Federico Granata. Seminar based on ICTP summer school on particle physics June 2015, Trieste. Ph.D. XXIX cicle 17/11/2015 Tutor: Carlo Oleari

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1 Federico Granata Ph.D. XXIX cicle 17/11/015 Tutor: Carlo Oleari Seminar based on ICTP summer school on particle physics 15-6 June 015, Trieste

2 (GBET) References J.M. Cornwall, D.N. Levin and G. Tiktopoulos, Phys. Rev. D 10, 1145 (1974) C.E. Vayonakis, Lett. Nuovo Cimento 17, 383 (1976) M.S. Chanowitz and M.K. Gaillard, Nucl. Phys. B 61, 379 (1985) GBET relates longitudinally polarized vector bosons to the scalar sector at high energies Electroweak symmetry breaking Massless bosons become massive by eating Goldstone bosons Massless boson transverse polarizations ɛ µ (q) ɛ ν (q) = g µν Massive boson transverse + 1 longitudinal polarization ɛ µ (q) ɛ ν (q) = g µν + q µ q ν /m

3 (GBET) Particle moving along z-axis: q µ = (E, 0, 0, q z ), E = q z + m Polarization vectors ɛ µ T (q) = 1 (0, 1, ±i, 0) ɛ µ L (q) = 1 m (q z, 0, 0, E) At high energies, ( ɛ µ L (q) = qµ m ) m + O E The polarization vector becomes increasingly parallel to the momentum With processes involving longitudinal bosons V L at high energies: Approximation ɛ µ L (q) q µ /m

4 (GBET) With processes involving longitudinal bosons V L at high energies: M µ ɛ µ L (q) 1 m M µq µ = EW Ward identities = GBET V L φ = ( 1 + O ( m E )) With more than one longitudinal boson: m [ ( M(p 1... p n, V L1... V Lm ) = i 1 Q V m )] k M(p1... p n, φ 1... φ m ) 1 + O E k=1 Precision 1% at E = 1 TeV

5 Example: the top quark decay p t b W + k q z-axis along W + direction p = m t k = 0 q = m W ɛ µ (q) = 1 (0, 1, i, 0) R 1 m (q, 0, 0, E) qµ m 0 1 (0, 1, i, 0) L 0 R M = i g m E b m t t m W cos (θ/) 0 i g E b m t sin (θ/) L ( ) ( ) Γ t = αw Eb 8 mt 1 + m t m t m W Enhancement proportional to m t /m W (visible even from matrix element) It appears clearer if we look back at the complete sum over polarizations

6 Example: the top quark decay Let s inspect the Goldstone boson production process: p t b φ + k q Yukawa coupling between Goldstone bosons and fermions: m t = y t v, v = m W g M = i y t Eb m t cos (θ/) = i g m t E b m t cos (θ/) m W The meaning of the GBET The enhancement of Γ t by a factor mt /m W reflects the fact that the coupling tv L comes from the Higgs sector, rather than the gauge sector

7 (GBET) Analysis of longitudinal gauge bosons = Details on the Higgs sector and on Electroweak Symmetry Breaking At the LHC W and Z bosons cannot be directly revealed because they decay Angular distributions of the decay products give information on the polarization of the vector boson: Longitudinal boson: mainly decays to the direction of motion sin θ Transverse boson: mainly decays to the direction of motion (1 ± cos θ) 0

8 Example 1: Signal extraction Study of channel H VV 4l By analyzing the vector bosons polarizations it is possible to extract signal from background Background: q q VV : gg VV : a Vector bosons couple to light particles mainly transverse modes (longitudinal ones suppressed by GBET) The GBET favors Higgs boson decay into longitudinal gauge bosons Looking at the directions of the decay products it is possible to extract the signal

9 Example : New Physics searches Suppose the existence of New Physics with different couplings in Higgs-EW sector: L a m W v HW µw µ + a m Z v HZ µz µ (a 1) W L W L W L W L scattering would grow with energy like ( 1 a ) E, becoming strongly coupled at high energies Need of New Physics contributions (composite Higgs, partial composite Higgs,... ) to restore the correct behaviour Information on the value of a can be obtained, once again, by analyzing the leptons coming from the W bosons decay up to now, no deviations from L SM : no Composite Higgs!

