Electroweak Baryogenesis and Sphaleron at LHC

Eibun Senaha (National Taiwan U) Jan. 11, 2017@U of Tokyo based on [1] C.-W. Chiang (Natl Taiwan U), K. Fuyuto (UMass-Amherst), E.S., 1607.07316 [PLB] [2] K. Funakubo (Saga U), K. Fuyuto, E.S., 1612.05431 Outline

• Motivation

• Electroweak baryogenesis (EWBG) in a nutshell

• EWBG with lepton ﬂavor violation

• Does a band structure affect (B+L)-changing processes?

• (B+L)-changing process in high-E collisions

• (B+L)-changing process at high-T

• Summary Introduction problems after the Higgs discovery Does the 125 GeV boson alone do the following jobs? V

v 1 t m ATLAS and CMS V WZ κ LHC Run 1 ? or

F −1

v 10 m F

- mass generation κ

b 10−2 τ

ATLAS+CMS SM Higgs boson 10−3 µ - EW symmetry breaking [M, ε] fit 68% CL 95% CL 10−4 10−1 1 10 102 Particle mass [GeV]

Figure 19: Best fit values as a function of particle mass for the combination of ATLAS and CMS data in the case of the parameterisation described in the text, with parameters defined as  m /v for the fermions, and as p m /v F · F V · V for the weak vector bosons, where v = 246 GeV is the vacuum expectation value of the Higgs field. The dashed Experiments will answer those(blue) line indicates thegrand predicted dependence on the particlequestions mass in the case of the SM Higgs boson. Thein solid the near future. (red) line indicates the best fit result to the [M, ✏] phenomenological model of Ref. [128] with the corresponding 68% and 95% CL bands.

6.3.2. Probing the lepton and quark symmetry Most importantly, thoseThe parameterisation experiments for this test is very similar to that of Section 6.3.1, which probes themay up- and down- also shed light type fermion symmetry. In this case, the free parameters are =  / , =  / , and  =   / , lq l q Vq V q qq q · q H where the latter term is positive definite, like uu. The quark couplings are mainly probed by the ggF process, the H and H bb decays, and to a lesser extent by the ttH process. The lepton couplings ! ! are probed by the H ⌧⌧ decays. The results are expected, however, to be insensitive to the relative ! on unsolved problems.sign of the couplings, because there is no sizeable lepton–quark interference in any of the relevant Higgs boson production processes and decay modes. Only the absolute value of the lq parameter is therefore considered in the fit. The results of the fit are reported in Table 19 and Fig. 22. The p-value of the compatibility between the datadark and the SM predictions matter is 79%. The likelihood scan for the lq parameter is shown in Fig. 23 If for the combination of ATLAS and CMS. Negative values for the parameter Vq are excluded by more than 4. Higgs is a window to new physics. Multi-Higgs baryogenesis45 Let us open the window!! Introduction problems after the Higgs discovery Does the 125 GeV boson alone do the following jobs? V

v 1 t m ATLAS and CMS V WZ κ LHC Run 1 ? or

F −1

v 10 m F

- mass generation κ

b 10−2 τ

ATLAS+CMS SM Higgs boson 10−3 µ - EW symmetry breaking [M, ε] fit 68% CL 95% CL 10−4 10−1 1 10 102 Particle mass [GeV]

6.3.2. Probing the lepton and quark symmetry Most importantly, thoseThe parameterisation experiments for this test is very similar to that of Section 6.3.1, which probes themay up- and down- also shed light type fermion symmetry. In this case, the free parameters are =  / , =  / , and  =   / , lq l q Vq V q qq q · q H where the latter term is positive definite, like uu. The quark couplings are mainly probed by the ggF process, the H and H bb decays, and to a lesser extent by the ttH process. The lepton couplings ! ! are probed by the H ⌧⌧ decays. The results are expected, however, to be insensitive to the relative ! on unsolved problems.sign of the couplings, because there is no sizeable lepton–quark interference in any of the relevant Higgs boson production processes and decay modes. Only the absolute value of the lq parameter is therefore considered in the fit. The results of the fit are reported in Table 19 and Fig. 22. The p-value of the compatibility between the datadark and the SM predictions matter is 79%. The likelihood scan for the lq parameter is shown in Fig. 23 If for the combination of ATLAS and CMS. Negative values for the parameter Vq are excluded by more than 4. Higgs is a window to new physics. Multi-Higgs baryogenesis45 Let us open the window!! Electroweak baryogenesis [Kuzmin, Rubakov, Shaposhnikov, PLB155,36 (‘85) ] Sakharov’s conditions

✤ B violation: anomalous (sphaleron) process (LH fermions) ✤ C violation: chiral gauge interaction

✤ CP violation: KM phase and/or other sources in beyond the SM

✤ Out of equilibrium: 1st order EW phase transition (EWPT) with expanding bubble walls

BAU can arise by the growing bubbles. EWBG in a nutshell [Kuzmin, Rubakov, Shaposhnikov, PLB155,36 (‘85) ] symmetric phase =0 H: Hubble constant h i (s) B >H

broken phase =0 h i6 (b) B H

broken phase =0 h i6 (1) (b) B

n = nL nL + nR nR =0 (1) Asymmetries arise (∵ CPV) but no BAU. B b ¯b b ¯b =0 =0 6 6 | {z } | {z } EWBG in a nutshell [Kuzmin, Rubakov, Shaposhnikov, PLB155,36 (‘85) ] symmetric phase =0 H: Hubble constant h i (s) B >H (2) broken phase =0 h i6 (1) (b) B

n = nL nL + nR nR =0 (1) Asymmetries arise (∵ CPV) but no BAU. B b ¯b b ¯b =0 =0 L 6 L R 6 R ∵ nB = nb n¯ + nb n¯ nB =0 (2) LH part changes ( sphaleron) -> BAU | {z b} | {z b }! 6 changed | {z } EWBG in a nutshell [Kuzmin, Rubakov, Shaposhnikov, PLB155,36 (‘85) ] symmetric phase =0 H: Hubble constant h i (s) B >H (2) broken phase =0 h i6 (1) (b) B

n = nL nL + nR nR =0 (1) Asymmetries arise (∵ CPV) but no BAU. B b ¯b b ¯b =0 =0 L 6 L R 6 R ∵ nB = nb n¯ + nb n¯ nB =0 (2) LH part changes ( sphaleron) -> BAU | {z b} | {z b }! 6 changed (3) If (b) H (2) broken phase =0 h i6 (1) (b) B

n = nL nL + nR nR =0 (1) Asymmetries arise (∵ CPV) but no BAU. B b ¯b b ¯b =0 =0 L 6 L R 6 R ∵ nB = nb n¯ + nb n¯ nB =0 (2) LH part changes ( sphaleron) -> BAU | {z b} | {z b }! 6 changed (3) If (b) cannot redo EWPT in lab. exp. =0 H: Hubble constant So, test Sakharov’criteria instead. h i (s) B >H (2) broken phase =0 h i6 (1) (b) B

n = nL nL + nR nR =0 (1) Asymmetries arise (∵ CPV) but no BAU. B b ¯b b ¯b =0 =0 L 6 L R 6 R ∵ nB = nb n¯ + nb n¯ nB =0 (2) LH part changes ( sphaleron) -> BAU | {z b} | {z b }! 6 changed (3) If (b)

Esph is proportional to the Higgs VEV

what we need is large Higgs VEV after the EWPT

EWPT has to be “strong” 1st order!! B-changing rate in the broken phase is

Esph

Esph is proportional to the Higgs VEV

what we need is large Higgs VEV after the EWPT

EWPT has to be “strong” 1st order!! Current status of EWBG Current status of EWBG - SM EWBG was excluded.

st ∵ No 1 -order PT for mh=125 GeV.

[Kajantie at al, PRL77,2887 (’96); Rummukainen et al, NPB532,283 (’98); Csikor et al, PRL82, 21 (’99); Aoki et al, PRD60,013001 (’99). Laine et al, NPB73,180(’99)] Current status of EWBG - SM EWBG was excluded.

st ∵ No 1 -order PT for mh=125 GeV. not satisﬁed

[Kajantie at al, PRL77,2887 (’96); Rummukainen et al, NPB532,283 (’98); Csikor et al, PRL82, 21 (’99); Aoki et al, PRD60,013001 (’99). Laine et al, NPB73,180(’99)]

- MSSM EWBG was excluded. ∵ light stop scenario is inconsistent with LHC data

[D. Curtin, P. Jaiswall, P. Meade., JHEP08(2012)005; T. Cohen, D. E. Morrissey, A. Pierce, PRD86, 013009 (2012); K. Krizka, A. Kumar, D. E. Morrissey, PRD87, 095016 (2013)] Current status of EWBG - SM EWBG was excluded.

st ∵ No 1 -order PT for mh=125 GeV. not satisﬁed

[Kajantie at al, PRL77,2887 (’96); Rummukainen et al, NPB532,283 (’98); Csikor et al, PRL82, 21 (’99); Aoki et al, PRD60,013001 (’99). Laine et al, NPB73,180(’99)]

- MSSM EWBG was excluded. ∵ light stop scenario is not satisﬁed inconsistent with LHC data

st ∵ No 1 -order PT for mh=125 GeV. not satisﬁed

[Kajantie at al, PRL77,2887 (’96); Rummukainen et al, NPB532,283 (’98); Csikor et al, PRL82, 21 (’99); Aoki et al, PRD60,013001 (’99). Laine et al, NPB73,180(’99)]

- MSSM EWBG was excluded. ∵ light stop scenario is not satisﬁed inconsistent with LHC data

[D. Curtin, P. Jaiswall, P. Meade., JHEP08(2012)005; T. Cohen, D. E. Morrissey, A. Pierce, PRD86, 013009 (2012); K. Krizka, A. Kumar, D. E. Morrissey, PRD87, 095016 (2013)]

- No enough data to exclude other models (2HDM, NMSSM etc). Anyway, collider test of EWBG will be done based on the B-preservation criteria: Current status of EWBG - SM EWBG was excluded.

st ∵ No 1 -order PT for mh=125 GeV. not satisﬁed

[Kajantie at al, PRL77,2887 (’96); Rummukainen et al, NPB532,283 (’98); Csikor et al, PRL82, 21 (’99); Aoki et al, PRD60,013001 (’99). Laine et al, NPB73,180(’99)]

- MSSM EWBG was excluded. ∵ light stop scenario is not satisﬁed inconsistent with LHC data

[D. Curtin, P. Jaiswall, P. Meade., JHEP08(2012)005; T. Cohen, D. E. Morrissey, A. Pierce, PRD86, 013009 (2012); K. Krizka, A. Kumar, D. E. Morrissey, PRD87, 095016 (2013)]

- No enough data to exclude other models (2HDM, NMSSM etc). Anyway, collider test of EWBG will be done based on the B-preservation criteria: Recent papers on EWBG

1612.04086 1611.05757

1612.00466 1612.01270

1609.09849

1611.02073

1609.07143 1601.01681 Sorry, this is incomplete list. C.-W. Chiang (Natl Taiwan U), K. Fuyuto (UMass-Amherst), E.S., 1607.07316 [PLB] Higgs decay with LFV

CMS: 1502.07400 [PLB] 2.4σexcess

Atlas: 1604.07730

What does lepton ﬂavor-violating (LFV) Higgs tell us?

2 Higgs doublets model (2HDM) is one of the simplest solutions.

- LFV comes from the off-diagonal entries of ρij. ρij ∈ ℂ ⇒ CPV - μ-τ ﬂavor violation can explain h->μτ and g-2. [Y. Omura, E.S., K.Tobe, JHEP052015028, PRD94,055019(2016)] Higgs decay with LFV

CMS: 1502.07400 [PLB] 2.4σexcess

Atlas: 1604.07730

What does lepton ﬂavor-violating (LFV) Higgs tell us?

2 Higgs doublets model (2HDM) is one of the simplest solutions.

CMS: 1502.07400 [PLB] 2.4σexcess

Atlas: 1604.07730

What does lepton ﬂavor-violating (LFV) Higgs tell us?

