Higgs or Higgsless? From a unitarity viewpoint

Transcription

1 Higgs or Higgsless? From a unitarity viewpoint KEK theory center mini-workshop: Composite Higgs/Higgsless models and related topics toward LHC era Masaharu Tanabashi (Nagoya U.) October 29,

2 Higgs sector of the standard model is known to be problematic. Is it possible to construct models without Higgs? 2

3 The role of the Higgs boson in the SM: Renormalizability : W and Z are gauge bosons (universality of weak interaction). Explicit breaking of electroweak gauge symmetry makes the theory non-renormalizable. We need, at least, one Higgs boson so as to feed W and Z masses through spontaneous breaking. Unitarity : The longitudinal W boson (W L ) scattering amplitude grows as the CM energy increases. If there is no Higgs boson, it eventually violates the unitarity. 3

4 Life without a Higgs 4

5 Renormalizability : New physics (cutoff scale of SM) is believed to exist at TeV. In principle, renormalizability is not a primary issue in this sense. However, the lack of renormalizability usually implies a loss of robust predictability. How can we ensure the consistency with the existing precision electroweak measurements without introducing a Higgs boson then? July 2008 Theory uncertainty Δα (5) had = ± ± incl. low Q 2 data m Limit = 154 GeV U 0 m t = ± 2.9 GeV m H = GeV m W prel. Δχ 2 3 T 0 Γ ll 2 1 Excluded Preliminary m H [GeV] -0.2 sin 2 θ lept eff 68 % CL S m H m t 5

6 Unitarity: B.W.Lee, C.Quigg, and H.B.Thacker In the standard model, a Higgs boson (scalar resonance) unitarizes the W L W L scattering amplitude: a c a c a c im(w a LW b L W c LW d L)= + W + h +crossed. b d b d b d For E M W M(W a LW b L W c LW d L)= s v 2 2 MH MH 2 s δab δ cd + t v 2 2 MH M 2 H t δac δ bd + u v 2 2 MH MH 2 u δad δ bc, with M 2 H = λv 2, v 250 GeV. W L W L scattering amplitude remains perturbative even at high energy scale s 1 TeV thanks to the light Higgs exchange. How can we ensure the W L W L scattering unitarity (or calculability) without introducing a Higgs boson? 6

7 Unitarity in the WW Scattering 7

8 Unitarity in the W L W L scatteing Let us consider the longitudinally polarized W (W L ). It s polarization vector p ɛ μ (L) = E E M W p, E2 = p 2 + MW 2, ɛ (L)μ ɛ μ (L) = 1 p grows for large E M W. Naive power counting in E suggests M(W a LW b L W c LW d L) ɛ (L)μ 4 E4 M 4 W Unitarity seems to be violated in the high energy W L W L scattering. 8

9 For simplicity, we consider the g Y =0case. (Z = W 3 ) Two Feynman diagrams contributing to the WW scattering im gauge (ab cd) = a b c d a + W Contribution from the 4W vertex (terms proportional to δ ab δ cd ) { E 4 g WWWW (1 + cos θ)[3 cos θ 2 M W 2 E ] 2 M 4 W b (1 cos θ)[3 + cos θ 2 M 2 W E 2 ] c d + crossed. } 9

10 t-channel W exchange (terms proportional to δ ab δ cd ) { E 4 gwww 2 (1 cos θ)[3 + cos θ 2 M } W 2 2 E ]+1 W ( cos θ)m E 2 M 4 W u-channel W exchange (terms proportional to δ ab δ cd ) { E 4 gwww 2 (1 + cos θ)[3 cos θ 2 M } W 2 2 E ]+1 W (1 11 cos θ)m E 2 M 4 W Each diagram behaves E 4 /MW 4 at high energy. The leading E 4 /MW 4 term cancels in the amplitude thanks to the Ward-Takahashi identity of the gauge symmetry. g WWWW = g 2 WWW = g 2 W 10

