Beyond Perturbative QFT. 6 9 April 2017 Hellenic Particle Physics Society, Ioannina, Greece

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1 Beyond Perturbative QFT Apostolos Pilaftsis School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom 6 9 April 2017 Hellenic Particle Physics Society, Ioannina, reece Based on A.P. and D. Teresi, Nucl. Phys. B 874 (2013) 594; Nucl. Phys. B906 (2016) 381 & arxiv: Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

2 Motivation: New Era of High Precision QFT Improved Lattice Techniques Higher Order Loop Calculations in Perturbative 1PI QFT Improved Solutions to Schwinger Dyson Equations On-Shell Amplitude Techniques eometric S-Matrix Approach in N = 4 SYM Theories. Other Analytical Non-Perturbative Approaches?

3 Outline: The Standard Theory of Electroweak Symmetry Breaking: SM The 2PI Effective Action and Field-Theoretic Problems Symmetry-Improved 2PI (SI2PI) Formalism SI2PI Effective Potential Exact R Invariance of SI2PI Effective Potential R Improved SI2PI Effective Potential Conclusions and Future Directions

4 The Standard Theory of Electroweak Symmetry Breaking Higgs Mechanism in SM: SU(3) c SU(2) L U(1) Y SU(3) c U(1) em [P. W. Higgs 64; F. Englert, R. Brout 64;. S. uralnik, C. R. Hagen, T. W. B. Kibble 64] V(φ) φ φ Higgs potential V(φ) V(φ) = m 2 φ φ + λ(φ φ) 2. round state: ( ) m 2 0 φ = 2λ 1 carries weak charge, but no electric charge and colour.

5 The Standard Theory of Electroweak Symmetry Breaking Higgs Mechanism in SM: SU(3) c SU(2) L U(1) Y SU(3) c U(1) em [P. W. Higgs 64; F. Englert, R. Brout 64;. S. uralnik, C. R. Hagen, T. W. B. Kibble 64] V(φ) φ φ Higgs potential V(φ) V(φ) = m 2 φ φ + λ(φ φ) 2. round state: ( ) m 2 0 φ = 2λ 1 carries weak charge, but no electric charge and colour. Custodial Symmetry of the SM with g = Y f = 0 and V(φ): [e.g., P. Sikivie, L. Susskind, M. B. Voloshin, V. I. Zakharov 80.] Φ ( φ, iσ 2 φ ) Φ U L ΦU C, with U L SU(2) L, U C SU(2) C, and SU(2) L SU(2) C /Z 2 SO(4)

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7 The Coleman-Weinberg Effective Potential [S. Coleman and E. Weinberg 73; based on J. oldstone, A. Salam, S. Weinberg 62,. Jona-Lasinio, 64.] Loopwise Expansion of the Scalar Potential: V eff (φ) = V(φ) i d 4 ( k k 2 2 (2π) 4 ln V ) (φ) k 2 The Coleman Weinberg Effective Potential is based on the One-Particle-Irreducible (1PI) formalism + O( 2 ) breaks Classical Scale Symmetries = Dimensional Transmutation

8 The Coleman-Weinberg Effective Potential [S. Coleman and E. Weinberg 73; based on J. oldstone, A. Salam, S. Weinberg 62,. Jona-Lasinio, 64.] Loopwise Expansion of the Scalar Potential: V eff (φ) = V(φ) i d 4 ( k k 2 2 (2π) 4 ln V ) (φ) k 2 + O( 2 ) The Coleman Weinberg Effective Potential is based on the One-Particle-Irreducible (1PI) formalism breaks Classical Scale Symmetries = Dimensional Transmutation

9 The SM Coleman-Weinberg Effective Potential at NNLO [. Degrassi, S. Di Vita, J. Elias-Miro, J.R. Espinosa,.F. iudice,. Isidori, A. Strumia, JHEP1208 (2012) 098; F. Bezrukov, M. Y. Kalmykov, B.A. Kniehl, M. Shaposhnikov, JHEP1210 (2012) 140.] Higgs Potential versus Variations in Top Mass M t by 0.1 MeV [Analysis includes the Multi-Critical scenario: D.L. Bennett, H.B. Nielsen, IJMA9 (1994) 5155]

