Symmetry and Geometry of Generalized Higgs Sectors

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Transkrypt:

Symmetry and Geometry of Generalized Higgs Sectors Ryo Nagai Tohoku University in collaboration with M. Tanabashi (Nagoya U.), Y. Uchida (Nagoya U.), and K. Tsumura (Kyoto U.) PPP2018 @ YITP, Aug. 6-10, 2018

Outline Introduction Higgs Effective Field Theory Perturbative Unitarity vs. EWPTs Summary and Outlook

We found a h(125)! The 125GeV Higgs boson was discovered at the LHC.

h(125) = hsm? Higgs coupling measurements V h(125) κf κv V

h(125) = hsm? Higgs coupling measurements κf κ f 2.5 ATLAS-CONF-2017-047 SM prediction ATLAS Preliminary κ F κf 2 CMS Preliminary CMS-PAS-HIG-17-031 -1 35.9 fb (13 TeV) 2 Best fit Combined 68% CL Combined 95% CL H γ γ 68% CL -1 s = 13 TeV, 36.1 fb H γ γ and H ZZ* 4l m H = 125.09 GeV 1.5 1.5 H ZZ* 4l 68% CL 1 1 0.5 0.5 H bb H ZZ H ττ H γγ Best fit SM expected 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 κ V κv H WW Combined 1 σ region 2σ region 0.5 1 1.5 2 κ V κv ~10% deviation from κv, F = 1 is still allowed.

h(125) = hsm? Higgs coupling measurements M.Peskin (2014) 5% 4% W hww 5% 4% Z hzz 3% 2% CMS-1 LHC CMS-2 ILC 3% 2% CMS-1 LHC CMS-2 ILC 1% 1% LHC CMS 250 500 500up 10001000up LHC CMS 250 500 500up 10001000up ~1% precise measurement is expected to be realized.

What if κv 1? WLWL WLWL Longitudinal Longitudinal Longitudinal Longitudinal

What if κv 1? WLWL WLWL Energy Energy Energy Energy Unitarity is violated?

What if κv 1? WLWL WLWL h s M ' v 2 1 kv 2 WL W L!W L W E2 L M 2 (1 k2 V) Gauge boson Higgs boson

What if κv 1? WLWL WLWL h s M ' v 2 1 kv 2 WL W L!W L W E2 L M 2 (1 k2 V) The SM higgs boson ( κv =1 ) keeps the theory perturbatively unitary.

What if κv 1? Λ[ ] Perturbative unitarity violation - - - - Δκ If κv 1, perturbative unitarity in WLWL scattering seems to be violated at certain energy scale.

What if κv 1? ff ff h S ' 1 12 1 k2 V ln L2 M 2 h Electroweak Precision Tests (EWPTs) at LEP : S = 0.03 ± 0.10 Global fitting by Gfitter Group

What if κv 1? Λ[ ] Perturbative unitarity violation EWPTs - - - - Δκ κv 1 causes not only perturbative unitarity violation but also inconsistency with EWPTs.

What if κv 1? Higgs Sector h 125

What if κv 1? Higgs Sector π125 Composite?

What if κv 1? Higgs Sector h 125 H a H ±. Elementary. But mixed state?

Outline Introduction Higgs Effective Field Theory Perturbative Unitarity vs. EWPTs Summary and Outlook

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Higgs EFT Q. How can we formulate the Higgs sector? SU(2)W doublet filed : 4 = 3(NGBs) + 1(higgs) Examples : F = 1 p 2 exp i v pa t a 0 v + h L SM =(D µ F) (D µ F)+ L SMEFT =(D µ F) (D µ F)+ C L 2 F F (D µ F) (D µ F)+ What if h(125) 2 F? e.g. Composite Higgs models Roles of PU in extended Higgs scenarios

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sha1_base64="groxcp7noqij2vzm5ouzu5rcgza=">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</latexit> CCWZ method Systematic way for constructing model with G H Coleman-Wess-Zumino (1969) Coleman-Callen-Wess-Zumino (1969) General Lagrangian for NGBs arising through EWSB : L p = v2 4 Tr[D µu D µ U] * We here assume custodial symmetry for simplicity. NGB fields : Gauge bosons : U = e i v pa t a D µ U = µ U iguw µ + ig Y B µ U

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CCWZ method General Lagrangian for NGBs and singlet scalar(s) L = v2 4 Tr[D µu D µ U] 1 + 2 Â h k h WW h v + Â kww hh0 h,h 0! hh 0 v 2 + Examples: Nagai-Tanabashi-Tsumura (2016) L SM = v2 4 Tr[D µu D µ hv h2 U] 1 + 2 + v 2 L SM+S = v2 4 Tr[D µu D µ U] 1 + 2c L MCSM = v2 4 Tr[D µu D µ U] 1 + 2 1 hv Hv h2 H2 + 2s + c2 + s2 v2 v 2 + 2cshH v 2 v 2 h 8 f 2 v + 1 v 2 h 2 f 2 v 2 +

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CCWZ method General Lagrangian for NGBs and singlet scalar(s) L = v2 4 Tr[ µu µ U] 1 + 2 Â h k h WW h v + Â kww hh0 h,h 0! hh 0 v 2 + = 1 2 µ(p a hh 0 ) 1 + 2 Â k h WW h v + Â k hh0 WW hh0 v 2 (d ab + O(p 2 )) 0 0 1! µ 0 B @ p a h h 0. 1 C A = 1 2 g ij(f) µ f i µ f j NGBs and Higgs field(s) define coordinates on a scalar manifold.

