Solutions to gauge hierarchy problem SS 10, Uli Haisch 1
Quantum instability of Higgs mass So far we considered only at RGE of Higgs quartic coupling (dimensionless parameter). Higgs mass has a totally different behavior. It is highly dependent on UV physics, which leads to so-called gauge-hierarchy problem Higgs-boson mass receives one-loop corrections from graphs containing scalars, gauge bosons, and fermions h S V F h h h h F h 2
Fine-tuning in SM Even for a low UV cutoff scale of 5 TeV a careful balancing act (% fine-tuning) is required to maintain a desired output gauge bosons Λ = 5 TeV Higgs tuned input 0 1 0.5-0.5 small Higgs-boson mass in face of oneloop SM corrections that read: top -1-1.5-2 δm 2 h G F Λ 2 4π 2 2 ( 6M 2 W +3M 2 Z 12m 2 t +3M 2 h) 3
How to stabilize Higgs potential We saw that description of electroweak symmetry breaking (EWSB) with a Higgs boson suffers from several instabilities at quantum level (triviality and stability bound of quartic coupling, hierarchy problem of Higgs mass) Extra structures (particles and/or symmetries) are needed to stabilize Higgs potential. To keep radiative corrections under control, one can use essentially two different tools, namely the spin trick or the Goldstone theorem 4
Spin trick: Basics A particle of spin s has 2 s + 1 polarizations unless it travels with speed of light (c), in which case it can have less degrees of freedom Conversely, if a symmetry decouples polarizations then particle moves with c and remains massless. For example, gauge invariance guarantees that spin-1 particles have no longitudinal polarizations and chiral symmetry ensures that spin-1/2 particles have only one helicity. In both cases dangerous radiative corrections are absent due to the symmetries But trick not directly applicable to spin-0 particles 5
Spin trick: At work To apply spin trick to scalars one has to enlarge four-dimensional (4D) Poincaré symmetry (Coleman- Mandula and Haag-Lopuszanski-Sohnius theorems) First possibility consists in embedding 4D Poincaré algebra into a superalgebra. Then supersymmetry (SUSY) between fermions and bosons allows to extend spin trick to scalar particles Second way is to use an extra dimension (XD). After compactification of XD, five-dimensional (5D) gauge field decomposes into a 4D gauge and a 4D scalar field. Symmetry between vectors and scalars then allows to extend spin trick to spin-0 particles 6
Spin trick: SUSY F S h h h h F Considering one-loop corrections to Higgs mass from a Dirac fermion F and N S complex scalars S, one has: L = λ F h FF λ S h 2 S 2 +... = δm 2 h = λ F 2 8π 2 Λ2 + N S λ S 16π 2 Λ2 +... 7
Spin trick: SUSY We see that Λ 2 terms in latter relation neatly cancel if N S = 2 and λ S = λ F 2. Cancellation of Λ 2 corrections to scalar masses is not only possible, but actually unavoidable, once one assumes that there is a symmetry relating fermions and bosons But SUSY cannot be exact symmetry in nature, since otherwise we should have seen superpartners e L and e R of electron (selectrons) with masses of 511 KeV, but we didn t. In order not to reintroduce a strong UV dependence into renormalized scalar masses, SUSY 2 2 has to be broken softly (M h = M soft λ/(16 π 2 ) ln Λ/M soft ) 8
Little hierarchy problem Actually in minimal supersymmetric SM (MSSM) one has: M 2 h = M 2 Z cos 2 2β + 3g2 8π 2 m 4 t M 4 W ln m2 t m 2 t Mh [GeV] mh GeV 150 130 110 90 70 0.2 0.5 1 2 5 10 m t TeV m t [TeV] LEP limit tanβ 2 tanβ 5 tanβ 50 LEP-II bound of M h > 114.4 GeV requires that stop mass (m t ) should be larger than 1 TeV, reintroducing tuning at percent level (little hierarchy problem) 9
Spin trick: XD Basic idea of how to solve hierarchy problem in extra dimensions (XDs) via spin trick, consists in using 5 th component A 5 of 5D gauge field A M, whose mass is protected by 5D gauge invariance, to generate a light Higgs boson While A 5 is massless at tree level, including quantum corrections one finds that scalar receives a small mass at one-loop level. This way of stabilizing Higgs potential is called radiative symmetry breaking or Hosotani mechanism. In the following I briefly outline the basic steps of mechanism 10
Gauge fields in 5D Let us start by considering a SU(2) Yang-Mills theory in 5D with following action: [ S = Tr d 4 x dx 5 1 ] 4 F MNF MN = Tr d 4 x dx 5 [ 1 4 F µνf µν 1 2 F µ5f µ5 ] Here M, N = 0, 1, 2, 3, 5, while μ, ν = 0, 1, 2, 3 and A M A M τ i with τ i = σ i /2 (Pauli matrices) i 11
Gauge fields in 5D Now we compactify 5 th dimension to a circle, so that x 5 Rϕ with -π ϕ π, where R is radius of circle and Fourier expand gauge field: A µ = A (0) µ (x)+ n=1 ( ) A (n) µ (x)e inφ +h.c. x μ Component with superscript (0) is called zero mode, while those with (n) are called Kaluza-Klein (KK) excitations R x 5 Rϕ 12
