Top Seesaw, Custodial Symmetry and the 126 GeV (Composite) Higgs IAS Program on the Future of High Energy Physics Jan 21, 2015 based on arxiv:1311.5928, Hsin-Chia Cheng, Bogdan A. Dobrescu, JG JHEP 1408 (2014) 095 and arxiv:1406.6689, Hsin-Chia Cheng, JG JHEP 1410 (2014) 002
Introduction NJL Model, Top Condensation & Top Seesaw The Minimal Model Extension with Custodial Symmetry Conclusion
Why Composite Higgs? Hierarchy problem. One solution: no light fundamental scalar! Composite Higgs that no longer exists above the compositeness scale. No new physics at LHC yet! Small hierarchy? New strong dynamics at the compositeness scale. Usually predicts a heavy Higgs due to large quartic couplings, unless the Higgs mass is protected by some symmetry. Higgs boson as a pseudo Nambu-Goldstone boson (PNGB). (Holographic Higgs, Little Higgs...) Top Condensation Top Seesaw.
The Nambu-Jona-Lasinio Model Consider some theory at scale Λ with an effective four-fermion vertex L Λ = ψ L i / ψ L + ψ R i / ψ R + g2 Λ 2 (ψ L ψ R)(ψ L ψ R ). (1) Eq. (1) can be rewritten in the following form with an auxilliary field H L Λ = ψ L i / ψ L + ψ R i / ψ R + (gψ L ψ R H + h.c.) Λ 2 H H. (2) (A short review of the NJL model can be found in the appendix of arxiv:hep-ph/0203079 (C. T. Hill & E.H. Simmons).)
The Nambu-Jona-Lasinio Model Evolving down to scale µ with the fermion bubble approximation which generates kinetic and quartic terms of the H field and also gives a correction to the mass term. After normalizing the kinetic term of H, we have L µ = ψ L i / ψ L + ψ R i / ψ R + (ξψ L ψ R H + h.c.) + νh 2 m 2 HH H λ 2 (H H) 2, (3) where ξ 2 = 16π 2 N c log (Λ 2 /µ 2 ), λ = 32π 2 N c log (Λ 2 /µ 2 ). (4)
The Nambu-Jona-Lasinio Model One can think of H as a composite particle of the fermions, while Λ is the compositeness scale, at which the couplings are strong. The Yukawa coupling and the quartic coupling are related by λ = 2ξ 2. (5) m 2 H < 0 if g is large enough. (Spontaneous symmetry breaking!) If the theory is spontaneously broken, λ = 2ξ 2 implies m h = 2m f. (6) The results are subject to change when effects of other interactions are included.
Top Condensation ( ) tl Is it possible that H t R? b L Instead of the fermion bubble approximation, the full one-loop RG equations are used. [Phys. Rev. D 41, 16471660 (1990), (Bardeen, Hill, Lindner)] To get the right Electroweak VEV, top quark is too heavy unless the compositeness scale is extremely large. (Need the top Yukawa coupling to be very large at Λ and be 1 at weak scale.) Λ = 10 5 GeV m top 360 GeV. Λ = 10 19 GeV m top 220 GeV. m h m top. It doesn t work!
Top Condensation When a model doesn t work, we have two options. option 1: Give up.
Top Seesaw option 2: Modify the model until it works! Minimal modification: add a new vector-like top partner. The vector-like top partner mixes with the SM top and induces the top seesaw mechanism. With the top seesaw mechanism, one can have a large ( 1 ) Yukawa coupling while keeping the correct top mass ( 173 GeV ). A number of papers at the end of last century arxiv:hep-ph/9712319 (Dobrescu, Hill) arxiv:hep-ph/9809470 (Chivukula, Dobrescu, Georgi, Hill) arxiv:hep-ph/9908391 (Dobrescu)...
What s new? LHC has found the Higgs at around 126 GeV! the minimal model We found that by imposing an approximate U(3) L symmetry, the Higgs mass has a rather restricted range and we can easily obtain a 126 GeV Higgs. Constraint from T parameter forces this model to be fine-tuned. extension with custodial symmetry We could extend the model to embed custodial symmetry and reduce fine tuning. A 126 GeV Higgs is still easily obtained. Not minimal anymore!