2 Higgs doublets model (2HDM) is one of the simplest solutions.

- LFV comes from the off-diagonal entries of ρij. ρij ∈ ℂ ⇒ CPV - μ-τ ﬂavor violation can explain h->μτ and g-2. [Y. Omura, E.S., K.Tobe, JHEP052015028, PRD94,055019(2016)] EWBG with LFV [C-W. Chiang, K.Fuyuto, E.S., arXiv:1607.07316 (PLB)] benchmark point:

- Br(h->μτ)=0.84%

- g-2 favored region

mA ≳ mH

for Re(ρτμρμτ)>0. - EWBG-viable region

vC/TC > 1.17 EWBG with LFV [C-W. Chiang, K.Fuyuto, E.S., arXiv:1607.07316 (PLB)] benchmark point:

- Br(h->μτ)=0.84%

- g-2 favored region

mA ≳ mH

for Re(ρτμρμτ)>0. - EWBG-viable region

vC/TC > 1.17 EWBG with LFV [C-W. Chiang, K.Fuyuto, E.S., arXiv:1607.07316 (PLB)] benchmark point:

- Br(h->μτ)=0.84%

- g-2 favored region

mA ≳ mH

for Re(ρτμρμτ)>0. - EWBG-viable region

vC/TC > 1.17 EWBG with LFV [C-W. Chiang, K.Fuyuto, E.S., arXiv:1607.07316 (PLB)] benchmark point:

- Br(h->μτ)=0.84%

- g-2 favored region

mA ≳ mH

for Re(ρτμρμτ)>0. - EWBG-viable region

vC/TC > 1.17

Combined: 300 GeV ≲ mH ≲ mA ≲ 450 GeV A -> ττ ATLAS-CONF-2016-085

In our scenario: ATLAS Preliminary Observed ATLAS Preliminary Observed

)[pb] H/A → ττ, 95 % CL limits Expected )[pb] H/A → ττ, 95 % CL limits Expected τ 10 ±1σ τ 10 ±1σ τ -1 τ -1 s = 13 TeV, ≤ 13.3 fb ±2σ s = 13 TeV, ≤ 13.3 fb ±2σ For BAU → gluon-gluon fusion 2015, 3.2 fb-1 (Obs.) → b-associated production 2015, 3.2 fb-1 (Obs.)

1 1

- |ρτμ|=|ρμτ|=0.1-0.6, BR(H/A BR(H/A × × - |ρττ|=0.8-0.9. σ σ

10−1 10−1 -> probed by A->ττ.

τhadτhad (Exp.) τhadτhad (Exp.) −2 −2 - Br(A->ττ) also depends 10 τlepτhad (Exp.) 10 τlepτhad (Exp.) on other ρ couplings. 200 400 600 800 1000 1200 200 400 600 800 1000 1200

(model dependent) mA [GeV] mA [GeV] (a) gluon–gluon fusion (b) b-associated production

β 80 β 80 ATLAS Preliminary Observed ATLAS Preliminary Observed

tan Expected tan Expected H/A → ττ, 95 % CL limits H/A → ττ, 95 % CL limits 70 -1 ±1σ 70 -1 ±1σ s = 13 TeV, ≤ 13.3 fb 2 s = 13 TeV, ≤ 13.3 fb 2 mod+ ± σ ± σ m , M =1TeV -1 hMSSM scenario -1 60 h SUSY 2015, 3.2 fb (Obs.) 60 2015, 3.2 fb (Obs.)

ATLAS Run1, SM Higgs 50 50 boson couplings (Obs.) 40 40

30 30

20 20

τ τ (Exp.) τ τ (Exp.) 10 had had 10 had had τlepτhad (Exp.) τlepτhad (Exp.)

200 400 600 800 1000 1200 200 400 600 800 1000 1200

mA [GeV] mA [GeV] mod+ (c) mh scenario (d) hMSSM scenario Figure 5: The observed and expected 95% CL limits on the production cross section times branching fraction of a scalar particle decaying to a ⌧⌧ pair are shown for the combination of the ⌧lep⌧had and the ⌧had⌧had channels. For comparison, the expected limits for the individual channels, ⌧lep⌧had and ⌧had⌧had, are shown as well. The production mechanism of H/A ⌧⌧ is assumed to be (a) gluon–gluon fusion or (b) b-associated production. The observed ! mod+ and expected 95% CL limits on tan as a function of mA are shown in (c) for the MSSM mh scenario and (d) for the hMSSM scenario. In the case of the ⌧had⌧had channel, the mass range under study is 300 GeV–1.2TeV. In the case of the hMSSM scenario, exclusion limits are set also in the low tan and mA = 200 GeV region and around the mass value mA = 350 GeV. The exclusion limits are compared to the ATLAS 2015 H/A ⌧⌧ search result ! of Ref. [31]. For the hMSSM scenario, the exclusion arising from the SM Higgs boson coupling measurements of Ref. [110] is also shown.

19 A -> ττ ATLAS-CONF-2016-085

In our scenario: ATLAS Preliminary Observed ATLAS Preliminary Observed

1 1

- |ρτμ|=|ρμτ|=0.1-0.6, BR(H/A BR(H/A × × - |ρττ|=0.8-0.9. σ σ

10−1 10−1 -> probed by A->ττ.

τhadτhad (Exp.) τhadτhad (Exp.) −2 −2 - Br(A->ττ) also depends 10 τlepτhad (Exp.) 10 τlepτhad (Exp.) on other ρ couplings. 200 400 600 800 1000 1200 200 400 600 800 1000 1200

(model dependent) mA [GeV] mA [GeV] (a) gluon–gluon fusion (b) b-associated production

β 80 β 80 ATLAS Preliminary Observed ATLAS Preliminary Observed

ATLAS Run1, SM Higgs 50 50 boson couplings (Obs.) 40 40

30 30

20 20

τ τ (Exp.) τ τ (Exp.) 10 had had 10 had had τlepτhad (Exp.) τlepτhad (Exp.)

200 400 600 800 1000 1200 200 400 600 800 1000 1200

19 (B+L)-changing process and a band structure B+L violation - (B+L) is violated by a chiral anomaly in EW theories. Vacuum transition (instanton) [’t Hooft, PRL37,8 (1976), PRD14,3432 (1976)]

Transition rate at ﬁnite-E

instanton-based [Ringwald, NPB330,(1990)1, Espinosa, NPB343 (1990)310] E⤴ ⟹ σ(E)⤴

- But, instanton-based calculation is not valid at E>Esph

Bounce is more appropriate (transition between the ﬁnite-E states)

[Aoyama, Goldberg, Ryzak, PRL60, 1902 (’88)] -> Reduced model. [Funakubo, Otsuki, Takenaga, Toyoda, PTP87,663(’92), PTP89,881(’93)] [H. Tye, S. Wong, PRD92,045005 (’15)] Tye-Wong’s work [H. Tye, S. Wong, PRD92,045005 (2015)] F(E)

Tye-Wong

(instanton calculus) E0≃15 TeV

F(E) = 0 for E>Esph (Tye-Wong) ∵ a band structure

Q1: Can we observe the sphaleron process at LHC? Q2: Does the band affect sphaleron process at finite-T? mainly by the difficulty in constructing the classical configuration from the high-energy two-particle state, in contrast to the previous works in which the rate is suppressed by the instanton action. mainly by the difficulty in constructing the classical configuration from the high-energy Recently the issue has been revived by Tye and Wong[6]. Theirtwo-particle approch is state, similar in contrast to to the previous works in which the rate is suppressed by ours in that the noncontracting-loop parameter is regarded as thethe dynamical instanton variable action. to reduced the field theory to a quantum-mechanical problem They arguedRecently that the the issue energy has been revived by Tye and Wong[6]. Their approch is similar to eigenstate of the system with a periodic potential becomes a Bloch wave,ours in whose that the amplitude noncontracting-loop parameter is regarded as the dynamical variable to is not suppressed in contrast to that evaluated by use of the intanton-likereduced theconfigurations. field theory to a quantum-mechanical problem They argued that the energy eigenstate of the system with a periodic potential becomes a Bloch wave, whose amplitude In this bried report, we shall point out the some problems inis the not derivation suppressed in of contrast the to that evaluated by use of the intanton-like configurations. reduced quantum-mechanical system of [6], and propose the correct methodIn this of bried reduction. report, we shall point out the some problems in the derivation of the reduced quantum-mechanical system of [6], and propose the correct method of reduction.

2 The reduced model 2 The reduced model

2.1 classical mechanics 2.1 classical mechanics As discussed in [9], any static configuration of finite-energy doubletAs discussed scalar field in [9],Φ any(x) static configuration of finite-energy doublet scalar field Φ(x) 2 3 defines a map from S2 spanned by the spatial coordinate (θ, φ)todefinesS3 = a mapSU(2) from whichS spanned by the spatial coordinate (θ, φ)toS = SU(2) which 0 characterize0 the field at r = as a unitary transformation of the vacuum . A one- characterize the field at r = as a unitary transformation of the vacuum . A one- ∞ ! 1 " ∞ parameter! family1 " of such configurations connecting vacua, which cannot be contracted to parameter family of such configurations connecting vacua, which cannotapoint,isconstructedbyrealizingthefamilyofmapsasamapfrom be contracted to S3 to S3. Among apoint,isconstructedbyrealizingthefamilyofmapsasamapfromsuchS one-parameter3 to S3. Among families, that of the highest symmetry is expected to form the least such one-parameter families, that of the highest symmetry is expectedenergy to set form of configurations. the least The largest energy configurations along the parameter will energy set of configurations. The largestReduced energy configurations model alongbe a the saddle-point parameter configuration will with one negative mode. [Aoyama, Goldberg, Ryzak, PRL60, 1902 (1988)] The theory we concern is the SU(2) gauge-Higgs model with one scalar doublet, whose be a saddle-point configuration with[Funakubo, one negative Otsuki, Takenaga, mode. Toyoda, PTP87, 663lagrangian (1992), PTP89, is given 881 (1993) by ] The theory we concern is the SU(2)[H. Tye, gauge-Higgs S. Wong, PRD92,045005 model with(2015)] one scalar doublet, whose 2 2 lagrangian is given by 1 a aµν µ v SU(2)-gauge Higgs system (U(1)Y can be neglected) =SphaleronFµνF +(DµΦ)†D Φ λ Φ†Φ , (2.1) L −4 解を求める− # − 2 $ 1 v2 2 a aµν µ where Dµ = ∂µ + igAµ. The Manton’s ansatz for the noncontractible loop in the one- = FµνF +(DµΦ)†D Φ λ Φ†Φ saddle, point = least-energy(2.1) path 上のmaximum-energy configuration L −4 − # − 2 $doublet model is sphaleron v ˜ v 0 0 where Dµ =Let∂µ +usigA promoteµ. The μ Manton’s to a dynamical ansatz forvariable: the noncontractibleΦ loop(µ, r, θ in, φ)= the one-Φ = (1 h(r)) iµ + h(r)U(µ, θ, φ) , (2.2) Energy √2 √2 − e− cos µ 1 doublet model is least-energy% path! /gauge trf." = noncontractible! "& loop i 1 A (µ, r,vacuumθ, φ)= f(r)∂ U(µ, θ, φ) U − (µ, θ, φ), (2.3) i g i v ˜ μ ⇒v μ(t) 0 0 highest symmetry↕ config. Φ(µ, r, θ, φ)= Φ = (1 h(r)) iµ + h(r)U(µ, θ, φ) , (2.2) √ √ e− cosconfigurationµ where 1 NCS=1 2 2 % − ! space " ! "& μ(-∞)=0,i μ(+∞)=π: vacuum,1 eiµ(cos µ i sin µ cos θ) eiφ sin µ sin θ A (µ, r, θ, φ)= f(r)∂ U(µ, θ, φ) U − (µ, θ, φ), vacuum (2.3) i μ i U(µ, θ, φ)= iφ− iµ (tsph)=gπ/2: sphaleron ! e− sin µ sin θ e− (cos µ + i sin µ cos θ) " 2 2− = 1(cos µ +sin µ cos θ)+i [(τ1 sin φ + τ2 cos φ)sinµ sin θ + τ3 sin µ cos µ(1 cos θ)] where NCS=0 − 2 z 2 τ1y + τ2x z - We construct a reduced model by adopting Non-contractible= 1 cos loopµ + sin µ + i sin µ + τ3 sin µ cos µ 1 (2.4) iµ iφ (least energy !path) r " ' r ! − r "( a Manton’se (cos µansatz.i sin µ cos θ) e sin µ sin θ 2 ⋆ 4次元2 SU(2) gauge-Higgs doublet 系 ⋆ U(µ, θ, φ)= iφ− iµ = 1 cos µ +ˆz sin µ + i [(τ1yˆ + τ2xˆ)sinµ + τ3 sin µ cos µ (1 zˆ)] . e− sin µ sin θ e− (cos µ + i sin µ cos θ) − ! " ) * 2 2− 2 = 1(cos µ +sin µ cos θ)+i [(τ1 sin φ + τ2 cos φ)sinµ sin θ + τ3 sin µ cos µ(1 cos θ)] 2 −1 a aµν µ2 v z τ y + τ x z = FµνF +(DµΦ)† D Φ λ Φ†Φ 2 2 1 2 L −4 − − 2 = 1 cos µ + sin µ + i sin µ + τ3 sin µ cos µ 1 (2.4) ! " ! r " ' r ! − r "( 2 2 a τ a = 1 cos µ +ˆz sin µ + i [(τ1yˆ + τ2xˆ)sinµ + τ3 sin µ cos µ (1D Φzˆ)]=(. ∂ igA ) Φ, F = ∂ A ∂ A ig[A ,A ], A = A −µ µ − µ µν µ ν − ν µ − µ ν µ µ 2 ) * 2 —SphaleronTransition— 39/47 Comparison with Tye-Wong’s work Some differences between our work and Tye-Wong’s (TW’s).

