Radiative effects in decay of metastable vacua: a Green s function approach

Transcription

1 Radiative effects in decay of metastable vacua: a Green s function approach Peter Millington University Foundation Fellow, Technische Universität München p.w.millington@tum.de in collaboration with Björn Garbrecht arxiv: [hep-th]; accepted by PRD Kosmologietag 10 ZiF, Bielefeld University 8 May 2015

2 Introduction and motivation Numerous phenomenological examples across high-energy physics, astro-particle physics and cosmology: symmetry restoration at finite temperature and phase transitions in the early Universe [Kirzhnits, Linde, PLB42 (1972) 471; Dolan, Jackiw, PRD9 (1974) 3320; Weinberg, PRD9 (1974) 3357] first-order phase transitions may produce relic gravitational waves [Witten, PRD30 (1984) 272; Kosowsky, Turner, Watkins, PRD45 (1992) 4514; Caprini, Durrer, Konstandin, Servant, PRD79 (2009) ] potential role in the generation of the baryon asymmetry of the Universe [Morrissey, Ramsey-Musolf, New J.Phys.14 (2012) ; Chung, Long, Wang, PRD87 (2013) ] dynamics of both topological and non-topological defects, and other non-perturbative phenomena in non-linear field theories, e.g. domain walls, Q balls, oscillons, etc... vacuum stability of the SM and its extensions...

3 Introduction and motivation The perturbatively-calculated SM effective potential develops an instability at a scale GeV, given a 125 GeV Higgs boson. [Cabibbo, Maiani, Parisi, Petronzio, NPB158 (1979) 295; Sher, PR179 (1989) 273 & PLB317 (1993) 159; Isidori, Ridolfi, Strumia, NPB609 (2001) 387; Elias-Moro, Espinosa, Giudice, Isodori, Riotto, Strumia, PLB709 (2012) 222; Degrassi, Di Vita, Elias-Moro, Espinosa, Giudice, Isidori, Strumia, JHEP1208 (2012) 098; Alekhin, Djouadi, Moch, PLB716 (2012) 214;... ] [Buttazzo, Degrassi, Giardino, Giudice, Sala, Salvio, Strumia, JHEP1312 (2013) 089]

4 Introduction and motivation These predictions for the SM are subject to a number of uncertainties... experimental: determination of the top-quark pole mass [Bezrukov, Kalmykov, Kniehl, Shaposhnikov, JHEP1210 (2012) 140 [ ]; Masina, PRD87 (2013) ] phenomenological: impact of higher-dimension operators [Branchina, Messina, PRL111 (2013) ; Branchina, Messina, Platania, JHEP1409 (2014) 182 [ ]; Lalak, Lewicki, Olszewski, JHEP1405 (2014) 119; Branchina, Messina, Sher, PRD91 (2015) ; Eichhorn, Gies, Jaeckel, Plehn, Scherer, Sondenheimer, ]] theoretical: interpretation of the non-convexity of the effective potential [Weinberg, Wu, PRD36 (1987) 2474; Alexandre, Farakos, JPA41 (2008) ; Branchina, Faivre, Pangon, JPG36 (2009) ; Einhorn, Jones, JHEP0704 (2007) 051] implementation of RG improvement [Gies, Sondenheimer, EPJC75 (2015) 68] incorporation of the inhomogeneity of the solitonic background [Garbrecht, Millington, , cf. Goldstone, Jackiw, PRD11 (1975) 1486; e.g. of calculations in the homogeneous background, see Frampton, PRL37 (1976) 1378; PRD15 (1977) 2922; Camargo-Molina, O Leary, Porod, Staub, EPJC73 (2013) 2588] It is necessary to consider methods of finding tunnelling rates that account fully for radiative corrections within the inhomogeneous soliton background, particular if these radiative effects are dominant.

