Unitarized EFT for a strongly interacting BSM Electroweak Sector

Transcription

1 Unitarized EFT for a strongly interacting BSM Electroweak Sector Rafael L. Delgado In colaboration with: A.Castillo, A.Dobado, D.Espriu, M.J.Herrero, F.J.Llanes-Estrada and J.J.Sanz-Cillero Universidad Complutense de Madrid Presented at: Instituto de Física Corpuscular (IFIC), Valencia, 19th June 2017 arxiv: [hep-ph], accepted in EPJC; EPJC 77 no.4 (2017) 205; PRL 114 (2015) ; PRD 91 (2015) ; JHEP 1407 (2014) 149; JHEP 1402 (2014) 121; and J. Phys. G: Nucl. Part. Phys. 41 (2014) Rafael L. Delgado Unitarized EFT for a strongly interacting... 1 / 48

2 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting... 2 / 48

3 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 48

4 Standard lore The gauge bosons W ± and Z are massive. This is problematic: the massive terms are not gauge invariant. Gauge boson scattering amplitudes diverge with s at LO. Standard Model solution: Higgs-mechanism, which predicts the SM Higgs boson. Global symmetry breaking pattern: SU(2) L SU(2) R SU(2) C. In 2012, ATLAS and CMS find a GeV scalar resonance h, compatible with the Higgs of the SM. Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 48

5 Standard lore The gauge bosons W ± and Z are massive. This is problematic: the massive terms are not gauge invariant. Gauge boson scattering amplitudes diverge with s at LO. Standard Model solution: Higgs-mechanism, which predicts the SM Higgs boson. Global symmetry breaking pattern: SU(2) L SU(2) R SU(2) C. In 2012, ATLAS and CMS find a GeV scalar resonance h, compatible with the Higgs of the SM. Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 48

6 Standard lore The gauge bosons W ± and Z are massive. This is problematic: the massive terms are not gauge invariant. Gauge boson scattering amplitudes diverge with s at LO. Standard Model solution: Higgs-mechanism, which predicts the SM Higgs boson. Global symmetry breaking pattern: SU(2) L SU(2) R SU(2) C. In 2012, ATLAS and CMS find a GeV scalar resonance h, compatible with the Higgs of the SM. Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 48

7 Standard lore The gauge bosons W ± and Z are massive. This is problematic: the massive terms are not gauge invariant. Gauge boson scattering amplitudes diverge with s at LO. Standard Model solution: Higgs-mechanism, which predicts the SM Higgs boson. Global symmetry breaking pattern: SU(2) L SU(2) R SU(2) C. In 2012, ATLAS and CMS find a GeV scalar resonance h, compatible with the Higgs of the SM. Phys. Lett. B716, Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 48

8 Empirical situation New physics? 600 GeV GAP H (125.9 GeV, PDG 2013) W (80.4 GeV), Z (91.2 GeV) AND: No new physics!! If there is any... The SM until the Planck mass? Some issues: mass of neutrinos, gravity explanation (naturalness problem), astrophysical observation (dark matter, dark energy),... Four scalar light modes, a strong gap. Natural: further spontaneous symmetry breaking at f > v = 246 GeV? Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 48

9 Empirical situation New physics? 600 GeV GAP H (125.9 GeV, PDG 2013) W (80.4 GeV), Z (91.2 GeV) AND: No new physics!! If there is any... The SM until the Planck mass? Some issues: mass of neutrinos, gravity explanation (naturalness problem), astrophysical observation (dark matter, dark energy),... Four scalar light modes, a strong gap. Natural: further spontaneous symmetry breaking at f > v = 246 GeV? Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 48

10 Empirical situation New physics? 600 GeV GAP H (125.9 GeV, PDG 2013) W (80.4 GeV), Z (91.2 GeV) AND: No new physics!! If there is any... The SM until the Planck mass? Some issues: mass of neutrinos, gravity explanation (naturalness problem), astrophysical observation (dark matter, dark energy),... Four scalar light modes, a strong gap. Natural: further spontaneous symmetry breaking at f > v = 246 GeV? Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 48

11 Empirical situation New physics? 600 GeV GAP H (125.9 GeV, PDG 2013) W (80.4 GeV), Z (91.2 GeV) AND: No new physics!! If there is any... The SM until the Planck mass? Some issues: mass of neutrinos, gravity explanation (naturalness problem), astrophysical observation (dark matter, dark energy),... Four scalar light modes, a strong gap. Natural: further spontaneous symmetry breaking at f > v = 246 GeV? Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 48

12 Empirical situation New physics? 600 GeV GAP H (125.9 GeV, PDG 2013) W (80.4 GeV), Z (91.2 GeV) AND: No new physics!! If there is any... The SM until the Planck mass? Some issues: mass of neutrinos, gravity explanation (naturalness problem), astrophysical observation (dark matter, dark energy),... Four scalar light modes, a strong gap. Natural: further spontaneous symmetry breaking at f > v = 246 GeV? Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 48

13 Empirical situation: γγ physics Current efforts to measure γγ W + L W L and γγ Z LZ L channels. Only 2 events measured. Graphs from CMS 1. Wait for LHC Run II, CMS TOTEM and ATLAS AFP. Efforts for measuring γγ final states: SM Higgs decay channel. 1 JHEP 07 (2013) 116 Rafael L. Delgado Unitarized EFT for a strongly interacting... 5 / 48

14 - τ + τ - τ + τ - τ + τ - τ + τ - τ + τ - τ + τ Empirical situation: γγ physics Current efforts to measure γγ W + L W L and γγ Z LZ L channels. Only 2 events measured. Graphs from CMS 1. Wait for LHC Run II, CMS TOTEM and ATLAS AFP. Efforts for measuring γγ final states: SM Higgs decay channel. Events/100 GeV Data + - Inclusive W W tt Elastic γγ γγ + - W W CMS, (SM) -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ m(eµ) [GeV] Events/30 GeV s Data + - Inclusive W W tt Elastic γγ γγ + - W W Events/0.2 CMS, (SM) Data + - Inclusive W W tt Elastic γγ γγ + - W W E T [GeV] s CMS, (SM) -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ φ(eµ)/π -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ s 1 JHEP 07 (2013) 116 Rafael L. Delgado Unitarized EFT for a strongly interacting... 5 / 48

15 - τ + τ - τ + τ - τ + τ - τ + τ - τ + τ - τ + τ Empirical situation: γγ physics Current efforts to measure γγ W + L W L and γγ Z LZ L channels. Only 2 events measured. Graphs from CMS 1. Wait for LHC Run II, CMS TOTEM and ATLAS AFP. Efforts for measuring γγ final states: SM Higgs decay channel. Events/100 GeV Data + - Inclusive W W tt Elastic γγ γγ + - W W CMS, (SM) -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ m(eµ) [GeV] Events/30 GeV s Data + - Inclusive W W tt Elastic γγ γγ + - W W Events/0.2 CMS, (SM) Data + - Inclusive W W tt Elastic γγ γγ + - W W E T [GeV] s CMS, (SM) -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ φ(eµ)/π -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ s 1 JHEP 07 (2013) 116 Rafael L. Delgado Unitarized EFT for a strongly interacting... 5 / 48

16 - τ + τ - τ + τ - τ + τ - τ + τ - τ + τ - τ + τ Empirical situation: γγ physics Current efforts to measure γγ W + L W L and γγ Z LZ L channels. Only 2 events measured. Graphs from CMS 1. Wait for LHC Run II, CMS TOTEM and ATLAS AFP. Efforts for measuring γγ final states: SM Higgs decay channel. Events/100 GeV Data + - Inclusive W W tt Elastic γγ γγ + - W W CMS, (SM) -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ m(eµ) [GeV] Events/30 GeV s Data + - Inclusive W W tt Elastic γγ γγ + - W W Events/0.2 CMS, (SM) Data + - Inclusive W W tt Elastic γγ γγ + - W W E T [GeV] s CMS, (SM) -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ φ(eµ)/π -1 = 7 TeV, L = 5.05 fb + - Drell-Yan τ τ + - Diffractive W W W+jets Inelastic γγ s 1 JHEP 07 (2013) 116 Rafael L. Delgado Unitarized EFT for a strongly interacting... 5 / 48

17 Empirical situation: t t physics Initial t t states are important because of gluon fusion processes, with a large cross section at the LHC. The production of t t states is also a well studied experimental observable at the LHC. Top quark loop in the SM: VV states from gluon fusion Rafael L. Delgado Unitarized EFT for a strongly interacting... 6 / 48

18 Empirical situation: t t physics Initial t t states are important because of gluon fusion processes, with a large cross section at the LHC. The production of t t states is also a well studied experimental observable at the LHC. Top quark loop in the SM: VV states from gluon fusion Rafael L. Delgado Unitarized EFT for a strongly interacting... 6 / 48

19 Studied framework We consider a strongly interacting EWSBS, in contrast to the weakly interacting one of the SM. We study the processes VV VV, VV hh and hh hh, and extend the result to include γγ and t t states. Our LO scattering amplitudes within the EWSBS diverge, but are controlled by strongly interacting dynamics which respect unitarity. This situation is similar to low-energy QCD (hadron physics). In order to minimize our assumptions over the (hypothetical) underlying theory, we will use dispersion relations over a partial wave decomposition (the so called unitarization procedures); extend these unitarization procedures to the coupled-channels case; and consider an Effective Field Theory, computed at the NLO level (within the limits of the Equivalence Theorem), with three would-be Goldstone bosons ω and a Higgs-like boson h. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 48

20 Studied framework We consider a strongly interacting EWSBS, in contrast to the weakly interacting one of the SM. We study the processes VV VV, VV hh and hh hh, and extend the result to include γγ and t t states. Our LO scattering amplitudes within the EWSBS diverge, but are controlled by strongly interacting dynamics which respect unitarity. This situation is similar to low-energy QCD (hadron physics). In order to minimize our assumptions over the (hypothetical) underlying theory, we will use dispersion relations over a partial wave decomposition (the so called unitarization procedures); extend these unitarization procedures to the coupled-channels case; and consider an Effective Field Theory, computed at the NLO level (within the limits of the Equivalence Theorem), with three would-be Goldstone bosons ω and a Higgs-like boson h. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 48