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sha1_base64="n+emz/1r2ahoteh6/vyckos+pu8=">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</latexit> Generalized Higgs sector The EW sector in general extended Higgs scenarios can be expressed by the following form: L = 1 2 g ij(f) µ f i µ f j f i =(p a, h, h 0, ) NGBs Higgses The interaction between π (~ WL/ZL) and Higgs bosons is determined by the structure of gij (Φ) Alonso-Jenkins-Manohar (2016)

Outline Introduction Higgs Effective Field Theory Perturbative Unitarity vs. EWPTs Summary and Outlook

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sha1_base64="fjdde0s/ywh5zkjph6qnwett/ec=">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</latexit> <latexit sha1_base64="n+emz/1r2ahoteh6/vyckos+pu8=">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</latexit> Perturbative Unitarity The EW sector in general extended Higgs scenarios can be expressed by the following form: L = 1 2 g ij(f) µ f i µ f j f i =(p a, h, h 0, ) NGBs Higgses The interaction between π (~ WL/ZL) and Higgs bosons is determined by the structure of gij (Φ) Alonso-Jenkins-Manohar (2016) Perturbative Unitarity / EWPTs gij (Φ)??

Perturbative Unitarity High-energy behavior of WL/ZL ~ π. (Equivalence Theorem). VL VL π π ~ VL VL π π NGBs/Higgses scattering amplitudes can be expressed in terms of geometry of the scalar manifold.

Perturbative Unitarity Φi Φj Φk Φl Tree-level scattering amplitude: M fi f j!f k f l s 3 R iklj + R ilkj + t 3 R ijlk + R iljk + u 3 R ijkl + R ikjl Riemann curvature tensor

Perturbative Unitarity Φi Φj Φk Φl Tree-level scattering amplitude: M fi f j!f k f l s 3 R iklj + R ilkj + t 3 R ijlk + R iljk + u 3 R ijkl + R ikjl Perturbative unitarity condition: R ijkl = 0

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general extended Higgs scenarios can be expressed by the following form: L = 1 2 g ij(f) µ f i µ f j f i =(p a, h, h 0, ) NGBs Higgses The interaction between π (~ WL/ZL) and Higgs bosons is determined by the structure of gij (Φ) Alonso-Jenkins-Manohar (2016) Perturbative Unitarity / EWPTs gij (Φ)??

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EW corrections EW oblique corrections depend on not only geometry but also symmetry of the scalar manifold. The scalar manifold respects SU(2)W U(1)Y gauge sym. 4 Killing vectors For examples: w i a (a=1,2,3) SU(2)W symmetry w b a = v 2 d ab y b = v 2 d 3b y i U(1)Y symmetry 1 2 e abcp c + O(p 2 ) 1 2 e 3bcp c + O(p 2 )

EW corrections S-parameter at one-loop UV divergence: S log µ ( w j 3 ) ;i(ȳ i ) ;j Alonso-Jenkins-Manohar (2016) (w i a) ;j = f j wi a + G i kj wk a (y i ) ;j = f j yi + G i kj yk

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sha1_base64="s/3ychoign9kvaz0lflmnxrbsow=">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</latexit> EW corrections S-parameter at one-loop UV divergence: S log µ ( w j 3 ) ;i(ȳ i ) ;j Alonso-Jenkins-Manohar (2016) SU(2)W U(1)Y sym. [w a, w b ]= e abc w c [w a, y] =0 v i;j;k = R l kji v l (v = w a, y) S log µ e 3bc R i jc3( w j b ) ;i + e 3bc R i jbc(ȳ j ) ;i

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S-parameter at one-loop UV divergence: pp! f i f j f f! f i f j Φ i Φ j Φ i Φ j π π f f S log µ e 3bc R i jc3( w j b ) ;i + e 3bc R i jbc(ȳ j ) ;i

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sha1_base64="s/3ychoign9kvaz0lflmnxrbsow=">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</latexit> <latexit sha1_base64="s/3ychoign9kvaz0lflmnxrbsow=">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</latexit> EW corrections U-parameter at one-loop UV divergence: U log µ ( w j 1 ) ;i( w i 1 ) ;j ( w j 3 ) ;i( w i 3) ;j SU(2)W U(1)Y sym. [w a, w b ]= e abc w c [w a, y] =0 v i;j;k = R l kji v l (v = w a, y) U log µ e 1bc R i jbc( w j 1 ) ;i e 3bc R i jbc( w j 3 ) ;i

PU vs. EWPTs 1 L = gij (f) Dµ fi D µ f j 2 sha1_base64="u5no3xza5sgtqc77myxx/4s98jo=">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</latexit> sha1_base64="8qe9oe2dgok10+tqyh6fgljvxz8=">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</latexit> sha1_base64="fjdde0s/ywh5zkjph6qnwett/ec=">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</latexit> sha1_base64="n+emz/1r2ahoteh6/vyckos+pu8=">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</latexit> fi = (p a, h, h0, ) Perturbative Unitarity * M f i f j! f k f l E2 <latexit sha1_base64="xoi0ikfjqpuwowra/aw801ffxfm=">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</latexit> Rijkl EW oblique corrections Slog * <latexit 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<latexit sha1_base64="h1iux8+39+ih/wibyezkbnc7fw4=">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</latexit> Φi Φj j e3bc R jc3 (w b );i i +e3bc R jbc (y j );i <latexit 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e1bc R <latexit 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i j jbc ( w 1 );i i Φl <latexit 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jbc (w 3 );i

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