LHC SEARCHES FOR NEW PHYSICS BEYOND THE STANDARD MODEL
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The document discusses searches for new physics at the LHC using an effective field theory (EFT) approach. It outlines the talk which will cover EFT applications in the Higgs and electroweak sectors, including Higgs to ZZ* decays and diboson production. The EFT framework is introduced as a way to parameterize potential deviations from the standard model in a model-independent way, without assuming a specific new physics model. The most general EFT basis at dimension-6 contains 2499 operators, which can be reduced to 59 under certain flavor symmetries, using the so-called Warsaw basis.
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LHC SEARCHES FOR NEW PHYSICS BEYOND THE STANDARD MODEL
- 1. (PRECISION) NEW PHYSICS SEARCHES WITHINTHE LHC DATASET Raquel Gomez Ambrosio 1
- 2. OUTLINE OFTHETALK 1. Introduction and motivation: • LHC • EFT 2. EFT applications at LHC: State of the art 3. Part I: the Higgs Sector • Higgs to ZZ*: differential distributions and the complex mass scheme 4. Part II: the EW Sector • PP toVV: diboson production,VBS and what we can learn from that • Points for discussion: 2
- 3. INTRODUCTION LHC, EFT and all that 3
- 4. NO HINTS OF NEW PHYSICS AT LHC (YET) History of a crisis 4
- 5. NO HINTS OF NEW PHYSICS AT LHC (YET) History of a crisis 5
- 6. WHATTO DO IF NO NEW RESONANCES SHOW UP? Build new accelerators? Look for Long Lived Particles at LHC? Reexamine LHC and LEP data? Give up accelerators and look at the skies? Give up accelerators and look at quantum labs? 6
- 7. WHATTO DO IF NO NEW RESONANCES SHOW UP? Build new accelerators? Look for Long Lived Particles at LHC? Reexamine LHC and LEP data? Give up accelerators and look at the skies? Give up accelerators and look at quantum labs? One option for the meantime: make precise predictions for LHC and HL-LHC, accommodating for SM deviations 7
- 8. EFFECTIVE FIELDTHEORY Effective field theory: Decoupling heavy states from the light energy regime The first example was Fermi theory: ∼ 1 p2 + M2 → M2 ≫ p2 → 1 M2 ( 1 − p M2 + 1 2 ( p M2 ) 2 + … ) μ νe νμ e W p2 ≪ M2 W 8
- 9. EFFECTIVE FIELDTHEORY: SMEFTVS HEFT • Assuming linear representation for the Higgs, no new light particles, and SM symmetries (SMEFT): • The most general basis in dim-6 has 2499 Operators, 59 if we assume some flavour symmetries: Here we will use the so-called Warsaw Basis • The HEFT theory, a non-linear Higgs potential is studied (EChL), • Leading to a different perturbative counting of the effective Lagrangian ℒSMEFT = ℒSM + ∑ i ci Λ2 𝒪(6) i + ∑ j cj Λ4 𝒪(8) i + … Compatible with NLO corrections! (unlike kappa/anomalous approach) ℱ(H) = 1 + 2a H v + b ( H v ) 2 + … 9
- 10. THE SMEFT BASIS 1 : X3 QG fABC GA⌫ µ GB⇢ ⌫ GCµ ⇢ Q e G fABC e GA⌫ µ GB⇢ ⌫ GCµ ⇢ QW ✏IJK WI⌫ µ WJ⇢ ⌫ WKµ ⇢ Qf W ✏IJK f WI⌫ µ WJ⇢ ⌫ WKµ ⇢ 2 : H6 QH (H† H)3 3 : H4 D2 QH⇤ (H† H)⇤(H† H) QHD H† DµH ⇤ H† DµH 5 : 2 H3 + h.c. QeH (H† H)(¯ lperH) QuH (H† H)(q̄pur e H) QdH (H† H)(q̄pdrH) 4 : X2 H2 QHG H† H GA µ⌫GAµ⌫ QH e G H† H e GA µ⌫GAµ⌫ QHW H† H WI µ⌫WIµ⌫ QHf W H† H f WI µ⌫WIµ⌫ QHB H† H Bµ⌫Bµ⌫ QH e B H† H e Bµ⌫Bµ⌫ QHW B H† ⌧I H WI µ⌫Bµ⌫ QHf W B H† ⌧I H f WI µ⌫Bµ⌫ 6 : 2 XH + h.c. QeW (¯ lp µ⌫ er)⌧I HWI µ⌫ QeB (¯ lp µ⌫ er)HBµ⌫ QuG (q̄p µ⌫ TA ur) e H GA µ⌫ QuW (q̄p µ⌫ ur)⌧I e H WI µ⌫ QuB (q̄p µ⌫ ur) e H Bµ⌫ QdG (q̄p µ⌫ TA dr)H GA µ⌫ QdW (q̄p µ⌫ dr)⌧I H WI µ⌫ QdB (q̄p µ⌫ dr)H Bµ⌫ 7 : 2 H2 D Q (1) Hl (H† i ! D µH)(¯ lp µ lr) Q (3) Hl (H† i ! D I µH)(¯ lp⌧I µ lr) QHe (H† i ! D µH)(ēp µ er) Q (1) Hq (H† i ! D µH)(q̄p µ qr) Q (3) Hq (H† i ! D I µH)(q̄p⌧I µ qr) QHu (H† i ! D µH)(ūp µ ur) QHd (H† i ! D µH)( ¯ dp µ dr) QHud + h.c. i( e H† DµH)(ūp µ dr) 8 : (L̄L)(L̄L) Q`` (¯ lp µlr)(¯ ls µ lt) Q (1) qq (q̄p µqr)(q̄s µ qt) Q (3) qq (q̄p µ⌧I qr)(q̄s µ ⌧I qt) Q (1) `q (¯ lp µlr)(q̄s µ qt) Q (3) `q (¯ lp µ⌧I lr)(q̄s µ ⌧I qt) 8 : (R̄R)(R̄R) Qee (ēp µer)(ēs µ et) Quu (ūp µur)(ūs µ ut) Qdd ( ¯ dp µdr)( ¯ ds µ dt) Qeu (ēp µer)(ūs µ ut) Qed (ēp µer)( ¯ ds µ dt) Q (1) ud (ūp µur)( ¯ ds µ dt) Q (8) ud (ūp µTA ur)( ¯ ds µ TA dt) 8 : (L̄L)(R̄R) Qle (¯ lp µlr)(ēs µ et) Qlu (¯ lp µlr)(ūs µ ut) Qld (¯ lp µlr)( ¯ ds µ dt) Qqe (q̄p µqr)(ēs µ et) Q (1) qu (q̄p µqr)(ūs µ ut) Q (8) qu (q̄p µTA qr)(ūs µ TA ut) Q (1) qd (q̄p µqr)( ¯ ds µ dt) Q (8) qd (q̄p µTA qr)( ¯ ds µ TA dt) 8 : (L̄R)(R̄L) + h.c. Qledq (¯ lj per)( ¯ dsqtj) 8 : (L̄R)(L̄R) + h.c. Q (1) quqd (q̄j pur)✏jk(q̄k s dt) Q (8) quqd (q̄j pTA ur)✏jk(q̄k s TA dt) 8 : (L̄R)(L̄R) + h.c. Q (1) lequ (¯ lj per)✏jk(q̄k s ut) Q (3) lequ (¯ lj p µ⌫er)✏jk(q̄k s µ⌫ ut) Table 1: L6 of Refs. [222] as given in Ref. [204]. The flavour labels p, r, s, t on the Q operators are suppressed on the left hand side of the tables. 1 : X3 QG fABC GA⌫ µ GB⇢ ⌫ GCµ ⇢ Q e G fABC e GA⌫ µ GB⇢ ⌫ GCµ ⇢ QW ✏IJK WI⌫ µ WJ⇢ ⌫ WKµ ⇢ Qf W ✏IJK f WI⌫ µ WJ⇢ ⌫ WKµ ⇢ 2 : H6 QH (H† H)3 3 : H4 D2 QH⇤ (H† H)⇤(H† H) QHD H† DµH ⇤ H† DµH 5 : 2 H3 + h.c. QeH (H† H)(¯ lperH) QuH (H† H)(q̄pur e H) QdH (H† H)(q̄pdrH) 4 : X2 H2 QHG H† H GA µ⌫GAµ⌫ QH e G H† H e GA µ⌫GAµ⌫ QHW H† H WI µ⌫WIµ⌫ QHf W H† H f WI µ⌫WIµ⌫ QHB H† H Bµ⌫Bµ⌫ QH e B H† H e Bµ⌫Bµ⌫ QHW B H† ⌧I H WI µ⌫Bµ⌫ QHf W B H† ⌧I H f WI µ⌫Bµ⌫ 6 : 2 XH + h.c. QeW (¯ lp µ⌫ er)⌧I HWI µ⌫ QeB (¯ lp µ⌫ er)HBµ⌫ QuG (q̄p µ⌫ TA ur) e H GA µ⌫ QuW (q̄p µ⌫ ur)⌧I e H WI µ⌫ QuB (q̄p µ⌫ ur) e H Bµ⌫ QdG (q̄p µ⌫ TA dr)H GA µ⌫ QdW (q̄p µ⌫ dr)⌧I H WI µ⌫ QdB (q̄p µ⌫ dr)H Bµ⌫ 7 : 2 H2 D Q (1) Hl (H† i ! D µH)(¯ lp µ lr) Q (3) Hl (H† i ! D I µH)(¯ lp⌧I µ lr) QHe (H† i ! D µH)(ēp µ er) Q (1) Hq (H† i ! D µH)(q̄p µ qr) Q (3) Hq (H† i ! D I µH)(q̄p⌧I µ qr) QHu (H† i ! D µH)(ūp µ ur) QHd (H† i ! D µH)( ¯ dp µ dr) QHud + h.c. i( e H† DµH)(ūp µ dr) 8 : (L̄L)(L̄L) Q`` (¯ lp µlr)(¯ ls µ lt) Q (1) qq (q̄p µqr)(q̄s µ qt) Q (3) qq (q̄p µ⌧I qr)(q̄s µ ⌧I qt) Q (1) `q (¯ lp µlr)(q̄s µ qt) Q (3) `q (¯ lp µ⌧I lr)(q̄s µ ⌧I qt) 8 : (R̄R)(R̄R) Qee (ēp µer)(ēs µ et) Quu (ūp µur)(ūs µ ut) Qdd ( ¯ dp µdr)( ¯ ds µ dt) Qeu (ēp µer)(ūs µ ut) Qed (ēp µer)( ¯ ds µ dt) Q (1) ud (ūp µur)( ¯ ds µ dt) Q (8) ud (ūp µTA ur)( ¯ ds µ TA dt) 8 : (L̄L)(R̄R) Qle (¯ lp µlr)(ēs µ et) Qlu (¯ lp µlr)(ūs µ ut) Qld (¯ lp µlr)( ¯ ds µ dt) Qqe (q̄p µqr)(ēs µ et) Q (1) qu (q̄p µqr)(ūs µ ut) Q (8) qu (q̄p µTA qr)(ūs µ TA ut) Q (1) qd (q̄p µqr)( ¯ ds µ dt) Q (8) qd (q̄p µTA qr)( ¯ ds µ TA dt) Grzadkowski et al. (basis), Alonso et al. (representation) The most general basis in dim-6 has 2499 Operators, 59 if we assume some flavour symmetries: Here we will use the so-called Warsaw Basis 10