21 Studied framework We consider a strongly interacting EWSBS, in contrast to the weakly interacting one of the SM. We study the processes VV VV, VV hh and hh hh, and extend the result to include γγ and t t states. Our LO scattering amplitudes within the EWSBS diverge, but are controlled by strongly interacting dynamics which respect unitarity. This situation is similar to low-energy QCD (hadron physics). In order to minimize our assumptions over the (hypothetical) underlying theory, we will use dispersion relations over a partial wave decomposition (the so called unitarization procedures); extend these unitarization procedures to the coupled-channels case; and consider an Effective Field Theory, computed at the NLO level (within the limits of the Equivalence Theorem), with three would-be Goldstone bosons ω and a Higgs-like boson h. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 48

22 Studied framework We consider a strongly interacting EWSBS, in contrast to the weakly interacting one of the SM. We study the processes VV VV, VV hh and hh hh, and extend the result to include γγ and t t states. Our LO scattering amplitudes within the EWSBS diverge, but are controlled by strongly interacting dynamics which respect unitarity. This situation is similar to low-energy QCD (hadron physics). In order to minimize our assumptions over the (hypothetical) underlying theory, we will use dispersion relations over a partial wave decomposition (the so called unitarization procedures); extend these unitarization procedures to the coupled-channels case; and consider an Effective Field Theory, computed at the NLO level (within the limits of the Equivalence Theorem), with three would-be Goldstone bosons ω and a Higgs-like boson h. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 48

23 Studied framework We consider a strongly interacting EWSBS, in contrast to the weakly interacting one of the SM. We study the processes VV VV, VV hh and hh hh, and extend the result to include γγ and t t states. Our LO scattering amplitudes within the EWSBS diverge, but are controlled by strongly interacting dynamics which respect unitarity. This situation is similar to low-energy QCD (hadron physics). In order to minimize our assumptions over the (hypothetical) underlying theory, we will use dispersion relations over a partial wave decomposition (the so called unitarization procedures); extend these unitarization procedures to the coupled-channels case; and consider an Effective Field Theory, computed at the NLO level (within the limits of the Equivalence Theorem), with three would-be Goldstone bosons ω and a Higgs-like boson h. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 48

24 Studied framework We consider a strongly interacting EWSBS, in contrast to the weakly interacting one of the SM. We study the processes VV VV, VV hh and hh hh, and extend the result to include γγ and t t states. Our LO scattering amplitudes within the EWSBS diverge, but are controlled by strongly interacting dynamics which respect unitarity. This situation is similar to low-energy QCD (hadron physics). In order to minimize our assumptions over the (hypothetical) underlying theory, we will use dispersion relations over a partial wave decomposition (the so called unitarization procedures); extend these unitarization procedures to the coupled-channels case; and consider an Effective Field Theory, computed at the NLO level (within the limits of the Equivalence Theorem), with three would-be Goldstone bosons ω and a Higgs-like boson h. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 48

25 Studied framework We consider a strongly interacting EWSBS, in contrast to the weakly interacting one of the SM. We study the processes VV VV, VV hh and hh hh, and extend the result to include γγ and t t states. Our LO scattering amplitudes within the EWSBS diverge, but are controlled by strongly interacting dynamics which respect unitarity. This situation is similar to low-energy QCD (hadron physics). In order to minimize our assumptions over the (hypothetical) underlying theory, we will use dispersion relations over a partial wave decomposition (the so called unitarization procedures); extend these unitarization procedures to the coupled-channels case; and consider an Effective Field Theory, computed at the NLO level (within the limits of the Equivalence Theorem), with three would-be Goldstone bosons ω and a Higgs-like boson h. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 48

26 Strongly Interacting Effective Field Theory + Unitarity: similarity with hadronic physics Chiral Perturbation Theory plus Dispersion Relations. Simultaneous description of ππ ππ and πk πk πk πk up to MeV including resonances. Lowest order ChPT (WeinbergTheorems) and even one-loop computations are only valid at very low energies. A. Dobado and J.R. Peláez: SLAC-PUB-8031, arxiv: v1; Phys. Rev. D56 (1997) Rafael L. Delgado Unitarized EFT for a strongly interacting... 8 / 48

27 Strongly Interacting Effective Field Theory + Unitarity: similarity with hadronic physics Chiral Perturbation Theory plus Dispersion Relations. Simultaneous description of ππ ππ and πk πk πk πk up to MeV including resonances. Lowest order ChPT (WeinbergTheorems) and even one-loop computations are only valid at very low energies. A. Dobado and J.R. Peláez: SLAC-PUB-8031, arxiv: v1; Phys. Rev. D56 (1997) Rafael L. Delgado Unitarized EFT for a strongly interacting... 8 / 48

28 Strongly Interacting Effective Field Theory + Unitarity: similarity with hadronic physics Chiral Perturbation Theory plus Dispersion Relations. Simultaneous description of ππ ππ and πk πk πk πk up to MeV including resonances. Lowest order ChPT (WeinbergTheorems) and even one-loop computations are only valid at very low energies. A. Dobado and J.R. Peláez: SLAC-PUB-8031, arxiv: v1; Phys. Rev. D56 (1997) Rafael L. Delgado Unitarized EFT for a strongly interacting... 8 / 48

29 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting... 9 / 48

30 Equivalence Theorem For s M 2 h, M2 W, M2 Z (100 GeV)2, longitudinal modes of gauge bosons can be identified with the would-be Goldstones. For instance, T (W a L W b L W c L W d L ) = T (ωa ω b ω c ω d ) + O ( M W / s ) The EWSBS behaves as if the would-be Goldstone bosons were physical states. The non-gauged Lagragian can be used directly to compute scattering amplitudes. During the 90 s, the limits of applicability of this theorem were studied in detail, leading to the conclusion that it is valid for chiral Lagrangians, like those used in this presentation: works from W.B.Kilgore, P.B.Pal, X.Zhang, A.Dobado, J.R.Peláez, M.T.Urdiales, H.-J.He, Y.-P.Kuang, D.Espriu, J.Matias, J.F.Donoghue, J.Tandean,... Rafael L. Delgado Unitarized EFT for a strongly interacting... 9 / 48

31 Equivalence Theorem For s M 2 h, M2 W, M2 Z (100 GeV)2, longitudinal modes of gauge bosons can be identified with the would-be Goldstones. For instance, T (W a L W b L W c L W d L ) = T (ωa ω b ω c ω d ) + O ( M W / s ) The EWSBS behaves as if the would-be Goldstone bosons were physical states. The non-gauged Lagragian can be used directly to compute scattering amplitudes. During the 90 s, the limits of applicability of this theorem were studied in detail, leading to the conclusion that it is valid for chiral Lagrangians, like those used in this presentation: works from W.B.Kilgore, P.B.Pal, X.Zhang, A.Dobado, J.R.Peláez, M.T.Urdiales, H.-J.He, Y.-P.Kuang, D.Espriu, J.Matias, J.F.Donoghue, J.Tandean,... Rafael L. Delgado Unitarized EFT for a strongly interacting... 9 / 48

32 Equivalence Theorem For s M 2 h, M2 W, M2 Z (100 GeV)2, longitudinal modes of gauge bosons can be identified with the would-be Goldstones. For instance, T (W a L W b L W c L W d L ) = T (ωa ω b ω c ω d ) + O ( M W / s ) The EWSBS behaves as if the would-be Goldstone bosons were physical states. The non-gauged Lagragian can be used directly to compute scattering amplitudes. During the 90 s, the limits of applicability of this theorem were studied in detail, leading to the conclusion that it is valid for chiral Lagrangians, like those used in this presentation: works from W.B.Kilgore, P.B.Pal, X.Zhang, A.Dobado, J.R.Peláez, M.T.Urdiales, H.-J.He, Y.-P.Kuang, D.Espriu, J.Matias, J.F.Donoghue, J.Tandean,... Rafael L. Delgado Unitarized EFT for a strongly interacting... 9 / 48

33 Equivalence Theorem For s M 2 h, M2 W, M2 Z (100 GeV)2, longitudinal modes of gauge bosons can be identified with the would-be Goldstones. For instance, T (W a L W b L W c L W d L ) = T (ωa ω b ω c ω d ) + O ( M W / s ) The EWSBS behaves as if the would-be Goldstone bosons were physical states. The non-gauged Lagragian can be used directly to compute scattering amplitudes. During the 90 s, the limits of applicability of this theorem were studied in detail, leading to the conclusion that it is valid for chiral Lagrangians, like those used in this presentation: works from W.B.Kilgore, P.B.Pal, X.Zhang, A.Dobado, J.R.Peláez, M.T.Urdiales, H.-J.He, Y.-P.Kuang, D.Espriu, J.Matias, J.F.Donoghue, J.Tandean,... Rafael L. Delgado Unitarized EFT for a strongly interacting... 9 / 48

34 Equivalence Theorem For s M 2 h, M2 W, M2 Z (100 GeV)2, longitudinal modes of gauge bosons can be identified with the would-be Goldstones. For instance, T (W a L W b L W c L W d L ) = T (ωa ω b ω c ω d ) + O ( M W / s ) The EWSBS behaves as if the would-be Goldstone bosons were physical states. The non-gauged Lagragian can be used directly to compute scattering amplitudes. During the 90 s, the limits of applicability of this theorem were studied in detail, leading to the conclusion that it is valid for chiral Lagrangians, like those used in this presentation: works from W.B.Kilgore, P.B.Pal, X.Zhang, A.Dobado, J.R.Peláez, M.T.Urdiales, H.-J.He, Y.-P.Kuang, D.Espriu, J.Matias, J.F.Donoghue, J.Tandean,... Rafael L. Delgado Unitarized EFT for a strongly interacting... 9 / 48

35 Equivalence Theorem full comp. ET A(ww->hh) s (GeV) Comparison between the full LO ωω hh (cos θ = 3) and that computed through the ET. The SM is used here. Work in collaboration with S. Moretti, to test a modified version of MadGraph. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

36 Non-linear Electroweak Chiral Lagrangian We have no clue of what, how or if new physics... Non-linear EFT 2 for VV scattering at NLO level, minimally coupled to hh, where L = v 2 4 g(h/f ) Tr[(D µu) D µ U] µh µ h V (h), g(h/v) = 1 + 2a h v + b ( h v ) V (h) = V 0 + M2 h 2 h2 + λ n h n n=3 D µ U = µ U + iŵµu iu ˆB µ. M h and λ n are subleading in chiral counting. 2 Yellow Report: *C.Grojean, A.Falkowski, M.Trott, B.Fuks, *G.Buchalla, T.Plehn, G.Isidori, K.Tackmann, L.Brenner,...; CERN M. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

