difference between gauge-invariant and ordinary (= not gauge invariant) Barr-Zee contribution to electron EDM is about 5 - 8 % 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 Di ff erence between d e (gauge-inv. BZ) and d e (ordinary BZ) tan β = 10 Type II Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 17 /21
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 Electron EDM ● Contour Plot of
Heavy Higgs mass YbF, WN Heavy Higgs mass [T.Abe, J.Hisano, T.K , K.Tobioka, (2013)] Exclude Future Prospects ThO Fr Future Prospects Current exp. bound (90% CL) Electron EDM ● eEDM vs Heavy Higgs scale tan β = 10 tan β = 10 ● Type X ~ Type II , Type Y ~ Type I Feb 13, 2014 BURI 2014 Teppei KITAHARA -Univ. of Tokyo 19 /21
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