- 11. THE SMEFT BASIS Bosonic operators 1 : X3 QG fABC GA⌫ µ GB⇢ ⌫ GCµ ⇢ Q e G fABC e GA⌫ µ GB⇢ ⌫ GCµ ⇢ QW ✏IJK WI⌫ µ WJ⇢ ⌫ WKµ ⇢ Qf W ✏IJK f WI⌫ µ WJ⇢ ⌫ WKµ ⇢ 2 : H6 QH (H† H)3 3 : H4 D2 QH⇤ (H† H)⇤(H† H) QHD H† DµH ⇤ H† DµH 5 : 2 H3 + h.c. QeH (H† H)(¯ lperH) QuH (H† H)(q̄pur e H) QdH (H† H)(q̄pdrH) 4 : X2 H2 QHG H† H GA µ⌫GAµ⌫ QH e G H† H e GA µ⌫GAµ⌫ QHW H† H WI µ⌫WIµ⌫ QHf W H† H f WI µ⌫WIµ⌫ QHB H† H Bµ⌫Bµ⌫ QH e B H† H e Bµ⌫Bµ⌫ QHW B H† ⌧I H WI µ⌫Bµ⌫ QHf W B H† ⌧I H f WI µ⌫Bµ⌫ 6 : 2 XH + h.c. QeW (¯ lp µ⌫ er)⌧I HWI µ⌫ QeB (¯ lp µ⌫ er)HBµ⌫ QuG (q̄p µ⌫ TA ur) e H GA µ⌫ QuW (q̄p µ⌫ ur)⌧I e H WI µ⌫ QuB (q̄p µ⌫ ur) e H Bµ⌫ QdG (q̄p µ⌫ TA dr)H GA µ⌫ QdW (q̄p µ⌫ dr)⌧I H WI µ⌫ QdB (q̄p µ⌫ dr)H Bµ⌫ 7 : 2 H2 D Q (1) Hl (H† i ! D µH)(¯ lp µ lr) Q (3) Hl (H† i ! D I µH)(¯ lp⌧I µ lr) QHe (H† i ! D µH)(ēp µ er) Q (1) Hq (H† i ! D µH)(q̄p µ qr) Q (3) Hq (H† i ! D I µH)(q̄p⌧I µ qr) QHu (H† i ! D µH)(ūp µ ur) QHd (H† i ! D µH)( ¯ dp µ dr) QHud + h.c. i( e H† DµH)(ūp µ dr) 8 : (L̄L)(L̄L) Q`` (¯ lp µlr)(¯ ls µ lt) Q (1) qq (q̄p µqr)(q̄s µ qt) 8 : (R̄R)(R̄R) Qee (ēp µer)(ēs µ et) Quu (ūp µur)(ūs µ ut) 8 : (L̄L)(R̄R) Qle (¯ lp µlr)(ēs µ et) Qlu (¯ lp µlr)(ūs µ ut) Grzadkowski et al. (basis), Alonso et al. (representation) 11
- 12. THE SMEFT BASIS Bosonic operators 1 : X3 QG fABC GA⌫ µ GB⇢ ⌫ GCµ ⇢ Q e G fABC e GA⌫ µ GB⇢ ⌫ GCµ ⇢ QW ✏IJK WI⌫ µ WJ⇢ ⌫ WKµ ⇢ Qf W ✏IJK f WI⌫ µ WJ⇢ ⌫ WKµ ⇢ 2 : H6 QH (H† H)3 3 : H4 D2 QH⇤ (H† H)⇤(H† H) QHD H† DµH ⇤ H† DµH 5 : 2 H3 + h.c. QeH (H† H)(¯ lperH) QuH (H† H)(q̄pur e H) QdH (H† H)(q̄pdrH) 4 : X2 H2 QHG H† H GA µ⌫GAµ⌫ QH e G H† H e GA µ⌫GAµ⌫ QHW H† H WI µ⌫WIµ⌫ QHf W H† H f WI µ⌫WIµ⌫ QHB H† H Bµ⌫Bµ⌫ QH e B H† H e Bµ⌫Bµ⌫ QHW B H† ⌧I H WI µ⌫Bµ⌫ QHf W B H† ⌧I H f WI µ⌫Bµ⌫ 6 : 2 XH + h.c. QeW (¯ lp µ⌫ er)⌧I HWI µ⌫ QeB (¯ lp µ⌫ er)HBµ⌫ QuG (q̄p µ⌫ TA ur) e H GA µ⌫ QuW (q̄p µ⌫ ur)⌧I e H WI µ⌫ QuB (q̄p µ⌫ ur) e H Bµ⌫ QdG (q̄p µ⌫ TA dr)H GA µ⌫ QdW (q̄p µ⌫ dr)⌧I H WI µ⌫ QdB (q̄p µ⌫ dr)H Bµ⌫ 7 : 2 H2 D Q (1) Hl (H† i ! D µH)(¯ lp µ lr) Q (3) Hl (H† i ! D I µH)(¯ lp⌧I µ lr) QHe (H† i ! D µH)(ēp µ er) Q (1) Hq (H† i ! D µH)(q̄p µ qr) Q (3) Hq (H† i ! D I µH)(q̄p⌧I µ qr) QHu (H† i ! D µH)(ūp µ ur) QHd (H† i ! D µH)( ¯ dp µ dr) QHud + h.c. i( e H† DµH)(ūp µ dr) 8 : (L̄L)(L̄L) Q`` (¯ lp µlr)(¯ ls µ lt) Q (1) qq (q̄p µqr)(q̄s µ qt) 8 : (R̄R)(R̄R) Qee (ēp µer)(ēs µ et) Quu (ūp µur)(ūs µ ut) 8 : (L̄L)(R̄R) Qle (¯ lp µlr)(ēs µ et) Qlu (¯ lp µlr)(ūs µ ut) This basis is strongly correlated! (as we will see today…)
- 13. BOTTOM-UP EFT • Assuming linear representation for the Higgs, no new light particles, SM symmetries, etc: • Amplitudes and cross-sections: we discuss a possible strategy for a global analysis including all the dimension-six operators relevant to this process. 2 SMEFT: notations and conventions In this work, the bottom-up approach to EFT is used. The SM Lagrangian is extended with higher dimensional operators, consistent with the known SM symmetries. Further, we assume a linear representation for the physical Higgs field, in the form an SU(2) doublet. Such a theory is commonly known as SMEFT: LSMEFT = LSM + c(5) ⇤ O(5) + 1 ⇤2 X i c (6) i O (6) i + X j X k 1 ⇤2+k c (6+k) j O (6+k) j . (2.1) At dimension-five, dim = 5, there is only one possible operator, from Ref.[53], which doesn’t enter the process studied in this work. At dim = 6 , the complete basis has 59 operators in the flavour universal case and 2499 in the most general one. In this work we will use a parametrisation of the former commonly known as the Warsaw basis, from Ref. [54]. A general method to construct higher dimensional bases using Hilbert series was pro- posed in Ref. [55]. In the context of VBS, some subsets of dim = 8 operators affecting quartic gauge couplings have been proposed in Refs. [56, 57] Other EFT bases There are additional dimension-six bases, other than the Warsaw basis. It is quite common to use the SILH basis, from Ref.[1] in Higgs phenomenology, however it is not optimised for multiboson processes. Instead, there is a VBS-dedicated basis, typically known as the HISZ basis, from Ref.[58]. 13 where the O (6) i represent a basis of dim = 6 operators built from SM fields and respecting the known gauge symmetries14 , and ci are the Wilson coefficients of the theory. Further, the SMEFT amplitudes and cross sections can be parametrised as AEFT = ASM + g0 ⇤2 A6 + g02 ⇤4 A8 + . . . (91) EFT ⇠ |ASM |2 + 2 g0 ⇤2 ASM A6 + g0 2 ⇤4 ⇣ 2ASM A8 + |A6|2 ⌘ + . . . (92) Here, we assume the linear contribution (red) of the EFT effects to be leading. Analysis of the dim = 6 quadratic terms and the dim = 8 interference terms (both in blue) will be subject of further studies. In particular, dim = 8 are commonly associated with quartic gauge couplings and such contribution, albeit subleading, would represent some added value to the linear dim = 6 prediction. Linear EFT Quadratic + dim-8
- 14. EFT APPLICATIONS AT LHC 14 Fits and Precision Physics
- 15. HIGGS MEASUREMENTS: SIGNAL STRENGTHS CMS-HIG-17-031 σ(prod) × BR(decay) (σ × BR)SM = μi 15
- 16. Fits of the Higgs Sector: State of the art J. Ellis et al. arXiv:1803.03252 Englert et al. arXiv:1511.05170 Higgs data, Run I, SILH basis Higgs and EW data, Run II, Warsaw basis 16
- 17. FITS OFTHE HIGGS SECTOR How accurate are signal strengths? HXSWGYR4 • Shall we consider moving to a more precise parametrisation? • Simplified andTemplate Cross-Sections (STXS)? 17
- 18. Fits of the Higgs Sector: State of the art 18 ATL-PHYS-PUB-2019-042
- 19. ANALYTIC CALCULATIONS 19 An example: H to 4 leptons
- 20. 20 SM SMEFT Abstract: draft from September 18, 2019 Keywords: Effective Field Theory, Scattering Amplitudes, Phenomenology Figure 1: Leading order Feynman diagrams for H ! eēµµ̄ in SMEFT. Crossed vertices represent contributions dimension six operators. HTO 4 LEPTONS
- 21. NEW PHYSICS IN HTO 4 LEPTONS 21 Look at signal + background !