37 Non-linear Electroweak Chiral Lagrangian We have no clue of what, how or if new physics... Non-linear EFT 2 for VV scattering at NLO level, minimally coupled to hh, where L = v 2 4 g(h/f ) Tr[(D µu) D µ U] µh µ h V (h), g(h/v) = 1 + 2a h v + b ( h v ) V (h) = V 0 + M2 h 2 h2 + λ n h n n=3 D µ U = µ U + iŵµu iu ˆB µ. M h and λ n are subleading in chiral counting. 2 Yellow Report: *C.Grojean, A.Falkowski, M.Trott, B.Fuks, *G.Buchalla, T.Plehn, G.Isidori, K.Tackmann, L.Brenner,...; CERN M. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

38 Non-linear Electroweak Chiral Lagrangian We have no clue of what, how or if new physics... Non-linear EFT 2 for VV scattering at NLO level, minimally coupled to hh, where L = v 2 4 g(h/f ) Tr[(D µu) D µ U] µh µ h V (h), g(h/v) = 1 + 2a h v + b ( h v ) V (h) = V 0 + M2 h 2 h2 + λ n h n n=3 D µ U = µ U + iŵµu iu ˆB µ. M h and λ n are subleading in chiral counting. 2 Yellow Report: *C.Grojean, A.Falkowski, M.Trott, B.Fuks, *G.Buchalla, T.Plehn, G.Isidori, K.Tackmann, L.Brenner,...; CERN M. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

39 Non-linear Electroweak Chiral Lagrangian We need the parameterization of the U(ω a ) SU(2) L SU(2) R /SU(2) C coset. In either case, whatever the non linear term is, Two choices have been used: Spherical parameterization U(x) = 1 + i τ a ω a (x) v U(x) = 1 1 ω2 (x) v 2 + O ( ω 2). + i τ a ω a (x) v Exponential parameterization (here, a cross-check for EWSBS+γγ) ( U(x) = exp i τ a π a ) (x) v Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

40 Non-linear Electroweak Chiral Lagrangian We need the parameterization of the U(ω a ) SU(2) L SU(2) R /SU(2) C coset. In either case, whatever the non linear term is, Two choices have been used: Spherical parameterization U(x) = 1 + i τ a ω a (x) v U(x) = 1 1 ω2 (x) v 2 + O ( ω 2). + i τ a ω a (x) v Exponential parameterization (here, a cross-check for EWSBS+γγ) ( U(x) = exp i τ a π a ) (x) v Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

41 Non-linear Electroweak Chiral Lagrangian We need the parameterization of the U(ω a ) SU(2) L SU(2) R /SU(2) C coset. In either case, whatever the non linear term is, Two choices have been used: Spherical parameterization U(x) = 1 + i τ a ω a (x) v U(x) = 1 1 ω2 (x) v 2 + O ( ω 2). + i τ a ω a (x) v Exponential parameterization (here, a cross-check for EWSBS+γγ) ( U(x) = exp i τ a π a ) (x) v Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

42 Non-linear Electroweak Chiral Lagrangian We need the parameterization of the U(ω a ) SU(2) L SU(2) R /SU(2) C coset. In either case, whatever the non linear term is, Two choices have been used: Spherical parameterization U(x) = 1 + i τ a ω a (x) v U(x) = 1 1 ω2 (x) v 2 + O ( ω 2). + i τ a ω a (x) v Exponential parameterization (here, a cross-check for EWSBS+γγ) ( U(x) = exp i τ a π a ) (x) v Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

43 EFT for VV scattering, minimally coupled to hh Since we are considering scattering processes within the EWSBS, the covariant derivate reduces to D µ U = µ U. Define V µ (D µ U)U. The next counterterms are needed for the NLO computation of the VV scattering, minimally coupled to hh L 4 = a 4 [Tr(V µ V ν )][Tr(V µ V ν )] + a 5 [Tr(V µ V µ )][Tr(V ν V ν )] + d v 2 ( µh µ h) Tr[(D ν U) D ν U] + e v 2 ( µh ν h) Tr[(D µ U) D ν U] + g v 4 ( µh µ h)( ν h ν h). Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

44 EFT for VV scattering, minimally coupled to hh Since we are considering scattering processes within the EWSBS, the covariant derivate reduces to D µ U = µ U. Define V µ (D µ U)U. The next counterterms are needed for the NLO computation of the VV scattering, minimally coupled to hh L 4 = a 4 [Tr(V µ V ν )][Tr(V µ V ν )] + a 5 [Tr(V µ V µ )][Tr(V ν V ν )] + d v 2 ( µh µ h) Tr[(D ν U) D ν U] + e v 2 ( µh ν h) Tr[(D µ U) D ν U] + g v 4 ( µh µ h)( ν h ν h). Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

45 EFT for VV scattering, minimally coupled to hh Since we are considering scattering processes within the EWSBS, the covariant derivate reduces to D µ U = µ U. Define V µ (D µ U)U. The next counterterms are needed for the NLO computation of the VV scattering, minimally coupled to hh L 4 = a 4 [Tr(V µ V ν )][Tr(V µ V ν )] + a 5 [Tr(V µ V µ )][Tr(V ν V ν )] + d v 2 ( µh µ h) Tr[(D ν U) D ν U] + e v 2 ( µh ν h) Tr[(D µ U) D ν U] + g v 4 ( µh µ h)( ν h ν h). Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

46 EFT for VV scattering, minimally coupled to hh Since we are considering scattering processes within the EWSBS, the covariant derivate reduces to D µ U = µ U. Define V µ (D µ U)U. The next counterterms are needed for the NLO computation of the VV scattering, minimally coupled to hh L 4 = a 4 [Tr(V µ V ν )][Tr(V µ V ν )] + a 5 [Tr(V µ V µ )][Tr(V ν V ν )] + d v 2 ( µh µ h) Tr[(D ν U) D ν U] + e v 2 ( µh ν h) Tr[(D µ U) D ν U] + g v 4 ( µh µ h)( ν h ν h). Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

47 EFT for VV scattering, minimally coupled to hh Since we are considering scattering processes within the EWSBS, the covariant derivate reduces to D µ U = µ U. Define V µ (D µ U)U. The next counterterms are needed for the NLO computation of the VV scattering, minimally coupled to hh L 4 = a 4 [Tr(V µ V ν )][Tr(V µ V ν )] + a 5 [Tr(V µ V µ )][Tr(V ν V ν )] + d v 2 ( µh µ h) Tr[(D ν U) D ν U] + e v 2 ( µh ν h) Tr[(D µ U) D ν U] + g v 4 ( µh µ h)( ν h ν h). Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

48 EFT for VV scattering, minimally coupled to hh Since we are considering scattering processes within the EWSBS, the covariant derivate reduces to D µ U = µ U. Define V µ (D µ U)U. The next counterterms are needed for the NLO computation of the VV scattering, minimally coupled to hh L 4 = a 4 [Tr(V µ V ν )][Tr(V µ V ν )] + a 5 [Tr(V µ V µ )][Tr(V ν V ν )] + d v 2 ( µh µ h) Tr[(D ν U) D ν U] + e v 2 ( µh ν h) Tr[(D µ U) D ν U] + g v 4 ( µh µ h)( ν h ν h). Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

49 EFT for VV scattering, minimally coupled to hh Using the spherical parameterization for the SU(2) coset and neglecting the couplings with photons and quarks, we have the next Lagrangian describing VV VV, VV hh and hh hh processes: L = [ 1 + 2a h ( ) ] h 2 v + b µ ω a µ ω b v 2 ( δ ab + ωa ω b ) v 2 + 4a 4 v 4 µω a ν ω a µ ω b ν ω b + 4a 5 v 4 µω a µ ω a ν ω b ν ω b + 2d v 4 µh µ h ν ω a ν ω a + 2e v 4 µh µ ω a ν h ν ω a µh µ h + g v 4 ( µh µ h) 2 Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

50 Extension to γγ states Since coupling with photons are considered 3, the covariant derivative is defined as D µ U = µ U + iŵ µ U iu ˆB µ. The photon field A arises from the couplings with Ŵ µν and ˆB µν through a rotation to the physical basis; an anomalous three-particle coupling may appear h c W v ŴµνŴ µν h c B v ˆB ˆBµν µν = c γ h 2 v e2 A µν A µν The next additional NLO counterterms are needed, L 4 = a 1 Tr(U ˆB µν U Ŵ µν ) + ia 2 Tr(U ˆB µν U [V µ, V ν ]) ia 3 Tr(Ŵ µν [V µ, V ν ]) 3 Work in collaboration with M.J.Herrero and J.J.Sanz-Cillero, JHEP1407 (2014) 149. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

51 Extension to γγ states Since coupling with photons are considered 3, the covariant derivative is defined as D µ U = µ U + iŵ µ U iu ˆB µ. The photon field A arises from the couplings with Ŵ µν and ˆB µν through a rotation to the physical basis; an anomalous three-particle coupling may appear h c W v ŴµνŴ µν h c B v ˆB ˆBµν µν = c γ h 2 v e2 A µν A µν The next additional NLO counterterms are needed, L 4 = a 1 Tr(U ˆB µν U Ŵ µν ) + ia 2 Tr(U ˆB µν U [V µ, V ν ]) ia 3 Tr(Ŵ µν [V µ, V ν ]) 3 Work in collaboration with M.J.Herrero and J.J.Sanz-Cillero, JHEP1407 (2014) 149. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

52 Extension to γγ states Since coupling with photons are considered 3, the covariant derivative is defined as D µ U = µ U + iŵ µ U iu ˆB µ. The photon field A arises from the couplings with Ŵ µν and ˆB µν through a rotation to the physical basis; an anomalous three-particle coupling may appear h c W v ŴµνŴ µν h c B v ˆB ˆBµν µν = c γ h 2 v e2 A µν A µν The next additional NLO counterterms are needed, L 4 = a 1 Tr(U ˆB µν U Ŵ µν ) + ia 2 Tr(U ˆB µν U [V µ, V ν ]) ia 3 Tr(Ŵ µν [V µ, V ν ]) 3 Work in collaboration with M.J.Herrero and J.J.Sanz-Cillero, JHEP1407 (2014) 149. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

