The Higgs boson as a window to Beyond the Standard Model Physics Roberto Contino Università di Roma La Sapienza
1. what have we discovered so far...... and why we need an EWSB sector
The physics discovered so far can be powerfully classified according to a gauge principle, except for the terms responsible for the particles masses L SM = L 0 + L mass L 0 = 1 4 B µνb µν 1 4 W a µνw a µν 1 4 G µνg µν + 3 j=1 ( Ψ(j) L i DΨ(j) L ) (j) + Ψ R i DΨ(j) R L mass =MW 2 W µ + W µ + 1 2 M Z 2 Z µ Z µ { ū (i) L M iju u (j) (i) R + d L M ijd d (j) R i,j + ē(i) L M e ije (j) R } + ν(i) L M ijν ν (j) R + h.c. Mass terms are not invariant under the local SU(2) L xu(1) Y symmetry
Mass terms are responsible for the inconsistency of the theory at high energies: The scattering of longitudinal W s and Z s violates unitarity at high energy θ The optical theorem E p 1 s Im (A(θ = 0)) = σ tot(w W anything) requires for each partial wave: physical amplitudes bound within the unitarity circle Im(a l ) Im(a l (s)) = a l (s) 2 + a in l (s) 2 Re(a l (s)) 1 2 Re(a l )
The amplitude for scattering of longitudinal W s and Z s grows with the energy and eventually violates the unitarity bound: Ex: A(W + L W L W + L W L ) = g2 2 4M 2 W (s + t) each longitudinal polarization gives a factor E ( ɛ µ L = pµ E + O M W M W ) W L W L + Z,γ + W L Z,γ W L Unitarity is violated at s Λ = 1.2 TeV
The breaking of gauge invariance is in fact a fake: the symmetry is just hidden Σ = exp (iσ a χ a /v) D µ Σ = µ Σ ig 2 σ a 2 W a µ Σ + ig 1 Σ σ 3 2 B µ ρ = 1 follows from a larger global SU(2)LxSU(2)R approximate invariance broken only by Σ U L Σ U R g 1 and λ u λ d [ ] L mass = v2 4 Tr (D µ Σ) (D µ Σ) v 2 +a v 2 Tr [ Σ D µ Σ T 3] 2 i,j (ū (i) L (i) d L )Σ ( λ u ij u (j) R λ d ij d(j) R ) + h.c. In fact, an additional term that breaks the LR symmetry has been omitted as ρ exp 1 The SU(2)LxU(1)Y symmetry is now manifest, although non-linearly realized Σ U L Σ U Y U L (x) = exp(i α a L(x)σ a /2) U Y (x) = exp(i α Y (x)σ 3 /2) In the unitary gauge Σ = 1, L mass is equal to the original mass Lagrangian with ρ M 2 W M 2 Z cos2 θ W = 1
Violation of unitarity, thanks to the Equivalence Theorem, W µ L = χ ( ( M 2 1 + O W E 2 )) can be traced back to the scattering of the Goldstone bosons: χ + µ µ χ + χ χ A(χ + χ χ + χ ) = 1 (s + t) v2 and it is linked to the non-renormalizability of the Lagrangian We need a new EWSB sector that acts as a UV completion of the EW chiral Lagrangian and restores unitarity
11. Restoring unitarity by adding new field: the Higgs model
The most economical EWSB sector consists of 1 scalar field singlet under the SU(2) L xsu(2) R (and the local SU(2) L xu(1) Y ) symmetry : L EW SB = v2 4 Tr [ D µ Σ D µ Σ ] + a v 2 h Tr [ D µ Σ D µ Σ ] + b 1 4 h2 Tr [ D µ Σ D µ Σ ] + V (h) a and b are free parameters For a=1 the scalar exchange unitarizes the WW scattering χ + χ h For b=1 also the inelastic channels respect unitarity χ + χ A(χ + χ χ + χ ) = 1 v 2 [ s χ + χ h h χ + χ a s2 s m 2 h ] + (s t) h h
a=b=1 defines the Higgs Model, whose Lagrangian can be rewritten in the standard form in terms of the SU(2)L doublet H = 1 2 e iσa χ a /v ( 0 ) v + h Unitarity of the Higgs Model can be traced back to its renormalizability There is an unbroken custodial symmetry SO(3) preserved by the Higgs vev that leads to ρ = 1 H = ( ) w1 + i w 2 w 3 + i w 4 H H = i (w i ) 2 V (H H) is SO(4)~SU(2)LxSU(2)R invariant H H = v 2 breaks SO(4) SO(3)
1. Unitarity bound on Higgs mass A(W + L W L W + L W L ) = g2 2 m 2 h 4M 2 W [ s s m 2 h + t t m 2 h ] g2 2 m 2 h 4M 2 W Unitarity of WW scattering requires: m h 8π 2 5G F 780 GeV The heavier the Higgs is, the more strongly it couples and the broader it is λ 4 = m2 h 2v 2 Γ(h W + W ) 1 16π m 3 h v 2 As a general rule: the later unitarity is cured, the more strongly coupled the EWSB sector will be