Introducing a new vector-like quark We introduce a new SU(2) W -singlet vector-like quark, χ of electric charge +2/3. The following fermions form composite scalars due to strong interactions at scale Λ, which approximately preserves U(3) L U(2) R chiral symmetry. t R χ R t L b L H t H χ χ L φ t φ χ 2 doubles + 2 singlets, collective labels: ( ) ( ) Ht Hχ Φ t =, Φ χ =, Φ = ( ) Φ t Φ χ. (7) φ t φ χ
Yukawa couplings and scalar potential Yukawa couplings ξ 1. We will use ξ 3.6 as a reference value. ) L Yukawa = ξ (ψ 3L χ L Φ ( tr χ R ) + H.c. (8) Scalar potential V Φ = λ [ 1 2 Tr (Φ Φ) 2] + λ ( 2 2 Tr[Φ Φ]) + M 2 Φ Φ Φ. (9) 2 The quartic couplings λ 1 and λ 2 are related to the Yukawa coupling ξ. Fermion bubble approximation: λ 1 /(2ξ 2 ) = 1, λ 2 = 0. λ One loop RG running: 1 2ξ 0.4, λ 2 2 λ 1 0.2. We use 0.4 λ 1 2ξ 2 1, 0.2 λ 2 0. λ 1 (10) A U(3) L U(2) R invariant scalar potential won t produce the correct phenomenology at low energy!
Scalar potential We introduce additional explicit U(2) R breaking effects in the mass term which distinguish t R and χ R. SM gauge invariant mass terms at scale Λ L mass = µ χt χ L t R µ χχ χ L χ R + H.c. (11) map to tadpole terms for the SU(2) W -singlet scalars below Λ. The effective potential of the scalar sector has the following form V scalar = λ [ 1 2 Tr (Φ Φ) 2] + λ ( ) 2 2 Tr[Φ Φ] 2 +Mtt 2 Φ t Φt + M2 χχφ χφ χ + (MχtΦ 2 χφ t + H.c.) (0, 0, C χt)φ t (0, 0, C χχ)φ χ + H.c. (12)
Chiral symmetry breaking scale Neutral scalars may develop VEVs Φ t = 1 v t 0, Φ χ = 1 χ v 0 (13) 2 u t 2 u χ where v 2 v 2 t + v 2 χ, u 2 u 2 t + u 2 χ, f u 2 + v 2. (14) f is the chiral symmetry breaking scale. We expect Λ 4πf for f to be natural. (But we have v f which requires tuning!) We choose the basis in which v t = 0, v χ = v and define the angle γ as u t = u sin γ, u χ = u cos γ. (15) sin γ 1!
Top Seesaw Mechanism t and χ forms a 2 2 mass matrix ξ ( ) ( 0 t L, χ L 2 us γ ) ( v tr uc γ χ R ) + H.c. (16) The mass of the top quark is suppressed by s γ so that ξ can be much larger than one. m t ξ v sγ 2 s γ yt ξ. (17) We can obtain the correct top mass while keeping the compositeness scale relatively small. The heaver eigenstate is the top partner and has mass m t ξ f 2. (18)
Higgs mass The 126 GeV Higgs boson is the lightest mass eigenstate of the 4 CP-even neutral scalars. It is a PNGB of the approximate U(3) L /U(2) L symmetry breaking. Keeping the leading order terms in v 2 /f 2 and s γ( u t u ), M 2 h λ 1 2ξ 2 ( 1 + λ 1m 2 t ξ 2 M 2 H ± ) 1 y 2 t v 2. (19) With 0.4 λ 1 1, we have M 2ξ 2 h 185 GeV. (Numerical study gives M h 175 GeV.) Strong connection between M h and m top in top seesaw models! The loop contribution of the EW gauge bosons produces additional U(3) L breaking effects and further reduces the Higgs mass. We cut off this contribution by M ρ.
We can have a 126 GeV Higgs! f TeV 10 8 6 4 T 0.1 T 0.2 M h GeV Λ 1 2 Ξ 2 0.7 M Ρ 0 M Ρ 3 f M Ρ 5 f Ξ 3.6 2 0 2 4 6 8 10 12 M H ± f M h = 126 GeV can be obtained with reasonable parameters in our model. Main constraint comes from the T parameter (fermion loops). 68% bound T 0.1 corresponds to f 4.3 TeV (for ξ = 3.6). 95% bound T 0.15 corresponds to f 3.5 TeV (for ξ = 3.6). The heavy states are too heavy to be probed at the LHC! This is related to the fact that U(3) L does not contain a custodial SU(2) symmetry.