53 Extension to γγ states Since coupling with photons are considered 3, the covariant derivative is defined as D µ U = µ U + iŵ µ U iu ˆB µ. The photon field A arises from the couplings with Ŵ µν and ˆB µν through a rotation to the physical basis; an anomalous three-particle coupling may appear h c W v ŴµνŴ µν h c B v ˆB ˆBµν µν = c γ h 2 v e2 A µν A µν The next additional NLO counterterms are needed, L 4 = a 1 Tr(U ˆB µν U Ŵ µν ) + ia 2 Tr(U ˆB µν U [V µ, V ν ]) ia 3 Tr(Ŵ µν [V µ, V ν ]) 3 Work in collaboration with M.J.Herrero and J.J.Sanz-Cillero, JHEP1407 (2014) 149. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

54 Extension to γγ states Since coupling with photons are considered 3, the covariant derivative is defined as D µ U = µ U + iŵ µ U iu ˆB µ. The photon field A arises from the couplings with Ŵ µν and ˆB µν through a rotation to the physical basis; an anomalous three-particle coupling may appear h c W v ŴµνŴ µν h c B v ˆB ˆBµν µν = c γ h 2 v e2 A µν A µν The next additional NLO counterterms are needed, L 4 = a 1 Tr(U ˆB µν U Ŵ µν ) + ia 2 Tr(U ˆB µν U [V µ, V ν ]) ia 3 Tr(Ŵ µν [V µ, V ν ]) 3 Work in collaboration with M.J.Herrero and J.J.Sanz-Cillero, JHEP1407 (2014) 149. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

55 Extension to γγ states Since coupling with photons are considered 3, the covariant derivative is defined as D µ U = µ U + iŵ µ U iu ˆB µ. The photon field A arises from the couplings with Ŵ µν and ˆB µν through a rotation to the physical basis; an anomalous three-particle coupling may appear h c W v ŴµνŴ µν h c B v ˆB ˆBµν µν = c γ h 2 v e2 A µν A µν The next additional NLO counterterms are needed, L 4 = a 1 Tr(U ˆB µν U Ŵ µν ) + ia 2 Tr(U ˆB µν U [V µ, V ν ]) ia 3 Tr(Ŵ µν [V µ, V ν ]) 3 Work in collaboration with M.J.Herrero and J.J.Sanz-Cillero, JHEP1407 (2014) 149. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

56 Extension to t t states Lagrangian additions 4 : L = i Q Q vg(h) [ Q L UH QQ R + h.c.]. This expression, for the heaviest quark generation, expands to 5 { L Y = G(h) 1 ω2 ( Mt v 2 t t + M b bb ) + iω0 ( Mt tγ 5 t M b bγ 5 b ) v + i 2ω + (M b t L b R M t t R b L ) + i 2ω ( ) } Mt bl t R M b br t L v v Two NLO counterterms needed for renormalization, M t L 4 = g t v 4 µω a µ ω b t t + g t M t v 4 µh µ ht t 4 Work in collaboration with A.Castillo, arxiv: [hep-ph], accepted in EPJC. 5 G(h) = 1 + c 1(h/v) + c 2(h/v) , V tb very close to unity Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

57 Extension to t t states Lagrangian additions 4 : L = i Q Q vg(h) [ Q L UH QQ R + h.c.]. This expression, for the heaviest quark generation, expands to 5 { L Y = G(h) 1 ω2 ( Mt v 2 t t + M b bb ) + iω0 ( Mt tγ 5 t M b bγ 5 b ) v + i 2ω + (M b t L b R M t t R b L ) + i 2ω ( ) } Mt bl t R M b br t L v v Two NLO counterterms needed for renormalization, M t L 4 = g t v 4 µω a µ ω b t t + g t M t v 4 µh µ ht t 4 Work in collaboration with A.Castillo, arxiv: [hep-ph], accepted in EPJC. 5 G(h) = 1 + c 1(h/v) + c 2(h/v) , V tb very close to unity Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

58 Extension to t t states Lagrangian additions 4 : L = i Q Q vg(h) [ Q L UH QQ R + h.c.]. This expression, for the heaviest quark generation, expands to 5 { L Y = G(h) 1 ω2 ( Mt v 2 t t + M b bb ) + iω0 ( Mt tγ 5 t M b bγ 5 b ) v + i 2ω + (M b t L b R M t t R b L ) + i 2ω ( ) } Mt bl t R M b br t L v v Two NLO counterterms needed for renormalization, M t L 4 = g t v 4 µω a µ ω b t t + g t M t v 4 µh µ ht t 4 Work in collaboration with A.Castillo, arxiv: [hep-ph], accepted in EPJC. 5 G(h) = 1 + c 1(h/v) + c 2(h/v) , V tb very close to unity Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

59 Chiral counting s 5/2 s 2 s 3/2 s s 1/2 s NLO NLO LO LO 0 M t Mt 2 Mt 3 Mt 4 EWSBS alone EWSBS+t t M t s 5/2 s 2 s 3/2 s s 1/2 s NLO LO NLO LO 0 α α 2 α 3 α 4 EWSBS alone EWSBS+γγ α Note the usage of the chiral counting from 6. 6 G.Buchalla and O.Catà, JHEP07 (2012) 101; G.Buchalla, O.Catà and C.Krause, Phys.Lett.B731 (2014) 80; S.Weinberg, Physica A96 (1979) 327. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

60 Chiral counting s 5/2 s 2 s 3/2 s s 1/2 s NLO NLO LO LO 0 M t Mt 2 Mt 3 Mt 4 EWSBS alone EWSBS+t t M t s 5/2 s 2 s 3/2 s s 1/2 s NLO LO NLO LO 0 α α 2 α 3 α 4 EWSBS alone EWSBS+γγ α Note the usage of the chiral counting from 6. 6 G.Buchalla and O.Catà, JHEP07 (2012) 101; G.Buchalla, O.Catà and C.Krause, Phys.Lett.B731 (2014) 80; S.Weinberg, Physica A96 (1979) 327. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

61 Chiral counting s 5/2 s 2 s 3/2 s s 1/2 s NLO NLO LO LO 0 M t Mt 2 Mt 3 Mt 4 EWSBS alone EWSBS+t t M t s 5/2 s 2 s 3/2 s s 1/2 s NLO LO NLO LO 0 α α 2 α 3 α 4 EWSBS alone EWSBS+γγ α Note the usage of the chiral counting from 6. 6 G.Buchalla and O.Catà, JHEP07 (2012) 101; G.Buchalla, O.Catà and C.Krause, Phys.Lett.B731 (2014) 80; S.Weinberg, Physica A96 (1979) 327. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

62 Particular cases of the theory a 2 = b = 0 Higgsless ECL, now experimentally discarded. J.Gasser and H.Leutwyler, Annal.Phys.158,142; Nucl.Phys.B250,465&517 a 2 = 1 v 2 /f 2, b = 1 2v 2 /f 2 SO(5)/SO(4) Minimal Composite Higgs Model (MCHM) K.Agashe, R.Contino and A.Pomarol, Nucl.Phys.B719, 165 S.De Curtis, S.Moretti, K.Yagyu, E.Yildirim, JHEP1204 (2012) 042 a 2 = b = v 2 /f 2 Dilaton models E.Halyo, Mod.Phys.Lett.A8, 275; W.D.Goldberg et al, PRL a 2 = b = 1 Standard Model: without non-perturbative interactions Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

63 Particular cases of the theory a 2 = b = 0 Higgsless ECL, now experimentally discarded. J.Gasser and H.Leutwyler, Annal.Phys.158,142; Nucl.Phys.B250,465&517 a 2 = 1 v 2 /f 2, b = 1 2v 2 /f 2 SO(5)/SO(4) Minimal Composite Higgs Model (MCHM) K.Agashe, R.Contino and A.Pomarol, Nucl.Phys.B719, 165 S.De Curtis, S.Moretti, K.Yagyu, E.Yildirim, JHEP1204 (2012) 042 a 2 = b = v 2 /f 2 Dilaton models E.Halyo, Mod.Phys.Lett.A8, 275; W.D.Goldberg et al, PRL a 2 = b = 1 Standard Model: without non-perturbative interactions Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

64 Particular cases of the theory a 2 = b = 0 Higgsless ECL, now experimentally discarded. J.Gasser and H.Leutwyler, Annal.Phys.158,142; Nucl.Phys.B250,465&517 a 2 = 1 v 2 /f 2, b = 1 2v 2 /f 2 SO(5)/SO(4) Minimal Composite Higgs Model (MCHM) K.Agashe, R.Contino and A.Pomarol, Nucl.Phys.B719, 165 S.De Curtis, S.Moretti, K.Yagyu, E.Yildirim, JHEP1204 (2012) 042 a 2 = b = v 2 /f 2 Dilaton models E.Halyo, Mod.Phys.Lett.A8, 275; W.D.Goldberg et al, PRL a 2 = b = 1 Standard Model: without non-perturbative interactions Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

65 Particular cases of the theory a 2 = b = 0 Higgsless ECL, now experimentally discarded. J.Gasser and H.Leutwyler, Annal.Phys.158,142; Nucl.Phys.B250,465&517 a 2 = 1 v 2 /f 2, b = 1 2v 2 /f 2 SO(5)/SO(4) Minimal Composite Higgs Model (MCHM) K.Agashe, R.Contino and A.Pomarol, Nucl.Phys.B719, 165 S.De Curtis, S.Moretti, K.Yagyu, E.Yildirim, JHEP1204 (2012) 042 a 2 = b = v 2 /f 2 Dilaton models E.Halyo, Mod.Phys.Lett.A8, 275; W.D.Goldberg et al, PRL a 2 = b = 1 Standard Model: without non-perturbative interactions Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

66 Experimental bounds on low-energy constants As it would require measuring the coupling of two Higgses, there is no experimental bound over the value of b parameter 7. Over a, at a confidence level of 2σ (95%), CMS a (0.87, 1.14) ATLAS a (0.93, 1.34) Fit of Buchalla et. al a (0.80, 1.16) 7 Giardino, P.P., Aspects of LHC phenom., PhD Thesis (2013), Università di Pisa 8 Eur. Phys. J. C75 (2015), Report No. ATLAS-CONF G. Buchalla, O. Cata, A. Celis, and C. Krause, Eur.Phys.J. C76 (2016) no.5, 233 Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