10 Metastability of the SM Vacuum [D. Buttazzo,. Degrassi, P.P. iardino,.f. iudice, F. Sala, A. Salvio, and A. Strumia, JHEP1312 (2013) 089] But, Planckian physics may modify stability predictions by many orders! [V. Branchina, E. Messina, PRL111 (2013) ]

11 The 2PI Effective Action and Field-Theoretic Problems [J.M. Cornwall, R. Jackiw, E. Tomboulis, PRD10 (1974) 2428] Connected enerating Functional of 2PI Effective Action: W[J,K] = i ln Dφ i exp [ ( i S[φ] + Jxφ i i x + 1 )] 2 Kij xyφ i xφ j y, where S[φ] = x L[φ] is the classical action of a O(N) theory.

12 The 2PI Effective Action and Field-Theoretic Problems [J.M. Cornwall, R. Jackiw, E. Tomboulis, PRD10 (1974) 2428] Connected enerating Functional of 2PI Effective Action: W[J,K] = i ln Dφ i exp [ ( i S[φ] + Jxφ i i x + 1 )] 2 Kij xyφ i xφ j y, where S[φ] = x L[φ] is the classical action of a O(N) theory. Legendre transform of W[J,K] with respect to J and K: δw[j,k] δj i x φ i x, δw[j,k] δk ij xy = 1 2 ( i ij xy + φ i xφ j y), to get the 2PI effective action Γ[φ, ] = W[J,K] J i xφ i x 1 2 Kij xy ( i ij xy + φ i xφ j y).

13 The 2PI Effective Action: Γ[φ, ] = S[φ] + i 2 Tr ln 1 + i 2 Tr( 0 1 ) iγ (2) 2PI [φ, ], where Γ (2) 2PI [φ, ] =

14 The 2PI Effective Action: Γ[φ, ] = S[φ] + i 2 Tr ln 1 + i 2 Tr( 0 1 ) iγ (2) 2PI [φ, ], where Γ (2) 2PI [φ, ] = Equations of Motion: δγ[φ, ] δφ δγ[φ, ] δ = 0 = 0 1 = Hartree-Fock: =

15 Artist s impression of the infinite HF-term! Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

16 Field-Theoretic Problems addressed with 2PI Systematic formal resummation of high-order graphs: Rigorous Derivation of Schwinger Dyson Equations Thermal Masses in the high-t Regime Finite-Width Effects within Quantum Loops The IR Problem of the Coleman-Weinberg Effective Potential Non-Equilibrium QFT, through Kadanoff Baym equations. [For instance, P. Millington, AP, PRD88 (2013) ]

17 Field-Theoretic Problems addressed with 2PI Systematic formal resummation of high-order graphs: Rigorous Derivation of Schwinger Dyson Equations Thermal Masses in the high-t Regime Finite-Width Effects within Quantum Loops The IR Problem of the Coleman-Weinberg Effective Potential Non-Equilibrium QFT, through Kadanoff Baym equations. [For instance, P. Millington, AP, PRD88 (2013) ] BUT Truncations of 2PI lead to residual violations of symmetries, e.g. global or local symmetries. Erroneous First-Order Phase Transition in O(N) Theories oldstone Bosons become Massive Erroneous Thresholds for the Resummed Higgs-Boson Propagator.