5 Semi-classical tunnelling rate Archetype: Euclidean φ 4 theory with tachyonic mass µ 2 > 0 L = 1 2! ( µφ ) 2 1 2! µ2 φ ! gφ ! λφ4 + U 0 [for self-consistent numerical studies, see Bergner, Bettencourt, PRD69 (2004) ; Bergner, Bettencourt, PRD69 (2004) ; Baacke, Kevlishvili, PRD71 (2005) ; Baacke, Kevlishvili, PRD75 (2007) ] Non-degenerate minima: ϕ = v ± ± v 3g 2λ, v2 = 6µ2 λ U(ϕ) v + v ϕ

6 Semi-classical tunnelling rate The classical equation of motion 2 ϕ + U (ϕ) = 0 is analogous to a particle moving in a potential U (ϕ). There exists a solution the Coleman bounce satisfying [Coleman, PRD15 (1977) 2929; Callan, Coleman, PRD16 (1977) 1762; Coleman Subnucl. Ser. 15 (1979) 805; Konoplich, Theor. Math. Phys. 73 (1987) 1286] ϕ x 4 ± = + v, ϕ x 4 = 0 = 0, ϕ x = + v U(ϕ) v + v ϕ

7 Semi-classical tunnelling rate In hyperspherical coordinates, the boundary conditions are ϕ r = + v, dϕ/dr r = 0 = 0, with the bounce corresponding to the kink [Dashen, Hasslacher & Neveu, PRD 10 (1974) 4114; ibid. 4130; ibid. 4138] ϕ(r) = v tanh γ(r R), γ = µ/ 2. ϕ(r) true vacuum + v v false vacuum γ(r R) The bounce looks like a bubble of radius R = 12λ/g/v.

8 Semi-classical tunnelling rate The tunnelling rate Γ is calculated from the path integral Z[0] = [dφ]e S[φ]/, Γ/V = 2 Im Z[0] /V /T [see Callan, Coleman, PRD16 (1977) 1762] Expanding around the kink φ = ϕ + 1/2 ˆφ, the spectrum of the operator G 1 (ϕ) δ2 S[φ] = (4) + U (ϕ) δφ 2 φ = ϕ contains four zero eigenvalues (translational invariance of the bounce) and one negative eigenvalue (dilatations of the bounce). Writing B S[ϕ]: Z[0] = i λ 0 det (5) G 1 (ϕ) 2 e B/ (VT) 2( B 4(4γ2 2π ) ) 5 det (5) G 1 (v) 1/2

9 Quantum-corrected bounce Leading quantum corrections to both the bounce and the tunneling rate can be calculated from the 1PI effective action Γ[φ] = ln Z[J] + d 4 x J(x)φ(x) [Jackiw, PRD9 (1974) 1686] But in anticipation of the non-positive-definite eigenspectrum of fluctuations, we want to evaluate the functional integral in Z[J] by expanding around the quantum-corrected bounce ϕ (1), i.e. δγ[φ] = 0 δφ(x) φ=ϕ (1) δs[φ] = J(x) 0 δφ(x) φ=ϕ (1) Note, in the usual standard evaluation of the 1PI effective action, the physical limit is J(x) 0.

10 Quantum-corrected bounce At order, the quantum-corrected bounce ϕ (1) (x) satisfies 2 ϕ (1) (x) + U (ϕ (1) ; x) + Π(ϕ; x)ϕ(x) = 0 including the tadpole correction Π(ϕ; x) = λ G(ϕ; x, x) 2 It follows that the self-consistent choice of J(x) for this method of evaluation is J(x) = Π(ϕ; x)ϕ(x) [see Gabrecht, Millington, ] Expanding around ϕ (1) = φ δϕ, δϕ = ϕ (1) ϕ, we find Γ[ϕ (1) ] = S[ϕ (1) ] + iπ B (2) [ϕ] + 2 ln λ 0 det (5) G 1 (ϕ) (VT) 2( B 4(4γ2 2π ) ) 5 det (5) G 1 (v) 1/2

11 Quantum-corrected bounce The tunnelling rate per unit volume, at order 2, is ( ) 2 [ )] B Γ/V = (2γ) 5 λ exp 1 (B + B (1) + 2 B (2) + 2 B (2) 2π Expanded in δϕ(x) ϕ (1) (x) ϕ(x): one-loop corrections captured by the functional determinant: B (1) = 1 ( ) 2 tr(5) ln G 1 (ϕ) ln G 1 (v) two-loop corrections (i) B (2) due to the action of the corrected bounce and (ii) B (2) due to the corrections to the one-loop fluctuation determinant: B (2) = 1 d 4 x ϕ(x)π(ϕ; x)δϕ(x) = 2B (2) 2

12 Approximations 1. Thin-wall approximation µr 1: drop the damping term 1 µr d dr 1 µr d dr. 2. Planar-wall approximation: sum over discrete angular momenta replaced by an integral over linear momenta, i.e. j(j + 2) µ 2 R 2 k2 µ 2. z z R