67 Experimental bounds on low-energy constants As it would require measuring the coupling of two Higgses, there is no experimental bound over the value of b parameter 7. Over a, at a confidence level of 2σ (95%), CMS a (0.87, 1.14) ATLAS a (0.93, 1.34) Fit of Buchalla et. al a (0.80, 1.16) κ f CMS Preliminary fb -1 (8 TeV) fb (7 TeV) κ f CMS Preliminary 19.7 fb (8 TeV) fb (7 TeV) Observed SM Higgs H WW H ZZ H γγ 95% C.L. H ττ H bb κ V κ V 7 Giardino, P.P., Aspects of LHC phenom., PhD Thesis (2013), Università di Pisa 8 Eur. Phys. J. C75 (2015), Report No. ATLAS-CONF G. Buchalla, O. Cata, A. Celis, and C. Krause, Eur.Phys.J. C76 (2016) no.5, 233 Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

68 Experimental bounds on low-energy constants As it would require measuring the coupling of two Higgses, there is no experimental bound over the value of b parameter 7. Over a, at a confidence level of 2σ (95%), CMS a (0.87, 1.14) ATLAS a (0.93, 1.34) Fit of Buchalla et. al a (0.80, 1.16) κ f CMS Preliminary fb -1 (8 TeV) fb (7 TeV) κ f CMS Preliminary 19.7 fb (8 TeV) fb (7 TeV) Observed SM Higgs H WW H ZZ H γγ 95% C.L. H ττ H bb κ V κ V 7 Giardino, P.P., Aspects of LHC phenom., PhD Thesis (2013), Università di Pisa 8 Eur. Phys. J. C75 (2015), Report No. ATLAS-CONF G. Buchalla, O. Cata, A. Celis, and C. Krause, Eur.Phys.J. C76 (2016) no.5, 233 Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

69 Experimental bounds on low-energy constants As it would require measuring the coupling of two Higgses, there is no experimental bound over the value of b parameter 7. Over a, at a confidence level of 2σ (95%), CMS a (0.87, 1.14) ATLAS a (0.93, 1.34) Fit of Buchalla et. al a (0.80, 1.16) κ f CMS Preliminary fb -1 (8 TeV) fb (7 TeV) κ f CMS Preliminary 19.7 fb (8 TeV) fb (7 TeV) Observed SM Higgs H WW H ZZ H γγ 95% C.L. H ττ H bb κ V κ V 7 Giardino, P.P., Aspects of LHC phenom., PhD Thesis (2013), Università di Pisa 8 Eur. Phys. J. C75 (2015), Report No. ATLAS-CONF G. Buchalla, O. Cata, A. Celis, and C. Krause, Eur.Phys.J. C76 (2016) no.5, 233 Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

70 Experimental bounds on low-energy constants, NLO a 4 -a 5 α ATLAS 20.3 fb -1, s = 8 TeV pp W ± W ± jj K-matrix unitarization confidence intervals 68% CL 95% CL expected 95% CL Standard Model Direct constraint over a 4 -a 5 from ATLAS Collaboration 11 α 4 11 Taken from ref. [PRL113 (2014) ]. Note that CMS [PRL114 (2015) ] gives a constraint in terms of F S0 /Λ 4 and F S1 /Λ 4 parameters, which have no direct translation to the a 4 and a 5 ones [arxiv: , [hep-ph] ]. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

71 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

72 Partial wave decomposition EWSBS alone (+eventually t t) A IJ (s) = 1 32πK 1 1 dx P J (x)a I [s, t(s, x), u(s, x)] Matrix element from partial wave decomposition A I (s, t, u) = 16πK (2J + 1)P J [x(s, t)]a IJ (s) J=0 Helicity partial waves for EWSBS+γγ F λ 1λ 2 1 4π IJ (s) = 64π 2 dω A λ 1λ 2 K 2J + 1 I (s, Ω)Y J,λ1 λ 2 (Ω) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

73 Partial wave decomposition EWSBS alone (+eventually t t) A IJ (s) = 1 32πK 1 1 dx P J (x)a I [s, t(s, x), u(s, x)] Matrix element from partial wave decomposition A I (s, t, u) = 16πK (2J + 1)P J [x(s, t)]a IJ (s) J=0 Helicity partial waves for EWSBS+γγ F λ 1λ 2 1 4π IJ (s) = 64π 2 dω A λ 1λ 2 K 2J + 1 I (s, Ω)Y J,λ1 λ 2 (Ω) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

74 Partial wave decomposition EWSBS alone (+eventually t t) A IJ (s) = 1 32πK 1 1 dx P J (x)a I [s, t(s, x), u(s, x)] Matrix element from partial wave decomposition A I (s, t, u) = 16πK (2J + 1)P J [x(s, t)]a IJ (s) J=0 Helicity partial waves for EWSBS+γγ F λ 1λ 2 1 4π IJ (s) = 64π 2 dω A λ 1λ 2 K 2J + 1 I (s, Ω)Y J,λ1 λ 2 (Ω) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

75 EWSBS partial waves The form of the partial wave is A IJ (s) = A (0) IJ Which will be decomposed as A (0) IJ = Ks + A(1) IJ + O [ (s/v 2 ) 3]. ( A (1) IJ = B(µ) + D log s s + E log µ 2 µ 2 As A IJ (s) must be scale independent, B(µ) = B(µ 0 ) + (D + E) log µ2 µ 2 0 = B 0 + p 4 a 4 (µ) + p 5 a 5 (µ). ) s 2. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

76 EWSBS partial waves The form of the partial wave is A IJ (s) = A (0) IJ Which will be decomposed as A (0) IJ = Ks + A(1) IJ + O [ (s/v 2 ) 3]. ( A (1) IJ = B(µ) + D log s s + E log µ 2 µ 2 As A IJ (s) must be scale independent, B(µ) = B(µ 0 ) + (D + E) log µ2 µ 2 0 = B 0 + p 4 a 4 (µ) + p 5 a 5 (µ). ) s 2. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

77 EWSBS partial waves The form of the partial wave is A IJ (s) = A (0) IJ Which will be decomposed as A (0) IJ = Ks + A(1) IJ + O [ (s/v 2 ) 3]. ( A (1) IJ = B(µ) + D log s s + E log µ 2 µ 2 As A IJ (s) must be scale independent, B(µ) = B(µ 0 ) + (D + E) log µ2 µ 2 0 = B 0 + p 4 a 4 (µ) + p 5 a 5 (µ). ) s 2. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

78 Partial wave decomposition: γγ states The form of the partial wave is P IJ,Λ (s) = P (0) IJ,Λ + O(α2 em) + O(α em s 2 ). Note that γγ with J = 2, Λ = ±2 also couples with the EWSBS, following P (0) I 0,0 αs P(0) I 2,±2 α Based on a collaboration with profs. M.J.Herrero and J.J.Sanz-Cillero: JHEP1407 (2014) 149. Partial waves, unitarization and study of the parameter space: Eur.Phys.J. C77 (2017) no.4, 205. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

79 Partial wave decomposition: γγ states The form of the partial wave is P IJ,Λ (s) = P (0) IJ,Λ + O(α2 em) + O(α em s 2 ). Note that γγ with J = 2, Λ = ±2 also couples with the EWSBS, following P (0) I 0,0 αs P(0) I 2,±2 α Based on a collaboration with profs. M.J.Herrero and J.J.Sanz-Cillero: JHEP1407 (2014) 149. Partial waves, unitarization and study of the parameter space: Eur.Phys.J. C77 (2017) no.4, 205. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

80 Partial wave decomposition: γγ states The form of the partial wave is P IJ,Λ (s) = P (0) IJ,Λ + O(α2 em) + O(α em s 2 ). Note that γγ with J = 2, Λ = ±2 also couples with the EWSBS, following P (0) I 0,0 αs P(0) I 2,±2 α Based on a collaboration with profs. M.J.Herrero and J.J.Sanz-Cillero: JHEP1407 (2014) 149. Partial waves, unitarization and study of the parameter space: Eur.Phys.J. C77 (2017) no.4, 205. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

81 Partial wave decomposition: t t states The form of the partial wave is Q IJ (s) = Q (0) IJ + Q (1) IJ + O [ M t s 2 s/v 6] + O [ M 2 t s/v 4], which will be decomposed as Q (0) IJ Q (1) IJ = = K Q sm t ( B Q (µ) + E Q log s µ 2 As A IJ (s) must be scale independent, ) s sm t. B Q (µ) = B Q (µ 0 ) + E Q log µ2 µ 2 0 = B 0 + p g g t (µ). Based on a collaboration with A.Castillo: arxiv: [hep-ph], accepted in EPJC. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

82 Partial wave decomposition: t t states The form of the partial wave is Q IJ (s) = Q (0) IJ + Q (1) IJ + O [ M t s 2 s/v 6] + O [ M 2 t s/v 4], which will be decomposed as Q (0) IJ Q (1) IJ = = K Q sm t ( B Q (µ) + E Q log s µ 2 As A IJ (s) must be scale independent, ) s sm t. B Q (µ) = B Q (µ 0 ) + E Q log µ2 µ 2 0 = B 0 + p g g t (µ). Based on a collaboration with A.Castillo: arxiv: [hep-ph], accepted in EPJC. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

83 Partial wave decomposition: t t states The form of the partial wave is Q IJ (s) = Q (0) IJ + Q (1) IJ + O [ M t s 2 s/v 6] + O [ M 2 t s/v 4], which will be decomposed as Q (0) IJ Q (1) IJ = = K Q sm t ( B Q (µ) + E Q log s µ 2 As A IJ (s) must be scale independent, ) s sm t. B Q (µ) = B Q (µ 0 ) + E Q log µ2 µ 2 0 = B 0 + p g g t (µ). Based on a collaboration with A.Castillo: arxiv: [hep-ph], accepted in EPJC. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

84 Partial wave decomposition: t t states The form of the partial wave is Q IJ (s) = Q (0) IJ + Q (1) IJ + O [ M t s 2 s/v 6] + O [ M 2 t s/v 4], which will be decomposed as Q (0) IJ Q (1) IJ = = K Q sm t ( B Q (µ) + E Q log s µ 2 As A IJ (s) must be scale independent, ) s sm t. B Q (µ) = B Q (µ 0 ) + E Q log µ2 µ 2 0 = B 0 + p g g t (µ). Based on a collaboration with A.Castillo: arxiv: [hep-ph], accepted in EPJC. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

85 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

86 Unitarity for partial waves Unit. cond. for S matrix: SS = 1, plus analytical properties of matrix elements, plus time reversal invariance, Unitarity condition for partial waves Im A IJ,pi k 1 (s) = {a,b} 1 4m2 q [A IJ,pi q s i,ab (s)][a IJ,qi,ab k i (s)] Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