Extending the model to embed custodial symmetry If we extend the model to embed the SU(2) C custodial symmetry, the chiral symmetry breaking scale f may be reduced significantly. A naive extension is to introduce Bottom Seesaw with the addition of a bottom-partner ( embedding Sp(4) ). However, this suffers from the constraint on Z b b branching ratio. arxiv:hep-ph/9908330 (Collins, Grant, Georgi) arxiv:hep-ph/0108041 (He, Hill, Tait) The SM prediction for Z b b branching ratio is 0.8σ smaller than the measured value. If b mixes with a heavy singlet, the Z b b branching ratio is further reduced. This constrains the bottom partner to be very heavy, which makes it very hard to reduce f.
Extending the model to embed custodial symmetry We surrendered to the following paper, which constructed a custodial symmetry that also protects the Zb L bl coupling. arxiv:hep-ph/0605341 (Agashe, Contino, Da Rold, Pomarol) The Zb L bl coupling is protected under two conditions The new physics needs to be invariant under an O(4) global symmetry; b L has the same charge under SU(2) L and SU(2) R. In additional to the singlet top-partner χ, we introduce a pair of vector-like EW doublet quarks, (X, T ). ( ) tl X L transforms as (2, 2) under SU(2) L SU(2) R. b L T L (X, T ) has hypercharge +7/6. X has electric charge +5/3, T has electric charge +2/3. (X, T ) can have a large Dirac mass term which maps to the tadpole term of the corresponding scalars.
Particles and couplings X R T R t R χ R t L σ tx σtt 0 σtt 0 φ 0 tχ b L σ bx σ bt σ bt φ bχ X L σxx 0 σ + XT σ + Xt φ + Xχ T L σ TX σtt 0 σtt 0 φ 0 T χ χ L σ χx σχt 0 σχt 0 φ 0 χχ collective label Σ X Σ T Σ t Φ χ L Yukawa = ξψ L ΦΨ R + H.c., (20) V = λ 1 2 Tr[(Φ Φ) 2 ] + λ 2 2 (Tr[Φ Φ]) 2 +M 2 X Σ X Σ X + M 2 X Σ T Σ T + M 2 t Σ t Σt + M2 χφ χφ χ c X σ 0 XX c X σ 0 TT c χtσ 0 χt c χχφ 0 χχ + H.c. (21) Too complicated!
The low energy theory Assuming all the degrees of freedom in Σ X,T,t are heavy, we integrate them out, keeping terms up to O( 1 ). M 2 ( ) ( φ 0 Two EW doublets tχ φ + ) φ Xχ bχ φ 0, and one singlet φ 0 χχ, with VEVs v t, v T T χ and u χ. We add in additional mass terms to break U(5) L down to SO(5). We Include the contribution from EW gauge boson loops. The masses of the X, T quarks explicitly break custodial symmetry. We calculated the T -parameter and found it is acceptable in a large region of the parameter space.
We can have a 126 GeV Higgs with f 1 TeV! tan Β 1.5 1.4 1.3 1.2 1.1 150 T 0.15 T 0.1 100 126 T 0 T 0.06 f 1 TeV M h GeV 50 ξ = 3.6, λ 1 = 0.7, λ 2ξ 2 2 = 0, f = 1 TeV, λ 1 M ρ = 3f, M ΣX,T,t = 10 TeV. tan β = v t v T, m X is the mass of the heavy quark X with charge +5/3. We can get a 126 GeV Higgs with f 1 TeV! Smaller f less tuning (v 2 /f 2 can be improved to 5%, compared to 0.5% in the minimal model). T 0.11 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 m X TeV Need m X f to have the correct Higgs mass. (current LHC bound: m X 0.8 TeV)
Conclusion The Top Seesaw Model is a modification of Top Condensation by introducing a new vector like top partner. It addresses the origin of both electroweak symmetry breaking and top Yukawa coupling. The Higgs mass is related to the top mass and has a rather restricted range, M h 175 GeV, and one can easily obtain a 126 GeV Higgs. Constraint from T parameter requires the chiral symmetry breaking scale to be much higher than the electroweak scale, which requires tuning. By introducing a pair of vector-like EW doublet quarks (X, T ) and extending the scalar sector, we can embed custodial symmetry in the model and bring down the chiral symmetry breaking scale. Crossroads! More tuning or more complexity? The LHC is not enough. We need the 100 TeV collider!
