MadGraph/MadEvent with spin-2 particles: Graviton plus monojet and dijet production at the LHC Qiang Li ITP, Universit¨ at Karlsruhe PSI, Dec.18, 2008 Based on EPJC 56, 435 (2008), JHEP 0804:019 (2008), and ongoing work With Prof. Kaoru Hagiwara, Dr. Stefan Karg, Prof. Michael Kr¨ amer, Dr. Partha Konar, Dr. Kentarou Mawatari, Prof. Dieter Zeppenfeld Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 1 / 52
Outline Introduction Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 2 / 52
Outline Introduction HELAS Subroutines for Spin-2 particles Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 2 / 52
Outline Introduction HELAS Subroutines for Spin-2 particles MadGraph/MadEvent for Extra Dimension models Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 2 / 52
Outline Introduction HELAS Subroutines for Spin-2 particles MadGraph/MadEvent for Extra Dimension models Graviton plus two jets production Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 2 / 52
Outline Introduction HELAS Subroutines for Spin-2 particles MadGraph/MadEvent for Extra Dimension models Graviton plus two jets production NLO QCD corrections to Graviton monojet production Summary Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 2 / 52
Outline Summary Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 2 / 52
Introduction Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 3 / 52
Extra Dimensions (ED) Search for extra dimensions has been one of the major objects at the LHC, since its physical effects can appear at the TeV energy scale. Two major classes of Extra Dimensions models: “Flat” (factorizable) ED Large ED (ADD) (Arkani-Hamed, Dvali & Dimopoulos) TeV − 1 ED (variation of ADD) Universal ED(UED) (Appelquist, Cheng & Dobrescu) ... “Warped” (non-factorizable) ED Randall-Sundrum(RS) model ... Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 4 / 52
ADD Model (Arkani-Hamed, Dvali & Dimopoulos) Assuming the δ -extra dimensions are compacted into δ torus with the same radius r,the metrics in ADD Model are given by: ds 2 = ( η µν + h µν ) dx µ dx ν − r 2 d Ω 2 δ + ..., From dimensional analysis, the scalar curvature [ R (4+ δ ) ] = 2, thus the D-dimensional(D=4+ δ ) Einstein-Hilbert (EH) action is � � δ +2 d 4+ δ x | g (4+ δ ) | [ R (4+ δ ) ] , S D = − M s compared with 4-dimensional EH action � � 2 d 4 x | g (4) | [ R (4) ] , S 4 = − M Pl we have 2 δ +2 (2 π r ) δ = r δ M δ +2 Pl = M M s s Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 5 / 52
ADD Model: Large extra dimensions If r is quite small, at the order of Planck length, then M s ∼ M Pl . The effects of extra dimensions will be negligible. But... The Seattle Experiment probed directly gravity and tested Newton’s law only at Submillimeter level (0.2mm). Thus the length of extra dimensions can be much larger than the Planck length. Possibility of TeV scale extra dimensions: If δ = 1 and M s ∼ 1TeV, → r ∼ 10 15 cm, excluded, If δ = 2 and r < 0 . 2mm, → M s > 1 . 5TeV, If δ > 2 and M s ∼ TeV, → r < 10 − 6 cm, thus it will be difficult to probe extra dimensions by direct gravity test: High Energy Colliders. Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 6 / 52
ADD Model: Kaluza-Klein (KK) tower In ADD model, there is an infinite tower of 4D KK modes. L int = − 1 � ( h ( � n ) ) µν T µν , M Pl � n n -th KK mode h ( � n ) The mass of the � µν is | � n | / r . � M s � M s � δ +2 � 2 /δ ∆ m ∼ 1 2 12 δ − 31 r = M s ∼ 10 eV . δ TeV M Pl For M s = 1TeV and δ =4, 6 and 8, ∆ m =20KeV, 7MeV and 0.1GeV, respectively. Thus for δ ≤ 6, the KK tower can be looked as continuous. Mass density function: 2 m δ − 1 dm , with S δ − 1 = 2 π δ/ 2 M n | δ − 1 d | � Pl d � n = S δ − 1 | � n | = S δ − 1 Γ ( δ/ 2 ) . M 2+ δ s Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 7 / 52