2. Bounds on the Higgs mass through the quartic coupling The physical Higgs mass is set by the quartic coupling, which is a running parameter m 2 h = 2λ 4 v 2 16π 2 d d log Q λ 4 = 24 λ 2 4 ( 3g 2 + 9g 2 12yt 2 ) λ4 + 3 8 g 4 + 3 4 g 2 g 2 + 9 8 g4 6 yt 4 + For a too heavy Higgs, the first term dominates and drives to a Landau pole at large energy scales λ 4 Triviality Bound m 2 h 4π2 v 2 3 log(λ/v) largest scale of validity of the theory (cutoff scale) For a too small Higgs, the last term dominates and drives negative at large energy scales λ 4 vacuum stability Bound m 2 h 3y4 t v 2 4π 2 log(λ/v)
perturbativity up to the Planck scale requires m h < 180 GeV Λ = 10 19 GeV from: T. Hambye, K. Riesselmann Phys Rev, D55 (1997) 7255
3. UV instability of the Higgs mass term [ aka: the Hierarchy Problem ] The Higgs mass term receives quadratically divergent corrections from SM loops, making the physical mass highly sensitive to the value of the cutoff scale: δm 2 h = [ 6 y 2 t 3 4 ( 3 g 2 2 + g 2 1) 6 λ4 ] Λ 2 8π 2 The larger Λ, the less natural a light Higgs is The cutoff might be low: the Higgs model should be perhaps regarded as a parameterization rather than a dynamical explanation of EWSB
3. LEP constraints on m h and If the Higgs model is considered as an effective field theory, New Physics effects are encoded in higher order (non-renormalizable) operators : L = L SM + i,p c i Λ p O(4+p) i O (6) i { (H T a H)W a µνb µν, H D µ H 2, ( Ψγ µ Ψ) 2, ( Ψγ µ Ψ)(H D µ H) } S parameter T parameter (Δρ) oblique corrections 4-fermion operators vertex corrections Notice: those above are flavor and CP-conserving operators
The EW precision measurents performed at LEP, SLD and Tevatron can be used to test the Higgs Model and put constraints on m h and
Global fit to the Higgs Model [ from the LEP EWWG ] The dependence upon the Higgs mass is logarithmic and comes in through loop effects On the whole the Standard Model performs rather well, with a clear indication for a light Higgs m h 154 GeV @ 95% CL!" 2 6 5 4 July 2008 m t = 172.4 (1.2) GeV Theory uncertainty!# had =!# (5) 0.02758±0.00035 0.02749±0.00012 incl. low Q 2 data 3 2 1 Excluded Preliminary 0 30 100 300 m H [GeV] m Limit = 154 GeV yet the fit is not entirely satisfactory: χ 2 P~4.5% P~15% χ 2 min ndf χ 2 min ndf (all) = 28.0 17 (only high) = 18.2 13 m h [GeV]
Best fit m h = 87 +36 27 GeV is the result of a tension between leptonic and hadronic asymmetries July 2008 Only the hadronic asymmetries (and the NuTeV result) push for a high Higgs mass! Z " 0 had R 0 l A 0,l fb A l (P # ) R 0 b R 0 c A 0,b fb A 0,c fb A 0,b removing fb from the fit gives a better χ 2 but also : m h = 55 +30 20 GeV A b A c A l (SLD) sin 2 $ lept eff (Q fb ) m W *! W * Q W (Cs) sin 2 $%%(e % e % MS ) sin 2 $ W (&N) g 2 L (&N) g 2 R (&N) Constraint from the global fit *preliminary 10 10 2 10 3 M H [GeV]
The fit improves allowing for a (large) δg Rb from New Physics 0.12 0.11 g Rb 0.1 0.09 0.08 SM 68.3 95.5 99.5 % CL 0.07-0.46-0.45-0.44-0.43-0.42-0.41 g Lb However: very few examples of New Physics leading to a large and very small positive δg Rb δg Lb
Moral: The Higgs boson is light, unless New Physics is of a very special kind 0.01 0.009 68%,90%,99% CL 0.008 0.007 Ε 1 0.006 0.005 0.004 0.003 100 GeV 1 TeV 0.003 0.004 0.005 0.006 0.007 0.008 Ε 3
If the Higgs is light: bound on each individual coefficient is strong [ LEP paradox ] Possible conclusions: 1. more than one operator contribute 2. coefficients ci are small ( = New Physics is weakly coupled)
111. Curing the UV sensitivity of the Higgs model
Fact: mass of fermions and gauge bosons are UV-stable because protected by a symmetry Strategy: relating the Higgs to fermions or gauge fields to gain the symmetry protection CHIRAL SYMMETRY (fermion protection) SUSY h ) ( h h GAUGE SYMMETRY (gauge protection) GAUGE-HIGGS UNIFICATION [requires extra dimensions] h = A 5