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Phenomenology of the extended model The X (+5/3) and T (+2/3) quarks are the lightest new states in the model and carry color. They are the first particles to be discovered if this model is realized in nature. To unravel the underlying theory we would need to find other states and study their properties. If X and T are excluded up to a few TeV, the extended model is as fine-tuned as the minimal model! The Higgs couplings are roughly given by 1 v 2 /(2f 2 ) times the SM value (deviation 3% for f = 1 TeV), which can be probed by a future e + e collider.
Composite Higgs vs. Composite Higgs Holographic composite Higgs: (Agashe, Contino, Pomarol,...) Global symmetry is preserved by the Strong Sector. Explicit breaking comes from coupling to SM particles. Higgs mass is related to the top partner masses which cut off the radiative contributions. Top seesaw models Explicit breaking comes from fermion mass terms (similar to QCD). Higgs mass is related to top mass through the top seesaw mechanism.
Using RGE to estimate λ 1 /(2ξ 2 ) and λ 2 /λ 1 The Yukawa coupling ξ and the quartic couplings λ 1, λ 2 are related. V quartic = λ [ 1 2 Tr (Φ Φ) 2] + λ ( 2 2 Tr[Φ Φ]). (22) 2 Fermion bubble approximation: λ 1 /(2ξ 2 ) = 1, λ 2 = 0. One loop RG running: λ 1 2ξ 2 0.4, λ 2 λ 1 0.2. coupling ratios 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 10 5 GeV Λ1 2Ξ 2 Ξ 5, 20 Λ2 Λ1 Ξ 5, 20 0 1 2 3 4 5 Μ log 10 1 GeV We can t fully trust RGE! Instead, we use 0.4 λ 1 2ξ 2 1, 0.2 λ 2 λ 1 0. (23)
U(3) L breaking from electroweak interactions Additional explicit U(3) L breaking effects can feed into the mass and quartic terms through loops. SU(2) W U(1) Y gauge interactions explicit break U(3) L! We assume this contribution is cut off by M ρ, presumably the mass of some vector state in the theory. In leading order of s γ and v 2 /f 2, the 1-loop corrections from EW gauge bosons are given by (µ m t ξf / 2) M 2 h ( λ 1 + λ 2 4 3(g 2 1 + 3g 2 2) 16π 2 log ( Mρ µ ) 3(g2 1 + 3g2) 2 64π 2 M 2 ρ f 2 ) v 2. (24) The contribution reduces the Higgs mass. With not too large M ρ ( 5f ), we can still get the correct Higgs mass.
T parameter The heavy fermion t can give a large contribution to the T parameter, which is related to the fact that U(3) L does not contain a custodial SU(2) symmetry. The main contribution is captured by fermion loops [ 3 T = slm 4 2 16π 2 αv 2 t + 2s2 L(1 sl) 2 m2 t m2 t m 2 t m 2 t ( ) ] m 2 t ln sl(2 2 sl)m 2 2 mt 2 t, (25) which can be rewritten in terms of ξ and f as T [ 3 v 2 ξ 2 16π 2 αf 2 2 + 4m2 t ln ( ξf ) 2mt 2 2mt ]. (26)
Constraint from T parameter 10 T 8 0.025 f TeV 6 0.05 0.1 4 0.2 2 0.4 2.0 2.5 3.0 3.5 4.0 4.5 Ξ Assuming S 0, we have 68% bound T 0.1 corresponds to f 4.3 TeV (for ξ = 3.6) 95% bound T 0.15 corresponds to f 3.5 TeV (for ξ = 3.6) Large f implies large fine tuning (roughly measured by v 2 /f 2 ) new particles are heavy
Upper bound on Higgs mass 1.0 0.9 100 126 170 150 140 Λ1 2Ξ 2 0.8 0.7 0.6 0.5 0.4 90 126 90 126 M h GeV 150 f 4 TeV Ξ 3.6 MΡ 0 MΡ 3 f MΡ 5 f 2 4 6 8 M H ± f Mh GeV 130 120 110 100 90 Λ1 2Ξ 2 0.7 Ξ 3.6, f 4 TeV, Λ2 0 Ξ 3.0, f 4 TeV, Λ2 0 Ξ 4.5, f 4 TeV, Λ2 0 Ξ 3.6, f 10 TeV, Λ2 0 Ξ 3.6, f 4 TeV, Λ2 0.2Λ1 MΡ 3 f 80 0 2 4 6 8 M H f The Higgs mass in the leading order is sensitive to λ 1 /(2ξ 2 ), M H ±/f and M ρ/f. The dependences on f, ξ and λ 2 is mild. A larger Higgs mass occurs for larger M H ±/f, λ 1 /(2ξ 2 ) and smaller M ρ/f. M h 175 GeV.
