Symmetry and Geometry of Generalized Higgs Sectors

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1 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.) YITP, Aug. 6-10, 2018

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

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

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

5 h(125) = hsm? Higgs coupling measurements κf κ f 2.5 ATLAS-CONF SM prediction ATLAS Preliminary κ F κf 2 CMS Preliminary CMS-PAS-HIG 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 = GeV H ZZ* 4l 68% CL H bb H ZZ H ττ H γγ Best fit SM expected κ V κv H WW Combined 1 σ region 2σ region κ V κv ~10% deviation from κv, F = 1 is still allowed.

6 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 up up LHC CMS up up ~1% precise measurement is expected to be realized.

7 What if κv 1? WLWL WLWL Longitudinal Longitudinal Longitudinal Longitudinal

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

9 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

10 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.

11 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.

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

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

14 What if κv 1? Higgs Sector h 125

15 What if κv 1? Higgs Sector π125 Composite?

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

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

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19 <latexit sha1_base64="+abjcqlzz61c9ljqkj1u2nbl6zw=">aaago3icjvtnbtnaej4ucmvq2oiesfwqoqaiinpubsokpaousfz630rdnnkcjwpvcdy1hzqu9scbcy/agrnihbcpwyux4nanqihjkbhwyhacrg1ab2x5dzwz3zc/o3y19l0ozuwgn3lm7ln8+del1svly5fhjyavrewtrdpi1wn5lblrtsphe4fyjb3yfxuhfhaz6ov16s4zy19vcxl5rwal7oriq2m7gvf3hdtgvwxifv9oee94xdqokmvfo10zqxmtls72qu5ppxzpqzyaxsleohjobdta89hotm0u71jcf/v42ujv4xqbsqw0o1o52mkcw+27ilf9rkggijiybrwuh8wl5zbio5wjaisxuqzocuwuuidutdcahhknft4catacbxjogoaaypr9schcexpkeieo2wkfoomsr3ybmrajegn0sfg7g6ulbwqzbprskh4f2tuuoy67yz/kxg12tqg+m1wiiuogzsdhissufnk39okdsq/sm/vb/gzjwxcygjq4v1osccvjb24s//4nqpd/hzue4aau4cowy+yemnpi7mec1eqhiisniczfae+/pvx+vfrut9kh9hprfm8o2besngj/cj4uiqv3q7ko49pbrizadp9cbeo5vi9gf31syaxoinqbnanetymofqwl+iity+0/eahxmt2cdej9f3rur70kwhnilobsgf+wp1k29rbc+ixmkohaphuljkqxvd6j6e7nk+p6ajrstb7om4/xxfcgnk8wh0sxtaezwtbxczhjtaade/iu4ahtwaxmdisu/tdsjvowyjy4u/65to6y6uj0ei1kvkqz9pjrvyo/bs5joo1mdu8t+0pkram1qnq6qzk+eevyw2fypcyvqr6szvglzxvyikw2cfgdfm2ug20ww6oxbidd0dqlr/hvwr78v54u1mzkzvyql84w5p92/6ojcbnuwtt28rhmw3nygfxm7xtulhctdz1fzl/il+vxuterxbdzfy5d+a2/jtvwxw==</latexit> <latexit sha1_base64="+abjcqlzz61c9ljqkj1u2nbl6zw=">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</latexit> <latexit sha1_base64="+abjcqlzz61c9ljqkj1u2nbl6zw=">aaago3icjvtnbtnaej4ucmvq2oiesfwqoqaiinpubsokpaousfz630rdnnkcjwpvcdy1hzqu9scbcy/agrnihbcpwyux4nanqihjkbhwyhacrg1ab2x5dzwz3zc/o3y19l0ozuwgn3lm7ln8+del1svly5fhjyavrewtrdpi1wn5lblrtsphe4fyjb3yfxuhfhaz6ov16s4zy19vcxl5rwal7oriq2m7gvf3hdtgvwxifv9oee94xdqokmvfo10zqxmtls72qu5ppxzpqzyaxsleohjobdta89hotm0u71jcf/v42ujv4xqbsqw0o1o52mkcw+27ilf9rkggijiybrwuh8wl5zbio5wjaisxuqzocuwuuidutdcahhknft4catacbxjogoaaypr9schcexpkeieo2wkfoomsr3ybmrajegn0sfg7g6ulbwqzbprskh4f2tuuoy67yz/kxg12tqg+m1wiiuogzsdhissufnk39okdsq/sm/vb/gzjwxcygjq4v1osccvjb24s//4nqpd/hzue4aau4cowy+yemnpi7mec1eqhiisniczfae+/pvx+vfrut9kh9hprfm8o2besngj/cj4uiqv3q7ko49pbrizadp9cbeo5vi9gf31syaxoinqbnanetymofqwl+iity+0/eahxmt2cdej9f3rur70kwhnilobsgf+wp1k29rbc+ixmkohaphuljkqxvd6j6e7nk+p6ajrstb7om4/xxfcgnk8wh0sxtaezwtbxczhjtaade/iu4ahtwaxmdisu/tdsjvowyjy4u/65to6y6uj0ei1kvkqz9pjrvyo/bs5joo1mdu8t+0pkram1qnq6qzk+eevyw2fypcyvqr6szvglzxvyikw2cfgdfm2ug20ww6oxbidd0dqlr/hvwr78v54u1mzkzvyql84w5p92/6ojcbnuwtt28rhmw3nygfxm7xtulhctdz1fzl/il+vxuterxbdzfy5d+a2/jtvwxw==</latexit> <latexit sha1_base64="+abjcqlzz61c9ljqkj1u2nbl6zw=">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</latexit> Higgs EFT Q. How can we formulate the Higgs sector? SU(2)W doublet filed : 4 = 3(NGBs) + 1(higgs) F = 1 p 2 exp i v pa t a 0 v + h NGBs 125GeV higgs Roles of PU in extended Higgs scenarios