Method for band A0 Sphaleron mass structure WKB w/ 3 this work A0≠0 μ-dependent connection formulas Schroedinger eq. A0=0 μ-independent Tye-Wong numerically

We use a Manton’s ansatz with fully gauge inv. Classical action: -> no div. issue

c.f., TW’s: Msph = 17.1 TeV. With same normalization, Msph(ours) -> 23.0 TeV. Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180

Band gaps still exist E>Esph due to nonzero reﬂection rate. Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? Esph ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180

# of band

Band gaps still exist E>Esph due to nonzero reﬂection rate. Transition factor tunneling factor instanton calculus band picture

0 sum of band widths up to E Δ(E) ≃ -50 energy (E)

-100 (E) ∆ 10 log Band picture: -150 - State of density is restricted.

-200 - Exponential suppression at

E≪Esph is due to the tiny -250 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 band width. E [GeV] N.B. Δ(E) is not exactly 1 at slightly above Esph. Q. Does the band structure affect the (B+L)-changing process in high-E collisions? In a pure SU(2) theory, one ﬁnds ESph =9.11 TeV, and it is estimated that incorporating the U(1) of the Standard Model reduces this by 1%. Here we follow [14] in assuming a ∼ nominal value of ESph = 9 TeV, while presenting some numerical results for the alternative choices ESph =8, 10 TeV. Later, we also use a recast of early Run-2 searches for microscopic black holes to constrain the sphaleron transition rate as a function of ESph. As was discussed in detail in [14], the Bloch wave function for the periodic potential (2.2) is straightforwardly obtained, and the corresponding conducting (pass) bands can be calculated, as well as their widths and the gaps between the bands. The lowest- lying bands are very narrow, but the widths increase with the heights of the bands. Av- JHEP04(2016)086 eraging over the energies E1,2 of the colliding quark partons yields a strong suppression at E +E E , which corresponds to the exponential suppression found in a conventional 1 2 ≪ Sph tunnelling calculation. However, this suppression decreases as E + E E , and there 1 2 → Sph is no suppression for E + E E . 1 2 ≥ Sph The result of the analysis in [14] can be summarized in the partonic cross-section

4π σ(∆n = 1) exp c S(E) , (2.4) ± ∝ α ! W " where E is the centre-of-mass energy of the parton-parton collision, c 2 and the suppres- ∼ sion factor S(E) is shown in ﬁgure 8 of [14]. As seen there, it rises from the value S(E)= 1 − in the low-energy limit (E E ) to S(E) = 0 for energies E E , with very simi- ≪ Sph ≥ Sph lar results being found in [14] for calculations based on the work of [2] and [23]. For the purpose of our numerical calculations, we approximate S(E) at intermediate energies by

S(E) = (1 a)Eˆ + aEˆ2 1 for 0 Eˆ 1 , (2.5) − − ≤ ≤ where Eˆ E/E and a = 0.005. ≡ Sph − The overall magnitude of eq. (2.4) is not given. We speculate that the relevant scale should be proportional to the non-perturbativeLHC analysis electro-weak cross-section for q-q scattering, EW σqq . Analogously to the fact that the inelastic[J.Ellis andp- pK.Sakurai,cross-section JHEP04(2016)086] is given roughly by 1/m2 , we take σEW 1/m2 . Our cross-section formula is, thus, given as ∼ π Δ(B+L)qq≠0 ∼processW in the band picture:

1 d ab 4π σ(∆n = 1) = 2 dE L p exp c S(E) , (2.6) ± mW dE αW #ab $ ! " where p is anc unknown≃2, p: unknown factor parameter (that might well depend3 on the subprocess energy E) and d ab dEL is the partonHere, luminosityS(E) is approximated function of by the a collidingﬁtting function. quarks a and b, which are obtained from the parton distribution functions at a momentum fraction x, fa(x), evaluated at the appropriate energy scale E:

ln √τ d ab 2E − y y L = dyf (√τe )f (√τe− ), (2.7) dE E2 a b CM $ln √τ 2 2 where ECM is the centre-of-mass energy of the p-p collision and τ = E /ECM.

3S.-H. Henry Tye and Sam S.C. Wong, private communication and to appear.

–3– JHEP04(2016)086 JHEP04(2016)086 5 5. . 4 2 and . (4.1) 0 < , for (4.2) , the ∼ T c/s | n S η , T ∆ | Mass [TeV] antiquarks S 5. On the 100 TeV 33 TeV ¯ . s u/d 2 3l7q c/ = 1. As seen in < | p 20 and η |

JHEP04(2016)086> erence between these T ff p ]. We leave the investigation ) (solid curve), and for the = 2 and 26 7 – We also simulate the decays c 2.5 6 24 , 20 GeV and , and 3 colours of 6 . The di τ , 4 ) plane for > collisions: the final states should 1 sphaleron ν 5 displays histograms of the number 4 is negligible. ,p − / ¯ T 7 q. ≤ 4 τ p Sph = ud c ¯ q 8 for SR8 for the = 9 TeV, E . , given by ( 0 5 10 15 20 25 ≤ n ) plane, as in figure 0 0 ¯ µ q ∆ 0.1 ,p 0.3 0.2 S ν ∼

= +1 sphaleron-induced ¯

0.15 0.05 0.25 q = 9 TeV. Normalised Events Normalised and Sph Sph 1 transitions, as seen in n ¯ E q µ/ − ∆ n=+1 process E 1 processes leading to 10-particle ﬁnal states Sph 1 transitions in ATLAS event , ¯ = q Δ E uu − − e n –7– ¯ q distribution for this possibility ν . Left panel: the exclusion in the = ) with qqqqqqqqqqq, ∆ = , reaching ¯ T ℓ n 8 shows the distribution in the sphericity, , but for –6– = 1. Sph 1 transitions. e/ 6 H n ∆ ¯ ℓ E 2.6 p > 4 | ∆ ¯ ℓ ℓℓℓ n , satisfying the nominal acceptance cuts. As expected, ∆ Mass [TeV] | ) electroweak bosons [ µ → → W N α 2 for the nominal / . = 2 and + = 1 in ( – 11 – 0 0.1-0.01 be realized? qq qq -1 c e (1 = +1 process that leads to 14-particle ﬁnal states: p ≃ ). We accept only particles with ≃ N ), as functions of O p τ n T = shows the simulated ∆ ,H LHC analysis LHC ]. 6 jet n lep and 27 ). One should also consider processes with other values of = 9 TeV, N 8 TeV and with tails extending to lower masses, corresponding to events Q. Can p = 2 and , excluding 4.1 W 8 shows histograms of the numbers of top quarks in the sphaleron-induced Sph c 1 transitions obtained by recasting the ATLAS 2015 search for microscopic ∼

antiquark, one antilepton from each generation, three ¯

, 4 are within the widths of the lines. We recall that the overall normalization − 4

E = 8 and 10 TeV (dot-dashed and dashed lines, respectively). The variations in t

=9TeV: ¯

= d

displays some more properties of the ﬁnal states in sphaleron-induced transi- p<0.2 ≤

p<0.01 3/fb of data at 13 TeV. The variation in the exclusion for 1

n n=-1 process: process: n=-1 n=+1 process: process: n=+1 erent cuts in ( sph

4 c , for LHC collisions at 13 and 14 TeV (left panel, blue and red histograms, 14 TeV 13 TeV ﬀ

∆ u/ Sph ∼

Δ Δ

= 9 TeV. Consequently, the 95% CL exclusion in the (

≤ . Left panel: normalized invariant-mass distributions for the observable ﬁnal-state parti-

3l7q 3, with smaller numbers of charged leptons in events with ﬁnal state neutrinos

antiquarks, for a total of 10 ﬁnal state particles. The initial and ﬁnal states are current LHC data current

Sph

= 9 TeV,

. Right panel: acceptances for sphaleron-induced

@E=E

. Left panel: acceptances as in right panel of ﬁgure

- LHC Run2 w/ 100fb - LHC Run2 w/ -

≤

0 2 4 6 8 10 12

) plane of

The upper right panel of ﬁgure

Figure

0.1

,p 0.2

= +1 transitions, as seen in the left= panel +1 of transitions ﬁgure is correspondingly stronger than for

Sph . The energy dependence of the total cross section for sphaleron transitions for the nominal

0.15 0.05 0.25

lep Similarly, there would be even stronger exclusions for , the leading-order sphaleron-induced processes do not involve electroweak bosons. Normalised Events Normalised 7

is quite uncertain. Sph

n n E E states with two top quarks. There are relatively few final states with no top quarks, and and 0.6, respectively. Theof lower charged left leptons, panelN of figure and/or charged leptons outsidepanel of the figure nominal acceptancefinal states. range. The most Finally, common the outcome is lower to right observe just one top quark, followed by final state partons satisfying ourother chose hand, acceptance the cuts: visibleacceptance multiplicity cuts may and/or be if reduced there if are some neutrinos final-state in particles thesphaleron-induced fail final final state. these statestograms, with respectively. 10 Both and distributions 14 are final-state relatively particles broad, being as peaked red at and blue his- next simplest being the whose simulation yieldsnominal the multiplicities blue is histograms visiblenominal in in value if the figure gluon upper radiation or left other panel. higher-order QCD The processes yield multiplicity additional may final- exceed the being peaked at with (multiple) neutrino emission.tions As for seen collisions in at 33 the and right (particularly) panel, 100 TeV the extend corresponding totions: much distribu- larger the invariant masses. red histogramsdiscussed are for above the ( Figure 3 cles in sphaleron-induced transitions inrespectively). LHC collisions Right at panel: 13and and 100 TeV corresponding 14 TeV (green invariant-mass (blue andnominal and distributions pink red choices for histograms, histograms, future respectively). colliders These distributions at are 33 calculated forthe our left panel, the invariant-mass distributions for 13 and 14 TeV are quite similar, both ∆ nominal ∆ the right panel of figure quarks. The left panelas of a figure blue histogram,transitions. which Correspondingly, the is acceptances shifted in to the larger ATLAS search values regions than are for higher for the Figure 8 transitions to 14-particle final states.but Right for panel: sphaleron-induced the transitions exclusion in to the 14-particle ( final states. p Figure 7 selections with di ( black holes using The normalized invariant-mass distributions for the observable final-state particles are There are suggestions that the baryon and lepton number violating processes areWe enhanced use if our fermions own code to simulate sphaleron-induced processes. There is also a public Monte Carlo tool , leading to transitions of the form 5 6 tograms, respectively), for our nominal choices are produced associated withof many this possibility for future work. to simulate sphaleron events [ of heavy particles ( Neutrinos are removed from the list of observable particles. shown in figure respectively) and for future colliders at 33 and 100 TeV (right panel, green and pink his- A priori Since the dominant processescontain are a induced single by ¯ and three constrained so that the total electricwith charge is the conserved. momenta of We final-state make particles parton-level given simulations, by phase space. member of each electroweak doublet,t/b i.e., Figure 2 choices outlying choices the curves for 1 factor Δ(B+L)≠0 process [Funakubo, Otsuki, Takenaga, Toyoda, PTP87, 663 (1992), PTP89, 881 (1993)] transition amplitude:

path integral using coherent state |φ,π> ∵ appropriate for describing classical conﬁguration

- tunneling suppression appears in the intermediate process. - overlap issue: suppressions from and <φ,π|i>.

This point is not properly discussed in the work of Tye and Wong. Δ(B+L)≠0 process [Funakubo, Otsuki, Takenaga, Toyoda, PTP87, 663 (1992), PTP89, 881 (1993)] transition amplitude:

path integral using coherent state |φ,π> ∵ appropriate for describing classical conﬁguration

- tunneling suppression appears in the intermediate process. - overlap issue: suppressions from and <φ,π|i>.

This point is not properly discussed in the work of Tye and Wong. overlap factor

inner product between n particle state and coherent state:

1.0

2 2 0.8 - cross section ∝ |α1| …|αn| 0.6

2 0.4 - |α| has a peat at k=mW. 0.2

100 200 300 400 Sphaleron at LHC

W Case1: 2 -> sphaleron Sphaleron

For |p1|=|p2|≃Esph/2

Creation of sphaleron from the 2 energetic particles is difﬁcult. Sphaleron at LHC

W Case1: 2 -> sphaleron Sphaleron

For |p1|=|p2|≃Esph/2

Creation of sphaleron from the 2 energetic particles is difﬁcult.

Case2: 2 -> n W -> sphaleron 80 W bosons

W Sphaleron

n≃80 since Esph/√2mW ⫶ phase space factor: W difﬁcult to produce about 80 W bosons. How about high-T?