Examples on Feynman rules a P1 M Pl δ ab C µν,ρσ P ρ i 1 P σ − 2 p 2 b a c p 1 − g s M Pl f abc C µν,ρσ P σ 2 p 2 b b,s c,r k 2 k 3 � g s M Pl f abc C µν, ls ( k 1 r − k 2 r ) + C µν, lr ( k 3 s − k 1 s ) k 1 a,l � + C µν, sr ( k 2 l − k 3 l ) + F µν, lsr ( k 1 , k 2 , k 3 ) C µν,ρσ = η µρ η νσ + η µσ η νρ − η µν η ρσ , F µν,ρσλ ( k 1 , k 2 , k 3 ) = η µρ η σλ ( k 2 − k 3 ) ν + η µσ η ρλ ( k 3 − k 1 ) ν + η µλ η ρσ ( k 1 − k 2 ) ν + ( µ ↔ ν ) . Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 8 / 52
Constraints and Collider Phenomenology Supernova (SN1987A) cooling constraints: M s > 30TeV ( δ =2) and 4TeV ( δ =3). However, the constraints can be relaxed easily without large change on collider phenomenology. Virtual KK-graviton exchange and direct KK-graviton production (Missing Energy): LEP and Tevatron datas lead to M s > 1.2TeV( δ =3) and 0.83TeV( δ =6). Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 9 / 52
Graviton production with jet(s) At the LHC, Graviton production with monojet has been studied and found to have strong ability to probe higher extra dimension scale: jet+missing Energy It is natural and important to study the Graviton dijets production: more soft jets, more information Moreover, to improve the theoretical accuracy, NLO QCD corrections needed. ∆ R jj > 0 . 7, | η j | < 4 . 5, P miss > 500 GeV, CTEQ6L1 T µ r = µ f = P j T / 2: 0.093pb for δ = 3, 0.049pb for δ = 4 µ r = µ f = 2 P j T : 0.058pb for δ = 3, 0.03pb for δ = 4 See also on next page the scale dependence of P miss distributions for √ √ T µ r = µ f = 3 ˆ s , ˆ s / 3 Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 10 / 52
Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 11 / 52
q q Gn q q q q Gn Gn Gn g q g q Gn Gn g q g g q g Gn g g g g g g Gn Gn g g g g g g Large number of subprocesses and Feynman diagrams: = ⇒ Helicity amplitude method, MadGraph/MadEvent. Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 12 / 52
New HELAS Subroutines for spin-2 particles Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 13 / 52
New HELAS Subroutines HELAS (Helicity Amplitudes Subroutines) [MURAYAMA, WATANABE, HAGIWARA, 1992] is a set of Fortran77 subroutines which make it easy to compute the helicity amplitudes of an arbitrary tree-level Feynman diagram with a simple sequence of CALL SUBROUTINE statements. Calculating steps of Helicity amplitude: 1. Getting the external particles’ wave functions; 2. Computing the off-shell lines; 3. Calculating the helicity amplitude. Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 14 / 52
Tensor’s wavefunctions ǫ ( ∗ ) µν ( P , M ): TXXXXX(P, M, λ (=0, ± 1, ± 2), IF = ± , TWF) ǫ + µ ǫ + 1 2 ( ǫ + µ ǫ 0 ν + ǫ 0 µ ǫ + ν ) , 1 6 ( ǫ + µ ǫ − ν + ǫ − µ ǫ + ν + 2 ǫ 0 µ ǫ 0 1 2 ( ǫ − µ ǫ 0 ν + ǫ 0 µ ǫ − ν , ν ), ν ), √ √ √ ǫ − µ ǫ − ν . HELAS Subroutines for graviton’s interaction: 1. SST: SSTXXX, HSTXXX, USSXXX; 2. FFT: IOTXXX, FTIKXX, FTOXXX, UIOXXX; 3. FFVT: IOVTXX, FVTIXX, FVTOXX, JIOTXX, UIOVXX; 4. VVT: VVTXXX, JVTXXX, UVVXXX; 5. GGGT: GGGTXX, JGGTXX, UGGGXX; 6. GGGGT: GGGGTX, JGGGTX, UGGGGX. Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 15 / 52
Example: FVTIXX(fi, vc, tc, g, fmass, fwidth, fvti) ( � k ) + m f [2 η µν η ρσ − η µρ η νσ − η µσ η νρ ] γ σ ( g 1 P L + g 2 P R ) ψ ( k 1 ) k 2 − m 2 f + im f Γ f × ǫ ρ ( k 2 ) ǫ µν ( k 3 ) , Gauge invariance checking: q ¯ q → ST : SSTXXX, HSTXXX; q ¯ q → gT : IOTXXX, FTIXXX, FTOXXX, IOVTXX, VVTXXX; gg → gT : GGGTXX, JVTXXX; q ¯ q → ggT : FVTIXX, FVTOXX, JIOTXX, UIOXXX, UVVXXX, UIOTXX; gg → ggT : JGGGXX, UGGGXX, GGGGXX; Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 16 / 52
MadGraph/MadEvent for Extra Dimension models Qiang Li (ITP, Karlsruhe University) MadGraph/MadEvent with spin-2 particles PSI, Dec.18, 2008 17 / 52
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