18 QCD Phase Transition in the Standard 2PI Formalism [N. Petropoulos, J. Phys. 25 (1999) 2225] Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

19 Pertinent Literature to the oldstone-symmetry Problem. Baym,. rinstein, Phys. Rev. D 15 (1977) Amelino-Camelia, Phys. Lett. B 407 (1997) 268. N. Petropoulos, J. Phys. 25 (1999) Y. Nemoto, K. Naito, M. Oka, Eur. Phys. J. A 9 (2000) 245. J. T. Lenaghan, D. H. Rischke, J. Phys. 26 (2000) 431. H. van Hees, J. Knoll, Phys. Rev. D 66 (2002) J. Baacke, S. Michalski, Phys. Rev. D 67 (2003) Y. Ivanov, F. Riek, H. van Hees, J. Knoll, Phys. Rev. D 72 (2005) E. Seel, S. Struber, F. iacosa, D. H. Rischke, Phys. Rev. D 86 (2012) Markó, U. Reinosa, Z. Szép, Phys. Rev. D 87 (2013)

20 Symmetry-Improved 2PI Formalism [AP, D. Teresi, NPB874 (2013) 594] Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

21 Symmetry-Improved 2PI Formalism [AP, D. Teresi, NPB874 (2013) 594] Equivalence between 1PI and 2PI Effective Actions to All Orders: Γ 1PI [φ] = Γ[φ, (φ)], with δγ[φ, (φ)] δ = 0.

22 Symmetry-Improved 2PI Formalism [AP, D. Teresi, NPB874 (2013) 594] Equivalence between 1PI and 2PI Effective Actions to All Orders: Γ 1PI [φ] = Γ[φ, (φ)], with δγ[φ, (φ)] δ = 0. 1PI Ward Identity (e.g. for O(2)): δγ 1PI [φ] δφ i x T a ij φ j x = 0 = v x δ 2 Γ 1PI [φ] = δγ1pi [φ] δ y δ x δh 0.

23 Symmetry-Improved 2PI Formalism [AP, D. Teresi, NPB874 (2013) 594] Equivalence between 1PI and 2PI Effective Actions to All Orders: Γ 1PI [φ] = Γ[φ, (φ)], with δγ[φ, (φ)] δ = 0. 1PI Ward Identity (e.g. for O(2)): δγ 1PI [φ] δφ i x T a ij φ j x = 0 = v x δ 2 Γ 1PI [φ] = δγ1pi [φ] δ y δ x δh 0. Replace: δ 2 Γ 1PI [φ] = 1, xy, δ y δ x to obtain the Symmetry-Improved Equations of Motion: δγ[v, ] δ H/ = 0, v 1 (k = 0,v) = 0.

24 O(2) Hartree Fock Equations of Motion: HF Approximation: Γ (2) HF [ H, ] = + + H H Ansatz: 1 H/ (k) = k2 M 2 H/ + iε Equations of Motion: H MH 2 = 3λv 2 m 2 + (δλ A 1 +2δλ B 1 )v 2 δm (3λ+δλ A 2 +2δλ B 2 ) i H (k) + (λ+δλ A 2) k k i (k), M 2 = λv 2 m 2 + δλ A 1v 2 δm (λ+δλ A 2) i H (k) + (3λ+δλ A 2 +2δλ B 2 ) k k i (k), vm 2 = 0.

25 T-independent Resummed Counterterms in the HF approximation: [AP, D. Teresi, NPB874 (2013) 594; arxiv: ] δλ A 1 = δλ A 2 = LλA ǫ λ A (2 4LλB ǫ 1 2L(λA +2λ B ) ǫ ) + 4λ B (1 LλB ) ǫ + 4L2 λ B (λ A +λ B ) ǫ 2, δλ B 1 = δλ B 2 = LλB ǫ 2λ B 1 2LλB ǫ, δm 2 1 = Lm2 ǫ 2(λ A +λ B ) 1 2L(λA +λ B ) ǫ, with L /(16π 2 ), and λ A = λ A 1 = λ A 2 and λ B = λ B 1 = λ B 2.

26 Second-Order Phase Transition in the HF Approximation [AP, D. Teresi, NPB874 (2013) 594] ev v M H ev v ev M H 100 M H, M 50 M v T ev

27 Symmetry-Improved 2PI Effective Potential Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

28 Symmetry-Improved 2PI Effective Potential Symmetry-Improved Effective Potential Ṽeff(φ) from 1PI Ward Identity: φ 1 (k = 0;φ) = dṽeff(φ) dφ.