- 22. NEW PHYSICS IN HTO 4 LEPTONS The exact calculation can be used to improve the global EFT fit, and also compared with individual analyses, in order to label eventual deviations. Many anomalous couplings studies have been performed, for example: CMS- HIG-17-034. (But results are sometimes hard to recycle…) 22
- 23. H to 4 leptons in SMEFT: Shifts to the SMEFT Lagrangian at order v2/Λ2 to V (H) = λ " H† H − 1 2 v2 #2 − CH ' H† H (3 , (5.3) yielding the new minimum ⟨H† H⟩ = v2 2 " 1 + 3CH v2 4λ # ≡ 1 2 v2 T , (5.4) on expanding the exact solution (λ − ) λ2 − 3CH λv2)/(3CH ) to first order in CH . The shift in the vacuum expectation value (VEV) is proportional to CHv2, which is of order v2/Λ2. The scalar field can be written in unitary gauge as H = 1 √ 2 * 0 [1 + cH,kin] h + vT + , (5.5) where cH,kin ≡ " CH! − 1 4 CHD # v2 , vT ≡ " 1 + 3CHv2 8λ # v. (5.6) The coefficient of h in Eq. (5.5) is no longer unity, in order for the Higgs boson kinetic term to be properly normalized when the dimension-six operators are included. The kinetic terms L = (DµH† )(Dµ H) + CH! ' H† H ( ! ' H† H ( + CHD ' H† Dµ H (∗ ' H† DµH ( , (5.7) and the potential in Eq. (5.3) yield6 L = 1 2 (∂µh)2 − cH,kin v2 T , h2 (∂µh)2 + 2vh(∂µh)2 - − λv2 T " 1 − 3CHv2 2λ + 2cH,kin # h2 23
- 24. H to 4 leptons in SMEFT: Shifts to the SMEFT Lagrangian 24 The definition of the gauge fields and the gauge couplings are affected by the dimension-six terms. The relevant dimension-six Lagrangian terms are L(6) = CHGH† HGA µνGAµν + CHW H† HWI µνWIµν + CHBH† HBµνBµν + CHW BH† τI HWI µνBµν + CGfABC GAν µ GBρ ν GCµ ρ + CW ϵIJK WIν µ WJρ ν WKµ ρ . (5.15) In the broken theory, the X2H2 operators contribute to the gauge kinetic energies, LSM + L(6) = − 1 2 W+ µν Wµν − − 1 4 W3 µν Wµν 3 − 1 4 Bµν Bµν − 1 4 GA µν GAµν + 1 2 v2 T CHG GA µν GAµν , + 1 2 v2 T CHW WI µνWIµν + 1 2 v2 T CHBBµνBµν − 1 2 v2 T CHW BW3 µνBµν , (5.16) so the gauge fields in the Lagrangian are not canonically normalized, and the last term in Eq. (5.16) leads to kinetic mixing between W3 and B. The mass terms for the gauge bosons from LSM and L(6) are L = 1 4 g2 2v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.17) The gauge fields need to be redefined, so that the kinetic terms are properly normalized and diagonal. The first step is to redefine the gauge fields GA µ = GA µ ! 1 + CHGv2 T " , WI µ = WI µ ! 1 + CHW v2 T " , Bµ = Bµ ! 1 + CHBv2 T " . (5.18) The modified coupling constants are g3 = g3 ! 1 + CHG v2 T " , g2 = g2 ! 1 + CHW v2 T " , g1 = g1 ! 1 + CHB v2 T " , (5.19) so that the products g3GA µ = g3GA µ , etc. are unchanged. This takes care of the gluon terms. The electroweak terms are ! " LSM + L(6) = − 1 2 W+ µν Wµν − − 1 4 W3 µν Wµν 3 − 1 4 Bµν Bµν − 1 4 GA µν GAµν + 1 2 v2 T CHG GA µν GAµν , + 1 2 v2 T CHW WI µνWIµν + 1 2 v2 T CHBBµνBµν − 1 2 v2 T CHW BW3 µνBµν , (5.16) so the gauge fields in the Lagrangian are not canonically normalized, and the last term in Eq. (5.16) leads to kinetic mixing between W3 and B. The mass terms for the gauge bosons from LSM and L(6) are L = 1 4 g2 2v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.17) The gauge fields need to be redefined, so that the kinetic terms are properly normalized and diagonal. The first step is to redefine the gauge fields GA µ = GA µ ! 1 + CHGv2 T " , WI µ = WI µ ! 1 + CHW v2 T " , Bµ = Bµ ! 1 + CHBv2 T " . (5.18) The modified coupling constants are g3 = g3 ! 1 + CHG v2 T " , g2 = g2 ! 1 + CHW v2 T " , g1 = g1 ! 1 + CHB v2 T " , (5.19) so that the products g3GA µ = g3GA µ , etc. are unchanged. This takes care of the gluon terms The electroweak terms are L = − 1 2 W+ µν Wµν − − 1 4 W3 µν Wµν 3 − 1 4 Bµν Bµν − 1 2 ! v2 T CHW B " W3 µνBµν + 1 4 g2 2 v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.20) The mass eigenstate basis is given by [65] # W3 µ $ = # 1 −1 2 v2 T CHW B 1 2 $ # cos θ sin θ $ # Zµ $ , (5.21) 2 4 4 4 2 + 1 2 v2 T CHW WI µνWIµν + 1 2 v2 T CHBBµνBµν − 1 2 v2 T CHW BW3 µνBµν , (5.1 so the gauge fields in the Lagrangian are not canonically normalized, and the last term Eq. (5.16) leads to kinetic mixing between W3 and B. The mass terms for the gauge boson from LSM and L(6) are L = 1 4 g2 2v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.1 The gauge fields need to be redefined, so that the kinetic terms are properly normalize and diagonal. The first step is to redefine the gauge fields GA µ = GA µ ! 1 + CHGv2 T " , WI µ = WI µ ! 1 + CHW v2 T " , Bµ = Bµ ! 1 + CHBv2 T " . (5.1 The modified coupling constants are g3 = g3 ! 1 + CHG v2 T " , g2 = g2 ! 1 + CHW v2 T " , g1 = g1 ! 1 + CHB v2 T " , (5.1 so that the products g3GA µ = g3GA µ , etc. are unchanged. This takes care of the gluon term The electroweak terms are L = − 1 2 W+ µν Wµν − − 1 4 W3 µν Wµν 3 − 1 4 Bµν Bµν − 1 2 ! v2 T CHW B " W3 µνBµν + 1 4 g2 2 v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.2 The mass eigenstate basis is given by [65] # W3 µ Bµ $ = # 1 −1 2 v2 T CHW B −1 2 v2 T CHW B 1 $ # cos θ sin θ − sin θ cos θ $ # Zµ Aµ $ , (5.2 trick: 2 4 4 4 2 + 1 2 v2 T CHW WI µνWIµν + 1 2 v2 T CHBBµνBµν − 1 2 v2 T CHW BW3 µνBµν , (5.16) so the gauge fields in the Lagrangian are not canonically normalized, and the last term in Eq. (5.16) leads to kinetic mixing between W3 and B. The mass terms for the gauge bosons from LSM and L(6) are L = 1 4 g2 2v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.17) The gauge fields need to be redefined, so that the kinetic terms are properly normalized and diagonal. The first step is to redefine the gauge fields GA µ = GA µ ! 1 + CHGv2 T " , WI µ = WI µ ! 1 + CHW v2 T " , Bµ = Bµ ! 1 + CHBv2 T " . (5.18) The modified coupling constants are g3 = g3 ! 1 + CHG v2 T " , g2 = g2 ! 1 + CHW v2 T " , g1 = g1 ! 1 + CHB v2 T " , (5.19) so that the products g3GA µ = g3GA µ , etc. are unchanged. This takes care of the gluon terms. The electroweak terms are L = − 1 2 W+ µν Wµν − − 1 4 W3 µν Wµν 3 − 1 4 Bµν Bµν − 1 2 ! v2 T CHW B " W3 µνBµν + 1 4 g2 2 v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.20) The mass eigenstate basis is given by [65] # W3 µ Bµ $ = # 1 −1 2 v2 T CHW B −1 v2 T CHW B 1 $ # cos θ sin θ − sin θ cos θ $ # Zµ Aµ $ , (5.21)