Lower bound on Higgs mass 88 Mh min GeV 86 84 82 80 78 2 4 6 8 f TeV M h min as a funtion of f for ξ = 3.6, allowed by the condition λ h > 0 at scale m t ξf / 2. Higgs mass is restricted by 80 M h 175 GeV.
The low energy theory Assuming all the degrees of freedom in Σ X,T,t are heavy, we can integrate them out. At the lowest order, this is done by replacing the heavy fields with their VEVs. V = λ 1 2 Tr[(Φ Φ) 2 ] + λ 2 2 (Tr[Φ Φ]) 2 + M 2 Φ χ Φ χφ χ C χχ(φ χ + φ χ), (27) where 0 0 0 φ 0 t 0 0 0 φ φ 0 t b w Φ = 0 0 φ + φ b 2 X, Φ χ = w 0 0 φ 0 φ + X 2 T φ 0. (28) u 0 0 2 t φ 0 T χ φ 0 χ Φ χ contains two EW doublets and one singlet. Approximate U(5) L /U(4) L breaking gives 9 PNGBs! Large w explicitly breaks U(5) L to U(3) L, going back to the original model!
Explicit breaking down to O(5) To solve this problem, we introduce mass terms that explicitly break U(5) L to O(5). V U(5) = 1 ( ) 2 K 2 Tr[Σ Σ ] + A 2 χ, (29) where K 2 > 0 and ( φ 0 Σ t φ b φ + X φ 0 T ), (30) Σ 1 (Σ ɛσ ɛ T ) = 1 ( φ 0 t φ 0 T 2 2 φ X + φ b φ + X + ) φ+ b φ 0 t + φ 0. (31) T Eq. (29) lifts up the masses of A χ and one linear combination of the two doublets. Custodial symmetry approximately holds if K 2 λ 1 w 2. 4 pngbs from O(5)/O(4) breaking which form the light SM-like Higgs doublet.
Low energy potential The scalar potential becomes Two EW doublets ( φ 0 t and u χ. V = λ 1 2 Tr[(Φ Φ) 2 ] + λ 2 + 1 2 K 2 ( Tr[Σ Σ ] + A 2 χ φ b v ( 246 GeV) and tan β 2 (Tr[Φ Φ]) 2 + M 2 Φ χ Φ χφ χ ) C χχ(φ χ + φ χ). (32) ) ( ) φ + X φ 0, and one singlet φ 0 χ, with VEVs v t, v T T v 2 = v 2 t + v 2 T, tan β vt v T > 1. (33) Chiral symmetry breaking scale f = vt 2 + vt 2 + u2 χ. (34)
Extended top seesaw Fermion mass terms come from the Yukawa couplings. L ξ ( ) 0 0 v t t R tl T L χ L 0 w v T T R ξw X L X R. (35) 2 u t 0 u χ χ 2 R X has charge +5/3 and does not mix. t, T and χ (charge +2/3) form a 3 3 mass matrix. Denote mass eigenstates t 1, t 2, t 3, we have m X = ξw 2, m t2 m X, m t3 ξf 2. (36) Current experimental constraint: m X > 800 GeV at 95% CL (CMS). t 1 is the top quark mtop 2 ξ2 vt 2 2 where y t is defined as m 2 top y 2 t v 2 /2. ut 2 ut sin β yt f 2 f ξ. (37)
Electroweak precision constraints Masses of the X and T quarks explicitly break custodial symmetry. In a suitable range of the X, T masses, the T parameter can be small and consistent with the EW measurements. It is reasonable to assume the custodial symmetry is a good symmetry in the UV. S parameter receives contribution from heavy vector states, estimated to be Ŝ m2 W m 2 ρ, (38) where Ŝ = g2 /(16π) S and m ρ is the mass scale of the heavy vector state. The S parameter is within experimental constraint if m ρ is not too small. For m ρ = 3 TeV, S 0.08.