87 Unitarity for partial waves Unit. cond. for S matrix: SS = 1, plus analytical properties of matrix elements, plus time reversal invariance, LC Re(s) Im(s) x x x x x x x x x x x RC Unitarity condition for partial waves Im A IJ,pi k 1 (s) = {a,b} 1 4m2 q [A IJ,pi q s i,ab (s)][a IJ,qi,ab k i (s)] Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

88 Unitarity for partial waves Unit. cond. for S matrix: SS = 1, plus analytical properties of matrix elements, plus time reversal invariance, LC Re(s) Im(s) x x x x x x x x x x x RC Unitarity condition for partial waves Im A IJ,pi k 1 (s) = {a,b} 1 4m2 q [A IJ,pi q s i,ab (s)][a IJ,qi,ab k i (s)] Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

89 Unitarity for partial waves Unit. cond. for S matrix: SS = 1, plus analytical properties of matrix elements, plus time reversal invariance, q 1a e z p 2 θ θ ϕ k 2 p 1 e x k 1 q 2b e y Unitarity condition for partial waves Im A IJ,pi k 1 (s) = {a,b} 1 4m2 q [A IJ,pi q s i,ab (s)][a IJ,qi,ab k i (s)] Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

90 Unitarization procedures for elastic processes A IAM (s) = A N/D (s) = A IK (s) = A K 0 (s) = where [A (0) (s)] 2 A (0) (s) A (1) (s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) g(s)a L( s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) + g(s)a L(s), A 0(s) 1 ia 0 (s), g(s) = 1 π ( B(µ) D + E A L (s) = πg( s)ds 2 A R (s) = πg(s)es 2 ) s + log µ 2 PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

91 Unitarization procedures for elastic processes A IAM (s) = A N/D (s) = A IK (s) = A K 0 (s) = where [A (0) (s)] 2 A (0) (s) A (1) (s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) g(s)a L( s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) + g(s)a L(s), A 0(s) 1 ia 0 (s), g(s) = 1 π ( B(µ) D + E A L (s) = πg( s)ds 2 A R (s) = πg(s)es 2 ) s + log µ 2 PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

92 Unitarization procedures for elastic processes A IAM (s) = A N/D (s) = A IK (s) = A K 0 (s) = where [A (0) (s)] 2 A (0) (s) A (1) (s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) g(s)a L( s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) + g(s)a L(s), A 0(s) 1 ia 0 (s), g(s) = 1 π ( B(µ) D + E A L (s) = πg( s)ds 2 A R (s) = πg(s)es 2 ) s + log µ 2 PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

93 Unitarization procedures for elastic processes A IAM (s) = A N/D (s) = A IK (s) = A K 0 (s) = where [A (0) (s)] 2 A (0) (s) A (1) (s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) g(s)a L( s), A (0) (s) + A L (s) 1 A R(s) A (0) (s) + g(s)a L(s), A 0(s) 1 ia 0 (s), g(s) = 1 π ( B(µ) D + E A L (s) = πg( s)ds 2 A R (s) = πg(s)es 2 ) s + log µ 2 PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

94 Matricial versions of the methods [ 1 F IAM (s) = F (s)] (0) F N/D (s) = [ ] [ F (0) (s) F (1) (s) [ ( ) 1 1 F R (s) F (0) 1 (s) + 2 G(s)F L( s) F IK (s) = [1 + G(s) N 0 (s)] 1 N 0 (s), where G(s), F L (s), F R (s) and N 0 (s) are defined as G(s) = 1 ( B(µ)(D + E) 1 + log s ) π µ 2 F L (s) = πg( s)ds 2 F R (s) = πg(s)es 2 N 0 (s) = F (0) (s) + F L (s) 1 F (s)] (0), ] 1 N 0 (s), PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

95 Matricial versions of the methods [ 1 F IAM (s) = F (s)] (0) F N/D (s) = [ ] [ F (0) (s) F (1) (s) [ ( ) 1 1 F R (s) F (0) 1 (s) + 2 G(s)F L( s) F IK (s) = [1 + G(s) N 0 (s)] 1 N 0 (s), where G(s), F L (s), F R (s) and N 0 (s) are defined as G(s) = 1 ( B(µ)(D + E) 1 + log s ) π µ 2 F L (s) = πg( s)ds 2 F R (s) = πg(s)es 2 N 0 (s) = F (0) (s) + F L (s) 1 F (s)] (0), ] 1 N 0 (s), PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

96 Matricial versions of the methods [ 1 F IAM (s) = F (s)] (0) F N/D (s) = [ ] [ F (0) (s) F (1) (s) [ ( ) 1 1 F R (s) F (0) 1 (s) + 2 G(s)F L( s) F IK (s) = [1 + G(s) N 0 (s)] 1 N 0 (s), where G(s), F L (s), F R (s) and N 0 (s) are defined as G(s) = 1 ( B(µ)(D + E) 1 + log s ) π µ 2 F L (s) = πg( s)ds 2 F R (s) = πg(s)es 2 N 0 (s) = F (0) (s) + F L (s) 1 F (s)] (0), ] 1 N 0 (s), PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

97 Extension to γγ and t t scattering Basic assumption EWSBS is strongly interacting. γγ and t t are perturbative. Coupling with photons, controlled by α = e 2 /4π s/v 2. Coupling with top quarks, controlled by M t s/v 2 s/v 2. Perturbative unitarization: ωω {γγ, t t} P = ÃIJ A (0) P (0) IJ Perturbative unitarization: {ωω, hh} {γγ, t t} ( ) P = F (F R (0)) ( 1 P (0) ) R (0) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

98 Extension to γγ and t t scattering Basic assumption EWSBS is strongly interacting. γγ and t t are perturbative. Coupling with photons, controlled by α = e 2 /4π s/v 2. Coupling with top quarks, controlled by M t s/v 2 s/v 2. Perturbative unitarization: ωω {γγ, t t} P = ÃIJ A (0) P (0) IJ Perturbative unitarization: {ωω, hh} {γγ, t t} ( ) P = F (F R (0)) ( 1 P (0) ) R (0) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

99 Extension to γγ and t t scattering Basic assumption EWSBS is strongly interacting. γγ and t t are perturbative. Coupling with photons, controlled by α = e 2 /4π s/v 2. Coupling with top quarks, controlled by M t s/v 2 s/v 2. Perturbative unitarization: ωω {γγ, t t} P = ÃIJ A (0) P (0) IJ Perturbative unitarization: {ωω, hh} {γγ, t t} ( ) P = F (F R (0)) ( 1 P (0) ) R (0) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

100 Extension to γγ and t t scattering Basic assumption EWSBS is strongly interacting. γγ and t t are perturbative. Coupling with photons, controlled by α = e 2 /4π s/v 2. Coupling with top quarks, controlled by M t s/v 2 s/v 2. Perturbative unitarization: ωω {γγ, t t} P = ÃIJ A (0) P (0) IJ Perturbative unitarization: {ωω, hh} {γγ, t t} ( ) P = F (F R (0)) ( 1 P (0) ) R (0) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

101 Extension to γγ and t t scattering Basic assumption EWSBS is strongly interacting. γγ and t t are perturbative. Coupling with photons, controlled by α = e 2 /4π s/v 2. Coupling with top quarks, controlled by M t s/v 2 s/v 2. Perturbative unitarization: ωω {γγ, t t} P = ÃIJ A (0) P (0) IJ Perturbative unitarization: {ωω, hh} {γγ, t t} ( ) P = F (F R (0)) ( 1 P (0) ) R (0) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

102 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

103 Usability channel of unitarization procedures IJ Method of choice Any N/D IK IAM Any N/D IK The IAM method cannot be used when A (0) = 0, because it would give a vanishing value. The N/D and the IK methods cannot be used if D + E = 0, because in this case computing A L (s) and A R (s) is not possible. The naive K-matrix method, A K 0 (s) = A 0(s) 1 ia 0 (s), fails because it is not analytical in the first Riemann sheet and, consequently, it is not a proper partial wave compatible with microcausality. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

104 Usability channel of unitarization procedures IJ Method of choice Any N/D IK IAM Any N/D IK The IAM method cannot be used when A (0) = 0, because it would give a vanishing value. The N/D and the IK methods cannot be used if D + E = 0, because in this case computing A L (s) and A R (s) is not possible. The naive K-matrix method, A K 0 (s) = A 0(s) 1 ia 0 (s), fails because it is not analytical in the first Riemann sheet and, consequently, it is not a proper partial wave compatible with microcausality. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

105 Usability channel of unitarization procedures IJ Method of choice Any N/D IK IAM Any N/D IK The IAM method cannot be used when A (0) = 0, because it would give a vanishing value. The N/D and the IK methods cannot be used if D + E = 0, because in this case computing A L (s) and A R (s) is not possible. The naive K-matrix method, A K 0 (s) = A 0(s) 1 ia 0 (s), fails because it is not analytical in the first Riemann sheet and, consequently, it is not a proper partial wave compatible with microcausality. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

106 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

107 Scalar-isoscalar channels From left to right and top to bottom, elastic ωω, elastic hh, and cross channel ωω hh, for a = 0.88, b = 3, µ = 3 TeV and all NLO parameters set to 0. PRL 114 (2015) , PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

108 Vector-isovector channels From our ref 12. We have taken a = 0.88 and b = 1.5, but while for the left plot all the NLO parameters vanish, for the right plot we have taken a 4 = 0.003, known to yield an IAM resonance according to the Barcelona group PRD 91 (2015) PRD 90 (2014) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

109 Scalar-isotensor channels (IJ = 20) From our ref 14. From left to right, a = 0.88, a = We have taken b = a 2 and the NLO parameters set to zero. Both real and imaginary part shown. Real ones correspond to bottom lines at left and upper at low E at right. 14 PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

110 Isotensor-scalar channels (IJ = 02) a = 0.88, b = a 2, a 4 = 2a 5 = 3/(192π), all the other NLO param. set to zero. PRD 91 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

111 Reson. in W L W L W L W L due to a 4 and a 5, ours a = 0.90, b = a 2 PRD 91 (2015) From left, clockwise, IJ = 00, 11, 20 Excluding resonances M S < 700 GeV, M V < 1.5 TeV Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

112 Reson. in W L W L W L W L due to a 4 and a 5, Barcelona CROSS-CHECK: Espriu, Yencho, Mescia PRD88, PRD90, At right, exclusion regions include resonances with M S,V < 600 GeV. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