- 25. H to 4 leptons in SMEFT: Shifts to the SMEFT Lagrangian 25 The neutral fields undergo an additional rotation: so that the products g3GA µ = g3GA µ , etc. are unchanged. This takes care of the gluon terms. The electroweak terms are L = − 1 2 W+ µν Wµν − − 1 4 W3 µν Wµν 3 − 1 4 Bµν Bµν − 1 2 ! v2 T CHW B " W3 µνBµν + 1 4 g2 2 v2 T W+ µ W− µ + 1 8 v2 T (g2W3 µ − g1Bµ)2 + 1 16 v4 T CHD(g2W3 µ − g1Bµ)2 . (5.20) The mass eigenstate basis is given by [65] # W3 µ Bµ $ = # 1 −1 2 v2 T CHW B −1 2 v2 T CHW B 1 $ # cos θ sin θ − sin θ cos θ $ # Zµ Aµ $ , (5.21) – 18 – c̄w = ĉw ✓ 1 + v̂2 T 4 CHD + ŝwv̂2 T 2 ˆ cw CHWB ◆ , s̄2 w = 1 c̄2 w . (A.14) With this notation, the relation between the weak-basis fields Aa µ and the mass basis fields Ãa µ is Aa µ = Mab Rbc Ãc µ . (A.15) In terms of the input parameters in (2.7), the explicit definitions of the photon and Z-boson fields in terms of the weak-basis fields is W3 µ Bµ ! = 0 @ ĉw + 1 4 ĉwv̂2 T ⇣ CHD + 4ŝw ĉw CHWB ⌘ ŝw ĉ2 wv̂2 T 4ŝw ⇣ CHD + 4ŝw ĉw CHWB ⌘ ŝw + ĉ2 wv̂2 T 4ŝw CHD ĉw + ĉwv̂2 T 4 CHD 1 A Zµ Aµ ! . (A.16) Furthermore, the dimension-6 SMEFT covariant derivative in the mass basis is given by Dµ = @µ i e ŝw 1 + ĉ2 wv̂2 T 4ŝ2 w CHD + ĉwv̂2 T ŝw CHWB W+ µ ⌧+ + Wµ ⌧ i e ĉwŝw ✓ 1 + (2ĉ2 w 1)v̂2 T 4ŝ2 w CHD + ĉwv̂2 T ŝw CHWB ◆ ⌧3 ŝ2 wQ + e ✓ ĉwv̂2 T 2ŝw CHD + v̂2 T CHWB ◆ Q Zµ ieQAµ , (A.17) where Q = ⌧3 + Y and ⌧± = (⌧1 ± i⌧2)/ p 2. g2 2 HW B g2 2 so that sin θ = g1 # g1 2 + g2 2 ! 1 + v2 T 2 g2 g1 g2 2 − g1 2 g2 2 + g1 2 CHW B " , cos θ = g2 # g1 2 + g2 2 ! 1 − v2 T 2 g1 g2 g2 2 − g1 2 g2 2 + g1 2 CHW B " . (5.23) The photon is massless, as it must be by gauge invariance, since U(1)Q is unbroken. The W and Z masses are M2 W = g2 2v2 T 4 , M2 Z = v2 T 4 (g1 2 + g2 2 ) + 1 8 v4 T CHD(g1 2 + g2 2 ) + 1 2 v4 T g1g2CHW B. (5.24) The covariant derivative is Dµ = ∂µ + i g2 √ 2 $ W+ µ T+ + W− µ T− % + i gZ $ T3 − s2 Q % Zµ + i e Q Aµ, (5.25) where Q = T3 + Y , and the effective couplings are given by e = g1g2 # g2 2 + g1 2 ! 1 − g1g2 g2 2 + g1 2 v2 T CHW B " = g2 sin θ − 1 2 cos θ g2 v2 T CHW B, gZ = & g2 2 + g1 2 + g1g2 # g 2 + g 2 v2 T CHW B = e sin θ cos θ ! 1 + g1 2 + g2 2 2g1g2 v2 T CHW B " ,
- 26. H to 4 leptons in SMEFT: Shifts to the SMEFT Lagrangian t, physical parameters: ē, mH, MW , MZ, mf , Ci . (2 xpressions from [8], one has M2 W = ḡ2 2v2 T 4 , M2 Z = v2 T 4 (ḡ2 1 + ḡ2 2) + 1 8 v4 T CHD(ḡ2 1 + ḡ2 2) + 1 2 v4 T ḡ1ḡ2CHWB , ē = ḡ2s̄w 1 2 c̄wḡ2v2 T CHWB , s̄2 w = ḡ2 1 ḡ2 1 + ḡ2 2 + ḡ1ḡ2(ḡ2 2 ḡ2 1) (ḡ2 1 + ḡ2 2)2 v2 T CHWB . (2 quantities appear as couplings in the covariant derivative in the broken pha after rotation to the mass eigenbasis; in particular ḡ2 governs the charged cur while ē is the electric charge. Manipulating the above expressions we can write 1 v = ē 2M s̄ ✓ 1 + ĉw 2ŝ CHWBv̂2 T ◆ , (2 M2 W = ḡ2 2v2 T 4 , M2 Z = v2 T 4 (ḡ2 1 + ḡ2 2) + 1 8 v4 T CHD(ḡ2 1 + ḡ2 2) + 1 2 v4 T ḡ1ḡ2CHWB , ē = ḡ2s̄w 1 2 c̄wḡ2v2 T CHWB , s̄2 w = ḡ2 1 ḡ2 1 + ḡ2 2 + ḡ1ḡ2(ḡ2 2 ḡ2 1) (ḡ2 1 + ḡ2 2)2 v2 T CHWB . The barred quantities appear as couplings in the covariant derivative in the the theory after rotation to the mass eigenbasis; in particular ḡ2 governs the couplings while ē is the electric charge. Manipulating the above expressions 1 vT = ē 2MW s̄w ✓ 1 + ĉw 2ŝw CHWBv̂2 T ◆ , where we have defined v̂T ⌘ 2MW ŝw ē ; ŝ2 w ⌘ 1 M2 W M2 Z , ĉ2 w ⌘ 1 ŝ2 w , such that the hatted quantities are the usual definitions in the SM. In e we denote parameters multiplying the Wilson coefficients by the hatted qu consistent to O(1/⇤2 NP ), and is the notation which will be adopted through is possible to re-express vT and s̄w in terms of the gauge boson masses, and q from them. In particular, the quantity s̄w can be expressed as 26 Important: To what extent do these shifts hold if we go to quadratic orders? (for example, ) c̄2 w + s̄2 w ≠ 1
- 27. 27 Not completely unrelated: Complex mass scheme See for example the prophecy4f manual [2]. Since the SMEFT Lag it is better understood in terms of the NLO-Complex mass schem found for this is [3]. µ2 v = m2 v i vmv we denote the SMEFT parameters with bars, and the SM ones w confusing at first but common in the literature, µ2 z = m2 z + imz z = m2 z ✓ 1 + i z mz ◆ where mz and z are calculated from the SMEFT corrections to th particular m2 z = m2 z + m2 z = m2 z + v2 ⇤2 ✓ m2 z 2 cHD + 2 mz mw sw cHWB ◆ z = mz 16⇡ |cSM HZll;1 + cEFT HZll;1|2 + |cSM HZll;2 + cEFT HZll;2|2 2 ( z/mz) = |cSM HZll;1 + cEFT HZll;1|2 + |cSM HZll;2 + cEFT HZll;2|2 |cSM HZll; 32⇡ Only rigorous way to define unstable particles in propagators
- 28. 28 Not completely unrelated: Complex mass scheme See for example the prophecy4f manual [2]. Since the SMEFT Lag it is better understood in terms of the NLO-Complex mass schem found for this is [3]. µ2 v = m2 v i vmv we denote the SMEFT parameters with bars, and the SM ones w confusing at first but common in the literature, µ2 z = m2 z + imz z = m2 z ✓ 1 + i z mz ◆ where mz and z are calculated from the SMEFT corrections to th particular m2 z = m2 z + m2 z = m2 z + v2 ⇤2 ✓ m2 z 2 cHD + 2 mz mw sw cHWB ◆ z = mz 16⇡ |cSM HZll;1 + cEFT HZll;1|2 + |cSM HZll;2 + cEFT HZll;2|2 2 ( z/mz) = |cSM HZll;1 + cEFT HZll;1|2 + |cSM HZll;2 + cEFT HZll;2|2 |cSM HZll; 32⇡ Only rigorous way to define unstable particles in propagators !!