29 Symmetry-Improved 2PI Effective Potential Symmetry-Improved Effective Potential Ṽeff(φ) from 1PI Ward Identity: φ 1 (k = 0;φ) = dṽeff(φ) dφ. Solution: Ṽ eff (φ) = = φ 0 φ v dφφ 1 (k = 0;φ) + Ṽeff(φ = 0) dφφ 1 (k = 0;φ) + P(T,µ), where P(T, µ) is the thermodynamic pressure = hydrostatic pressure, i.e. it satisfies Baym s thermodynamic consistency. [. Baym, PR127 (1962) 1391] Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

30 SI2PI Effective Potential in the HF Approximation [AP, D. Teresi, NPB874 (2013) 594] Re V eff 10 4 ev T 430 ev T 420 ev T 410 ev Λ m 125 ev T 400 ev Im V eff 10 4 ev T 430 ev T 420 ev T 410 ev T 400 ev Λ m 125 ev Φ ev Φ ev Note: ImṼeff(φ) < 0, for 0 < φ < v = Vacuum instability for the concave part of the potential. [E.J. Weinberg, A.-Q. Wu, PRD36 (1987) 2474]

31 Equations of Motion including Sunset Diagrams: 1 H (p) = p2 (3λ+δλ A 1 +2δλ B 1 )v 2 + m 2 + δm 2 1 ( i H H H ), 1 (p) = p2 (λ+δλ A 1)v 2 + m 2 + δm 2 1 ( i v 1 (0) = 0. H H + + ), Symmetry-Improved 2PI Effective Potential Ṽeff(φ): φ 1 (k = 0;φ) = dṽeff(φ) dφ.

32 2PI Fractal Resummation Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

33 2PI Fractal Resummation Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

34 2PI Fractal Resummation Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

35 Higgs Selfenergy in 2PI [AP, D. Teresi, NPB874 (2013) 594] 150 s M H 2 ev 2 2 Re M H s H M H 2 Im M H s ev s ev U H* H U H* H* H H

36 oldstone Selfenergy in 2PI [AP, D. Teresi, NPB874 (2013) 594] s ev s H M H Re M Im M s ev s ev H U * H H U * H*

37 Exact R Invariance of SI2PI Effective Potential Finite R running of 2PI quartic couplings in the Sunset Approximation: µ d dµ λ = 2ǫλ, µ d dµ λa 1 = 2ǫλ A 1 + 4L(λ B 1 λ A 2 +λ A 1λ A 2 +λ A 1λ B 2 +2λ 2 ) 16L 2 λ 2 (3λ A 2 +λ B 2 ), µ d dµ λb 1 = 2ǫλ B 1 + 4L(λ B 1 λ B 2 +4λ 2 ) 32L 2 λ 2 λ B 2, µ d dµ λa 2 = 2ǫλ A 2 + 4Lλ A 2(λ A 2 +2λ B 2 ), µ d dµ λb 2 = 2ǫλ B 2 + 4L(λ B 2 ) 2, with L /(16π 2 ). 1PI R running: µ d dµ λ 1PI = 2ǫλ 1PI + 20Lλ 2 1PI +O(L2 ) Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

38 R-Running of 2PI Quartic Couplings in the Sunset Approximation [A.P., D. Teresi, arxiv: ] PI [ev]

39 4V( ) / Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis R-Invariant SI2PI Effective Potential in the Sunset Approximation [A.P., D. Teresi, arxiv: ] PI, = * PI, = SI2PI,all [ev]

40 R Improved SI2PI Effective Potential SI2PI Effective Potential: Ṽeff(φ,λ(µ = φ));µ = φ) φ = φ 1 (k = 0;φ). R-SI2PI Effective Potential: d dφṽeff(φ,λ(µ = φ);µ = φ) = Ṽeff + Ṽeff λ φ λ µ + Ṽeff µ=φ µ=φ µ=φ µ }{{ µ=φ } = 0 in the SI2PI Approximations through NNLO: Ṽeff λ V 1PI λ, Ṽeff µ V 1PI µ.