At high temperatures, the overlap suppressions do not exist.

∵

Particles with momenta O(mW) are abundant in thermal bath.

sizable overlap with the classical conﬁguration

Q. Does the band structure affect electroweak baryogengesis? B preservation criteria

modiﬁed? B preservation criteria

modiﬁed?

If yes, modiﬁed! B preservation criteria

modiﬁed?

If yes, modiﬁed!

EWBG-viable region must be re-analyzed!! Vacuum decay rate at ﬁnite-T

Ordinary case: [Afﬂeck, PRL46,388 (1981)]

≃14 GeV

≃0.42 ≃0.51 Band case: Impact of band For simplicity, we use the band structure obtained before.

blue: ordinary case red: band case

For T=100 GeV, Γ/ΓA = 1.06. How about B-number preservation criteria? Impact of band For simplicity, we use the band structure obtained before. typical EWBG region

blue: ordinary case red: band case

For T=100 GeV, Γ/ΓA = 1.06. How about B-number preservation criteria? B preservation criteria

Γwith the band effect is ⇒ B preservation criteria

Γwith the band effect is ⇒

zero mode factor B preservation criteria

Γwith the band effect is ⇒

zero mode factor ⋍ 4.4 (MSSM) B preservation criteria

Γwith the band effect is ⇒

band effect

zero mode factor ⋍ 4.4 (MSSM) B preservation criteria

Γwith the band effect is ⇒

band effect

zero mode factor ⋍ 4.4 (MSSM) B preservation criteria

Γwith the band effect is ⇒

band effect

zero mode factor ⋍ 4.4 (MSSM)

Band effect has little effect on the B preservation criteria. Summary

• We have discussed EWBG with LFV in 2HDM.

• some parameter space can explain h->μτ, muon g-2, and BAU. 300 GeV ≲ mH ≲ mA for Re(ρτμρμτ)>0. • We also discussed the band effect on the sphaleron processes at T=0 and T≠0.

• Even though the tunneling suppression disappears at E≳Esph, sphaleron process in high-E collisions suffers from the overlap suppression. -> the process is unlikely to occur.

• T≃100 GeV, sphaleron process is virtually unaffected. -> no impact on EWBG. Backup Baryon Asymmetry of the Universe (BAU)

❒ Our Universe is baryon-asymmetric.

[PDG2016]

❒ Sakharov criteria (’67)

(1) Baryon number (B) violation (2) C and CP violation (3) Out of equilibrium BAU must arise - After inﬂation - Before Big-Bang Nucleosynthesis (T≃O(1) MeV). 2

II. MODEL lepton sector are

Y = ¯liL(Y1Φ1 + Y2Φ2)ij ejR +h.c. The 2HDM is an extension of the SM by adding one ad- −L yi 1 ditional Higgs doublet. Such a two-Higgs doublets struc- e¯iL δij sβ α + ρij cβ α ejRh ∋ √2 − √2 − ture is motivated by some UV theories such as supersym- ! $ metric theories. The most general Higgs potential at the yi 1 +¯eiL δij cβ α ρij sβ α ejRH renormalizable level is √2 − − √2 − ! $ i + e¯iLρijejRA +h.c., (4) V0(Φ1, Φ2) √2 2 2 2 = m Φ†Φ + m Φ†Φ (m Φ†Φ +h.c.) 1 1 1 2 2 2 − 3 1 2 where i, j are flavor indices, Y1,2 are general 3-by-3 com- λ1 2 λ2 2 plex matrices, and + (Φ†Φ ) + (Φ†Φ ) + λ (Φ†Φ )(Φ†Φ ) 2 1 1 2 2 2 3 1 1 2 2 1 e e λ5 2 ρ = t y δ + (V †Y V ) , (5) + λ (Φ†Φ )(Φ†Φ )+ (Φ†Φ ) ij β i ij L 2 R ij 4 1 2 2 1 2 1 2 − cβ ! with V e defined as the unitary matrices such that di- + λ6(Φ1†Φ1)+λ7(Φ2†Φ2) (Φ1†Φ2)+h.c. . (1) L,R SM $ agonalize the charged lepton Yukawa couplings Y = " # e SM e (Y1cβ + Y2sβ), i.e., VL†Y VR = YD = diag(ye,yµ,yτ ). 2 2 By this diagonalization, the mass terms of the charged Hermiticity of V0 requires that m1, m2, λ1, λ2, λ3 and 2 leptons are given by mi = yiv/√2 with i = e, µ, τ. λ4 are all real while m3, λ5, λ6 and λ7 are generally complex, with one of them being possibly made real by a Non-diagonal elements of ρij are the sources of tree- field redefinition of either Higgs doublet. We parametrize level FCNH processes. In the literature, the so-called Chen-Sher ansatz [20] for ρij is adopted in order to avoid Φi (i =1, 2) as experimental constraints. In our analysis, however, we will not assume it to obtain a parameter space that can + accommodate both h µτ and g 2 anomalies. In φi (x) iθi → − Φi(x)=e 1 , (2) general, Y1,2 cannot be uniquely determined even though vi + hi(x)+iai(x) % √2 ( YD is known. In our analysis, we make some working & ' assumption on Y1,2 for baryogenesis, and determine the structure of FCNH couplings ρ at T = 0. For later use, iθi where vie are the VEV’s. For simplicity, we assume YUJI OMURA, EIBUN SENAHA, and KAZUHIRO TOBEiφij PHYSICAL REVIEW D 94, 055019 (2016) we define ρ = ρ e . μτ τμ B. The muon anomalousij magnetic ij m m ρe ρe δa μ τ that CP is not violated by the Higgs potential and the moment (muon g − 2) | | μ ¼ 16π2 m2 2 m2 We have shown that the μ − τ flavor violating Yukawa 2 h 3 2 mH 3 A 3 cβα log m2 − 2 sβα log m2 − 2 log m2 − 2 couplings in the general 2HDM explain the CMS excess in × ð τ Þ ð τ Þ − τ ; VEV’s, leading to θi = 0. Here we express v1,2 in terms m2 þ m2 m2 the h → μτ decay mode. Here we consider the contribution h H A of the μ − τ flavor violating interaction to the muon 15 3 of polar coordinates, v1 = v cos β and v2 = v sin β with anomalous magnetic moment (muon g − 2). ð Þ Exp The discrepancy between the measuredIII. value (ahμ ) and µτ, (g 2) AND EDM µ μτ τμ SM where we have assumed that ρ ρ is real, for simplicity. v 246 GeV. In the following, we will use the shorthand the standard model predictionbosons (a toμ ) ofδa the muonand gd− 2→arehas given by − e e We parametrize the sphaleron energy as E (T )= µ µ We will discuss the effect of an imaginary part of these sph ≃ been reported in Ref. [48]. For example, Ref. [57] suggests 4πv(T ) (T )/g with g being the SU(2) gauge coupling notation sβ =sinβ, cβ = cos β and tβ = tan β. the following result: Yukawa couplings later. We note that a degeneracy of all E 2 2 L mµmτ Re(ρµτ ρτµ)neutral Higgs bosons suppresses the extra contributionconstant. to Eq. (12) is then rewritten as δaµ = 2 the muon g − 2, as seen in Eq. (15). ExpHereSM we outline−1016π some important consequences of aμ − aμ 26.1 8.0 × 10 : 13 The so-called Barr-Zee–type two-loop diagramsv can(T ) g When discussing phenomenology, it is useful to use the ¼ð Æ Þ 2 ð Þ contribute2 to the muon g 2 if diagonal elements of ρ 2 cβ αf(rh) sβ αf(rH ) f(rA−) f > 42.97 + log terms ζsph(T ) . (13) Ref. [17] to make− this+ are paper− nonzero. In the self-contained. parameter space, we(7) are consideringT 4π (T ) ≡ Higgs (Georgi) basis [19] in which only one Higgs dou- Here we consider whether the extra× contributionsm2 induced here,m such2 contributions− m2 are always subdominant and E by the μ τ flavor violating interactions!μτ can accommodateh H A " & ' − h-> and muon g-2 numerically unimportant. In the cases of the flavor- this muon g The2 anomaly. Higgs The effective decay operator for tothe µ and τ in the currentOne model can see that occursζsph(T ) is mostly controlled by (T ). blet develops the VEV and the Nambu-Goldstone bosons − dµ 1 changing muon or tau decays, however, they can compete E muon g − 2 is expressed by = Arg(ρµτ ρτµwith)δa theµ , one-loop contributions and can(8) play a significantThe logarithmic corrections in the bracket come from 0 In 2HDM,at tree levelite is− through2easymµ to µaccommodaterole.-τ Detailsinteractions will be discussed below. not (for only earlier h-> studies,μτ(b )but (G ,G ) are decoupled from the physical states: | | the prefactor of ΓB . To our best knowledge, an exten- ± v μν In Fig. 2, we show numerical results for the extra L 2 μ¯ Lσ μRFμν H:c: 14 sive study on the prefactor in the 2HDM is still missing. muonsee g-2. Refs.¼ Λwhere [21]),þ [Y. and Omura,ð itsÞ contribution branchingE.S., K.Tobe, to muon g −JHEP052015028,2 ratio(δaμ) as a function takes of cβα thePRD94,055019 form (2016)] In the minimal supersymmetric SM case, on the other We note that the chirality of muon is flipped in the operator.1 2 ln rφ hand, the zero mode factors of the fluctuations about Therefore, if there is a large chiralityf(r flipφ)= induced by− the 2 +3 rφ 2(1 rφ) 1 rφ − 2 2the2 sphaleron typically amount to about 10% [24]. This + h->new physics, μτ it can enhance the extra contributions to the # $ G − m−h( ρµτ + ρτµ is)c subdominantβ α and, therefore, we will neglect them in muon g − 2 [58]. Feynman diagrams for the one-loop 1 3 | | | | − corrections involving the neutralBr( Higgsh bosons andµ thelnτμr)=φ− , (9) our numerical, analysis(6) for simplicity. Φ′ = cβΦ1 + sβΦ2 = 1 , rφ≃ 1 − 2 1 0 τ flavor violating Yukawa couplings are described→≪ in Fig. 1. 