- 29. SZ 12SZ 34 s2 w Im a bit surprised that these expressions dont depend on sw and cw? expanded the full expression in terms of Mus and then substituted back t ?? The corrections to the squared amplitude follow from these in the expect |A|2 ! |Â|2 (1 + 2g6Re ( A)) O(g2 6). The amplitudes can then computed directly using the complex mass defi µ̂V = m̂V 1 ˆ2 V m̂2 V !1/4 exp i 2 arctan ˆV m̂V !! 4 The complex mass scheme in SMEFT In the complex mass scheme, the gauge boson masses are substituted by co See for example the prophecy4f manual [2]. Since the SMEFT Lagrangia it is better understood in terms of the NLO-Complex mass scheme. Th found for this is [3]. µ2 v = m2 v i vmv we denote the SMEFT parameters with bars, and the SM ones with ha confusing at first but common in the literature, µ2 z = m2 z + imz z = m2 z ✓ 1 + i z mz ◆ H to 4 leptons in SMEFT: Complex mass scheme For the proper treatment of unstable particles one has to introduce a complex mass: heme in SMEFT he gauge boson masses are substituted by complex quantities. f manual [2]. Since the SMEFT Lagrangian has some shifts, ms of the NLO-Complex mass scheme. The best reference I µ2 v = m2 v i vmv (4.1) eters with bars, and the SM ones with hats, which is a bit in the literature, = m2 z + imz z = m2 z ✓ 1 + i z mz ◆ (4.2) d from the SMEFT corrections to the Z mass and width. In v2 ⇤2 ✓ m2 z 2 cHD + 2 mz mw sw cHWB ◆ 1|2 + |cSM HZll;2 + cEFT HZll;2|2 2 where m2 z is the shift to the 1 2 ZµZµ term in the SMEFT Lagrangian, after we ha the B0 µ and B3 µ fields to remove the Z-A transition. or in our new notation, with (x) = x (x), m2 z = m2 z(1 + m2 z) = m2 z ✓ 1 + v2 ⇤2 ✓ cHD 2 + 2 mw cHWB mz sw ◆◆ z = mz 16⇡ |cSM HZll;1 + cEFT HZll;1|2 + |cSM HZll;2 + cEFT HZll;2|2 2 ( z/mz) = |cSM HZll;1 + cEFT HZll;1|2 + |cSM HZll;2 + cEFT HZll;2|2 |cSM HZll;1|2 + |cSM HZll;2|2 1 now IMPORTANT: We have to decide if we keep m and m as real inside the µ2 z = m2 z + (m2 z) | {z } 2R 0 B B B @ 1 + i 2 6 6 6 4 z mz + ✓ z mz ◆ | {z } 2R 3 7 7 7 5 1 C C C A = µ̂2 z + (m2 z) | {z } 2R + i (m2 z · z/mz) | {z } 2C (4.5) (a · b) = a · b + a · b . on, with (x) = x (x), (µ2 ) = 1 + (m2 ) + i z/mz · ( z/mz) (4.6) low the mass shift to be complex correct. I think one should define the mass as the SM mass with the everything complex according to the rules in the previous subsection, ex mass as a complex function of complex functions. 3 0 2 31 mw = mw + mw, m2 w = m2 w w = w + w w = . . . (4.10) (4.11) µ2 w = µ2 w (1 i( w/mw)) (4.12) There is a further important detail to mention1, namely the difference between µ2 z, a complex quantity which appears in the propagator as defined in eq. (4.1). And µ⇤ zµz, which is real and appears when squaring Feynman diagrams where the coupling is proportional to µz. µ⇤ zµz = |µz|2 6= |µ2 z| (4.13) I want to write this very clear for us, to understand. I will have a look at some GP paper well this is well explained 1 Obviously this problem is not inherent to the EFT and it already appears when using the complex mass scheme in higher order calculations for the SM EW sector. However in the latter case we find that µ ! 1 whereas in the former we expect µ ! 0. – 6 – Which also receives corrections from SMEFT: The pole mass doesn’t change but the known relations do (like in NLO) 29
- 30. CHOICE OF INPUT PARAMETERS Different Schemes lead to different SMEFT predictions (interpretations?) already at LO • Different implementations available in the MC generators: • ’Mw scheme’: {Mw, Mz, GF} • ‘Alpha scheme’: { , Mz, GF} • The optimal scheme for analytic calculations seems to be: { , Mz, Mw} α α as has already been done in (3.5). In contrast, in the GF -scheme, which wa calculating the partial NLO results of [55], one employs 1 p 2 1 v2 T = GF 1 p 2 C (3) Hl ee + C (3) Hl µµ ! + 1 2 p 2 ✓ C ll µeeµ + C ll eµµe ◆ . We have found the choice (3.6) to be particularly convenient for the full NLO calc it involves only parameters which appear in the Lagrangian, and no tree-level d four-fermion operators contributing to muon decay is introduced. Of course, matter to convert the results obtained here to other renormalization schemes finite shifts between input parameters are known completely to NLO in SMEFT – calculations needed to trade vT for GF as in (3.7) have been obtained in [49]. The NLO counterterm is obtained by interpreting the external fields and p (3.4) and (3.5) as bare ones, which are then replaced by renormalized ones befo the resulting expression to NLO in the couplings. The bare and renormalized fiel through wavefunction renormalization factors according to h(0) = p Zhh = ✓ 1 + 1 Zh ◆ h , which we split up as M (0) L = M (4,0) L + M (6,0) L , (3.2) where the superscripts (4, 0) and (6, 0) refer to the dimension-4 and dimension-6 contributions respectively. In order to express results in terms of our choice of input parameters (2.7), it is convenient to introduce v̂T ⌘ 2MW ŝw e , ĉ2 w ⌘ M2 W M2 Z , ŝ2 w ⌘ 1 ĉ2 w . (3.3) These hatted quantities are defined in terms of masses and couplings as in the SM. After rotation to the mass basis following the steps in appendix A one finds M (4,0) L = mb v̂T , (3.4) M (6,0) L = mbv̂T CH2 CHD 4 ✓ 1 ĉ2 w ŝ2 w ◆ + ĉw ŝw CHWB v̂T mb C⇤ bH p 2 . (3.5) Our notation is such that CfH is the coefficient which contributes to the hff coupling after rotating to the mass basis. Its precise definition in terms of the coefficients multiplying the weak-basis operators in table 3 can be found in appendix A.4. The LO decay amplitude (as well as the NLO counterterm derived from it) depends on the choice of input parameters. Using those given in (2.7) requires that we eliminate vT according to the relation [55] 1 vT = 1 v̂T ✓ 1 + v̂2 T ĉw ŝw CHWB + ĉw 4ŝw CHD ◆ , (3.6)
- 31. H to 4 leptons in SMEFT: Complex mass scheme H) = 3 ge + Z Z 12 34 SZ 12SZ 34 + Z w s2 w W s2 w (3.18) at these expressions dont depend on sw and cw? is it because you ression in terms of Mus and then substituted back the mus for sw/cw e squared amplitude follow from these in the expected way, |A|2 ! |Â|2 (1 + 2g6Re ( A)) O(g2 6). (3.19) hen computed directly using the complex mass defined by µ̂V = m̂V 1 ˆ2 V m̂2 V !1/4 exp i 2 arctan ˆV m̂V !! (3.20) mass scheme in SMEFT cheme, the gauge boson masses are substituted by complex quantities. rophecy4f manual [2]. Since the SMEFT Lagrangian has some shifts, d in terms of the NLO-Complex mass scheme. The best reference I Shifts to the Amplitude A(1e , 2+ ē , 3µ , 4+ µ̄ , 5H) = 3 ge + 2 µZµ2 Z(SZ 12 + SZ 34) SZ 12SZ 34 µZ(2 s2 w + 2s4 w) s2 w(c2 w s2 w) + 2 µW s2 w(c2 w s2 w) (3.15) A(1e , 2+ ē , 3+ µ , 4µ̄ , 5H) = 3 ge + 2 µZµ2 Z(SZ 12 + SZ 34) SZ 12SZ 34 µZ(1 + 2s2 w) c2 w s2 w + 2 µW (c2 w s2 w) (3.16) A(1+ e , 2ē , 3µ , 4+ µ̄ , 5H) = 3 ge + 2 µZµ2 Z(SZ 12 + SZ 34) SZ 12SZ 34 µZ(1 + 2s2 w) c2 w s2 w + 2 µW (c2 w s2 w) (3.17) A(1+ e , 2ē , 3+ µ , 4µ̄ , 5H) = 3 ge + 2 µZµ2 Z(SZ 12 + SZ 34) SZ 12SZ 34 + µZ(2 + s2 w) s2 w 2 µW s2 w (3.18) Im a bit surprised that these expressions dont depend on sw and cw? is it because you expanded the full expression in terms of Mus and then substituted back the mus for sw/cw ?? Work in progress with S. Badger and M. Schoenherr 31
- 32. From the dimension six SMEFT we have, + cEFT HZZ;1 tr+ ⇣ µ / p2 ⌫ / p3 ⌘ + cEFT HZZ;2 tr ⇣ µ / p2 ⌫ / p3 ⌘ cEFT HZA;1 tr+ ⇣ µ / p2 ⌫ / p3 ⌘ + cEFT HZA;2 tr ⇣ µ / p2 ⌫ / p3 ⌘ cEFT HAA;1 tr+ ⇣ µ / p2 ⌫ / p3 ⌘ + cEFT HAA;2 tr ⇣ µ / p2 ⌫ / p3 ⌘ cEFT Zll;1 µ P+ + cEFT Zll;2 µ P + cEFT Zll;3 µ⌫ p1⌫P+ + cEFT Zll;4 µ⌫ p1⌫P cEFT HZll;1 µ P+ + cEFT HZll;2 µ P cEFT HZll;3 µ⌫ p2⌫P+ + cEFT HZll;4 µ⌫ p2⌫P (A.2) H to 4 leptons in SMEFT Work in progress with S. Badger and M. Schoenherr 32
- 33. The EFT corrections can also be presented in a relatively compact form using on-shell helicity amplitudes. We find: A(EFT) (1e , 2+ ē , 3µ , 4+ µ̄ , 5H) = h13i[24]c [EFT,1](1,1) H2e2µ + h13i[2|/ 1/ 3|4]c [EFT,2](1,1) H2e2µ + h1|/ 2/ 4|3i[24]c [EFT,3](1,1) H2e2µ (3.7) A(EFT) (1e , 2+ ē , 3+ µ , 4µ̄ , 5H) = h14i[23]c [EFT,1](1,2) H2e2µ + h14i[2|/ 1/ 4|3]c [EFT,2](1,2) H2e2µ + h1|/ 2/ 3|4i[23]c [EFT,3](1,2) H2e2µ (3.8) A(EFT) (1+ e , 2ē , 3µ , 4+ µ̄ , 5H) = h23i[14]c [EFT,1](1,2) H2e2µ + h23i[1|/ 2/ 3|4]c [EFT,2](2,1) H2e2µ + h2|/ 1/ 4|3i[14]c [EFT,3](2,1) H2e2µ (3.9) A(EFT) (1+ e , 2ē , 3+ µ , 4µ̄ , 5H) = h24i[13]c [EFT,1](2,2) H2e2µ + h24i[1|/ 2/ 4|3]c [EFT,2](2,2) H2e2µ + h2|/ 1/ 3|4i[13]c [EFT,3](2,2) H2e2µ (3.10) where c [EFT,1](i,j) H2e2µ = cEFT HllZ;icSM Zll;j SZ 12 + cEFT HllZ;jcSM Zll;i SZ 34 + cSM HZZ;1 ⇣ cEFT Zll;icSM Zll;j + cSM Zll;icEFT Zll;j ⌘ SZ 12SZ 34 (3.11) c [EFT,2](i,j) H2e2µ = cEFT HAA;1(cSM All;1)2 s12s34 + cEFT HAZ;1cSM All;1 cSM Zll;i SZ 12s34 + cSM Zll;j s12SZ 34 ! + cEFT HZZ;1cSM Zll;icSM Zll;j SZ 12SZ 34 (3.12) c [EFT,3](i,j) H2e2µ = cEFT HAA;2(cSM All;1)2 s12s34 + cEFT HAZ;2cSM All;1 cSM Zll;i SZ 12s34 + cSM Zll;j s12SZ 34 ! + cEFT HZZ;2cSM Zll;icSM Zll;j SZ 12SZ 34 (3.13) H to 4 leptons in SMEFT: Complex mass scheme Interference Amplitude 33
- 34. 34 On-shell decay of the H in its rest frame Preliminary
- 35. 35 On-shell decay of the H in its rest frame Preliminary
- 36. 36 QUADRATICTERMS where the O (6) i represent a basis of dim = 6 operators built from SM fields and respecting the known auge symmetries14 , and ci are the Wilson coefficients of the theory. Further, the SMEFT amplitudes nd cross sections can be parametrised as AEFT = ASM + g0 ⇤2 A6 + g02 ⇤4 A8 + . . . (91) EFT ⇠ |ASM |2 + 2 g0 ⇤2 ASM A6 + g0 2 ⇤4 ⇣ 2ASM A8 + |A6|2 ⌘ + . . . (92) Here, we assume the linear contribution (red) of the EFT effects to be leading. Analysis of the dim = 6 uadratic terms and the dim = 8 interference terms (both in blue) will be subject of further studies. In articular, dim = 8 are commonly associated with quartic gauge couplings and such contribution, albeit ubleading, would represent some added value to the linear dim = 6 prediction. Definition of the fiducial region Linear Quadratic + Linear + - + - Quadratic + + + - Double insertion + + - -
- 37. 37 On-shell decay of the H in its rest frame Linear Quadratic Very Preliminary!
