Finite Gluon Fusion Amplitude in the Gauge-Higgs Unification. Nobuhito Maru. (Chuo University) Based on N.M., MPL A (2008)

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1 Finite Gluon Fusion Amplitude in the Gauge-Higgs Unification Nobuhito Maru (Chuo University Based on N.M., MPL A 77 (008 /0/008 Towards New Developments in Field & String

2 Introduction One of the problems in in the Standard Model: Gauge Hierarchy Problem Quantum corrections to Higgs mass is sensitive to the cutoff scale of the theory = Higgs + W,Z,γ + Top δ m H Λ 6 π Too large!! (Natural cutoff scale is Planck/GUT scale

3 In GHU, extra component of higher dim gauge field is identified with Higgs where the (local mass term A y is forbidden by its gauge transformation ( ( A A + ε x, y + i ε x, y, A y y y y In other words, no local counter term is is allowed No quadratic divergence, finite regardless of the nonrenormalizability m Ay 6 π R

4 D-dim QED on Hatanaka, Inami Inami & Lim Lim (998 (998 5D Non-Abelian gauge theory on Gersdorff, Irges Irges & Quiros Quiros (00 (00 6D Non-Abelian gauge theory on Antoniadis, Benakli Benakli & Quiros Quiros (00 (00 6D Scalar QED on Lim, Lim, N.M. N.M. & Hasegawa (006 (006 5D QED on Explicit calculations of Higgs mass N.M. N.M. & Yamashita (006, Hosotani, N.M., N.M., Takenaga & Yamashita (007 (007 GHU has opened up a new avenue to solve the gauge hierarchy problem

5 Question: Is there any other finite physical observable? Natural to to guess in in the gauge-higgs sector of of the SM One of of the candidates: Gluon Fusion amplitude gg -> H Coefficient of of Dim 5 operator H a a ν HG G ν We can expect them to to be be finite due to to the higher dim. gauge invariance A A ε x, y i ε x, y, A + ( + ( y y y y This process are very important because the dominant Higgs production A y A G a G a ν y ν

6 ψ Explicit calculation in (D+-dim SU( gauge theory with a triplet fermion on S^/Z [ D ] m R = g dx dy xy xym v 4 s x ( ( ( ( ( D 4 D 0 0 H 4π n= sin π n ( nmr K π Rnm xy ( ( ( D ( D H π Rn mgs D Γ ( D [ D ] ( D D 5 i π vr ( ( sin πnmr π sin ( πnmr = cot ( πmr : D = 4, = ( mr : D = 5 n n = n = n= sin π nmr ( n D 4

7 Comments Even in in realistic compactifications, the finiteness remains unchanged since the information of compactification is is an IR nature At higher order loops, divergences from subdiagrams have to be renormalized by the ordinary procedures Two photon decay amplitude H -> -> γγ is is also finite (Main discovery mode in in case mh mh < 50GeV ν ( 0 H HFν F = 0 A y, γ = 0

8 Summary We have shown that the gluon fusion amplitude gg -> -> H is isfinite in in gauge-higgs unification in in any dimension regardless of its nonrenormalizablity (striking impact on collider physics!! Gauge invariant operator describing the amplitude is is forbidden by higher dimensional gauge symmetry This is is a new finite physical observable other than Higgs mass

9 Backup Slides

10 Gluon fusion is the dominant Higgs production process

11 SM contributions gg -> H C SM glue L eff = C hg G SM aν a glue ν ( m α SF 4m t t mh = v 8π m t 4 F x = x x sin x + x ( ( ( ( ( H -> γγ L eff = C SM γ hf ν F ν C = C + C C SM t W γ γ γ t γ ( 4 m ( α 4 m α F m m t 4 F m m =, = v m v m em t h W W em W h Cγ 8π t 8π W ( ( ( ( ( F x = + x+ x x sin x 7 x

12 L Consider (D+-dim SU( gauge theory on S^/Z ( MN M = Tr FMN F + iψγ DM Ψ MN M N N M 5 M N M M 5 M M M ( ψ, ψ, ψ T M Γ = [, ](, 0,,,, F = F F ig A A M N = D a a a ( : : Gell-Mann matrices D = ig A A = A λ λ Ψ= ( 5 γ, iγ A Boundary conditions: ( ( ( ( ( ( ( ( ( (+,+ only has massless mode ( ( ( ( ( ( ( ( ( ( +, + + ψ R(, ( ψ R( ( ψ ( ++, ++,,,, ++, ψl = ++, ++,,, A5 =,, ++,, Ψ= ψl ++, +,,, ++, ++, ++,, ψ, + +, + L R