16πΓh We impose Eq. (13) at a critical temperature T at √ v + h1′ + iG As shown in Fig. 1, the chirality is flipped in the internal C % 2 ( line of τ lepton in the diagram. Therefore,2 it2 induces the with rφ = mτ /mφ. Non-degeneracy of the neutral Higgs which the effective potential has two degenerate minima. (3) O mτ=mμ enhancement in the extra contributions to the + ð Þ masses and proper choice of the overall sign are essential In our analysis, TC and vC v(TC ) are determined us- & ' muon whereg − 2, comparedm withh the= one generated 125 by GeV the and Γh =4.1 MeV. It is easy to ≡ H flavor-diagonal Yukawafor coupling. obtaining We stress a su thatffi thecientμ − δaµ. In Ref. [22], the discrep- ing finite-temperature one-loop effective potential with τ flavor violating interactionancy of is essential (g 2) to obtainbetween such an the experimental value and the thermal resummation. As mentioned in Sec. III, sβ α 3 Φ2′ = sβΦ1 + cβΦ2 = 1 , accommodateμτ µ Br(τμ h µτ) 0.84% by taking ρµτ − h′ + iA enhancement. Note thatSM both prediction couplings− ρe wasand ρe estimatedshould as is close to 1 in the region of interest to us. In such a − % √2 2 ( be nonzero to flip the chirality in the internal τ lepton line. → ≃ | | ∼ Finally, theρ expression of the enhanced(0.1) extra andcontributionc (0.01). It hadcase, been we may simplify shown the analysis of EWPT to a one- τµ EXP SM β α 10 δaμ is given by δaµ = a a bosons=(26−.1 to8δ.0)aµ and10− dµ. are(10) givendimensional by problem with a singleWe order parametrize parameter as the sphaleron energy as Esph(T )= & ' muon| g-2| ∼ O µ − µ ±∼ O× that such parameter choices did not violatediscussed the in Refs. current [10, 12]. 4πv(T ) (T )/g with g being the SU(2) gauge coupling E 2 2 L As demonstrated in Ref. [17], this deviationmµmτ Re( couldρµτ beρτµ ac-) As is well-known, the extra heavyconstant. Higgs bosons Eq. (12) can is then rewritten as experimentalcommodated bounds for m ±δ onaµ[200= other, 500] GeV LFV with processes an ap- such as τ where h = c H + s h and h = s H + c h H,A,H ∈ 16π2 play a major role in enhancing vC /TC in the 2HDM. 1′ β α β α 2′ β α β α propriate mass hierarchy. From EWBG point of view, 2 →2 − − − − − µγ and τ µνν¯, etc [17]. Note2 that2 it isIn this suffi case,cientM form3/(sβcβ)mustnotexceedcer-v(T ) g2 with α being a mixing angle between the two CP-even this mass range of the heavy Higgs bosonscβ α hasf(r theh) rightsβ αtainf(rH values,) f depending(rA) ≡ on the magnitude of> quartic cou- 42.97 + log terms ζsph(T ) . (13) → − − T 4π (T ) ≡ eitherscaleρµ forτ realizingor ρτ theµ first-orderto be EWPT,nonzero though2 to the explain+ LFV 2 the h 2 µ, τ (7) FIG. 2. Numerical result for δa as a function ofplings;c and otherwise, the so-called non-decoupling eEffects × ! mh μ mHβα − mA " & ' Higgs fields (h1,2). In this paper, h is the 125 GeV Higgs interactions by themselves doBR h not→ μτ playfor m a central250 GeV (upper role figure) in and 350 GeV → ð Þ A ¼ would diminish, rendering a suppressed vC /TC.Phe- excessthe only. realization, If both as will be couplings shown(lower in figure). the Regions following coexist, where thesections. muon ong − 2 anomaly the in other hand, One can see that ζsph(T ) is mostly controlled by (T ). dEq.µ (13) is explained1 within 1σ, 2σ, and 3σ arenomenological shown. consequences of the non-decoupling effects E boson and assumed to be the lightest of them all. = Arg(Æ ρ Æ ρ )Æδa , (8) The logarithmic corrections in the bracket come from AppropriateFIG. 1. A Feynman diagramThe formass current neutral Higgs limitdifferences boson on contri-dµ isHere [23] we determine among the mass spectrum (mµτ ofhτ, heavy µm HiggsHµ,at bosonsmT A=) 0 are include needed. significant deviations in the h γγ de- we can have a one-loope diagram−2m that inducesμτ τμ g 2 and (b) butions to the muon g − 2. A photon is attached somewhere in the assuming λ4 λ5 0µ.5 in Eq. (10). We have assumed ρe ρe < → |0 with| ρμτ ¼ ρτμ¼to obtain the positive contribution tocayδa . width [25]− and the triple Higgsthe coupling prefactor [26]. of Γ . To our best knowledge, an exten- charged lepton line. 19e − e μ B Without any symmetries, both Higgs doublets can cou- electric dipole momentdµ < 1.9 10 (EDM)− ¼e cm , of muon,(11) denotedFor the evaluation by δa ofµ , we solvesive the studyequations on of the mo- prefactor in the 2HDM is still missing. | | where× E tion for the sphaleron with appropriateIn the boundary minimal condi- supersymmetric SM case, on the other ple to fermions. The relevant Yukawa interactions in the and dwhichµ, respectively. is about three orders055019-4 Contributions of magnitude larger than ofdµ the neutral Higgs 9 tions [27, 28]. Here, we use the tree-level Higgs potential estimated with Arg(ρµτ ρτµ)=1andδaµ =3.0 101− in 2 ln rφ hand, the zero mode factors of the fluctuations about f(r )= for simplicity.+3 r In this case, ζ may be underestimated Eq. (8). In what follows, we will clarifyφ the relationship×− 2 φ sph 2(1 rφ) 1 rφ − the sphaleron typically amount to about 10% [24]. This # by (10)% since$ (0) > (TC ). As will be shown be- between CP violation appearing in dµ and that− relevant − O E E is subdominant and, therefore, we will neglect them in to BAU. 1 3 low, this approximation does not affect our conclusion. − ln rφ ,A detailed analysis of ζsph(9)(T ) willour be given numerical elsewhere. analysis for simplicity. rφ≃ 1 − 2 ≪ In passing, recent studies show that ζ (T )=1.1 Wesph imposeC Eq.− (13) at a critical temperature TC at IV. BARYON NUMBER PRESERVATION2 2 1.2 in the real singlet-extended SMwhich [29] and theζsph eff(ectiveTC )= potential has two degenerate minima. with rφ = mτ /mφ. Non-degeneracy1.23 in of the the scale-invariant neutral Higgs 2HDM [13] for typical param- masses and proper choice of theeter overall sets. sign are essential In our analysis, TC and vC v(TC ) are determined us- The baryon number is generated via the sphaleron pro- ≡ for obtaining a sufficient δa . InIn Ref. Fig. [22], 1, v /T theand discrep-δa are showning finite-temperature in the (m ,m ) one-loop effective potential with cess outside the bubble (symmetric phase), and it canµ C C µ H A ancy of (g 2) between the experimentalplane. Here, we value take andmA = themH± tothermal avoid the resummation.electroweak As mentioned in Sec. III, sβ α survive if the sphaleron process is sufficientlyµ quenched − − ρ parameter constraint, and choose cβ α =0.006, ρτµ = inside the bubble (brokenSM prediction phase). To was this estimated end, the as is− close to 1| in| the region of interest to us. In such a ρµτ ,andφτµ + φµτ = π/4, as favored by the solution sphaleron rate in the broken phase, denoted as Γ(b)(T ), | | case, we may simplify the analysis of EWPT to a one- EXP SMB of (g 2)µ anomaly.10 For the remaining parameters, we must be smaller than the Hubbleδaµ = constant,aµ Ha(µT ).=(26 More .1 8.0)− 10− . (10) dimensional problem with a single order parameter as − ±set M =100GeV,tan× β =1andλ6 = λ7 =0asan explicitly, example. Contours of vC /TC arediscussed plotted with in the Refs. solid [10, 12]. As demonstrated in Ref. [17], thiscurves deviation in gray could with values be ac- of 1.0, 1.17,As 1.5, is well-known,2.0, 2.5, 3.0 the extra heavy Higgs bosons can (b) Esph(T )/T ΓB (T ) (prefactor)commodatede− for m ± [200from, 500] bottom GeV to with top, where an ap- 1.17 corresponds to ζ . Al- ≃ H,A,H ∈ play a majorsph role in enhancing vC /TC in the 2HDM. propriate mass2 hierarchy. Fromlowed EWBG 1σ,2σ pointand 3 ofσ regions view, of δaµ are shown by the2 2 plings;ζsph)andfavoredby otherwise, the so-called non-decoupling effects interactions by themselves19 do not play a central role in (g =110.75 in the 2HDM) and mP =1.22 10 GeV. (g 2)µ have an overlap if mA >mwouldH .Wenotethat diminish, rendering a suppressed vC /TC.Phe- ∗ the realization, as× will be shown in− the following sections. nomenological consequences of the non-decoupling effects The current limit on dµ is [23] at T = 0 include significant deviations in the h γγ de- → 19 cay width [25] and the triple Higgs coupling [26]. d < 1.9 10− e cm , (11) | µ| × For the evaluation of , we solve the equations of mo- tion for the sphaleron withE appropriate boundary condi- which is about three orders of magnitude larger than dµ 9 tions [27, 28]. Here, we use the tree-level Higgs potential estimated with Arg(ρµτ ρτµ)=1andδaµ =3.0 10− in Eq. (8). In what follows, we will clarify the relationship× for simplicity. In this case, ζsph may be underestimated by (10)% since (0) > (TC ). As will be shown be- between CP violation appearing in dµ and that relevant O E E to BAU. low, this approximation does not affect our conclusion. A detailed analysis of ζsph(T ) will be given elsewhere. In passing, recent studies show that ζ (T )=1.1 sph C − IV. BARYON NUMBER PRESERVATION 1.2 in the real singlet-extended SM [29] and ζsph(TC )= 1.23 in the scale-invariant 2HDM [13] for typical parameter sets. The baryon number is generated via the sphaleron pro- In Fig. 1, v /T and δa are shown in the (m ,m ) cess outside the bubble (symmetric phase), and it can C C µ H A plane. Here, we take m = m ± to avoid the electroweak survive if the sphaleron process is sufficiently quenched A H ρ parameter constraint, and choose cβ α =0.006, ρτµ = inside the bubble (broken phase). To this end, the − | | ρµτ ,andφτµ + φµτ = π/4, as favored by the solution sphaleron rate in the broken phase, denoted as Γ(b)(T ), | | B of (g 2)µ anomaly. For the remaining parameters, we must be smaller than the Hubble constant, H(T ). More − set M =100GeV,tanβ =1andλ6 = λ7 =0asan explicitly, example. Contours of vC /TC are plotted with the solid curves in gray with values of 1.0, 1.17, 1.5, 2.0, 2.5, 3.0 (b) Esph(T )/T Γ (T ) (prefactor)e− B ≃ from bottom to top, where 1.17 corresponds to ζsph. Al- 2 ζsph)andfavoredby 19 (g =110.75 in the 2HDM) and mP =1.22 10 GeV. (g 2)µ have an overlap if mA >mH .Wenotethat ∗ × − 2