- 38. BEYONDTHE HIGGS SECTOR: 38 electroweak processes image credit ATLAS collaboration
- 39. THE ELECTROWEAK SECTOR • Combining the families of ‘multiboson’ processes, one can get a great deal of extra information. • By this, we define 3 families of proceses: Multiboson production, Vector Boson Scattering andVector Boson Fusion 39
- 40. EFT FORVBS • Ongoing effort within theVBSCan COST action • Feel free to get in touch • Conveners of the “Combination WG”: R. Covarelli, Magdalena Slawinska, Despoina Sampsonidou • Others: Kristin Lohwasser, M. Pellen, P. Govoni, (Me)… • Young researchers encouraged 40
- 41. TGC AND QGC IN DIM-6 ZμZνW+ ρ W− σ ZμW+ ν W− ρ SM + 1 Λ2 (ℱ(CHD, CHW, CHB, CHℓ(3), C′ ℓℓ) + ℱ(CHWB, CW, pi)) Field and parameter shifts Vertex 𝒪HD = (H† DμH) * (H† DμH) 𝒪Hℓ(3) = (H† iDμH) (ℓ̄p γμ ℓr) 𝒪′ ℓℓ = (ℓ̄p γμ ℓr) (ℓ̄s γμ ℓt) 𝒪HB = (H† H) BμνBμν 𝒪W = ϵIJKWI μνWJ νρWK ρμ 𝒪HWB = (H† τI H) WI μνBμν 𝒪HW = (H† H) WI μνWμνI 41
- 42. VBS (ZZ) te the effects of including all the Warsaw-basis operators on the Z) cross-section. Some examples of the Feynman diagrams that re shown in figure 5. ynman diagrams contributing to the VBS(ZZ) process in dim = the dimension-six insertions. numerically, the following expressions for the bosonic contribu- on: 7 c̄HB 0.053 c̄H⇤ 0.0021 c̄g HB + 0.010 c̄Hd 1.84 c̄HD c̄Hl(3) 0.017 c̄Hq(1) + 5.61 c̄Hq(3) 0.033 c̄Hu + 0.59 c̄HW 41 c̄] HW 0.69 c̄HWB 0.022 c̄ ^ HWB + 0.23 c̄W 0.086 c̄f W . (5.1) In this section, we investigate the effects of including all the Warsaw-basis operators on the computation of the VBS(ZZ) cross-section. Some examples of the Feynman diagrams that contribute to the process are shown in figure 5. Figure 5: Some of the Feynman diagrams contributing to the VBS(ZZ) process in dim = 6 EFT. The blobs represent the dimension-six insertions. In particular we found, numerically, the following expressions for the bosonic contribu- tion to the total cross-section: EFT,bosonic SM ⇡ 1. + 0.047 c̄HB 0.053 c̄H⇤ 0.0021 c̄g HB + 0.010 c̄Hd 1.84 c̄HD 3.86 c̄Hl(3) 0.017 c̄Hq(1) + 5.61 c̄Hq(3) 0.033 c̄Hu + 0.59 c̄HW 0.0041 c̄] HW 0.69 c̄HWB 0.022 c̄ ^ HWB + 0.23 c̄W 0.086 c̄f W . (5.1) And for the fermionic contribution: EFT,fermionic SM ⇡ 1. 3.23 · 10 6 c̄dd 2.89 · 10 6 c̄ (1) dd 3.86 c̄ (1) `` + 0.0010 c̄ (1) qd + 1.80 · 10 20 c̄ (8) qd 1.93 c̄(1) qq 2.57 c̄(11) qq 14.3 c̄(3) qq 10.3 c̄(31) qq 0.0049 c̄(1) qu 2.51 · 10 20 c̄(8) qu + 0.00020 c̄ (1) ud + 1.62 · 10 21 c̄ (8) ud 0.0010 c̄uu 0.00099 c̄(1) uu . (5.2) Both of this expressions have been extracted from simple numerical analysis of relatively small Monte Carlo samples and hence are dominated by the MC uncertainty. Which we estimate of the order of 10% for each of the interference terms. The only purpose of displaying them here is to give an impression of the relative sensitivity of this process to the different EFT operators. computation of the VBS(ZZ) cross-section. Some examples of the Feynman diagrams that contribute to the process are shown in figure 5. Figure 5: Some of the Feynman diagrams contributing to the VBS(ZZ) process in dim = 6 EFT. The blobs represent the dimension-six insertions. In particular we found, numerically, the following expressions for the bosonic contribu- tion to the total cross-section: EFT,bosonic SM ⇡ 1. + 0.047 c̄HB 0.053 c̄H⇤ 0.0021 c̄g HB + 0.010 c̄Hd 1.84 c̄HD 3.86 c̄Hl(3) 0.017 c̄Hq(1) + 5.61 c̄Hq(3) 0.033 c̄Hu + 0.59 c̄HW 0.0041 c̄] HW 0.69 c̄HWB 0.022 c̄ ^ HWB + 0.23 c̄W 0.086 c̄f W . (5.1) And for the fermionic contribution: EFT,fermionic SM ⇡ 1. 3.23 · 10 6 c̄dd 2.89 · 10 6 c̄ (1) dd 3.86 c̄ (1) `` + 0.0010 c̄ (1) qd + 1.80 · 10 20 c̄ (8) qd 1.93 c̄(1) qq 2.57 c̄(11) qq 14.3 c̄(3) qq 10.3 c̄(31) qq 0.0049 c̄(1) qu 2.51 · 10 20 c̄(8) qu + 0.00020 c̄ (1) ud + 1.62 · 10 21 c̄ (8) ud 0.0010 c̄uu 0.00099 c̄(1) uu . (5.2) Both of this expressions have been extracted from simple numerical analysis of relatively small Monte Carlo samples and hence are dominated by the MC uncertainty. Which we estimate of the order of 10% for each of the interference terms. The only purpose of displaying them here is to give an impression of the relative sensitivity of this process to the different EFT operators. It is interesting to observe that the bosonic interference is generally positive, while the fermionic one is generally negative. This means that in the case that all the Wilson Moreover, in the available VBS(ZZ) analysis, the S/B discrimination was only achieved means of a boosted decision tree (BDT) and a Matrix Element (ME) discriminator, ribed in Refs. [79, 133]. Given this situation, it wouldn’t be sensible to add the EFT effects in the signal and in the much larger background. In the following we show a preliminary study of the T effects in the background and we discuss which regions and observables are best for observation of EFT effects. Characterization of the background this study, we generated a sample of events of the QCD component, i.e. O(g2 s ), to the ! ZZjj process. The cross-section for this component is much larger than the purely one, although this discrepancy can be minimised in the VBS fiducial region, yielding: µS/B = sig. bkg. VBS region ⇡ 0.25 (7.1) sensitivity to the different dim = 6 operators can be extracted numerically in the same as in equations (5.1)-(5.2), EFT,bkg SM,bkg ⇡ 1. 0.00073 c̄Hd 0.0036 c̄HD + 0.044 c̄HG 0.00016 c̄g HG 0.077 c̄ (3) Hl + 0.018 c̄ (1) Hq + 0.17 c̄ (3) Hq + 0.0065 c̄Hu + 0.035 c̄HWB 0.077 c̄ (1) `` (7.2) background is sensitive to less operators than the signal, and it is sensitive to other Figure 10: Some of the Feynman diagrams contributing to the VBS(ZZ) dominant back- ground process in dim = 6 EFT. The blobs represent the dimension-six insertions. Figure 10: Some of the Feynman diagrams contributing to the ground process in dim = 6 EFT. The blobs represent the dimens Background: arXiv: 1809.04189 (EPJC) 42 “Brute force” approach…
- 43. Generalisation:All operators, all channels Work in progress with K. Lohwasser By parametrising these 12 processes we could set bounds (ideally) to about 12 EFT operators. Moreover, if we identify di↵erent observables and phase space regions that have di↵erent operator dependence, we can improve the number of constrains. For example, with: µ = ZZ,V BS ZZ,SM = 1 + a0 · cW + b0 · cHW + d0 · cHW B + . . . and, µpT = pT (j1) pT (j1)SM bin3 = 1 + a1 · cW + b1 · cHW + d1 · cHW B + . . . and, µMM = M(Z1Z2) M(Z1Z2)SM bin3 = 1 + a2 · cW + b2 · cHW + d2 · cHW B + . . . we can solve the equation system: 0 @ µ µpT µMM 1 A = 0 @ a0 b0 d0 a1 b1 d1 a2 b2 d2 1 A 0 @ cW cHW cHW 1 A (3) where {a0, . . . d2} are the coefficients we have extracted numerically from our Monte Carlo studies. The fitting technology should be designed in such a way that is relatively easy to replace those coefficients in the future as the theoretical understanding and the MC productions improve. 43
- 44. Generalisation:All operators, all channels Work in progress with K. Lohwasser 44
- 45. VBS (ZZ) te the effects of including all the Warsaw-basis operators on the Z) cross-section. Some examples of the Feynman diagrams that re shown in figure 5. ynman diagrams contributing to the VBS(ZZ) process in dim = the dimension-six insertions. numerically, the following expressions for the bosonic contribu- on: 7 c̄HB 0.053 c̄H⇤ 0.0021 c̄g HB + 0.010 c̄Hd 1.84 c̄HD c̄Hl(3) 0.017 c̄Hq(1) + 5.61 c̄Hq(3) 0.033 c̄Hu + 0.59 c̄HW 41 c̄] HW 0.69 c̄HWB 0.022 c̄ ^ HWB + 0.23 c̄W 0.086 c̄f W . (5.1) certainty, but it points us to a region that the SM production becomes negligible are n (2.9) will have a dominant role. This will ns are very interesting themselves. We find ils of such distributions, which is something ur observables, but is not so pronounced as ases: pT (Z1) and pT (j1), we also reach the ut the EFT effects remain. arXiv: 1809.04189 (EPJC) 45