41 R-Improved SI2PI Effective Potential to NNLO Approximation [A.P., D. Teresi, arxiv: ]

42 Conclusions Preserving symmetries in loop-truncated 2PI actions is a longstanding problem Novel Approach to lobal Symmetries: = Symmetry-Improved 2PI Effective Action Massless oldstone Bosons 2nd-Order Phase Transition in the HF Approximation Smooth Thresholds consistent with Massless oldstone Bosons Consistent Resummation within Quantum Loops IR-Safe SI2PI Effective Potential [A.P., D. Teresi, Nucl. Phys. B906 (2016) 381.] The SI2PI Effective Potential is exactly R Invariant in the Sunset Approximation Reduced R Dependence of the R-SI2PI Effective Potential up to NNLO Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

43 Future Directions Extension: 2PI npi Effective Actions [3PI: M.J. Brown and I.B. Whittingham, PRD91 (2015) ] Extension to Local Symmetries: U(1), SU(N) Spontaneous Breaking of Local auge Symmetries Precision Computations of SM & BSM Effective Potentials.

44 Future Directions Extension: 2PI npi Effective Actions [3PI: M.J. Brown and I.B. Whittingham, PRD91 (2015) ] Extension to Local Symmetries: U(1), SU(N) Spontaneous Breaking of Local auge Symmetries Precision Computations of SM & BSM Effective Potentials. New Era of Analytical Non-Perturbative QFT

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46 Back-Up Slides Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

47 MS Renormalization in 2PI Naive Renormalization: = i (k) M 21 ǫ, Naive CT: δm2 =? M 2 1 ǫ. k But, M 2 = M 2 (T) = Temperature-dependent CT?!

48 MS Renormalization in 2PI Naive Renormalization: = i (k) M 21 ǫ, Naive CT: δm2 =? M 2 1 ǫ. k But, M 2 = M 2 (T) = Temperature-dependent CT?! What has gone wrong? [J.-P. Blaizot, E. Iancu, U. Reinosa, NPA736 (2004) 149; J. Berges et al, AP320 (2005) 344] δm 2 U U δλ Is there any systematic renormalization, e.g. in the MS scheme? [W.A. Bardeen, A. Buras, D. Duke, T. Muta, PRD18 (1978) 3998]

49 MS Renormalization in the 2PI Formalism Procedure: Isolate UV infinities in EoMs, e.g. by Dimensional Regularization. Require that the UV-finite part of EoMs be UV finite: (...) UV T fin H + (...) UV T fin + (...) UV v 2 + (...) UV 1! = 0 Cancel separately the UV infinities TH fin fin (T), T (T), v2 (T), 1. Check UV consistency: 4 2 = 8 Constraints, for 5 CTs : δm 2 1, δλ A 1, δλ B 1, δλ A 2, δλ B 2. This is a non-trivial check!

50 IR Divergences in the Coleman Weinberg Effective Potential Contributions to the SM effective potential: [S.P. Martin, PRD89 (2014) ] H m 2 log m 2 d dφ H log m 2 Π Π log m 2 d dφ Π Π 1 m 2 Π Π 1 m 2 d dφ Π Π ( 1 m 2 ) 2 Π Π m 2 = λφ 2 m 2 M 2 = m 2 +Π (1) k= = 0 at φ = v

51 IR Divergence in the Coleman Weinberg Potential 0 20 dv dφ ev Φ tree level 60 leading3 loop resummed Φ ev

52 SI2PI Approach to the oldstone-boson IR Problem [A.P., D. Teresi, Nucl. Phys. B906 (2016) 381.] More complete resummation: first-principle approach it takes into account the momentum-dependence of self-energy insertions more topologies no ad-hoc subtraction of m 2 log m2 contributions in Π [S.P. Martin, PRD90 (2014) ; J. Elias-Miro, J. Espinosa, T. Konstandin, JHEP1408 (2014) 034.] 1 φ dṽ eff dφ 1 (φ) k=0 + + [ ] 0 (φ) No IR divergences: the would-be divergent self-energies are 2PR