II. MODEL lepton sector are

Y = ¯liL(Y1Φ1 + Y2Φ2)ij ejR +h.c. The 2HDM is an extension of the SM by adding one ad- −L yi 1 ditional Higgs doublet. Such a two-Higgs doublets struc- e¯iL δij sβ α + ρij cβ α ejRh ∋ √2 − √2 − ture is motivated by some UV theories such as supersym- ! $ metric theories. The most general Higgs potential at the yi 1 +¯eiL δij cβ α ρij sβ α ejRH renormalizable level is √2 − − √2 − ! $ i + e¯iLρijejRA +h.c., (4) V0(Φ1, Φ2) √2 2 2 2 = m Φ†Φ + m Φ†Φ (m Φ†Φ +h.c.) 1 1 1 2 2 2 − 3 1 2 where i, j are flavor indices, Y1,2 are general 3-by-3 com- λ1 2 λ2 2 plex matrices, and + (Φ†Φ ) + (Φ†Φ ) + λ (Φ†Φ )(Φ†Φ ) 2 1 1 2 2 2 3 1 1 2 2 1 e e λ5 2 ρ = t y δ + (V †Y V ) , (5) + λ (Φ†Φ )(Φ†Φ )+ (Φ†Φ ) ij β i ij L 2 R ij 4 1 2 2 1 2 1 2 − cβ ! with V e defined as the unitary matrices such that di- + λ6(Φ1†Φ1)+λ7(Φ2†Φ2) (Φ1†Φ2)+h.c. . (1) L,R SM $ agonalize the charged lepton Yukawa couplings Y = " # e SM e (Y1cβ + Y2sβ), i.e., VL†Y VR = YD = diag(ye,yµ,yτ ). 2 2 By this diagonalization, the mass terms of the charged Hermiticity of V0 requires that m1, m2, λ1, λ2, λ3 and 2 leptons are given by mi = yiv/√2 with i = e, µ, τ. λ4 are all real while m3, λ5, λ6 and λ7 are generally complex, with one of them being possibly made real by a Non-diagonal elements of ρij are the sources of tree- field redefinition of either Higgs doublet. We parametrize level FCNH processes. In the literature, the so-called Chen-Sher ansatz [20] for ρij is adopted in order to avoid Φi (i =1, 2) as experimental constraints. In our analysis, however, we will not assume it to obtain a parameter space that can + accommodate both h µτ and g 2 anomalies. In φi (x) iθi → − Φi(x)=e 1 , (2) general, Y1,2 cannot be uniquely determined even though vi + hi(x)+iai(x) % √2 ( YD is known. In our analysis, we make some working & ' assumption on Y1,2 for baryogenesis, and determine the structure of FCNH couplings ρ at T = 0. For later use, iθi where vie are the VEV’s. For simplicity, we assume YUJI OMURA, EIBUN SENAHA, and KAZUHIRO TOBEiφij PHYSICAL REVIEW D 94, 055019 (2016) we define ρ = ρ e . μτ τμ B. The muon anomalousij magnetic ij m m ρe ρe δa μ τ that CP is not violated by the Higgs potential and the moment (muon g − 2) | | μ ¼ 16π2 m2 2 m2 We have shown that the μ − τ flavor violating Yukawa 2 h 3 2 mH 3 A 3 cβα log m2 − 2 sβα log m2 − 2 log m2 − 2 couplings in the general 2HDM explain the CMS excess in × ð τ Þ ð τ Þ − τ ; VEV’s, leading to θi = 0. Here we express v1,2 in terms m2 þ m2 m2 the h → μτ decay mode. Here we consider the contribution h H A of the μ − τ flavor violating interaction to the muon 15 3 of polar coordinates, v1 = v cos β and v2 = v sin β with anomalous magnetic moment (muon g − 2). ð Þ Exp The discrepancy between the measuredIII. value (ahμ ) and µτ, (g 2) AND EDM µ μτ τμ SM where we have assumed that ρ ρ is real, for simplicity. v 246 GeV. In the following, we will use the shorthand the standard model predictionbosons (a toμ ) ofδa the muonand gd− 2→arehas given by − e e We parametrize the sphaleron energy as E (T )= µ µ We will discuss the effect of an imaginary part of these sph ≃ been reported in Ref. [48]. For example, Ref. [57] suggests 4πv(T ) (T )/g with g being the SU(2) gauge coupling notation sβ =sinβ, cβ = cos β and tβ = tan β. the following result: Yukawa couplings later. We note that a degeneracy of all E 2 2 L mµmτ Re(ρµτ ρτµ)neutral Higgs bosons suppresses the extra contributionconstant. to Eq. (12) is then rewritten as δaµ = 2 the muon g − 2, as seen in Eq. (15). ExpHereSM we outline−1016π some important consequences of aμ − aμ 26.1 8.0 × 10 : 13 The so-called Barr-Zee–type two-loop diagramsv can(T ) g When discussing phenomenology, it is useful to use the ¼ð Æ Þ 2 ð Þ contribute2 to the muon g 2 if diagonal elements of ρ 2 cβ αf(rh) sβ αf(rH ) f(rA−) f > 42.97 + log terms ζsph(T ) . (13) Ref. [17] to make− this+ are paper− nonzero. In the self-contained. parameter space, we(7) are consideringT 4π (T ) ≡ Higgs (Georgi) basis [19] in which only one Higgs dou- Here we consider whether the extra× contributionsm2 induced here,m such2 contributions− m2 are always subdominant and E by the μ τ flavor violating interactions!μτ can accommodateh H A " & ' − h-> and muon g-2 numerically unimportant. In the cases of the flavor- this muon g The2 anomaly. Higgs The effective decay operator for tothe µ and τ in the currentOne model can see that occursζsph(T ) is mostly controlled by (T ). blet develops the VEV and the Nambu-Goldstone bosons − dµ 1 changing muon or tau decays, however, they can compete E muon g − 2 is expressed by = Arg(ρµτ ρτµwith)δa theµ , one-loop contributions and can(8) play a significantThe logarithmic corrections in the bracket come from 0 In 2HDM,at tree levelite is− through2easymµ to µaccommodaterole.-τ Detailsinteractions will be discussed below. not (for only earlier h-> studies,μτ(b )but (G ,G ) are decoupled from the physical states: | | the prefactor of ΓB . To our best knowledge, an exten- ± v μν In Fig. 2, we show numerical results for the extra L 2 μ¯ Lσ μRFμν H:c: 14 sive study on the prefactor in the 2HDM is still missing. muonsee g-2. Refs.¼ Λwhere [21]),þ [Y. and Omura,ð itsÞ contribution branchingE.S., K.Tobe, to muon g −JHEP052015028,2 ratio(δaμ) as a function takes of cβα thePRD94,055019 form (2016)] In the minimal supersymmetric SM case, on the other We note that the chirality of muon is flipped in the operator.1 2 ln rφ hand, the zero mode factors of the fluctuations about Therefore, if there is a large chiralityf(r flipφ)= induced by− the 2 +3 rφ 2(1 rφ) 1 rφ − 2 2the2 sphaleron typically amount to about 10% [24]. This + h->new physics, μτ it can enhance the extra contributions to the # $ G − m−h( ρµτ + ρτµ is)c subdominantβ α and, therefore, we will neglect them in muon g − 2 [58]. Feynman diagrams for the one-loop 1 3 | | | | − corrections involving the neutralBr( Higgsh bosons andµ thelnτμr)=φ− , (9) our numerical, analysis(6) for simplicity. Φ′ = cβΦ1 + sβΦ2 = 1 , rφ≃ 1 − 2 1 0 τ flavor violating Yukawa couplings are described→≪ in Fig. 1. 16πΓh We impose Eq. (13) at a critical temperature T at √ v + h1′ + iG As shown in Fig. 1, the chirality is flipped in the internal C % 2 ( line of τ lepton in the diagram. Therefore,2 it2 induces the with rφ = mτ /mφ. Non-degeneracy of the neutral Higgs which the effective potential has two degenerate minima. (3) O mτ=mμ enhancement in the extra contributions to the + ð Þ masses and proper choice of the overall sign are essential In our analysis, TC and vC v(TC ) are determined us- & ' muon whereg − 2, comparedm withh the= one generated 125 by GeV the and Γh =4.1 MeV. It is easy to ≡ H flavor-diagonal Yukawafor coupling. obtaining We stress a su thatffi thecientμ − δaµ. In Ref. [22], the discrep- ing finite-temperature one-loop effective potential with τ flavor violating interactionancy of is essential (g 2) to obtainbetween such an the experimental value and the thermal resummation. As mentioned in Sec. III, sβ α 3 Φ2′ = sβΦ1 + cβΦ2 = 1 , accommodateμτ µ Br(τμ h µτ) 0.84% by taking ρµτ − h′ + iA enhancement. Note thatSM both prediction couplings− ρe wasand ρe estimatedshould as is close to 1 in the region of interest to us. In such a − % √2 2 ( be nonzero to flip the chirality in the internal τ lepton line. → ≃ | | ∼ Finally, theρ expression of the enhanced(0.1) extra andcontributionc (0.01). It hadcase, been we may simplify shown the analysis of EWPT to a one- τµ EXP SM β α 10 δaμ is given by δaµ = a a bosons=(26−.1 to8δ.0)aµ and10− dµ. are(10) givendimensional by problem with a singleWe order parametrize parameter as the sphaleron energy as Esph(T )= & ' muon| g-2| ∼ O µ − µ ±∼ O× that such parameter choices did not violatediscussed the in Refs. current [10, 12]. 4πv(T ) (T )/g with g being the SU(2) gauge coupling E 2 2 L As demonstrated in Ref. [17], this deviationmµmτ Re( couldρµτ beρτµ ac-) As is well-known, the extra heavyconstant. Higgs bosons Eq. (12) can is then rewritten as experimentalcommodated bounds for m ±δ onaµ[200= other, 500] GeV LFV with processes an ap- such as τ where h = c H + s h and h = s H + c h H,A,H ∈ 16π2 play a major role in enhancing vC /TC in the 2HDM. 1′ β α β α 2′ β α β α ρμτ propriate massρ hierarchy.τμ From EWBG point of view, 2 →2 − − − − − µγ and τ µνν¯, etc [17]. Note2 that2 it isIn this suffi case,cientM form3/(sβcβ)mustnotexceedcer-v(T ) g2 with α being a mixing angle between the two CP-even this mass range of the heavy Higgs bosonscβ α hasf(r theh) rightsβ αtainf(rH values,) f depending(rA) ≡ on the magnitude of> quartic cou- 42.97 + log terms ζsph(T ) . (13) → − − T 4π (T ) ≡ eitherscaleρµ forτ realizingor ρτ theµ first-orderto be EWPT,nonzero though2 to the explain+ LFV 2 the h 2 µ, τ (7) FIG. 2. Numerical result for δa as a function ofplings;c and otherwise, the so-called non-decoupling eEffects × ! mh μ mHβα − mA " & ' Higgs fields (h1,2). In this paper, h is the 125 GeV Higgs interactions by themselves doBR h not→ μτ playfor m a central250 GeV (upper role figure) in and 350 GeV → ð Þ A ¼ would diminish, rendering a suppressed vC /TC.Phe- excessthe only. realization, If both as will be couplings shown(lower in figure). the Regions following coexist, where thesections. muon ong − 2 anomaly the in other hand, One can see that ζsph(T ) is mostly controlled by (T ). dEq.µ (13) is explained1 within 1σ, 2σ, and 3σ arenomenological shown. consequences of the non-decoupling effects E boson and assumed to be the lightest of them all. = Arg(Æ ρ Æ ρ )Æδa , (8) The logarithmic corrections in the bracket come from AppropriateFIG. 1. A Feynman diagramThe formass current neutral Higgs limitdifferences boson on contri-dµ isHere [23] we determine among the mass spectrum (mµτ ofhτ, heavy µm HiggsHµ,at bosonsmT A=) 0 are include needed. significant deviations in the h γγ de- we can have a one-loope diagram−2m that inducesμτ τμ g 2 and (b) butions to the muon g − 2. A photon is attached somewhere in the assuming λ4 λ5 0µ.5 in Eq. (10). We have assumed ρe ρe < → |0 with| ρμτ ¼ ρτμ¼to obtain the positive contribution tocayδa . width [25]− and the triple Higgsthe coupling prefactor [26]. of Γ . To our best knowledge, an exten- charged lepton line. 19e − e μ B Without any symmetries, both Higgs doublets can cou- electric dipole momentdµ < 1.9 10 (EDM)− ¼e cm , of muon,(11) denotedFor the evaluation by δa ofµ , we solvesive the studyequations on of the mo- prefactor in the 2HDM is still missing. | | where× E tion for the sphaleron with appropriateIn the boundary minimal condi- supersymmetric SM case, on the other ple to fermions. The relevant Yukawa interactions in the and dwhichµ, respectively. is about three orders055019-4 Contributions of magnitude larger than ofdµ the neutral Higgs 9 tions [27, 28]. Here, we use the tree-level Higgs potential estimated with Arg(ρµτ ρτµ)=1andδaµ =3.0 101− in 2 ln rφ hand, the zero mode factors of the fluctuations about f(r )= for simplicity.+3 r In this case, ζ may be underestimated Eq. (8). In what follows, we will clarifyφ the relationship×− 2 φ sph 2(1 rφ) 1 rφ − the sphaleron typically amount to about 10% [24]. This # by (10)% since$ (0) > (TC ). As will be shown be- between CP violation appearing in dµ and that− relevant − O E E is subdominant and, therefore, we will neglect them in to BAU. 1 3 low, this approximation does not affect our conclusion. − ln rφ ,A detailed analysis of ζsph(9)(T ) willour be given numerical elsewhere. analysis for simplicity. rφ≃ 1 − 2 ≪ In passing, recent studies show that ζ (T )=1.1 Wesph imposeC Eq.− (13) at a critical temperature TC at IV. BARYON NUMBER PRESERVATION2 2 1.2 in the real singlet-extended SMwhich [29] and theζsph eff(ectiveTC )= potential has two degenerate minima. with rφ = mτ /mφ. Non-degeneracy1.23 in of the the scale-invariant neutral Higgs 2HDM [13] for typical param- masses and proper choice of theeter overall sets. sign are essential In our analysis, TC and vC v(TC ) are determined us- The baryon number is generated via the sphaleron pro- ≡ for obtaining a sufficient δa . InIn Ref. Fig. [22], 1, v /T theand discrep-δa are showning finite-temperature in the (m ,m ) one-loop effective potential with cess outside the bubble (symmetric phase), and it canµ C C µ H A ancy of (g 2) between the experimentalplane. Here, we value take andmA = themH± tothermal avoid the resummation.electroweak As mentioned in Sec. III, sβ α survive if the sphaleron process is sufficientlyµ quenched − − ρ parameter constraint, and choose cβ α =0.006, ρτµ = inside the bubble (brokenSM prediction phase). To was this estimated end, the as is− close to 1| in| the region of interest to us. In such a ρµτ ,andφτµ + φµτ = π/4, as favored by the solution sphaleron rate in the broken phase, denoted as Γ(b)(T ), | | case, we may simplify the analysis of EWPT to a one- EXP SMB of (g 2)µ anomaly.10 For the remaining parameters, we must be smaller than the Hubbleδaµ = constant,aµ Ha(µT ).=(26 More .1 8.0)− 10− . (10) dimensional problem with a single order parameter as − ±set M =100GeV,tan× β =1andλ6 = λ7 =0asan explicitly, example. Contours of vC /TC arediscussed plotted with in the Refs. solid [10, 12]. As demonstrated in Ref. [17], thiscurves deviation in gray could with values be ac- of 1.0, 1.17,As 1.5, is well-known,2.0, 2.5, 3.0 the extra heavy Higgs bosons can (b) Esph(T )/T ΓB (T ) (prefactor)commodatede− for m ± [200from, 500] bottom GeV to with top, where an ap- 1.17 corresponds to ζ . Al- ≃ H,A,H ∈ play a majorsph role in enhancing vC /TC in the 2HDM. propriate mass2 hierarchy. Fromlowed EWBG 1σ,2σ pointand 3 ofσ regions view, of δaµ are shown by the2 2 plings;ζsph)andfavoredby otherwise, the so-called non-decoupling effects interactions by themselves19 do not play a central role in (g =110.75 in the 2HDM) and mP =1.22 10 GeV. (g 2)µ have an overlap if mA >mwouldH .Wenotethat diminish, rendering a suppressed vC /TC.Phe- ∗ the realization, as× will be shown in− the following sections. nomenological consequences of the non-decoupling effects The current limit on dµ is [23] at T = 0 include significant deviations in the h γγ de- → 19 cay width [25] and the triple Higgs coupling [26]. d < 1.9 10− e cm , (11) | µ| × For the evaluation of , we solve the equations of mo- tion for the sphaleron withE appropriate boundary condi- which is about three orders of magnitude larger than dµ 9 tions [27, 28]. Here, we use the tree-level Higgs potential estimated with Arg(ρµτ ρτµ)=1andδaµ =3.0 10− in Eq. (8). In what follows, we will clarify the relationship× for simplicity. In this case, ζsph may be underestimated by (10)% since (0) > (TC ). As will be shown be- between CP violation appearing in dµ and that relevant O E E to BAU. low, this approximation does not affect our conclusion. A detailed analysis of ζsph(T ) will be given elsewhere. In passing, recent studies show that ζ (T )=1.1 sph C − IV. BARYON NUMBER PRESERVATION 1.2 in the real singlet-extended SM [29] and ζsph(TC )= 1.23 in the scale-invariant 2HDM [13] for typical parameter sets. The baryon number is generated via the sphaleron pro- In Fig. 1, v /T and δa are shown in the (m ,m ) cess outside the bubble (symmetric phase), and it can C C µ H A plane. Here, we take m = m ± to avoid the electroweak survive if the sphaleron process is sufficiently quenched A H ρ parameter constraint, and choose cβ α =0.006, ρτµ = inside the bubble (broken phase). To this end, the − | | ρµτ ,andφτµ + φµτ = π/4, as favored by the solution sphaleron rate in the broken phase, denoted as Γ(b)(T ), | | B of (g 2)µ anomaly. For the remaining parameters, we must be smaller than the Hubble constant, H(T ). More − set M =100GeV,tanβ =1andλ6 = λ7 =0asan explicitly, example. Contours of vC /TC are plotted with the solid curves in gray with values of 1.0, 1.17, 1.5, 2.0, 2.5, 3.0 (b) Esph(T )/T Γ (T ) (prefactor)e− B ≃ from bottom to top, where 1.17 corresponds to ζsph. Al- 2 ζsph)andfavoredby 19 (g =110.75 in the 2HDM) and mP =1.22 10 GeV. (g 2)µ have an overlap if mA >mH .Wenotethat ∗ × − Baryon number density

is a function of ρττ, ρτμ and ρμτ

- Leading effect: ρττ

- Subheading effect: ρτμ and ρμτ .