113 Resonance from W L W L hh a = 1, b = 2, IAM, elastic chann. W L W L W L W L, red figure from 3D-printer Rafael L. Delgado, Antonio Dobado, Felipe J. Llanes-Estrada, Possible New Resonance from W L W L -hh Interchannel Coupling, PRL 114 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

114 Resonance from W L W L hh a = 1, b = 2, IAM, inelastic chann. W L W L hh, yellow figure from 3D-printer Rafael L. Delgado, Antonio Dobado, Felipe J. Llanes-Estrada, Possible New Resonance from W L W L -hh Interchannel Coupling, PRL 114 (2015) Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

115 Resonances in W L W L W L W L due to a and b parameters PRL & PRD 91 (2015) From left, clockwise, IJ = 00, 11, 20 Excluding resonances M S < 700 GeV, M V < 1.5 TeV Constraint over b even without data about W L W L hh and hh hh scattering processes. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

116 Motion of the resonance mass and width Dependence on b with a 2 = 1 fixed (upper curve) and for a = 1 ξ and b = 1 2ξ with ξ = v/f as in the MCHM (lower blue curve). PRL 114 (2015) Video, (a,b) param. space Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

117 Resonances in W L W L W L W L due to a and a 4 parameters b = a 2 PRD 91 (2015) From left, clockwise, IJ = 00, 11, 20 Excluding resonances M S < 700 GeV, M V < 1.5 TeV Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

118 Resonances in W L W L W L W L due to b, g, d and e parameters Effective Theory, PRD 91 (2015) , isoscalar channels (I = J = 0). Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

119 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

120 Conclusions: our work We have performed a comprehensive study of the unitarized amplitudes obtained from the non-linear Effective Chiral Lagrangian, which describes the EWSBS in the TeV region. We have carried out an NLO computation with a massless EFT. Studied 2 2 scattering processes within the EWSBS: {V L V L, hh} {V L V L, hh}. And (weak) couplings with γγ and t t states. We are ready for new strong interactions at the LHC. What are the prospects? Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

121 Conclusions: our work We have performed a comprehensive study of the unitarized amplitudes obtained from the non-linear Effective Chiral Lagrangian, which describes the EWSBS in the TeV region. We have carried out an NLO computation with a massless EFT. Studied 2 2 scattering processes within the EWSBS: {V L V L, hh} {V L V L, hh}. And (weak) couplings with γγ and t t states. We are ready for new strong interactions at the LHC. What are the prospects? Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

122 Conclusions: our work We have performed a comprehensive study of the unitarized amplitudes obtained from the non-linear Effective Chiral Lagrangian, which describes the EWSBS in the TeV region. We have carried out an NLO computation with a massless EFT. Studied 2 2 scattering processes within the EWSBS: {V L V L, hh} {V L V L, hh}. And (weak) couplings with γγ and t t states. We are ready for new strong interactions at the LHC. What are the prospects? Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

123 Conclusions: our work We have performed a comprehensive study of the unitarized amplitudes obtained from the non-linear Effective Chiral Lagrangian, which describes the EWSBS in the TeV region. We have carried out an NLO computation with a massless EFT. Studied 2 2 scattering processes within the EWSBS: {V L V L, hh} {V L V L, hh}. And (weak) couplings with γγ and t t states. We are ready for new strong interactions at the LHC. What are the prospects? Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

124 Conclusions: our work We have performed a comprehensive study of the unitarized amplitudes obtained from the non-linear Effective Chiral Lagrangian, which describes the EWSBS in the TeV region. We have carried out an NLO computation with a massless EFT. Studied 2 2 scattering processes within the EWSBS: {V L V L, hh} {V L V L, hh}. And (weak) couplings with γγ and t t states. We are ready for new strong interactions at the LHC. What are the prospects? Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

125 Conclusions: experimental issues, W + L W L prod. in the SM M(w+ w ) Entries Mean RMS Underflow 0 Overflow M Production of W + W (blue) vs. W + L W L (red) in the SM. x-axis in GeV and y-axis in events/33.3 GeV. s = 13 TeV, L = 10 fb 1. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

126 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

127 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

128 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

129 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

130 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

131 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

132 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

133 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

134 Conclusions: a-b parameter space New scalar particle + mass gap (for a 2 = b = 1, the weakly interacting SM) New physics would very likely imply strong interactions, in elastic W L W L and inelastic hh scattering. For a 2 = b 1, Strong elastic interactions. σ-like pole in the second Riemann sheet. Even if a 1, but we allow b > a 2, Strong dynamics resonating between the W L W L and hh channels. A new scalar pole in the sub-tev region. New constraint on b, even in the absence of data about W L W L hh and hh hh, just looking at the W L W L scattering. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

135 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

136 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

137 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

138 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

139 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

140 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

141 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

142 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

143 Conclusions, possible extensions SM unitarity. Higgsless model (now experimentally excluded) unitarity violation in WW scattering new physics. Higgs like boson found no unitarity violation? Depends on couplings. Ok with the present experimental bounds. Measurements at the LHC Run II, important. Conceivable additional extensions: fermion loops, non vanishing values for M H, M W, M Z, and a full computation without using the equivalence theorem. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

144 Table of contents 1 Introduction Motivation of the low-energy effective Lagrangian Effective Lagrangian 2 Partial wave decomposition and unitarization procedures Partial waves Unitarization procedures Methods of choice 3 Analysis of the parameter space 4 Conclusions 5 Other Research Interests Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

145 Other research interests: Collider Phenomenology of Kaluza-Klein Models Work with J.A.R.Cembranos, and A.Dobado, PRD 88 (2013) When dealing with a rigid brane (f M D ), the fact that gravitons propagate in the bulk 5D space produce a tower of KK excitations with increasing invariant mass. This mass roughly correspond to the eigen energy of a quantum oscillator confined in the two extra dimensions. When integrating over all the possible KK modes, we obtain an effective theory where two massive gravitons of a certain mass M D are emitted, although actually only one graviton (with an unknown invariant mass) is emitted. When the brane is flexible, the fluctuations of the brane give rise to the so called branons, which are the particle associated to such fluctuation. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

146 Other research interests: Collider Phenomenology of Kaluza-Klein Models Work with J.A.R.Cembranos, and A.Dobado, PRD 88 (2013) When dealing with a rigid brane (f M D ), the fact that gravitons propagate in the bulk 5D space produce a tower of KK excitations with increasing invariant mass. This mass roughly correspond to the eigen energy of a quantum oscillator confined in the two extra dimensions. When integrating over all the possible KK modes, we obtain an effective theory where two massive gravitons of a certain mass M D are emitted, although actually only one graviton (with an unknown invariant mass) is emitted. When the brane is flexible, the fluctuations of the brane give rise to the so called branons, which are the particle associated to such fluctuation. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

147 Other research interests: Collider Phenomenology of Kaluza-Klein Models Work with J.A.R.Cembranos, and A.Dobado, PRD 88 (2013) When dealing with a rigid brane (f M D ), the fact that gravitons propagate in the bulk 5D space produce a tower of KK excitations with increasing invariant mass. This mass roughly correspond to the eigen energy of a quantum oscillator confined in the two extra dimensions. When integrating over all the possible KK modes, we obtain an effective theory where two massive gravitons of a certain mass M D are emitted, although actually only one graviton (with an unknown invariant mass) is emitted. When the brane is flexible, the fluctuations of the brane give rise to the so called branons, which are the particle associated to such fluctuation. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

148 Other research interests: Collider Phenomenology of Kaluza-Klein Models Work with J.A.R.Cembranos, and A.Dobado, PRD 88 (2013) When dealing with a rigid brane (f M D ), the fact that gravitons propagate in the bulk 5D space produce a tower of KK excitations with increasing invariant mass. This mass roughly correspond to the eigen energy of a quantum oscillator confined in the two extra dimensions. When integrating over all the possible KK modes, we obtain an effective theory where two massive gravitons of a certain mass M D are emitted, although actually only one graviton (with an unknown invariant mass) is emitted. When the brane is flexible, the fluctuations of the brane give rise to the so called branons, which are the particle associated to such fluctuation. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

149 Other research interests: Collider Phenomenology of Kaluza-Klein Models Work with J.A.R.Cembranos, and A.Dobado, PRD 88 (2013) When dealing with a rigid brane (f M D ), the fact that gravitons propagate in the bulk 5D space produce a tower of KK excitations with increasing invariant mass. This mass roughly correspond to the eigen energy of a quantum oscillator confined in the two extra dimensions. When integrating over all the possible KK modes, we obtain an effective theory where two massive gravitons of a certain mass M D are emitted, although actually only one graviton (with an unknown invariant mass) is emitted. When the brane is flexible, the fluctuations of the brane give rise to the so called branons, which are the particle associated to such fluctuation. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

150 Other research interests: Collider Phenomenology of Kaluza-Klein Models In this work, the single photon channel is used to check both the KK graviton an branon models. In particular, the KK graviton case will be used to compare with the simulation of ref. Aad, G., Abajyan, T., The ATLAS Collaboration, Phys. Rev. Lett., 110, (2013) [hep-ex/ ] Then, the branon case will be checked against the ATLAS data ( s = 7 TeV and an integrated luminosity of 4.6 fb 1 ) in the single photon channel. We used the computer resources, technical expertise, and assistance provided by the Barcelona Supercomputing Centre (BSC) and the Tirant supercomputer support staff at Valencia. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

151 Other research interests: Collider Phenomenology of Kaluza-Klein Models In this work, the single photon channel is used to check both the KK graviton an branon models. In particular, the KK graviton case will be used to compare with the simulation of ref. Aad, G., Abajyan, T., The ATLAS Collaboration, Phys. Rev. Lett., 110, (2013) [hep-ex/ ] Then, the branon case will be checked against the ATLAS data ( s = 7 TeV and an integrated luminosity of 4.6 fb 1 ) in the single photon channel. We used the computer resources, technical expertise, and assistance provided by the Barcelona Supercomputing Centre (BSC) and the Tirant supercomputer support staff at Valencia. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

152 Other research interests: Collider Phenomenology of Kaluza-Klein Models In this work, the single photon channel is used to check both the KK graviton an branon models. In particular, the KK graviton case will be used to compare with the simulation of ref. Aad, G., Abajyan, T., The ATLAS Collaboration, Phys. Rev. Lett., 110, (2013) [hep-ex/ ] Then, the branon case will be checked against the ATLAS data ( s = 7 TeV and an integrated luminosity of 4.6 fb 1 ) in the single photon channel. We used the computer resources, technical expertise, and assistance provided by the Barcelona Supercomputing Centre (BSC) and the Tirant supercomputer support staff at Valencia. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