53 Numerical Estimates [A.P., D. Teresi, Nucl. Phys. B906 (2016) 381.] 1.0 scalar sector 1 loop 2 loop 0.8 leading 3 loop dv dφ ev resummed 2 loop SI2PI 1 Φ Φ ev

54 Quantum Effects from Chiral Fermions Γ (2) [φ, H,, + ] = Γ (2) scalar [φ, H,, + ] + 3i TrlnS α(0) [φ] i { t H + t 0 + t + } t t b 1 (k; φ) = (0) 1 (k; φ) [ ] (k;φ) (0) (k;φ) Ioannina 2017 Beyond Perturbative QFT A. Pilaftsis

55 Renormalization of Spurious Custodially Breaking Effects [A.P., D. Teresi, Nucl. Phys. B906 (2016) 381.] Φ j Φ j t R Q j t R Q j Φ i Φ j + i j Φ i i Φ j j λh 4 t (Φ Φ) 2 Q i t R Q i t R Φ i raphs contained in the 2PI resummation: Φ j Φ j Φ i t R Q j t R Q j Φ i + Φ i = CΦ i Φ j Φ i Φ j + CΦ i Φ j Φ i Φ j i Q i t R j i Q i t R j Cφ i j i j

56 raphs NOT contained in the 2PI resummation: Φ i Q i t R i j t R Φ j Φ i Q j Φ j + Q i t R Φ i Φ i j t R Q j Φ j Φ j = C Φ i Φ j Φ i Φ j i + C Φ i Φ j Φ iφ j Cφ j Φ j t R Q j Φ i Φ i Φ j + Φ j Φ i j t R Q j Φ i Φ j + (i j) i Q i t R i Q i t R = CΦ i Φ j Φ i Φ j + CΦ i Φ j Φ i Φ j + (i j) 2Cφ Renormalization condition for determining δλ cb : 1,0 (k = 0;v)! = 1,+ (k = 0;v)! = 0.

57 Numerical Estimates (with fermion effects included) [A.P., D. Teresi, Nucl. Phys. B906 (2016) 381.] 20 scalar fermion sector 15 dv dφ ev loop 2 loop 1 Φ leading 3 loop 5 resummed 2 loop SI2PI Φ ev

58 Scalar versus Fermion Contributions to Π (s) [A.P., D. Teresi, Nucl. Phys. B906 (2016) 381.]

59 Renormalization of Spurious Custodially Breaking Effects [A.P., D. Teresi, Nucl. Phys. B906 (2016) 381.] Φ j Φ j t R Q j t R Q j Φ i Φ j + i j Φ i i Φ j j λh 4 t (Φ Φ) 2 Q i t R Q i t R Φ i raphs contained in the 2PI resummation: Φ j Φ j Φ i t R Q j t R Q j Φ i + Φ i = CΦ i Φ j Φ i Φ j + CΦ i Φ j Φ i Φ j i Q i t R j i Q i t R j Cφ i j i j

60 raphs NOT contained in the 2PI resummation: Φ i Q i t R i j t R Φ j Φ i Q j Φ j + Q i t R Φ i Φ i j t R Q j Φ j Φ j = C Φ i Φ j Φ i Φ j i + C Φ i Φ j Φ iφ j Cφ j Φ j t R Q j Φ i Φ i Φ j + Φ j Φ i j t R Q j Φ i Φ j + (i j) i Q i t R i Q i t R = CΦ i Φ j Φ i Φ j + CΦ i Φ j Φ i Φ j + (i j) 2Cφ Renormalization condition for determining δλ cb : 1,0 (k = 0;v)! = 1,+ (k = 0;v)! = 0.