Gauge fields in 5D Working in almost axial gauge by choosing A 5 = A 5 action takes form S = Tr d 4 x =2πR Tr n=1 dx 5 [ 1 4 F µνf µν + 1 2 ( 5A µ ) 2 + 1 2 d 4 x [ 1 2 { 1 2 µ A (n) ν [ ] 2 µ A (0) ν ν A (0) 1 µ + 2 ν A (n) µ (0) (x) ] ( Dµ A (0) ) 2 5 ( µ A (0) ) 2 5 ] } 2 + n2 A (n) 2 R 2 µ + O(A 3 ) which shows that 5D theory is equivalent to a 4D theory with two massless zero modes and an infinite tower of massive KK modes, m 2 = n 2 /R 2 n 13
Gauge-boson spectrum 4/R 3/R 2/R 1/R 0 A (0) A (0) 5 µ scalar zero mode vector zero mode............ A (2) µ A (2) µ A (1) µ A (1) µ complex vectors KK modes 14
Gauge-Yukawa unification Realize that massless scalar A 5 could be something like a Higgs In order to proceed let us couple a 5D fermion to gauge field and see if we can get a Yukawa coupling. We first need a representation of 5D Clifford algebra, {Γ M, Γ N } = 2 η MN where η MN is 5D Minkowski metric. One finds Γ µ γ µ, Γ 5 iγ 5 which necessarily implies that a 5D fermion is a 4- component spinor Ψ 15
Gauge-Yukawa unification We KK decompose 5D spinor as follows Ψ = n= Ψ (n) (x)e inφ Plugging this into 5D Dirac action gives S = d 4 x dx 5 Ψ ( id M Γ M m ) Ψ = d 4 [ x dx 5 Ψ (idµ γ µ m) Ψ Ψγ 5 5 Ψ + igψa 5 γ 5 Ψ ] [ ( ] =2πR d 4 x Ψ (n) iγ µ µ m i n ) R γ 5 Ψ (n) + O(ΨAΨ) n= 16
Fermion spectrum 4/R...... 3/R...... 2/R 1/R m...... Ψ (1) Ψ (1) Ψ (0) Ψ (0) left right 17
Gauge-Yukawa unification Looking at decomposed action, we see that we have generated a tower of 4D Dirac fermions with masses m 2 n = m 2 + n2 and a Yukawa coupling, ig ΨA 5 γ 5 Ψ, with strength of gauge coupling (gauge-yukawa unification) Of course, an unattractive feature of our theory is that we have non-chiral 4D fermion zero modes due to 5D Lorentz invariance. Yet SM is a theory of chiral Weyl 2-component fermions. We will simply ignore this chirality problem in the following R 2 18
Radiative symmetry breaking Ψ A µ A µ A 5 A 5 Ψ A 5 A 5 A 5 A 5 Since a light scalar is unnatural in (non-susy) QFTs it is rather surprising to see a massless 4D scalar A 5 emerging from higher dimensions. Of course, we should again consider quantum corrections to classic picture and see what happens to scalar mass at oneloop level 19
Radiative symmetry breaking From a 4D viewpoint we would naively expect that scalar mass squared receives a correction of order g 2 16π 2 Λ2 suggesting that A 5 is naturally heavy But from a 5D viewpoint A 5 is massless because it is part of a 5D gauge field and thus protected by 5D gauge invariance Which viewpoint is correct? 20
Radiative symmetry breaking To answer question one has to compute effective potential for A 5. I only state the result. Calculating fermionic contribution one finds that A 5 would like to have a non-vanishing VEV, A 5 ~ 1/(gR) Gauge loops would however prefer a vacuum at A 5 = 0. If one includes sufficiently many copies of Ψ fermions will always dominate and A 5 will be indeed non-zero in vacuum. Since A 5 is a isovector this VEV will break SU(2) to U(1) and we have a model of EWSB where U(1) is electromagnetism and A 5 is Higgs (radiative symmetry breaking or Hosotani mechanism) 21
Radiative symmetry breaking Spectrum of particles of our theory is given by M γ =0, M W 1 R, M Ψ 1 R, M KK 1 R, M h 1 πr g 4π 1 R While we get a light Higgs, feature M KK M ~ W is problematic since such light KK states would have already been seen. Above line of reasoning shows however how spin trick can be applied (in principle) to scenarios with additional spatial dimensions 22
Scales in XD theory UV completion (EFT fails) E 2/(αR) Λ UV 5D EFT 1/R M KK M W v 4D effective field theory (EFT) M h α/( 4πR) 0 23
Goldstone theorem Recall that when a global symmetry is spontaneously broken, spectrum contains a massless spin-0 particle (Goldstone theorem) Again it seems difficult to invoke this trick to protect SM Higgs boson from radiative corrections because a Nambu-Goldstone boson (NBG) can, unlike SM Higgs field, only have derivative couplings (shift symmetry) So-called little Higgs theories have been designed to circumvent these problems. They provide realistic examples of Higgs boson realized as a (pseudo) NGB 24
Interlude: NGBs Consider a theory with a single complex scalar ϕ with potential V = V(ϕ*ϕ). Kinetic term ( μ ϕ*) ( μ ϕ) as well as V invariant under U(1) transformation φ e iα φ, α = const. If minimum of V is not at ϕ = 0 but at ϕ = f, U(1) is spontaneously broken in vacuum. We write φ(x) = 1 2 (f + r(x)) e iθ(x)/f where r(x) is massive radial mode and θ(x) is NGB 25
Interlude: NGBs Radial mode is invariant under U(1), whereas NGB shifts θ θ + α/f U(1) symmetry is said to be non-linearly realized Interestingly, shift symmetry ensures that in EFT obtained after integrating out heavy radial mode NGB cannot have a mass term 1 2 m2 θ 2 so that only derivative couplings with θ are allowed 26