153 Other research interests: Collider Phenomenology of Kaluza-Klein Models In this work, the single photon channel is used to check both the KK graviton an branon models. In particular, the KK graviton case will be used to compare with the simulation of ref. Aad, G., Abajyan, T., The ATLAS Collaboration, Phys. Rev. Lett., 110, (2013) [hep-ex/ ] Then, the branon case will be checked against the ATLAS data ( s = 7 TeV and an integrated luminosity of 4.6 fb 1 ) in the single photon channel. We used the computer resources, technical expertise, and assistance provided by the Barcelona Supercomputing Centre (BSC) and the Tirant supercomputer support staff at Valencia. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

154 Other research interests: Collider Phenomenology of Kaluza-Klein Models Our estimation of ATLAS analysis, applied to branon phenomenology CMS: Phys.Lett. B755 (2016) , full detector simulation Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

155 Other research interests: Gluon deconfinement Work with C.Hidalgo-Duque, F.J.Llanes-Estrada and A.Dobado, Few-Body Systems 54 (2013) Hypothetical deconfined gluon, produces a secondary collision at the beampipe or Internal Tracking System. Signature similar to a neutron. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

156 Other research interests: Gluon deconfinement Work with C.Hidalgo-Duque, F.J.Llanes-Estrada and A.Dobado, Few-Body Systems 54 (2013) Hypothetical deconfined gluon, produces a secondary collision at the beampipe or Internal Tracking System. Signature similar to a neutron. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

157 Other research interests: Gluon deconfinement Work with C.Hidalgo-Duque, F.J.Llanes-Estrada and A.Dobado, Few-Body Systems 54 (2013) Hypothetical deconfined gluon, produces a secondary collision at the beampipe or Internal Tracking System. Signature similar to a neutron. Rafael L. Delgado Unitarized EFT for a strongly interacting / 48

158 Rafael L. Delgado Unitarized EFT for a strongly.7 interacting / 48 Other research interests: Multistep Bose Einstein Condensation Work with P.Bargue no and F.Sols, PRE 86 (2012) Study of Bose Einstein Condensation with several energy hierarchies, E n (n z ) = ω ( n + 1 2) + εn 2 z

159 Rafael L. Delgado Unitarized EFT for a strongly.7 interacting / 48 Other research interests: Multistep Bose Einstein Condensation Work with P.Bargue no and F.Sols, PRE 86 (2012) Study of Bose Einstein Condensation with several energy hierarchies, E n (n z ) = ω ( n + 1 2) + εn 2 z

160 Rafael L. Delgado Unitarized EFT for a strongly.7 interacting / 48 Other research interests: Multistep Bose Einstein Condensation Work with P.Bargue no and F.Sols, PRE 86 (2012) Study of Bose Einstein Condensation with several energy hierarchies, E n (n z ) = ω ( n + 1 2) + εn 2 z

161 Backup Slides Rafael L. Delgado Unitarized EFT for a strongly interacting... 1 / 13

162 I) IAM method This method needs a NLO computation, t ω = tω 0, 1 tω 1 t ω 0 where [ ] [ ] [ ]) s s µ t1 ω = s (D 2 2 log µ 2 + E log µ 2 + (D + E) log µ 2 0 Rafael L. Delgado Unitarized EFT for a strongly interacting... 2 / 13

163 I) IAM method This method needs a NLO computation, t ω = tω 0, 1 tω 1 t ω 0 where [ ] [ ] [ ]) s s µ t1 ω = s (D 2 2 log µ 2 + E log µ 2 + (D + E) log µ 2 0 Rafael L. Delgado Unitarized EFT for a strongly interacting... 2 / 13

164 Check at tree level We have checked 15, for the tree level case, L = 1 ( 2 g(ϕ/f ) µω a µ ω b δ ab + ωa ω b ) v 2 ω µϕ µ ϕ 1 2 M2 ϕϕ 2 λ 3 ϕ 3 λ 4 ϕ g(ϕ/f ) = 1 + ( ϕ g n f n=1 ) n = 1 + 2α ϕ f ( ϕ ) 2 + β +.. f where a αv/f, b = βv 2 /f 2, and so one, the concordance with the methods 15 See J.Phys. G41 (2014) Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 13

165 Check at tree level We have checked 15, for the tree level case, L = 1 ( 2 g(ϕ/f ) µω a µ ω b δ ab + ωa ω b ) v 2 ω µϕ µ ϕ 1 2 M2 ϕϕ 2 λ 3 ϕ 3 λ 4 ϕ g(ϕ/f ) = 1 + ( ϕ g n f n=1 ) n = 1 + 2α ϕ f ( ϕ ) 2 + β +.. f where a αv/f, b = βv 2 /f 2, and so one, the concordance with the methods 15 See J.Phys. G41 (2014) Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 13

166 Check at tree level We have checked 15, for the tree level case, L = 1 ( 2 g(ϕ/f ) µω a µ ω b δ ab + ωa ω b ) v 2 ω µϕ µ ϕ 1 2 M2 ϕϕ 2 λ 3 ϕ 3 λ 4 ϕ g(ϕ/f ) = 1 + ( ϕ g n f n=1 ) n = 1 + 2α ϕ f ( ϕ ) 2 + β +.. f where a αv/f, b = βv 2 /f 2, and so one, the concordance with the methods 15 See J.Phys. G41 (2014) Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 13

167 Check at tree level We have checked 15, for the tree level case, L = 1 ( 2 g(ϕ/f ) µω a µ ω b δ ab + ωa ω b ) v 2 ω µϕ µ ϕ 1 2 M2 ϕϕ 2 λ 3 ϕ 3 λ 4 ϕ g(ϕ/f ) = 1 + ( ϕ g n f n=1 ) n = 1 + 2α ϕ f ( ϕ ) 2 + β +.. f where a αv/f, b = βv 2 /f 2, and so one, the concordance with the methods 15 See J.Phys. G41 (2014) Rafael L. Delgado Unitarized EFT for a strongly interacting... 3 / 13

168 II) K matrix so that, for t ω, T = T (1 J(s)T ) 1,, J(s) = 1 [ ] s π log Λ 2, t ω = t ω J(t ω t ϕ t 2 ωϕ) 1 J(t ω + t ϕ ) + J 2 (t ω t ϕ t 2 ωϕ), for β = α 2 (elastic case), t ω = t ω 1 Jt ω Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 13

169 II) K matrix so that, for t ω, T = T (1 J(s)T ) 1,, J(s) = 1 [ ] s π log Λ 2, t ω = t ω J(t ω t ϕ t 2 ωϕ) 1 J(t ω + t ϕ ) + J 2 (t ω t ϕ t 2 ωϕ), for β = α 2 (elastic case), t ω = t ω 1 Jt ω Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 13

170 II) K matrix so that, for t ω, T = T (1 J(s)T ) 1,, J(s) = 1 [ ] s π log Λ 2, t ω = t ω J(t ω t ϕ t 2 ωϕ) 1 J(t ω + t ϕ ) + J 2 (t ω t ϕ t 2 ωϕ), for β = α 2 (elastic case), t ω = t ω 1 Jt ω Rafael L. Delgado Unitarized EFT for a strongly interacting... 4 / 13

171 III)Large N N, with v 2 /N fixed. The amplitude A N to order 1/N is a Lippmann-Schwinger series, A N = A A NI 2 A + ANI 2 ANI 2 A... d 4 q i I (s) = (2π) 4 q 2 (q + p) 2 = 1 [ ] s 16π 2 log Λ 2 = 1 8π J(s) Note: actually, N = 3. For the (iso)scalar partial wave (chiral limit, I = J = 0), t ω N (s) = tω 0 1 Jt ω 0 Rafael L. Delgado Unitarized EFT for a strongly interacting... 5 / 13

172 III)Large N N, with v 2 /N fixed. The amplitude A N to order 1/N is a Lippmann-Schwinger series, A N = A A NI 2 A + ANI 2 ANI 2 A... d 4 q i I (s) = (2π) 4 q 2 (q + p) 2 = 1 [ ] s 16π 2 log Λ 2 = 1 8π J(s) Note: actually, N = 3. For the (iso)scalar partial wave (chiral limit, I = J = 0), t ω N (s) = tω 0 1 Jt ω 0 Rafael L. Delgado Unitarized EFT for a strongly interacting... 5 / 13

173 III)Large N N, with v 2 /N fixed. The amplitude A N to order 1/N is a Lippmann-Schwinger series, A N = A A NI 2 A + ANI 2 ANI 2 A... d 4 q i I (s) = (2π) 4 q 2 (q + p) 2 = 1 [ ] s 16π 2 log Λ 2 = 1 8π J(s) Note: actually, N = 3. For the (iso)scalar partial wave (chiral limit, I = J = 0), t ω N (s) = tω 0 1 Jt ω 0 Rafael L. Delgado Unitarized EFT for a strongly interacting... 5 / 13

174 IV) N/D (elastic scattering at tree level only β = α 2. See ref. J.Phys. G41 (2014) ). Ansatz t ω (s) = N(s) D(s), where N(s) has a left hand cut (and Im N(s > 0) = 0) D(s) has a right hand cut (and ID(s < 0) = 0); D(s) = 1 s π N(s) = s π 0 0 ds N(s ) s (s s iɛ) ds Im N(s ) s (s s iɛ) Rafael L. Delgado Unitarized EFT for a strongly interacting... 6 / 13

175 Coupled channels, tree level amplitudes f = 2v, β = α 2 = 1, λ 3 = M 2 ϕ/f, λ 4 = M 2 ϕ/f 2. OX axis: s in TeV 2. Rafael L. Delgado Unitarized EFT for a strongly interacting... 7 / 13

176 Tree level, modulus of t ω, K matrix All units in TeV. From top to bottom, f = 1.2, 0.8, 0.4 TeV Λ = 3 TeV µ = 100 GeV Rafael L. Delgado Unitarized EFT for a strongly interacting... 8 / 13

177 Im t ω in the N/D method, f = 1 TeV, β = 1, m = 150 GeV Rafael L. Delgado Unitarized EFT for a strongly interacting... 9 / 13

178 Re t ω and Im t ω, large N, f = 400 GeV Rafael L. Delgado Unitarized EFT for a strongly interacting / 13

179 Re t ω and Im t ω, large N, f = 4 TeV Rafael L. Delgado Unitarized EFT for a strongly interacting / 13