10 Example 3: Simplify perturbative calculations The calculation of processes involving V L at high energy can be simplified using the GBET: e + e W + L W L e + e φ + φ Single diagrams have bad high-energy beaviour, only the sum is ok Only one diagram, with good high-energy behaviour The GBET is even more useful for processes like W L W L W L W L : the associated process φφ φφ contains few diagrams and gives a very good approximation of the complete result

11 The GBET in my activity I have computed and implemented in the POWHEG BOX the EW corrections in Sudakov approximation to the associated production processes HV /HVj (with leptonic decay of V ) Sudakov approximation s mw ( N ) p k s m W, 1 < N < n 1 k=1 Consider only matrix elements that are not mass-suppressed: M ϕ1...ϕn 0 (p 1,..., p n ) E d = Neglect matrix elements with structure M ϕ1...ϕn 0 (p 1,..., p n ) m k W E d k

12 The GBET in my activity: HW production and decay Leading pole approximation: M production propagator decay i p V m V + iγ V m V Good approximation around W resonance δm qq HW Hlν i p W m W + iγ W m W [ ] δm qq HW λ M Wλ lν + M qq HW λ δm W λ lν λ i δm qq HW λ M W λ lν p W m W + iγ W m W δm W λ lν : no high-energy variables = can be neglected Sum over W polarizations: consider separately transverse and longitudinal W bosons λ

13 The GBET in my activity: HW production and decay The energy regime in which Sudakov corrections are computed is the same in which the GBET can be applied (E m W ) p 1 p p H q Matrix element for HW production M i gw q V CKMm W v L (p )γ µ u L (p 1 )ɛ µ (q ) p 1 p p H q Matrix element for Hφ production M i gw q V CKM v L (p )γ µ u L (p 1 )(q p H ) µ Check: HW L VS Hφ m W v L (p )γ µ u L (p 1 )ɛ µ L (q ) m W v L (p )γ µ u L (p 1 )q µ /m W = v L (p )γ µ u L (p 1 )q µ v L (p )γ µ u L (p 1 )(q p H ) µ = v L (p )γ µ u L (p 1 )(q p 1 p ) µ = v L (p )γ µ u L (p 1 )q µ

14 The GBET in my activity: HW production and decay The matrix element for the production of transverse W boson is mass-suppressed with respect to the longitudinal one at high energies = Can be neglected dσ/dp H T [pb/gev] p H T [GeV] LO L LO L+T LO L/(L+T) dσ/dy H [pb] LO L LO L+T LO L/(L+T) y H

15 The GBET in my activity: HW production and decay The mass-suppression of the transverse W boson is even more evident when inspecting NLO EW corrections in Sudakov approximation dσ/dp H T [pb/gev] p H T [GeV] NLO L NLO L+T NLO L/(L+T) dσ/dy H [pb] NLO L NLO L+T NLO L/(L+T) y H

16 Conclusions GBET relates longitudinally polarized gauge bosons to the scalar sector Old theorem, but extremely useful for the analysis of the upcoming LHC data at 13 and 14 TeV: - Signal extraction - New physics searches - Simplify calculations and increase computational speed In my research: - Up to now, used to obtain a faster code - In the future, the study of high-energy tails of the distributions will be useful for new physics searches at the LHC

17 Polarization vectors for generic q direction Particle moving in a generic direction: q µ = (E, q x, q y, q z ) E = q + m Cartesian polarization vectors ɛ µ 1 (q) = 1 ( 0, q xq z q T q, q ) yq z q, q T q ɛ µ (q) = 1 (0, q y, q x, 0) q T ɛ µ 3 (q) = E ( ) q m q E, q x, q y, q z Transverse and longitudinal polarization vectors ɛ µ T (q) = 1 [ɛ µ 1 (q) iɛµ (q)] ɛ µ L (q) = ɛ µ 3 (q)