SUSY in a nutshell: each SM particle has a superpartner with opposite statistics sfermions (scalar) ( ψ ψ ) ( h h higgsinos (fermion) ) ( AµÃ ) gauginos (fermion) UV instability canceled by the superpartners t R top t L,R tl internal consistency + anomaly cancellation require 2 Higgs doublets h comes along with: H ±, H, A dynamical explanation for EWSB: m 2 H driven negative by top loops via RG evolution Higgs must be light: m 2 h MZ 2 cos 2 2β + N c m 4 ( t m 2 2π 2 v 2 log t m 2 t ) 2 m h 135 GeV
1v. Restoring unitarity by adding an infinite number of new states
The QCD analogy: QCD is a remarkable example of a (strongly-interacting) sector which leads to a chiral Lagrangian in the low-energy limit QCD SU(2)L SU(2)R π SU(2)V For mq=0 QCD with 2 flavors has an SU(2)LxSU(2)R invariance L QCD = q=u,d q γ µ( i µ g 3 λ a G a µ) q SU(2)LxSU(2)R spontaneously broken to SU(2)V at low energy by the qq condensate L χ = f 2 π 4 Tr [ µ Σ µ Σ ] the three pions π a are the composite Goldstone bosons of the chiral symmetry breaking Σ(x) = e iσa π a (x)/f π f π = 92 MeV
Unitarity of ππ scattering is enforced by the contribution of the heavier vector and axial resonances : ρ n m ρ 800 MeV n π m π 140 MeV the pion lighter because is a (pseudo)-goldstone
After turning on the SU(2) L xu(1) Y gauging, the pions are in fact eaten to give mass to the W and Z: W a µ,b a µ _ _ QCD SU(2) L SU(2) R U(1) B = + π SU(2) V U(1) B +... G µν (q) = i q 2 1 1 g 2 2 Π(q2 ) (η µν qµ q ν ) q 2 U(1) Q unbroken: massless photon iπ µν (q) = d 4 q e iqx T {J µ (x)j ν (0)} Π µν (q) = Π(q 2 ) ( q 2 η µν q µ q ν) custodial symmetry ρ=1 Problems: 1) f π =92MeV M W = g f π /2 =30 MeV! 2) we do observe the pions!
The technicolor paradigm: [Weinberg, Susskind] QCD _ _ SU(2)L U(1)Y _ _ techni-qcd SU(2)L SU(2)R π SU(2)V f π = 93 MeV SU(2)L SU(2)R πtc SU(2)V F π = 246 GeV F π >> f π 1) W long, Z long mostly from πtc : M W g F π /2 =80 GeV 2) still a physical pion in the spectrum, mostly π
Naive Technicolor does not work Large corrections to the S parameter 0 J µ ρ = ɛ r µ f ρ m ρ W 3 ρ B L = S 32π W 3 µνb µν ( v S 16π m ρ ) 2 N π 0.01 0.009 68%,90%,99% CL 0.008 0.007 Ε 1 0.006 0.005 0.004 0.003 0.003 0.004 0.005 0.006 0.007 0.008 Ε 3 naive estimate for a scaled-up QCD-like dynamics
A possible solution to the EWPT problem: suppose the strong dynamics has also a light scalar bound state playing the role of the Higgs the composite Higgs partially unitarize h the WW scattering until a scale Λ = Λ ξ g hw W = g SM hw W (1 + O(ξ))... until the vector resonances take on n ρ n
Composite Higgs models [Georgi & Kaplan, `80s] Aµ (G SM ) BSM _ _ ψ Requirements: H G G 1) GSM G 2) H is a doublet of SU(2)L Example: SO(5) SO(4) ~ SU(2)L SU(2)R gives 4 real Goldstones: one SU(2)L doublet H
vector resonances can be heavier: smaller corrections to EWPO m ρ m h h m W the Higgs is composite: Planck/TeV Hierarchy solved
ξ = ( v f ) 2 new parameter compared to TC (fixed by the dynamics) H H m ρ 4πf N W 3 ρ B S 16π ( v m ρ ) 2 ξ N π ξ 0 [ f ] decoupling limit: All ρ s become heavy and one re-obtains the SM
LHC will tell us which path Nature has chosen! (The End)
Extra Slides
Field theories with a curved fifth dimension give an explicit realization of 4D composite Higgs models Resonances of the strong sector Kaluza-Klein excitations
Example of strong dynamics: a Randall-Sundrum setup UV brane graviton & light fermions AdS5 Higgs IR brane warp factor Scales depend on the position: translation in y 4D rescaling Solution to the Hierarchy Problem geography of wave functions in the bulk k M Pl k e 2kπR TeV ds 2 = e 2ky dx µ dx ν η µν dy 2 0 y πr
The holographic description { SM fields live here UV brane Bulk IR brane Higgs profile UV brane Bulk + IR brane Elementary sector { ψ Aµ _ _ H G G Composite sector