13 Green s function Green s function: u ( ) ϕ(r ( ) )/v and m ( 1 + k 2 /4/γ 2) 1/2 [ ( ) m( G(u, u, m) = 1 ϑ(u u 1 u u )m 2 ) 2γm 1 + u 1 u ( ) (1 u)(1 + m + u) 1 3 (1 + m)(2 + m) ( 1 3 (1 u )(1 m + u ) (1 m)(2 m) ) + (u u ) ] The renormalization is performed using the effective potential of a homogeneous false-vacuum configuration. Renormalized tadpole correction: Π R (u) = 3λγ2 16π 2 [ 6 + ( 1 u 2)( 5 π 3 u 2 )]

14 Heat-kernel method [Diakonov, Petrov and Yung, PLB130 (1983) 385; Diakonov, Petrov, Yung, Sov.J.Nucl.Phys.39 (1984) 150 [Yad.Fiz.39 (1984) 240]; Konoplich, Theor.Math.Phys.73 (1987) 1286 [Teor.Mat.Fiz.73 (1987) 379; Vassilevich, Phys. Rept. 388 (2003) 279; Carson, McLerran, PRD41 (1990) 647; Carson, Li, McLerran, Wang, PRD42 (1990) 2127; Carson, PRD42 (1990) 2853] Functional determinant over the positive-definite modes: tr (5) ln G 1 (ϕ; x) = d 4 dτ x K(ϕ; x, x τ). τ The heat kernel is the solution to the heat-flow equation with K(ϕ; x, x 0) = δ (4) (x x ). But it s Laplace transform τ K(ϕ; x, x τ) = G 1 (ϕ; x)k(ϕ; x, x τ), 0 K(ϕ; x, x s) = 0 dτ e sτ K(ϕ; x, x τ) is just the Green s function with k 2 k 2 + s. [cf. direct integration of the Green s function: Baacke, Junker, MPLA8 (1993) 2869; PRD49 (1994) 2055; 50 (1994) 4227; Baacke, Daiber, PRD51 (1995) 795; Baacke, PRD78 (2008) ; Gel fand-yaglom thoerem JMP1 (1960) 48: Baacke, Kiselev PRD48 (1993); Dunne, Kirsten, JPA39 (2006) 11915; Dunne, JPA41 (2008) ]

15 Example To enhance the radiative effects, while remaining in a perturbative regime, we consider an N -field model: [see t Hooft, NPB72 (1974) 461] L N i = 1 [ ] 1 ( ) 2 1 µχ i + 2! 2! m2 χχ 2 i + λ 4 φ2 χ 2 i. For m 2 χ γ 2, the χ renormalized tadpole correction is Σ R (u) = λγ2 γ 2 [ ( )( u u 2)]. 8π 2 mχ 2 Dominant and 2 corrections:

16 Quantum-corrected bounce δϕ(u) = v γ du u G(u, u, m) ( ) k = 0 Π R (u ) + N Σ R (u ) 1 u 1 2 δφ 16π 2 /(3λ v) γ(r-r) N γ 2 /m 2 χ: 0 (solid), 0.5 (dashed), 1 (dash-dotted) and 1.5 (dotted)

17 Quantum-corrected bounce 100 (φ+δφ) 16π 2 /(3λ v) γ(r-r) N γ 2 /m 2 χ: 0 (solid), 0.5 (dashed), 1 (dash-dotted) and 1.5 (dotted) [Qualitative agreement with Bergner, Bettencourt, PRD69 (2004) ] reduction in bounce action increase in tunneling rate.

18 Concluding remarks Green s function approach to the calculation of radiative corrections to tunnelling rates from false vacua, including the functional determinant and corrections to the bounce itself. In the scalar toy model considered, this calculation could be performed analytically, by employing the thin-wall approximation and planar-wall approximation. The method is well-suited to numerical analysis and may be of particular use for radiatively-generated minima, e.g. for the instability of the SM, massless Coleman-Weinberg model of SSB or symmetry restoration at finite temperature. [Coleman & Weinberg, PRD 7 (1973) 1888]

19 Back-up slides Tunnelling rate B (1) ( 3λ = B B = 8π2 R 3 γ 3 λ 16π 2 )[ π γ 2 mχ 2 ( ) d 4 x ϕ(u) Π R (u) + N Σ R (u) δϕ(u) B (2) + B (2) = 1 2 ( ) = B 2 [ 3λ π ( π + 5 π N ] ) π γ 2 N mχ 2 14 ] γ 4 N 2 mχ 4