20 <latexit 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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=">aaagqxicjvtlbtnafl0tbep5tiuneiwioiawee1qgykevmggzr+evmqayhemivxhnn4eussbnkj8aatwicgbwmevwiifynfpadaiirgabweunsqpomwsmbm+c8+5j7l2ztenzxdiz2ly0ohmkantx6apnzh5amz27vqdzw5cxzz027td9zrmsdowzmk3ffouo67u2jvtrtw27qrzty50pco27vtdr262tazlnaxd86gqzp4v3zavsnxwnt00oratlr2jopv9lahouxu2jwqcr3zukczcjpkxzm9npqxp5cltnwzskaa2sblih2ysrh6edsricadbpba6f5lb55iixgawkrdqon3c2srbccynli3wb4m9ojw1scv2bcpui9faefexeoxvb87edmgspbz4it6kxffzvbnfxd8xmsgolumxey3gsqc68+zs6u8dub3491bjedokc6telok1ubn5qmec52qa4bbgmpb9dlaf767ewsmhf8ur8r15vhq74imyttq/9nflcuxfmcgawlvgaumksm5vqltg7piewikwmuw7zj3fn+pi7qizijj7kbvh/t8qfvoqxd1zwfwxfgv9vi5oa2bmobouvbpxvxoqwgv9giziqvx2rck+l6rsj7m3e90ujhyrpbg9xdct/uryhp5ngketfagkprnsydbbpppc0gxmal3jpy1vdy6l1rwb3sbvgeazm+zfsox2oa8y354nxpa1w1ymdhw2a6eb415uxxif2a9zlbfopadjpnx5+wgzqzguq4h4wmzh8h0/4qhgo0jfezdfft4wem/defb7gu6mkneef5zid1oc/l+ocqwrhzuf4vj8bvfo8iudonn0gs6hkddoke7repuq3bn6q+/pq+zt5lvmr+znbdo5kwdo0mdi/pkhckxobg==</latexit> <latexit 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] 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] 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] 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] Â h k h WW h v + Â kww hh0 h,h 0! hh 0 v 2 + = 1 2 µ(p a hh 0 ) Â k h WW h v + Â k hh0 WW hh0 v 2 (d ab + O(p 2 )) 0 0 1! µ 0 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)

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

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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)??

27 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.

28 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

29 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

30 <latexit 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sha1_base64="n+emz/1r2ahoteh6/vyckos+pu8=">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</latexit> EWPTs 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)??

31 <latexit 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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 )

32 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> 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

36 PU vs. EWPTs 1 L = gij (f) Dµ fi D µ f j 2 sha1_base64="u5no3xza5sgtqc77myxx/4s98jo=">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</latexit> 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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 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corrections Slog * <latexit 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