11 M gauge (ab cd) =δ ab δ cd g 2 W E 4 MW 4 M 2 W E 2 + = δab δ cd g 2 W E 2 M 2 W + We use E 2 = s/4, M 2 W = g2 W v2 /4 M gauge (ab cd) = s v 2 δab δ cd + t v 2 δac δ bd + u v 2 δad δ bc +. This form agrees with the low energy theorem (equivalence theorem) B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. Lett. 38, 883 (1977); Phys.Rev.D16, 1519 (1977). This is clear because NG boson W L (Higgs mechanism) 11

12 Unitarity If the W L W L scattering amplitude is completely given by the low energy theorem The probability of the W L W L scattering exceeds unity at the s =8πv 2 eneryg scale. unitarity violation Two possibilities Unitarity bound : 8πv 1.2TeV perturbative case The W L W L scattering behavior is modified thanks to the existence of particles lighter than the unitarity bound (predictability) non-perturbative casse The theory beoomes non-perturbative above the unitarity bound. The unitarity should be recovered in a non-perturbative manner. (predictability may be lost.) 12

13 Perturbative unitarity in the standard model Higgs sector Higgs exchange diagram im Higgs (ab cd) = a b h c d + crossed. M Higgs (ab cd) =g 2 hw W s 2 MW 4 Using the standard model Higgs relation 1 M 2 h s δab δ cd +. g hw W = M 2 W v 13

14 we notice that the s E 2 term cancels M(ab cd) =M gauge + M Higgs = s v 2 2 Mh Mh 2 s δab δ cd +. The amplitude agrees with the low energy theorem at s M 2 h = λv2. The amplitude approaches to a constant λ at the region s M 2 h = λv2. The theory is perturbative is the constant λ is sufficiently small. It is easy to extend this to multi-higgs models From the perturbative unitarity requirement of the W L W L scattering n g 2 h (n) WW = M 4 W v 2 hw W coupling perturbative unitarity Essence of a Higgs 14

15 Physics ensuring the unigtarity in the W L W L scattering O(E 4 ) cancellation gauge symmetry O(E 2 ) cancellation g WWWW = g 2 WWW n g 2 h (n) WW = M 4 W v 2 unitarity sum rules The h (n) WW coupling determimes the Higgs property of h (n). 15

16 Can a spin-1 resonance unitarize the W L W L scattering amplitude? a c a c im(w a LW b L W c LW d L)= + W + W 0 a b d b d c +crossed. Answer: Yes! if we suitably adjust WWW coupling. b d M(W a LW b L W c LW d L)= 1 3v 2 (s u) M 2 W M 2 W 2 W M +(s t) t M 2 W u Cancellation of bad high-energy behavior is achieved through exchange of massive spin-1 particle W.! δ ab δ cd + 16

17 Note, however, we need to introduce yet another massive vector particle W so as to unitarize the W L W L W L W L amplitude... A tower of massive vector particles: W, W, W, W, This situation is naturally realized in gauge theory with an extra dimension A tower of massive Kaluza-Klein modes Chivukula, Dicus and He ; Csaki, Grojean, Murayama, Pilo and Terning Gauge symmetry breaking through boundary conditions 17

18 Unitarity sum rules Spin-0 (Higgs) exchange case: g WWWW = g 2 WWW, g WWWW M 2 W =4 n g 2 h (n) WW Spin-1 exchange case (Higgsless): g WWWW = n g 2 WWW (n), 4g WWWW M 2 W =3 n g 2 WWW (n) M 2 W (n) If there exist spin-0 and spin-1 simultaneously (gaugephobic Higgs): Cacciapaglia et al. hep-ph/ , c.f. Hikasa and Igi g WWWW = n g 2 WWW (n), 4g WWWW M 2 W =3 n g 2 WWW (n) M 2 W (n) +4 n g 2 h (n) WW 18

19 Higgsless models 19

20 Gauge symmetry breaking through boundary conditions 5D gauge theory with an interval extra dimension y =0 y = l BC: Neumann(N)? Dirichlet(D)? A μ, A 5 1. y A μ (x, y) y=0 =0(N), y A μ (x, y) y=l =0(N) [NN] 0 massless spin-1 field: N N unbroken 4D gauge symmetry 2. y A μ (x, y) y=0 =0(N), A μ (x, y) y=l =0(D) [ND] 1 absence of massless spi-1 field: N 3 2 D 4D gauge sym is broken 20