13 L Consider (D+-dim SU( gauge theory on S^/Z ( MN M = Tr FMN F + iψγ DM Ψ MN M N N M 5 M N M M 5 M M M ( ψ, ψ, ψ T M Γ = [, ](, 0,,,, F = F F ig A A M N = D a a a ( : : Gell-Mann matrices D = ig A A = A λ λ Ψ= ( 5 γ, iγ A Boundary conditions: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( +, + + ψ R(, ( ψ R( ( ψ ( ++, ++,,,, ++, ψl = ++, ++,,, A5 =,, ++,, Ψ= ψl ++, +,,, ++, ++, ++,, ψ, + +, + L R SU( SU( x U(

14 L Consider (D+-dim SU( gauge theory on S^/Z ( MN M = Tr FMN F + iψγ DM Ψ MN M N N M 5 M N M M 5 M M M ( ψ, ψ, ψ T M Γ = [, ](, 0,,,, F = F F ig A A M N = D a a a ( : : Gell-Mann matrices D = ig A A = A λ λ Ψ= ( 5 γ, iγ A Boundary conditions: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( +, + + ψ R(, ( ψ R( ( ψ ( ++, ++,,,, ++, ψl = ++, ++,,, A5 =,, ++,, Ψ= ψl ++, +,,, ++, ++, ++,, ψ, + +, + L R Higgs is is identified with 0 mode of A5 A5 (KK (KK modes modes of of A5 A5are absorbed into into KK KK gauge gauge bosons

15 L Consider (D+-dim SU( gauge theory on S^/Z ( MN M = Tr FMN F + iψγ DM Ψ MN M N N M 5 M N M M 5 M M M ( ψ, ψ, ψ T M Γ = [, ](, 0,,,, F = F F ig A A M N = D a a a ( : : Gell-Mann matrices D = ig A A = A λ λ Ψ= ( 5 γ, iγ A Boundary conditions: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( +, + + ψ R(, ( ψ R( ( ψ ( ++, ++,,,, ++, ψl = ++, ++,,, A5 =,, ++,, Ψ= ψl ++, +,,, ++, ++, ++,, ψ, + +, + L R Chiral fermions are easily obtained

16 we obtain D-dim effective Lagrangian of fermion ( n n n n i ( ψ, ψ, ψ γ ψ n= ψ ( ( ( ( L 4D fermion = ψ B + W + W 0 ψ g ( ( ( B + ( ψ, ψ, ψ W W + 0 γ ψ ψ n n n n ( 0 0 B mn 0 0 ψ ( ψ, ψ, ψ 0 mn m ψ 0 m m n ψ g + g + itlγ tl + b ( iγ m b + ( t γ LbW + bγ LtW + ( t γ Lt + bγ Lb W g ( t Lt b Lb b + γ + γ γ Rb B 6 n g L 5 R 5 mn g m R π R 5 ( γ, ( + γ, =, =, = ( = gv M W

17 D-dim effective Lagrangian in terms of mass eigenbasis ( n ψ ( n ψ 0 0 ψ n n = i ( ψ, ψ, ψ γ ψ n= ψ = 0 ψ ψ 0 ψ ψ B + + W + W W ( n ψ g W B W B + ( ψ, ψ, ψ W + γ ψ ψ W B W B W + ( 4D ( ( L fermion mn 0 0 ψ ( ψ, ψ, ψ 0 mn + m 0 ψ 0 0 m m n ψ g + g + itlγ t + b ( iγ m b + ( t γ LbW + bγ LtW + ( t γ Lt + bγ Lb W + L g ( t Lt b Lb γ + γ bγ Rb B 6 n g5 L γ5, R + γ5, mn =, g =, m= gv = M R π R ( ( ( W

18 D-dim effective Lagrangian in terms of mass eigenbasis ( n ψ ( n ψ 0 0 ψ n n = i ( ψ, ψ, ψ γ ψ n= ψ = 0 ψ ψ 0 ψ ψ B + + W + W W ( n ψ g W B W B + ( ψ, ψ, ψ W + γ ψ ψ W B W B W + ( 4D ( ( L fermion mn 0 0 ψ ( ψ, ψ, ψ 0 mn + m 0 ψ 0 0 m m n ψ g + g + itlγ t + b ( iγ m b + ( t γ LbW + bγ LtW + ( t γ Lt + bγ Lb W + L g ( t Lt b Lb γ + γ bγ Rb B 6 n g5 L γ5, R + γ5, mn =, g =, m= gv = M R π R ( ( ( W

19 Fermion contributions to gluon fusion amplitude (D+-dim SU( gauge theory with a triplet fermion on S^/Z Essential Point for calculation KK spectrum: Mass splitting m m + = = n R n + m m + m v m Top Yukawa: Sign flipping R v Crucial for cancellation of divergences