- 46. Extra: access to four-fermion vertices q l1 l2 l3 l4 q qa qb Figure 1: Some examples of four-fermion vertices entering the VBS process. where AEF T has to be defined, for example for two dim. 6 operators {O (6) 1 , O (6) 2 }, A (6) EF T = ASM + c (6) 1 ⇤2 (A1,6) + c (6) 2 ⇤2 (A2,6) if we further add dim.8 operators, {O (8) 1 , O (8) 2 }, A (8) EF T = A (6) EF T + c (8) 1 ⇤4 (A1,8) + c (8) 2 ⇤4 (A2,8) putting all pieces together, |AEF T |2 =|ASM |2 + c (6) 1 ⇤2 2 A⇤ 1,6ASM + c (6) 2 ⇤2 2 A⇤ 2,6ASM + (2)
- 47. 47 WW production: diboson andVBS qq̄ → W+ W−
- 48. 48 WW production: diboson andVBS qq̄ → W+ W+ jj
- 49. EXAMPLE:WZVBS • Measured at LHC Run-2 • Unfolded distributions available from ATLAS (for the BKG) • Mixed EW-QCD corrections recently published: relatively large (16%) where the O (6) i represent a basis of dim = 6 operators built from SM fields and respecting the kn gauge symmetries14 , and ci are the Wilson coefficients of the theory. Further, the SMEFT amplit and cross sections can be parametrised as AEFT = ASM + g0 ⇤2 A6 + g02 ⇤4 A8 + . . . EFT ⇠ |ASM |2 + 2 g0 ⇤2 ASM A6 + g0 2 ⇤4 ⇣ 2ASM A8 + |A6|2 ⌘ + . . . Here, we assume the linear contribution (red) of the EFT effects to be leading. Analysis of the dim quadratic terms and the dim = 8 interference terms (both in blue) will be subject of further studie 49
- 50. Positive interference Negative interference cW cHW cHWB cHl3 cHq1 cHq3 cHu cll1 cqq1 cqq11 cqq3 cqq31 0.01 0.10 1 10 100 Sensitivity % EFT effect for ZW 50 Class 1: Ow Preliminary
- 51. Final Goal: Fits from unfolded distributions ATLAS 1902.05759 0 100 200 [fb/GeV] Z T p ∆ / fid. σ ∆ 1 − 10 Data MATRIX Sherpa 2.2.2 1.18 × Powheg+Pythia ATLAS -1 = 13 TeV, 36.1 fb s ν → Z ± W ℓ′ ℓℓ [fb] fid σ ∆ 10 [GeV] Z T p 0 100 200 Ratio to MATRIX 0.8 1 1.2 220 ≥ (a) 0 100 200 [fb/GeV] W T p ∆ / fid. σ ∆ 1 − 10 1 Data MATRIX Sherpa 2.2.2 1.18 × Powheg+Pythia ATLAS -1 = 13 TeV, 36.1 fb s ν → Z ± W ℓ′ ℓℓ [fb] fid σ ∆ 1 10 [GeV] W T p 0 100 200 Ratio to MATRIX 0.8 1 1.2 220 ≥ (b) [fb/GeV] WZ T m ∆ / fid. σ ∆ 2 − 10 1 − 10 1 Data MATRIX Sherpa 2.2.2 ATLAS -1 = 13 TeV, 36.1 fb s ν → Z ± W ℓ′ ℓℓ [fb] fid σ ∆ 1 10 2 10 (c) (W,Z) [fb] φ ∆ / fid. σ ∆ 10 2 10 Data MATRIX Sherpa 2.2.2 1.18 × Powheg+Pythia ATLAS -1 = 13 TeV, 36.1 fb s (d) With the extra* information from unfolded distributions, we can perform a bin-per-bin fit of the different EFT operators: 51 *Most of this maeasurements were not available until very recently!
- 52. SUMMARY • Have we made the best of LHC? And of the Higgs Sector? • An EFT analysis can parametrise LHC processes in the most model-independent way • Differential distributions and backgrounds contain a lot of info • An accurate EFT prediction can be recycled for the future 52
- 53. … AND CONCLUSIONS • Differential distributions + NLO EFT should become the state of the art for LHC-BSM predictions • The EW sector is in it’s infancy … there’s a lot to be measured yet • Someones background can be someone else’s signal ! • A HEFT analysis of VBS will be an interesting counterpart to the SMEFT • dim-6 + dim-8 combined approach also to follow 53
- 55. BACKUP 55 more EW EFT pheno and NLO EFT
- 56. 1 56
- 57. FUTURE PROSPECTS (HL-LHC) R. Covarelli, R. Gomez Ambrosio, for HL-HE-LHC yellow report Table 63: Different sensitivities to each of the Warsaw basis operators. The operators that are not listed do not intervene in the process, or do it in a negligible way. Each sensitivity ✏i is calculated as ✏i = | EF T SM SM |, and they include a standard EFT prefactor v 2 ⇤ 2 |⇤=1TeV which needs to be taken into account if substituting values for the ci in the table. NB: we quote the absolute value for the sensitivities ✏. Conclusions The VBS(ZZ) and QCD(ZZ) final states, still largely unexplored at the LHC, will be an important source of constraints on dim = 6 EFT operators at the HL-LHC. We have shown the impact that values of Wil- son coefficients still experimentally allowed have on differential distributions that are easily accessible experimentally in this channel. Fig. 89: Two generic simulations showing the EFT effects on key differential distributions: invariant mass of the di-jet system (left) and transverse momentum of the leading Z boson (right). We selected • VBS(ZZ) with leptonic decays: very good prospects for the future runs • LHC Run-2: O(10) events HL-LHC: O(100) events
- 58. 58
- 61. 61
- 62. 62
- 63. 63 1I
- 64. !!
- 65. NLO-EFT #2: two point functions Wave functions and counterterms We construct self-energies, Dyson resum them and require that all propagators are UV finite. In a second step we construct 3 -point (or higher) functions, check their dim-4-finiteness and remove the remaining dim-6 UV divergences by mixing the Wilson coefficients Wi: Renormalized Wilson coefficients are scale dependent and the logarithm of the scale can be resummed in terms of the LO coefficients of the anomalous dimension matrix tant and g4+2k = 1/( √ 2GF Λ2)k. For each process the dim = 4 LO defines N = 3 for H → γγ etc.). Furthermore, N6 = N for tree initiated processes and N − plitude is obtained by inserting wave-function factors and finite renormalizat s UV finite all relevant, on-shell, S-matrix elements. It is made in two steps: fi Φ = ZΦ Φren , p = Zp pren , (3 ms are defined by Zi = 1 + g2 16π2 ! dZ (4) i + g6 dZ (6) i " . (3 esum them and require that all propagators are UV finite. In a second step ns, check their O(4) -finiteness and remove the remaining O(6) UV divergences Wi = ∑ j Z W ij Wren j . (3 e scale dependent and the logarithm of the scale can be resummed in terms of mension matrix [11]. nt and g4+2k = 1/( √ 2GF Λ2)k. For each process the dim = 4 LO defines t = 3 for H → γγ etc.). Furthermore, N6 = N for tree initiated processes and N − tude is obtained by inserting wave-function factors and finite renormalizati UV finite all relevant, on-shell, S-matrix elements. It is made in two steps: fi Φ = ZΦ Φren , p = Zp pren , (3 s are defined by Zi = 1 + g2 16π2 ! dZ (4) i + g6 dZ (6) i " . (3 um them and require that all propagators are UV finite. In a second step check their O(4) -finiteness and remove the remaining O(6) UV divergences Wi = ∑ j Z W ij Wren j . (3 cale dependent and the logarithm of the scale can be resummed in terms of t nsion matrix [11]. for loop initiated ones. The full amplitude is obtained by inserting wave-functi counterterms. Renormalization makes UV finite all relevant, on-shell, S-matrix e we introduce counterterms Φ = ZΦ Φren , p = Zp pren , for fields and parameters. Counterterms are defined by Zi = 1 + g2 16π2 ! dZ (4) i + g6 dZ (6) i " . We construct self-energies, Dyson resum them and require that all propagators construct 3-point (or higher) functions, check their O(4) -finiteness and remove th mixing the Wilson coefficients Wi: Wi = ∑ j Z W ij Wren j . Renormalized Wilson coefficients are scale dependent and the logarithm of the s LO coefficients of the anomalous dimension matrix [11]. Our aim is to discuss Higgs couplings and their SM deviations which requires pr Definition The Higgs couplings can be extracted from Green’s functions in well- of the poles after extracting the parts which are 1P reducible. These are well-de both in production and in decays; from this perspective, VH production or vect B +c2 θ aφB +s2 θ aφW and cθ = cren θ etc. The last step in the UV-renormalizati on coefficients which cancels the remaining (dim = 6) parts. To this purpo Wi = ∑ j Z W ij Wren j , Z W ij = δij + g2 16π2 dZ W ij ΔUV . ed by requiring cancellation of the residual UV poles and we obtain