21 4D gauge sym and spectrum In addition to the massive spin-1 KK particles, we have 1. [NN]: massless spin-1 (unbroken gauge sym) photon 2. [ND]: absence of massless particle (broken gauge sym) 3. [DN]: absence of massless particle (broken gauge sym) W ±, Z 4. [DD]: massless spin-0 (gauge and global syms are broken) Applying this mechanism to EWSB, we can push up the unitarity vilation scale around 10TeV. (little Hierarchy) R. Sekhar Chivukula, D. A. Dicus and H. J. He, Unitarity of compactified five dimensional Yang-Mills theory, Phys. Lett. B 525, 175 (2002) [arxiv:hep-ph/ ]. C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Gauge theories on an interval: Unitarity without a Higgs, Phys. Rev. D 69, (2004) [arxiv:hep-ph/ ]. 21

22 Brane localized Higgs field (aka RS model) y =0 y = l A μ, A 5 BC: Neumann BC at both brane y A μ (x, y) y=0 =0(N), y A μ (x, y) y=l =0(N) [NN] Brane localized Higgs φ: S Higgs = dyδ(y l + ɛ) [ (D μ φ) (D μ φ) V (φ φ) ] (VEV v b ) dyδ(y l + ɛ)v 2 b A μ A μ KK mode equation for the gauge field [ 2 y + δ(y l + ɛ)g 2 vb 2 ] χ (n) (y) =Mnχ 2 (n) (y) 1-dim Schroedinger eq. with δ-function repulsive force 22

23 1. finite v b case 1 δ-function repulsive force at y = l N N brane affects the 3 2 wave-function form. 2. v b case 1 δ-function repulsive force at y = l brane N 3 2 N affects the effective boundary condition at the brane. 23

24 Remarks Brane localized Higgs with an infinite VEV. (equivalent) Dirichlet BC (Higgsless) The KK gauge boson spectrum remains finite around the compactification scale even in the infinite Higgs VEV limit. Higgsless models can be regarded as a variant of usual RS model. 24

25 Effective theory viewpoint Deconstruction 25

26 Deconstruction of boundary conditions Deconstruction (latticization) of extra dimension A M, M = μ, 5 U 2 U 3 U 4 U 5 A 1 μ A 2 μ A 3 μ A 4 μ A 5 μ moose diagram Arkani-Hamed, Cohen and Georgi ; Hill, Pokorski and Wang a : lattice spacing A j μ = A μ (x, y = ja) : gauge field at site j U j =exp(i link field. field. Z ja dya 5 (x, y)) : (j 1)a non-linear σ model Note: Moose model can be viewed as a generalization of Bando-Kugo-Yamawaki s Hidden Local Symmetry (HLS) model (Phys.Rep.164,217(1988)) + Georgi s vector symmetry model (NPB331,311(1990)). 26

27 Deconstructions of an interval in moose notation: [DD] G G G G #(U j )=#(A j μ)+1. [NN] G G G G G #(U j )=#(A j μ) 1. G [DN] G G G G G #(U j )=#(A j μ). [ND] G G G G G #(U j )=#(A j μ). which correspond to 5D gauge theories with an interval compactification: H.-J. He, hep-ph/ y = 0 y = l A μ, A 5 [DD] A μ (x, y) y=0 =0(D), A μ (x, y) y=l =0(D), [NN] 5 A μ (x, y) y=0 =0(N), 5 A μ (x, y) y=l =0(N), [DN] A μ (x, y) y=0 =0(D), 5 A μ (x, y) y=l =0(N), 5 A 5 (x, y) y=0 =0(N), 5 A 5 (x, y) y=l =0(N). A 5 (x, y) y=0 =0(D), A 5 (x, y) y=l =0(D). 5 A 5 (x, y) y=0 =0(N), A 5 (x, y) y=l =0(D). 27

28 Spectrum and 4D gauge symmetry: In addition to a tower of massive spin-1 KK-modes, we have 1. [DD]: massless spin-0 particle. 4D gauge sym. is all broken. 2. [NN]: massless spin-1 particle. unbroken 4D gauge sym. 3. [DN]: no massless particle. 4D gauge sym. is all broken. 4. [ND]: no massless particle. 4D gauge sym. is all broken. 28

29 Advantages for deconstruction in 5D Higgsless models Familiar language of spontaneous gauge symmetry breaking (gauged nonlinear σ model). Easier to understand the physics behind the delay of unitarity violation. Easier to calculate corrections to electroweak interactions. Allowing for arbitrary background 5D geometry, spatially dependent gauge couplings, and brane kinetic terms. Easier to perform loop analysis using well-known chiral perturbation method. 29

30 5D Higgsless model SU(2) R U(1) B L U(1) Y SU(2) L SU(2) R SU(2) V (y =0) (y = l) SU(2) L SU(2) R U(1) B L quarks/leptons Csaki et al. PRL92 (2004) ; Nomura JHEP 11 (2003) 050; Barbieri et al. PLB591 (2004) 141. and its deconstruction fermions SU(2) fermions U(1) Foadi, Gopalakrishna, Schmidt ; Casalbuoni, De Curtis and Dominici ; Chivukula, Simmons, He, Kurachi and M.T.; Georgi 5D Higgsless models are described by linear moose. c.f. BESS model (Casalbuoni et al., PLB (1985)) HLS model (Bando-Kugo-Yamawaki) & Vector limit model (Georgi) 30

31 Linear moose models General linear moose with localized J μ W and J μ Y : SU(2) g g g g g g 0 1 p N N+1 q U(1) g N+M g N+M+1 with L = 1 4 f N+M+1 X j=1 f 1 N+1 N+M+1 J μ W J μ Y f 2 j tr((d μ U j ) (D μ U j )) f N+M+1 X j=0 D μ U j = μ U j ia j 1 μ U j + iu j A j μ. 1 4g 2 j F j μνf jμν Low energy Fermi-constant: (EW symmetry is broken collectively) 2GF = 1 v 2 CC = N+1 X j=p+1 Low energy QED coupling: N+M+1 1 e = X 2 1 f 2 j, j=0 1 g 2 j. 1 v 2 NC = qx j=p+1 c.f. HLS model (Bando-Kugo-Yamawaki) 1 f 2 j. 31

32 Delay of unitarity violation For large N p, we are able to achieve delay of unitarity violation min(f j ) v 250 GeV, Note, however, 8π min(fj ) 8πv 1.2TeV. N< 100 if we assume perturbative gauge coupling : g 2 j /(4π) < 1. The perturbative unitarity is eventually violated at or below 10 TeV. ρ parameter ρ =1+Δρ v2 CC v 2 NC =1+v 2 q j=n+2 1 f 2 j 1. In this talk, we concentrate our attention on Case I models (q = N +1)inwhichΔρ =0is satisfied automatically. 32

33 The model we consider in this talk g 0 SU(2) g g g 1 N N+1 f 1 f N+1 J W μ J Y μ 33

34 General Sum Rules Chivukula, He, Kurachi, Simmons, M.T. Phys.Rev.D78, (2008) [arxiv: [hep-ph]] 34

35 General Sum Rules We try to genelarize the unitarity sum rule g WWWW = n g 2 WWW (n), 4g WWWW M 2 W =3 n g 2 WWW (n) M 2 W (n) Finite deconstruction Inelastic scattering nn mm Generalize to the transverse gauge boson scattering such as LL LT, LL TT, LT TT. important for the collider confirmation of the Higgsless scenario 35

36 We deduce the sum rules using the equivalence theorems M(L (n) L (n) L (m) L (m) ) M(π (n) π (n) π (m) π (m) ), E 4,E 2, (E 0 ) M(L (n) L (n) L (m) T (m) ) M(π (n) π (n) π (m) T (m) ), E 3,E 1 M(L (n) L (n) T (m) T (m) ) M(π (n) π (n) T (m) T (m) ), E 2,E 0 M(L (n) T (n) T (m) T (m) ) M(π (n) T (n) T (m) T (m) ), E 1 In the continuum limit (N ), M(π (n) π (n) π (m) π (m) )=0, M(π (n) π (n) π (m) T (m) )=0 36

37 Results G nnmm 4 = k G nnk 3 G mmk 3 = k G nmk 3 G nmk 3, 2(M 2 n + M 2 m)g nnmm 4 + k (G nmk 3 ) 2 [ (M 2 n M 2 m) 2 M 2 k 3M 2 k ] = 4 v 2 M 2 nm 2 m G nnmm 4,. Here G nmlk 4 g W(n) W (m) W (l) W (k), G nml 3 g W(n) W (m) W (l), G nmlk 4 g π(n) π (m) π (l) π (k), In the continuum limit, we have G nmlk 4 =0 37

38 These sum rules agree with the unitarity sum rule in the continuum limit G nnnn 4 = k G nnk 3 G nnk 3, 4M 2 ng nnnn 4 =3 k M 2 k (G nnk 3 ) 2, 38

39 Sum rules in the linear moose models W mass matrix M 2 W, π mass matrix M 2 W are given by (g Y =0), M 2 W = Q T Q, M 2 W = QQ T, where Q 1 2 g 0 f 1 g 1 f 1 g 1 f 2 g 2 f g N 1 f N g N f N. g N f N+1 39

40 Diagonalization M 1 M 2... = M diag W = RT Q T R = RT QR M N+1 The matrices R, R represent the KK gauge bson W (n) (NG boson π (n) ) wave function in the extra dimension. G nmk 3, G nmlk 4 aregivenbythekkgaugebosonwavefunction, G nmk 3 = j g j R j,n R j,m R j,k, G nmlk 4 = j g 2 j R j,n R j,m R l,k R j,k, while G nmlk 4 given by the NG boson wave function, G nmlk 4 = j v 2 f 2 j R j,n Rj,m Rl,k Rj,k. 40

41 A relation between W (n) and π (n) functions (WT identity) RM diag W = QR Completeness of the wave function δ j,j =(RR T ) j,j = k R j,k R j,k, δ j,j =( R R T ) j,j = k R j,k R j,k. Generalized sum rules WT identities + completeness relations 41

42 Example: Derivation of i G nmi 3 G lki 3 = G nmlk 4 i G nmi 3 G lki 3 = i j,j g j g j R j,n R j,m R j,i R j,nr j,mr j,i = j,j g j g j R j,n R j,m R j,nr j,mδ j,j = j g 2 j R j,n R j,m R j,n R j,m = G nmlk 4 42

43 Summary of the WT identities and the completeness relations G nmlk 4 = i G nmi 3 G lki 3, 4 Gnmlk v 2 4 = G nmlk 4 (Mn 2 + Mm 2 + Ml 2 + Mk 2 ) i (G nmi 3 G lki 3 + G nli 3 G mki 3 + G nki 3 G mli 3 )M 2 i, (M 2 n M 2 m)(m 2 l M 2 k ) i G nmi 3 G lki 3 M 2 i = i (G nki 3 G mli 3 G nli 3 G mki 3 )M 2 i, ( G 4 =0in the continuum limit). c.f., Sakai and Uekusa, Prog.Theo.Phys. 118 (2007) 315 [hep-th/ ] 43

44 Low Energy Effective Theory in 4D How can we ensure the consistency with the existing precision electroweak measurements? 44

45 A three-site Higgsless model Chivukula, Coleppa, Di Chiara, Simmons, He, Kurachi and M.T., PRD (2006); See also Bando, Kugo, Yamawaki s HLS model Phys.Rep.164,217(1988). SU(2) SU(2) U(1) gauge theory The gauge sector is precisely that of the BESS model. Casalbuoni et al., PLB (1985)) Fermion mass terms: L f = λf 1 ψl0 U 1 ψ R1 M ψ R1 ψ L1 f 2 ψl1 U 2 λ u λ d 1 0 u R2 d R2 For simplicity, we examine the case f 1 = f 2 = 2v andworkinthe limit 1 A+h.c.. g 0 g 1 1, g 2 g 1 1, and thus, g W g 0, g Y g 1. 45

46 Fermion mass matrix: (seesaw like) m 0 2 λv ε L 0 M m f 1 ε fr, ε L λ λ, ε fr λ f λ Light fermion mass: m f mm f M 2 + m 2 f = 2 λvεl ε fr 1+ε 2 fr and its eigenstate (or delocalization) ( ψ f,light L 1 ε2 L 2 ) ψ f L0 + ε Lψ f L1 where we assumed ε fr 1. Heavy (KK) fermion mass: M f,kk M 2 + m 2 f = 2 λv 1+ε 2 fr 46

47 For M v, we can integrate out the heavy KK-fermion. The fermion delocalization effect can then be replaced by an operator L f = x 1 ψl (i /DU 1 U 1 )ψ L, x 1 ε 2 L, ε L = 2λv M ψ L is a left-hand fermion at site-0, D μ ψ L = μ ψ L ig 0 W 0μ ψ L. S-parameter S = 4π g 2 1 ( 1 2g2 1 g 2 0 vanishes in the ideal delocalization limit: x 1 ) x 1 = g2 0, g 2g1 2 W ff =0. c.f. Anichini, Casalbuoni, and De Curtis, PLB (1995). 47

48 Tree level is not enough... 48

49 Fermionic one-loop corrections to T parameter Chivukula, Coleppa, Di Chiara, Simmons, He, Kurachi and M.T., PRD (2006) αt 1 16π 2 m 4 t M 2 v 2 = 1 ε 4 tr M 2 16π 2 v 2. Assuming ideal delocalization of fermions, we find M αt=2.5*10^ αt=5*10^ M W 49

50 Allowed region in S-T depends on the reference Higgs mass M H,ref. S S BSM S SM (M H,ref ), T T BSM T SM (M H,ref ) αt < ( )form H,ref = 340GeV (1000GeV). Which M H,ref should we use? We need to evaluate bosonic one-loop diagrams in order to get more precise bounds. 50

51 One loop constraint from precision electroweak measurements (95%CL): M GeV M W 380 GeV M GeV M W 500 GeV TeV 3.0 TeV g W ff g W TeV 3.0 TeV g W ff g W The cutoff dependence is small. Tiny (but non-zero) W ff coupling. T. Abe, S. Matsuzaki, and M.T., PRD78, (2008) 51

52 M (GeV) Λ = 4.3 TeV Λ = 3.0 TeV M W (GeV) M W > 380MeV is required by the ZWW measurement at LEP2. The cutoff Λ should satisfy Λ < 4πf 1 =4πf 2 =4.3TeV, which implies M W < 600GeV 52

53 LHC phenomenology of W 53

54 W production cross sections through W WZ vertex: H.-J. He et al., arxiv:

55 (a) (b) 100fb 1 55

56 W production cross sections through W ff vertex: T. Ohl and C. Speckner, arxiv: u ν l u u W + l W + d W + Z u W + Z e d ū d e events per bin (50 bins total) final state: lν l jj m W = 380 GeV, ε L = m W = 500 GeV, ε L = m W = 600 GeV, ε L = m bulk = 3.5 TeV for m W = 380 GeV, 500 GeV m bulk = 4.3 TeV for m W = 600 GeV 100 events per bin (50 bins total) final state: lljj m W = 380 GeV, ε L = m W = 500 GeV, ε L = m W = 600 GeV, ε L = m bulk = 3.5 TeV for m W = 380 GeV, 500 GeV m bulk = 4.3 TeV for m W = 600 GeV total invariant mass [GeV] total invariant mass [GeV] 100fb 1 56

57 Summary Higgsless theory is an interesting alternative to the standard model Higgs, achieving tree level unitarity at 1TeV. We analyzed an effective theory (three site Higgsless model) at one-loop level and found the model is consistent with the available precicion electroweak measurements. The allow ranges of the KK gauge boson coupling g W ff, the KK gauge boson mass M W,and the KK quark/lepton masses M are severely constrained, however. The KK gauge boson W will be discovered at LHC with L =20 30 fb 1. Although, in the case of flavor universal KK-fermion mass, FCNC is protected by GIM mechanism, more study on the flavor physics should be done in the Higgsless theory. 57