2HDM with LFV explains h->μτ, muon g-2, and BAU. Sphaleron ■ A static saddle point solution w/ ﬁnite energy of the gauge-Higgs system. [N.S. Manton, PRD28 (’83) 2019]

Energy

sphaleron B =0 Esph Instanton: quantum tunneling Sphaleron: thermal ﬂuctuation instanton

-1 0 1 Ncs

B+L anomaly 3 jµ = g2Tr(F F˜µ ) g2B B˜µ , µ B+L 162 2 µ 1 µ µ µjB L =0, g2 2 B =3N N = 2 d3x Tr F A g A A A CS CS 32⇥2 ijk ij k 3 2 i j k ⇤ ⇥ Eigenvalue problem Hamiltonian:

Band energy is determined by solving [N.L.Balazs, Ann.Phys.53,421 (1969)]

with 3 connection formulas depending on energy.

over-barrier parabolic potential

linear potential hhh coupling in the 2HDM [update of Kanemura, Okada, E.S., PLB606,(2005)361] 450 h 400 strong 1st-order EWPT h 350 300

h 250 - 1st-order EWPT is induced 200 by heavy Higgs bosons. 150 100 - hVV and hff can be SM-like. 50 0 0 20 40 60 80 100 120 140 - Δλhhh>(15-20)%

450

400

350

300

250

200

150

100

0 0 20 40 60 80 100 120 140

Figure 1: Contour plot of ∆λhhh/λhhh and ϕC /TC in the 2HDM.

1 22 ATLAS ATLAS Simulation Preliminary 25 Simulation Preliminary 20 s=14 TeV, 3000 fb-1 s=14 TeV, 3000 fb-1 18 H(bb)H( γγ ) t H(t γγ ) H(bb)H( γγ ) t H(t γγ ) 16 bbH( γγ ) t Xt 20 bbH( γγ ) t Xt Z(bb)H( γγ ) bb γγ Z(bb)H( γγ ) bb γγ 14 Events/10 GeV Others Others Events/2.5 GeV 12 15 10 8 10 6 2 Higgs Self-Coupling Phenomenology 4 5 2 Higgs boson pair production from gluon fusion can be described at leading order (LO) by the Feyn- 0 man diagrams shown in Figure0 1. Only the diagram on the left hand side includes a contribution from 50 100 150the 200 triple Higgs 250 coupling, whereas50 in the 100 case of 150the diagram 200 on the right 250 hand side the self-coupling constantm does[GeV] not play a role. Both diagrams contain fermionic loops and are dominated by the con- bb m γγ [GeV] tribution fromhhh the top quark. Therecoupling is a relative minus sign betweenat the twoLHC contributions, resulting (a) mbb in destructive interference that eﬀectively reduces(b) m theγγ total Higgs pair production cross section in the Standard Model. 1 Figure 7: The distributions of mbb (a) and mγγ (b) for 3000 fb− after applying all the selection criteria except the mbb (a) and mγγ (b) mass cuts. The individual shapes of the contributions are obtained using the events surviving the event selection before the mass criteria and angular cuts are applied, but normalized to the number of expected events after the full event selection. The ttX contribution includes tt¯( 1 lepton) and tt¯γ, while ‘Others’ includes cc¯γγ, bb¯γ j, bbjj¯ and jjγγ. ≥ Figure 1: Feynman diagrams describing Higgs pair production from gluon fusion at LO.

σ / σSM as a function of λ / λSM 40 ATLAS SM Simulation Preliminary

2 σ 8 35 / ATL-PHYS-PUB-2014-019 LO σ ATL-PHYS-PUB-2015-046LO NLO 10 NLO

HH) [pb] 1.5 HH) [pb] 7 NNLO NNLO 30 → →

(pp (pp 61 σ 1 σ 25 -1 105 0.5 20 -2 95% CL upper limit on 104 0 15 ATLAS Simulation-10 -5 Preliminary 0 5 10 -10 -5 0 5 10 -1 SM 3 SM s = 14 TeV: 3000 fb λHHH / λHHH λHHH / λHHH 10 2 Exp. 95% CL Exp. 95% CLs ± 1 σ -1 5 Figure 2: The dependence of the inclusive± 1 σ Higgs pair production± cross 2 σ section at √s = 14 TeV on L dt = 3000 fb ± 2 σ 1 had-had selection ∫ λHHH, on the left with a linear y-scale and with a log y-scale on thelep-had right. e selection The LO and NLO values are 0 lep-had µ selection s = 14 TeV

Projected limit on the total HH yield (events) -2obtained 0 with 2 the HPAIR 4 program 6 [9], and 8 for NNLO 10 the results0 from Ref. [4, 5] are used. λ / λ −10 −5 0 5 10 15 SM λ / λSM

Figure 8: 95% CLs limit (dashed line)This on effect the can size be of seen an in additional FigureFigure2 (left), HH 9: 95% contributio where Confidence di-Higgsn summed Level cross upper sections with limit the for on the diff crosserent section values of the the HH bb⌧⌧ assuming Standard Model Access to λhhh of 2HDM at the LHC is challenging. ! expected Standard Model HH yield,self-coupling as wellλ asHHH theare shown,1/2σ uncertainty atcouplings LO, next-to-leading is bands, shown overlaidin order the dashed (NLO), on line the and with number next-to-next-to- its 68% and 95%leading error bands. order The solid black, dashed blue and dotted ± violet lines show a fit of the expected number of events normalised by the SM number of events for dierent HHH of predicted total (box + self-coupling)(NNLO). HH A value events of asλHHH a function= 0 corresponds of the λ/ toλSM theratio case where after application there is no self-coupling of of the Higgs ⌧ ⌧ ⌧ ⌧ the analysis cuts (solid line). boson, and thus the amplitude of theafter left the diagram selection in for Figure the had1 vanishes.had, lep had Forelectron this case and the muon cross channels. section is enhanced by approximately a factor of two compared to the Standard Model [10, 11]. The cross section decreases with increasing values of8 the Conclusions self-coupling up to a value of 2.44 times the Standard Model SM value (λHHH) where the cross section is at its minimum. Figure 2 (right) shows that the cross-section is never zero. For larger values of λHHH the cross-section increases again. Due to the (approximately) parabolic shape of the cross-section,Cut and measuring count studies only the have total been cross performed section on for Monte the pair Carlo production simulation under several eective assumptions of the Higgs trilinear self-coupling constant HHH for three dierent channels on the HH bb⌧⌧ final process does not allow12 the value of the self coupling constant to be inferred but the degeneracy could ! be removed by further measurementsstate. of In its each dependence case, the on selection kinematical has been varia optimisedbles. to maximise the signal to background ratio, considering Figure 2 also shows that themost differences of the betweenreducible cross-section and irreducible predictions backgrounds. at diff Aerent parametrisation order in of the ATLAS detector has been pQCD are large. The NNLO valuesused are to used estimate in the the remainder impact of of this its performance note. in rejecting the backgrounds. As a final result, the expected significance for detecting the signal has been calculated combining the dierent channels under a 3% luminosity uncertainty and a 3% background modelling uncertainty assumption for the2 main backgrounds. Under such conditions, it is expected that the signal would have a significance of 0.60, while if the eective Higgs trilinear self-coupling HHH is twice the SM prediction, a significance of 0.40 could be reached and if it is zero, the significance would be 0.84. Assuming we have Standard Model data, we can also set an upper limit of 4.3 (HH bb¯⌧+⌧ ) at 95% Confidence ⇥ ! Level on the signal cross section. Finally, we can project an exclusion at 95% Confidence Level of BSM HH production with HHH/ 4 and HHH/ 12. SM  SM

17 22 ATLAS ATLAS Simulation Preliminary 25 Simulation Preliminary 20 s=14 TeV, 3000 fb-1 s=14 TeV, 3000 fb-1 18 H(bb)H( γγ ) t H(t γγ ) H(bb)H( γγ ) t H(t γγ ) 16 bbH( γγ ) t Xt 20 bbH( γγ ) t Xt Z(bb)H( γγ ) bb γγ Z(bb)H( γγ ) bb γγ 14 Events/10 GeV Others Others Events/2.5 GeV 12 15 10 8 10 6 2 Higgs Self-Coupling Phenomenology 4 5 2 Higgs boson pair production from gluon fusion can be described at leading order (LO) by the Feyn- 0 man diagrams shown in Figure0 1. Only the diagram on the left hand side includes a contribution from 50 100 150the 200 triple Higgs 250 coupling, whereas50 in the 100 case of 150the diagram 200 on the right 250 hand side the self-coupling constantm does[GeV] not play a role. Both diagrams contain fermionic loops and are dominated by the con- bb m γγ [GeV] tribution fromhhh the top quark. Therecoupling is a relative minus sign betweenat the twoLHC contributions, resulting (a) mbb in destructive interference that eﬀectively reduces(b) m theγγ total Higgs pair production cross section in the Standard Model. 1 Figure 7: The distributions of mbb (a) and mγγ (b) for 3000 fb− after applying all the selection criteria except the mbb (a) and mγγ (b) mass cuts. The individual shapes of the contributions are obtained using the events surviving the event selection before the mass criteria and angular cuts are applied, but normalized to the number of expected events after the full event selection. The ttX contribution includes tt¯( 1 lepton) and tt¯γ, while ‘Others’ includes cc¯γγ, bb¯γ j, bbjj¯ and jjγγ. ≥ Figure 1: Feynman diagrams describing Higgs pair production from gluon fusion at LO.

σ / σSM as a function of λ / λSM 40 ATLAS SM Simulation Preliminary

2 σ 8 35 / ATL-PHYS-PUB-2014-019 LO σ ATL-PHYS-PUB-2015-046LO NLO 10 NLO

HH) [pb] 1.5 HH) [pb] 7 NNLO NNLO 30 → →

Projected limit on the total HH yield (events) -2obtained 0 with 2 the HPAIR 4 program 6 [9], and 8 for NNLO 10 the results0 from Ref. [4, 5] are used. λ / λ −10 −5 0 5 10 15 SM λ / λSM

17 Chapter 9 Summary

A summary of all model independent coupling precisions is given in Table 9.1.

Table 9.1. Summary ofHiggs expected accuraciescouplingsgi/gi formeasurements@ILC model independent determinations of the Higgs boson couplings. The theory errors are Fi/Fi =0.1%.Fortheinvisiblebranchingratio,thenumbersquotedare95% conﬁdence upper limits. ILC white paper, 1310.0763 ILC(250) ILC(500) ILC(1000) ILC(LumUp) Ôs (GeV) 250 250+500 250+500+1000 250+500+1000 1 L(fb≠ )250250+500250+500+10001150+1600+2500 ““ 18 % 8.4 % 4.0 % 2.4 % gg 6.4 % 2.3 % 1.6 % 0.9 % WW 4.8 % 1.1 % 1.1 % 0.6 % ZZ 1.3 % 1.0 % 1.0 % 0.5 % tt¯ –14%3.1%1.9% b¯b 5.3 % 1.6 % 1.3 % 0.7 % + · · ≠ 5.7 % 2.3 % 1.6 % 0.9 % cc¯ 6.8 % 2.8 % 1.8 % 1.0 % + µ µ≠ 91% 91% 16 % 10 % T (h) 12 % 4.9 % 4.5 % 2.3 % hhh –83%21%13% BR(invis.) < 0.9 % < 0.9 % < 0.9 % < 0.4 %

For the purpose of comparing ILC coupling precisions with those of other facilities we present the coupling errors in Table 9.2.