Zb b coupling The SM prediction for Z b b branching ratio is 0.8σ smaller than the measured value (most recent Gfitter results). The SM prediction for A b FB (forward-backward asymmetry) is 2.5σ larger than the measured value. The two discrepancies together prefer g ZbL bl (g ZbL bl ) SM, g ZbR br > (g ZbR br ) SM. (39) Corrections to Zb L bl are suppressed in our model. It is possible to embed b R in some representation under the custodial symmetry that enhances the Zb R br coupling (not discussed here).
Higgs mass Estimation of coupling ratios from RGE 0.35 λ 1 2ξ 2 1, 0.15 λ 2 λ 1 0. (40) 3 CP-even neutral scalars, the lightest eigenstate is the Higgs boson. M 2 h λ 1v 2 2f 2 (u2 t w 2 v T v t + v T ) (41) λ 1 2ξ ( yt 2 2 sin 2 β m2 X 2 f 2 1 + tan β )v 2. (42) To obtain the correct Higgs mass one would need m X f.
Other effects Similar to the minimal model, the corrections from EW gauge boson loops reduce the Higgs mass and we cut them off at M ρ. Including O(1/M 2 ) corrections from heavy scalars Σ Σ Φ χ λ Σ/Σ λ Φ χ M 2 h λ 1v 2 2f 2 Φ χ Φ χ [ ] ut 2 (1 λ 1f 2 ) w 2 v T (1 λ 1f 2 ). (43) 2MΣ 2 t v t + v T 2MΣ 2 X,T The Higgs mass decreases as M Σt decreases or M ΣX,T increases (and vice versa).
Dependence of Higgs mass on other parameters Λ1 2Ξ 2 1.0 M h GeV 0.9 150 0.8 0.7 0.6 126 0.5 100 50 0.4 0 1 2 3 4 5 6 7 M Ρ f M t TeV 20 18 16 14 12 10 8 6 M h GeV 140 130 126 120 100 6 8 10 12 14 16 18 0 M X,T TeV ξ = 3.6, λ 2 /λ 1 = 0, f = 1 TeV, tan β = 1.25 and m X = 0.9 TeV. Left: fixing M ΣX,T,t = 10f. Right: fixing λ 1 /(2ξ 2 ) = 0.7 and M ρ = 3f.
Need m X f to have the correct Higgs mass! 1.8 1.6 m X f 0.6 0.7 0.8 0.9 1.0 1.1 M Ρ f 5 T 0.15 T 0.1 0 f 1.5 TeV 0 f = 1.5 TeV, ξ = 3.6, λ 2 /λ 1 = 0, M ΣX,T,t = 10f. Plot of M ρ/f with the Higgs boson mass fixed at 126 GeV. tan Β 1.4 1.2 7 T 0.06 T 0.11 7 5 0 T 0 Λ 1 2 Ξ 2 1 0.7 0.35 0.8 1.0 1.2 1.4 1.6 1.8 m X TeV Can not have M ρ < 0! m X f is generally true for different values of f. m X > 800 GeV means f can not be much smaller than 1 TeV.
T= -0.11 Introduction NJL Model, Top Condensation & Top Seesaw The Minimal Model Extension with Custodial Symmetry Conclusion Hgg coupling β 1.5 1.4 1.3 1.2 T= 0.15 T= 0.1 0.9698 0.97 T= 0 f = 1 TeV c g c g SM 0.9696 1.1 0.9701 T= -0.06 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 ( ) ξ = 3.6, λ 1 /(2ξ 2 ) = 0.7, λ 2 /λ 1 = 0, f = 1 TeV, M ρ = 3f, M ΣX,T,t = 10f. Corrections are 3% for f = 1 TeV.