- 66. NLO-EFT #2: two point functions Wave functions and counterterms Φ = ZΦ Φren , p = Zp pren , fields and parameters. Counterterms are defined by Zi = 1 + g2 16π2 ! dZ (4) i + g6 dZ (6) i " . construct self-energies, Dyson resum them and require that all propaga nstruct 3-point (or higher) functions, check their O(4) -finiteness and remov xing the Wilson coefficients Wi: Wi = ∑ j Z W ij Wren j . normalized Wilson coefficients are scale dependent and the logarithm of th coefficients of the anomalous dimension matrix [11]. r aim is to discuss Higgs couplings and their SM deviations which require finition The Higgs couplings can be extracted from Green’s functions in w he poles after extracting the parts which are 1P reducible. These are well h in production and in decays; from this perspective, VH production or v nt and g4+2k = 1/( √ 2GF Λ2)k. For each process the dim = 4 LO defines t = 3 for H → γγ etc.). Furthermore, N6 = N for tree initiated processes and N − tude is obtained by inserting wave-function factors and finite renormalizati UV finite all relevant, on-shell, S-matrix elements. It is made in two steps: fi Φ = ZΦ Φren , p = Zp pren , (3 s are defined by Zi = 1 + g2 16π2 ! dZ (4) i + g6 dZ (6) i " . (3 um them and require that all propagators are UV finite. In a second step check their O(4) -finiteness and remove the remaining O(6) UV divergences Wi = ∑ j Z W ij Wren j . (3 cale dependent and the logarithm of the scale can be resummed in terms of t nsion matrix [11]. for loop initiated ones. The full amplitude is obtained by inserting wave-functi counterterms. Renormalization makes UV finite all relevant, on-shell, S-matrix e we introduce counterterms Φ = ZΦ Φren , p = Zp pren , for fields and parameters. Counterterms are defined by Zi = 1 + g2 16π2 ! dZ (4) i + g6 dZ (6) i " . We construct self-energies, Dyson resum them and require that all propagators construct 3-point (or higher) functions, check their O(4) -finiteness and remove th mixing the Wilson coefficients Wi: Wi = ∑ j Z W ij Wren j . Renormalized Wilson coefficients are scale dependent and the logarithm of the s LO coefficients of the anomalous dimension matrix [11]. Our aim is to discuss Higgs couplings and their SM deviations which requires pr Definition The Higgs couplings can be extracted from Green’s functions in well- of the poles after extracting the parts which are 1P reducible. These are well-de both in production and in decays; from this perspective, VH production or vect Discussion point: After we use different operators to remove residual dim-6 divergences, the interpretation of single operator effects becomes ill-defined
- 67. NLO-EFT #4 : Dyson Propagators n resummed propagators are crucial for discussing several issues, from renormalization to Ward-Slavnov T) identities [67–69]. Consider the W or Z self-energy; in general we have Σ VV µν (s) = g2 16π2 ! DVV (s)δµν + PVV (s) pµ pν " . corresponding partially resummed propagator is Δ̂ VV µν = − δµν s− M2 V + g2 16π2 DVV + g2 16π2 PVV pµ pν % s− M2 V + g2 16π2 DVV & % s− M2 V + g2 16π2 DVV − g2 16π2 PVV s & . nly consider the case where V couples to a conserved current; furthermore, we start by including one-p ucible (1PI) self-energies. Therefore the inverse propagators are defined as follows: H partially resummed propagator is given by g−2 Δ̂−1 HH(s) = −g−2 ZH % s− M2 H & − 1 16π2 ΣHH . A partially resummed propagator is given by g−2 Δ̂−1 AA(s) = −g−2 s ' ZA − 1 16π2 ΠAA ( . W partially resummed propagator is given by g−2 Δ̂−1 WW(s) = −g−2 ZW # s− M2 $ − 1 16π2 DWW . Z partially resummed propagator is given by Σ VV µν (s) = 16π2 DVV (s)δµν + PVV (s) pµ pν . e corresponding partially resummed propagator is Δ̂ VV µν = − δµν s− M2 V + g2 16π2 DVV + g2 16π2 PVV pµ pν % s− M2 V + g2 16π2 DVV & % s− M2 V + g2 16π2 DV only consider the case where V couples to a conserved current; furthermore, we star ducible (1PI) self-energies. Therefore the inverse propagators are defined as follows: • H partially resummed propagator is given by g−2 Δ̂−1 HH(s) = −g−2 ZH % s− M2 H & − 1 16π2 ΣHH . • A partially resummed propagator is given by g−2 Δ̂−1 AA(s) = −g−2 s ' ZA − 1 16π2 ΠAA ( . • W partially resummed propagator is given by g−2 Δ̂−1 WW(s) = −g−2 ZW # s− M2 $ − 1 16π2 DWW . • Z partially resummed propagator is given by F. ex. Higgs: transition is given by S ZA µν + S ZAct µν S ZA ct µν = g2 16π2 Σ ZAct µν ΔUV , s given in Eq.(56) and Σ ZAct µν = s dZ (4) AZ δµν + g6 ! s dZ (6) AZ δµν − aAZ " dZ (4) Z + dZ (4) A # pµ pν $ . mmed propagator is given by G−1 f (p) = Zf % i/ p + mf & Zf − Sf , ounterterms are Zf = ZRf γ− + ZLf γ+ , Zf = ZLf γ+ + ZRf γ− γ± = 1 2 " 1 ± γ5 # , ZIf = 1 − 1 2 g2 16π2 ! dZ (4) If + g6 dZ (6) If ΔUV $ , mf = Mf ' 1 + g2 16π2 dZmf ΔUV F. ex. fermions: −A transition is given by S ZA µν + S ZAct µν S ZA ct µν = g2 16π2 Σ ZAct µν ΔUV , A ν is given in Eq.(56) and Σ ZAct µν = s dZ (4) AZ δµν + g6 ! s dZ (6) AZ δµν − aAZ " dZ (4) Z + dZ (4) A # pµ pν $ . esummed propagator is given by G−1 f (p) = Zf % i/ p + mf & Zf − Sf , e counterterms are Zf = ZRf γ− + ZLf γ+ , Zf = ZLf γ+ + ZRf γ− γ± = 1 2 " 1 ± γ5 # , ZIf = 1 − 1 2 g2 16π2 ! dZ (4) If + g6 dZ (6) If ΔUV $ , mf = Mf ' 1 + g2 16π2 dZmf ΔUV ( , f denotes the renormalized fermion mass and I = L,R. We have introduced counterterms for fields
- 68. NLO-EFT #5 : Finite Renormalization We discuss 3 schemes for the physical parameters For the EFT coefficients MS-bar scheme is the only option ld represents an exception, and here we use Aµ = ZA Aren µ + ZAZ Zren µ , ZAZ = g2 16π2 " dZ (4) AZ + g6 dZ (6) AZ # ΔUV . rmion fields ψ are written by means of bare left-handed and right-handed chiral fie d for renormalized fields. oduce M2 = ZM M2 ren ZM = 1 + g2 16π2 " dZ (4) M + g6 dZ (6) M # ΔUV p = Zp pren Zp = 1 + g2 16π2 " dZ (4) p + g6 dZ (6) p # ΔUV . nterterms is given in Appendix C. It is worth noting that the insertion of dim = 6 o introduces UV divergences in Δ (6) f , Eq.(59), that are proportional to s and cannot y enter wave-function renormalization factors and will be cancelled at the level of e reducible transitions ch that there is a Z−A vertex of O(g )
- 69. NLO-EFT #5 : Finite Renormalization We discuss 3 schemes for the physical parameters • On-shell renormalization: the Complex Pole scheme we replace the conventional on-shell mass renormalization equation with the associate pression for the complex pole m2 0 = M2 OS ! 1 + g2 16π2 Re ΣVV;fin " M2 OS # $ =⇒ m2 0 = sV ! 1 + g2 16π2 ΣVV ;fin " M2 OS # $ , (95 here sV is the complex pole associated to V. In this Section we will discuss on-shell finite renormalization; afte moval of UV poles we have replaced m0 → mren etc. and we introduce MV ren = MV ;OS + g2 ren 16π2 % dZ (4) MV + g6 dZ (6) MV & (96 d require that s = MV;OS is a zero of the real part of the inverse V propagator, up toO(g2g6 ). Therefore we introduc M2 ren = M2 W;OS ' 1 + g2 ren 16π2 % dZ (4) MW + g6 dZ (6) MW (& , M2 Hren = M2 H ;OS ' 1 + g2 ren 16π2 % dZ (4) MH + g6 dZ (6) MH &( , cren θ = cW ' 1 + g2 ren 16π2 % dZ (4) cθ + g6 dZ (6) cθ &( , (97 plex Pole scheme we replace the conventional on-shell mass renormalization equation with the asso for the complex pole m2 0 = M2 OS ! 1 + g2 16π2 Re ΣVV;fin " M2 OS # $ =⇒ m2 0 = sV ! 1 + g2 16π2 ΣVV ;fin " M2 OS # $ , the complex pole associated to V. In this Section we will discuss on-shell finite renormalization; UV poles we have replaced m0 → mren etc. and we introduce MV ren = MV ;OS + g2 ren 16π2 % dZ (4) MV + g6 dZ (6) MV & that s = MV;OS is a zero of the real part of the inverse V propagator, up toO(g2g6 ). Therefore we intr M2 ren = M2 W;OS ' 1 + g2 ren 16π2 % dZ (4) MW + g6 dZ (6) MW (& , M2 Hren = M2 H ;OS ' 1 + g2 ren 16π2 % dZ (4) MH + g6 dZ (6) MH &( , cren θ = cW ' 1 + g2 ren 2 % dZ (4) cθ + g6 dZ (6) cθ &( , In the Complex Pole scheme we replace the conventional on-shell mass renormalization e expression for the complex pole m2 0 = M2 OS ! 1 + g2 16π2 Re ΣVV;fin " M2 OS # $ =⇒ m2 0 = sV ! 1 + g2 16π2 ΣVV ;fi where sV is the complex pole associated to V. In this Section we will discuss on-shell removal of UV poles we have replaced m0 → mren etc. and we introduce MV ren = MV ;OS + g2 ren 16π2 % dZ (4) MV + g6 dZ (6) MV & and require that s = MV;OS is a zero of the real part of the inverse V propagator, up toO(g2 M2 ren = M2 W;OS ' 1 + g2 ren 16π2 % dZ (4) MW + g6 dZ (6) MW (& , M2 Hren = M2 H ;OS ' 1 + g2 ren 16π2 % dZ (4) MH + g6 dZ (6) MH &( , cren θ = cW ' 1 + g2 ren 16π2 % dZ (4) cθ + g6 dZ (6) cθ &( , 2 2 2 2
- 70. Other examples in the literature • Charge renormalisation • Z to ff • Z & W poles • H to Z ff • H to VV • And others.… This paper and Pecjak ‘19 Dawson ‘18 Dawson, Giardino ‘19 This paper and Dawson ‘18 Buchalla ‘13 But not that many!