Table 9.2. Summary of expected accuracies gi/gi of Higgs boson couplings using, for each coupling, the fitting technique that most closely matches that used by LHC experiments. For gg,g“ ,gW ,gZ ,gb,gt,g· , T (h) the seven parameter HXSWG benchmark parameterization described in Section 10.3.7 of Ref. [206] is used. For the couplings gµ, ghhh and the limit on invisible branching ratio independent analyses are used. The charm coupling gc comes from our 10 parameter model independent fit. All theory errors are 0.1%.Fortheinvisiblebranchingratio,the numbers quoted are 95% confidence upper limits.

ILC(250) ILC(500) ILC(1000) ILC(LumUp) Ôs (GeV) 250 250+500 250+500+1000 250+500+1000 1 L(fb≠ )250250+500250+500+10001150+1600+2500 ““ 17 % 8.3 % 3.8 % 2.3 % gg 6.1 % 2.0 % 1.1 % 0.7 % WW 4.7 % 0.4 % 0.3 % 0.2 % ZZ 0.7 % 0.5 % 0.5 % 0.3 % tt¯ 6.4 % 2.5 % 1.3 % 0.9 % b¯b 4.7 % 1.0 % 0.6 % 0.4 % + · · ≠ 5.2 % 1.9 % 1.3 % 0.7 % T (h) 9.0 % 1.7 % 1.1 % 0.8 % + µ µ≠ 91 % 91 % 16 % 10 % hhh –83%21%13% BR(invis.) < 0.9 % < 0.9 % < 0.9 % < 0.4 % cc¯ 6.8 % 2.8 % 1.8 % 1.0 %

In the energy and luminosity scenarios discussed in this paper it was assumed that the luminosity upgrades at 250 and 500 GeV center of mass energy occurred after the energy upgrade at 1000 GeV. It is of interest to consider a scenario where the 250 GeV and 500 GeV luminosity upgrade running

135 Chapter 9 Summary

A summary of all model independent coupling precisions is given in Table 9.1.

For the purpose of comparing ILC coupling precisions with those of other facilities we present the coupling errors in Table 9.2.

135 BAU vs. electron EDM

C.-W. Chiang et al. / Physics Letters B 762 (2016) 315–320 319 in Fig. 1 and the heavy Higgs boson masses m 350 GeV and H = m m 400 GeV, leading to v /T 214.9GeV/99.2GeV A = H± = C C = = 2.17. As an ansatz for Y SM, we consider

√2me/v 00 Y SM 03.31 10 3 6.81i 10 4 , (18) = ⎛ × − − × − ⎞ 08.91i 10 3 3.70 10 3 × − × − ⎝ ⎠ which is diagonalized by 10 0 V e 0 0.365i 0.931i , (19) R − − = ⎛0 0.931 0.365 ⎞ − ⎝10 0⎠ V e 0 0.945i 0.327i . (20) L − − = ⎛0 0.327 0.945 ⎞ − ⎝ ⎠ Also, we take φττ π/2. Note that each of Y1,2 is fixed by e† SM e = V L Y V R Y D and Eq. (5) once ρij are given. = obs The solid lines represent the contours of Y B /Y 0.5(left) B = Fig. 3. Correlations between BAU and electron EDM in the ( ρττ , φττ) plane. The 29 | | 29 and 1.0(right). The dashed lines in black and red represent Br(h contours shown in white are de 2.0 10− e cm (left) and 3.0 10− e cm → | |obs= × × µτ ) 1.43% and 0.84%, respectively. The colored regions with the (right). On the other hand, Y B /Y 0.5(left) and 1.0 (right) are denoted by the = B = same color scheme as in Fig. 1 explain the (g 2)µ anomaly. In contours in black. − our setup, the dominant contribution to the CP-violating source 8 with φtt φττ gives Br(τ µγ ) 2 10 for ρττ 1. The im- term comes from SτL µR which is induced by the 32 elements of − φ≃ Y → ≃ × | | ≃ Y1 and Y2. We find that these off-diagonal elements have stronger pact of tt on B highly depends on the Yukawa structure of the dependences on ρττ than ρτµ and ρµτ . Under rather generous top quark. With our assumption, the top quark does not provide | | | | | | assumptions for the bubble wall profile, the generated BAU can dominant CP violation for the baryogenesis. reach its observed value for ρτµ 0.1 0.6and ρττ 0.8 Since ρττ and ρtt are complex, they may induce an elec- | | ≃ − | | ≃ − 0.9. Our main conclusion is that there is a parameter space that tron EDM through two-loop Barr–Zee diagrams, among which one is consistent with both experimental anomalies of h µτ and Higgs-photon mediated loop diagram is dominant. In Fig. 3, de → obs | | (g 2)µ as well as the observed BAU. and Y B /Y B are plotted as functions of ρττ and φττ. The con- − |29 | Some comments on the theoretical uncertainties are in order. tours in white represent de 2.0 10− e cm (left) and 3.0 29 | | = × × (1) We take the same size of %β as in Ref. [9] as a reference 10− e cm (right). de with ρττ 1and φττ π/2reaches a | | | | 29 | | = = value. However, quantitative studies of it in the 2HDM are still ab- maximal value of 3.5 10− e cm which is slightly smaller than 2 4 × 29 sent. In the MSSM, %β (10 10 ) depending on m [42]. the current bound on electron EDM, de < 8.7 10− e cm [53]. = O − − − A | | × In the next-to-MSSM, %β can reach (0.1) in some parameter This is due to the facts that the heavy Higgs boson couplings to the O space [43]. We should note that Y B is mostly subject to the un- electron is suppressed by cβ α and that the extra Yukawa coupling − certainties of %β among others since it is linearly proportional ρee is absent. It should be noted that the complex phases of ρτµ to %β . (2) Studies on v w can be found in Refs. [44–47], which and ρµτ do not contribute to de via the Higgs-photon mediated suggest that 0.1 ! v w ! 0.6in non-SUSY models. For stronger first- Barr–Zee diagrams since the internal photon line cannot change order EWPT, v w tends to be large and may reduce Y B to some the fermion flavors. As discussed in Sec. 3, on the other hand, the extent. In our analysis, we simply adopt the lowest value as the muon EDM is induced by those FCNH couplings at one-loop level, 22 most generous choice. A more precise determination of v us- giving rise to dµ 3 10− e cm for the current parameter set. w | | ≃ × ing our input parameters is definitely indispensable to obtaining We also find that τ µνν can give some constraints on ρτµ h → ¯ | | more precise Y . (3) As mentioned above, the treatment of ' is and ρµτ as well as mH and can be similar to the constraint B | | ± somewhat tricky. It is found that Y may change by a factor of coming from Br(h τµ) < 1.43. B → a few or more, depending on at which scale the k-dependent 'h is put in. (4) The VEV-insertion method used here is vulnerable 6. Conclusion to theoretical uncertainties (see, e.g., Refs. [5,48]) and may lead to an overestimated BAU compared to an all-order VEV resumma- We have studied electroweak baryogenesis in the general tion method [49–51]. However, a satisfactory formalism of the Y framework of the two-Higgs doublet model in light of the h µτ B → calculation beyond that is still not available, and hence the error and (g 2)µ anomalies. In this model, the heavy Higgs bosons − associated with the approximation cannot be properly quantified. with the appropriate µ-τ flavor violation can accommodate the In summary, the observed Y B can be marginally produced with above two anomalies. At the same time, these extra Higgs bosons the generous choice of the input parameters. However, a defini- can induce a strong first-order electroweak phase transition as re- tive statement cannot be made until the above-mentioned various quired for successful electroweak baryogenesis. theoretical uncertainties are fully under control. It is found that the µ-τ flavor-violating lepton sector has a We now turn to another experimental constraint. As ρττ great potential to generate sufficient baryon asymmetry of the | | increases, Br(τ µγ ) gets enhanced and, for ρττ 0.1, ex- Universe within the theoretical uncertainties. In this scenario, → | | " ceeds the current experimental upper bound, Br(τ µγ ) < 4.4 the interplay between ρτµ/ρµτ and ρττ is crucially important. 8 → × obs 10 [52]. However, as demonstrated in Ref. [17], an acciden- Our analysis suggests that Y B /Y 1for ρτµ 0.1 0.6 − B ≃ | | ≃ − tal cancellation between the one-loop and two-loop contributions and ρττ 0.8 0.9with (1) CP-violating phases. To sup- | | ≃ − O could occur in Br(τ µγ ) if the new top Yukawa coupling (de- press Br(τ µγ ), a cancellation mechanism has to be at work, → → noted as ρtt) took a nonzero value. In the current case, ρtt 0.5 which additionally predicts ρtt 0.5and φtt φττ. Since fu- | | ≃ | | ≃ ≃ Constraints in (cos(β-α), tanβ) plane arXiv:1509.00672

2HDM Type I ATLAS 2HDM Type II ATLAS

Obs. 95% CL Obs. 95% CL -1 s = 7 TeV, 4.5-4.7 fb-1 s = 7 TeV, 4.5-4.7 fb Best fit -1 Best fit -1 s = 8 TeV, 20.3 fb s = 8 TeV, 20.3 fb Exp. 95% CL Exp. 95% CL SM SM β β 10 10 tan tan 4 4 3 3 2 2

1 1

0.4 0.4 0.3 0.3 0.2 0.2

−1 10−1 10 −1 −0.8 −0.6−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 cos(β-α) cos(β-α) (a) Type I (b) Type II

2HDM Lepton-specific ATLAS 2HDM Flipped ATLAS Obs. 95% CL Obs. 95% CL s = 7 TeV, 4.5-4.7 fb-1 s = 7 TeV, 4.5-4.7 fb-1 Best fit Best fit -1 s = 8 TeV, 20.3 fb-1 s = 8 TeV, 20.3 fb Exp. 95% CL Exp. 95% CL SM SM β β 10 10 tan tan 4 4 3 3 2 2

1 1

0.4 0.4 0.3 0.3 0.2 0.2

10−1 10−1 −1 −0.8 −0.6−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 cos(β-α) cos(β-α) (c) Lepton-speciﬁc (d) Flipped

Figure 5: Regions of the [cos( ↵), tan ] plane of four types of 2HDMs excluded by ﬁts to the measured rates of Higgs boson production and decays. The likelihood contours where 2 ln ⇤ = 6.0, corresponding approximately to the 95% CL (2 std. dev.), are indicated for both the data and the expectation for the SM Higgs sector. The cross in each plot marks the observed best-ﬁt value. The light shaded and hashed regions indicate the observed and expected exclusions, respectively. The ↵ and parameters are taken to satisfy 0 ⇡/2 and 0 ↵ ⇡ without loss     of generality.

16 14 Constraint on μ-τ coupling 9 Summary 1502.07400 CMS 19.7 fb-1 (8 TeV) 1 | µ τ τ→ 3µ |Y 10-1

τ→ µ γ -2 ATLAS H→ττ 10 |Y µ τ Y observed τ µ |=m

µ m expected τ /v H→µτ 2 10-3 BR<0.1% BR<1% BR<10% BR<50%

10-4 10-4 10-3 10-2 10-1 1 |Y | µτ Figure 6: Constraints on the flavour-violating Yukawa couplings, Y and Y . The black | µt| | tµ| dashed lines are contours of (H µt) for reference. The expected limit (red solid line) B ! with one sigma (yellow) and two sigma (green) bands, and observed limit (black solid line) are derived from the limit on (H µt) from the present analysis. The shaded regions are B ! derived constraints from null searches for t 3µ (dark green) and t µg (lighter green). The ! ! yellow line is the limit from a theoretical reinterpretation of an ATLAS H tt search [4]. The ! light blue region indicates the additional parameter space excluded by our result. The purple diagonal line is the theoretical naturalness limit Y Y m m /v2. ij ji  i j significance of 2.4 s is observed, corresponding to a p-value of 0.010. The best fit branching +0.39 fraction is (H µt)=(0.84 0.37)%. A constraint of (H µt) < 1.51% at 95% confidence B ! B ! level is set. The limit is used to constrain the Yukawa couplings, p Y 2 + Y 2 < 3.6 10 3. | µt| | tµ| ⇥ It improves the current bound by an order of magnitude.

Acknowledgments We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); MoER, ERC IUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NIH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Re- public of Korea); LAS (Lithuania); MOE and UM (Malaysia); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS and RFBR (Russia); MESTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine);