CTP Approach to Leptogenesis

CTP Approach to Leptogenesis Matti Herranen a in collaboration with Martin Beneke a Björn Garbrecht a Pedro Schwaller b Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University a Institut für Theoretische Physik, Universität Zürich b 5. Kosmologietag, Bielefeld, 06.05.2010 M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 1 / 22

Outline Introduction Standard Approach to Leptogenesis Closed Time Path (CTP) Approach to Leptogenesis Conclusions and outlook M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 2 / 22

Why baryogenesis? to explain the excess of matter over antimatter in the universe: [WMAP 2003] n B = ( 6.1 +0.3 ) 0.2 10 10 n γ Challenge: Inflation washes out any initial baryon number. Some dynamical mechanism = baryogenesis is needed M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 3 / 22

Leptogenesis [Fukugita, Yanagida (1986)] The Standard Model is extended by adding heavy right-handed Majorana neutrinos N i, i = 1, 2,... (eg. see-saw models) L h β (l βφ) e Rβ Y αk(l α φ ) N k 1 2 N jm j N c j + h.c. M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 4 / 22

Leptogenesis [Fukugita, Yanagida (1986)] The Standard Model is extended by adding heavy right-handed Majorana neutrinos N i, i = 1, 2,... (eg. see-saw models) L h β (l βφ) e Rβ Y αk(l α φ ) N k 1 2 N jm j N c j + h.c. Key element: out-of-equilibrium L- and CP-violating Yukawa decays of the heavy Majorana neutrinos ε 1 Γ(N1 lφ) Γ(N1 lφ ) Γ(N 1 lφ) + Γ(N 1 lφ ) 1 X M 1 Im ([Y Y] 2 1j) 8π M j [Y Y] 11 j M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 4 / 22

Baryogenesis via Leptogenesis The Sakharov conditions (1967) must be fulfilled: Baryon number B violation C violation CP violation Out of equilibrium M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 5 / 22

Baryogenesis via Leptogenesis The Sakharov conditions (1967) must be fulfilled: Baryon number B violation B L broken by neutrino Majorana masses B + L broken by the electroweak Sphalerons C violation Left-chiral electroweak interactions CP violation Yukawa interactions between the charged leptons l α and the Majorana neutrinos N i Out of equilibrium Expansion of the Universe - decreasing temperature T Heavy neutrino decays become out of equilibrium when M 1 T M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 5 / 22

2. Standard Approach to Leptogenesis M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 6 / 22

Boltzmann Equations for Leptogenesis (n l n l ) + 3H(n l n l ) t d 3 k = (2π) 3 2 d 3 p d 3 q k 2 + Mi 2 (2π) 3 2 p (2π) 3 2 q (2π)4 δ 4 (k p q) { M N1 lφ 2 f Ni (k) M N1 lφ 2 f Ni (k) } M lφ N1 2 f l (p)f φ (q) + M lφ N1 2 f l(p)f φ (q) d 3 k d 3 l d 3 p d 3 q + (2π) 3 2 k (2π) 3 2 l (2π) 3 2 p (2π) 3 2 q (2π)4 δ 4 (k + l p q) { } M lφ lφ 2 f l(k))f φ (l) M lφ lφ 2 f l (k))f φ (l) Expressed in terms of vacuum matrix elements M α β M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 7 / 22

Vacuum Matrix Elements and RIS Subtraction Naive multiplication suggests asymmetry even in equilibrium: Γ lφ lφ 1 + 2ε The on-shell intermediate states of 2 2 processes are already accounted for by the 1 2 Real Intermediate State subtraction is needed M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 8 / 22

4. CTP Approach to Leptogenesis 1 1 related / complementary aspects in: Buchmüller, Fredenhagen (2000); De Simone, Riotto (2007); Garny, Hohenegger, Kartavtsev, Lindner (2009 & 2010); Anisimov, Buchmüller, Drewes, Mendizabal (2010) M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 9 / 22

Closed Time Path (CTP) Formalism a.k.a. Schwinger-Keldysh Formalism [Schwinger (1961); Keldysh (1964); Calzetta, Hu (1988)] In the heuristic Boltzmann equations, matrix elements are computed within the usual in-out framework: out S in time-ordered correlators: T[ψ(x) ψ(y)] In leptogenesis we want to calculate expectation values, e.g. charge density: j 0 (x) = ψ(x)γ 0 ψ(x), in a finite density medium M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 10 / 22

Closed Time Path (CTP) Formalism a.k.a. Schwinger-Keldysh Formalism [Schwinger (1961); Keldysh (1964); Calzetta, Hu (1988)] In the heuristic Boltzmann equations, matrix elements are computed within the usual in-out framework: out S in time-ordered correlators: T[ψ(x) ψ(y)] In leptogenesis we want to calculate expectation values, e.g. charge density: j 0 (x) = ψ(x)γ 0 ψ(x), in a finite density medium In-In generating functional on a Closed Time Path: Im t Path-ordered correlators: Re t ig C (x, y) = T C [ψ(x) ψ(y)] M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 10 / 22

Four propagators with respect to real time variable: ig + (u, v) = ig < (u, v) = ψ(v)ψ(u) ig + (u, v) = ig > (u, v) = ψ(u) ψ(v) ig ++ (u, v) = ig T (u, v) = T(ψ(u) ψ(v)) ig (u, v) = ig T (u, v) = T(ψ(u) ψ(v)) Free propagators in momentum space: ig < (p) = 2πδ(p 2 m 2 )(p/ + m) ˆϑ(p 0)f (p) ϑ( p 0)(1 f ( p)) ig > (p) = 2πδ(p 2 m 2 )(p/ + m) ˆ ϑ(p 0)(1 f (p)) + ϑ( p 0) f ( p) ig T i(p/ + m) (p) = p 2 m 2 + iε 2πδ(p2 m 2 )(p/ + m) ˆϑ(p 0)f (p) + ϑ( p 0) f ( p) ig T i(p/ + m) (p) = p 2 m 2 iε 2πδ(p2 m 2 )(p/ + m) ˆϑ(p 0)f (p) + ϑ( p 0) f ( p) f (p) and f (p) are the distribution functions of particles and antiparticles M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 11 / 22

Kadanoff-Baym Equations Schwinger-Dyson equations for (CTP) propagators: G G 0 = + Σ The Kadanoff-Baym equations are the <, > components: (i / m)g <,> Σ H G <,> Σ <,> G H = 1 ( Σ > G < Σ < G >) 2 M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 12 / 22

Kadanoff-Baym Equations Schwinger-Dyson equations for (CTP) propagators: G G 0 = + Σ The Kadanoff-Baym equations are the <, > components: (i / m)g <,> Σ H G <,> Σ <,> G H = 1 ( Σ > G < Σ < G >) 2 Approximations: Wigner transformation: G(k, x) = d 4 r e ik r G(x + r 2, x r 2 ), and gradient expansion to the lowest order (also in finite width Γ) The propagators are reduced to the free theory form Kinetic equations describe the evolution of f (p, t) and f (p, t): d dt γ 0iG <,> (k) = [ iσ > (k)ig < (k) iσ < (k)ig > (k) ] M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 12 / 22

Tree-Level Collision Terms in (Unflavoured) Leptogenesis The leading order self energies: l: N: iσ ab l (k) = Y i 2 d 4 k (2π) 4 d 4 k (2π) 4 (2π)4 δ (4) (k k k )P R ig ab Ni(k )P L i ba φ ( k ) d iσ ab 4 k d 4 k { Nij(k) = g w (2π) 4 (2π) 4 (2π)4 δ (4) (k k k ) Y i Yj P L ig ab l (k )P R i ab φ (k ) + Y i Y j C [ P L ig ba l ( k )P R ] T C i ba φ ( k ) } M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 13 / 22

Kinetic Equation for Lepton Asymmetry d dt (n l n l ) = d dt = d 4 k (2π) 4 γ 0iG <,> (k) d 4 k [ iσ > (2π) 4 (k)ig < (k) iσ < (k)ig > (k) ] = Y i 2 d 3 k d 3 k (2π) 3 2 k 2 (2π) 3 2 d 3 k k 2 + Mi 2 (2π) 3 2 k { (2π) 4 δ 4 (k k k ) 2k k (1 f Ni (k )) [f l (k)f φ (k ) f l (k) f φ (k )] } f Ni (k ) [(1 f l (k))(1 + f φ (k )) (1 f l (k))(1 + f φ (k ))] Washout contribution M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 14 / 22

KMS Relations [Kubo (1957); Martin, Schwinger (1959)] A useful symmetry for correlators in thermal equilibrium: ig > l,n (p) = eβp 0 ig < l,n (p), i > φ (p) = eβp 0 i < φ (p) Hold for (thermal) self energies as well: iσ > l,n (p) = eβp 0 iσ < l,n (p), iπ> φ (p) = eβp 0 iπ < φ (p) KMS relations imply the vanishing of the collision term in thermal equilibrium: iσ > (k)ig < (k) iσ < (k)ig > (k) = 0 }{{} ( e βp 0 iσ > (p))( e βp 0 ig < (p)) Within the CTP formalism, we get the vanishing of the asymmetry in equilibrium for free M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 15 / 22

Wave-function Contribution Orange cut Interference between two s-channel scatterings Blue cut Interference between loop and tree-level decays Z d 3 k (2π) 3 Cwf l (k) = 4 Im[Y1 2 Y2 2 ] Z M 1M 2 M1 2 M2 2 d 3 k q (2π) 3 2 k 2 + M1 2 δf N(k ) ΣNµ(k )Σ µ N (k ) g w where the thermal decay rate is Z Σ µ d 3 p d 3 q N(k) = g w (2π) 3 2 p (2π) 3 2 q (2π)4 δ 4 (k p q) p µ 1 f eq eq l (p) + fφ (q) When neglecting quantum-statistical corrections (i.e. (1 f l,n ) 1 and (1 + f φ ) 1), we recover the usual RIS M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 16 / 22

Vertex Contribution Orange cut Interference between s- and t-channel scatterings Blue cut Interference between loop and tree-level decays Z d 3 p Z (2π) 3 Cv l(p ) = 4 Im[Y1 2 Y2 2 d 3 k ] (2π) 3 2 p δf N1(k) V(k, M 2), k 2 + M1 2 Z d 3 p d 3 p V(k, M 2) = (2π) 3 2 p (2π) 3 2 p (2π)4 δ 4 (k p p ) p µ Γ µ(k, p, M 2) ˆ1 f l (p ) + f φ (p ) Thermal vertex function: Z Γ µ(k, p d 3 k d 3 k, M 2) = (2π) 3 2 k (2π) 3 2 k (2π)4 δ 4 (k k k ) k µ M 1M 2 (k p ) 2 M2 2 ˆ1 f l (k ) + f φ (k ) M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 17 / 22

Numerical Results: Effective CP-Violating Parameter Consider limit M 1 M 2 : ε wf eff ε wf T=0 = 16π g w d k k 2 k 0 Σ µ N1 (k)σ N1 µ(k) δf N1 (k) d k k 2 k µ Σ µ N1 (k) δf N1(k) k 0 10 1 0.1 0.01 Ε eff Ε T 0 1 Expect large effect for weak washout small effect for strong washout 0.001 0.001 0.01 0.1 1 10 z M 1 T M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 18 / 22

Lepton Asymmetry: Y l = n l n l s, Strong Washout 10 8 10 9 Y l 10 10 10 11 0.1 1 10 z M 1 T blue: thermal initial f N1 solid: full solution red: zero initial f N1 dashed: no thermal corrections in loops M 1 = 10 13 GeV, M 2 = 10 15 GeV, Y 1 = 5 10 2, Y 2 = 10 1, Im[Y 1Y 2 ] = Y 1Y 2 M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 19 / 22

Lepton Asymmetry: Y l = n l n l s, Moderate Washout 10 8 10 9 Y l 10 10 10 11 0.1 1 10 z M 1 T blue: thermal initial f N1 solid: full solution red: zero initial f N1 dashed: no thermal corrections in loops M 1 = 10 13 GeV, M 2 = 10 15 GeV, Y 1 = 2 10 2, Y 2 = 10 1, Im[Y 1Y 2 ] = Y 1Y 2 M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 20 / 22

Lepton Asymmetry: Y l = n l n l s, Weak Washout 10 8 10 9 Y l 10 10 10 11 0.1 1 10 z M 1 T blue: thermal initial f N1 solid: full solution red: zero initial f N1 dashed: no thermal corrections in loops M 1 = 10 13 GeV, M 2 = 10 15 GeV, Y 1 = 1 10 2, Y 2 = 10 1, Im[Y 1Y 2 ] = Y 1Y 2 M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 21 / 22

Conclusion and outlook Conclusions: First principle description of Leptogenesis within the CTP framework Alternative - perhaps more satisfactory - manifestation of CPT invariance in the kinetic equation when compared to the usual RIS subtraction scheme Full inclusion of quantum statistical corrections in loops and external states (contribute at the same level) M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 22 / 22

Conclusion and outlook Conclusions: First principle description of Leptogenesis within the CTP framework Alternative - perhaps more satisfactory - manifestation of CPT invariance in the kinetic equation when compared to the usual RIS subtraction scheme Full inclusion of quantum statistical corrections in loops and external states (contribute at the same level) Outlook: Flavor coherence effects, finite width effects, resonant Leptogenesis M. Herranen (RWTH Aachen) Bielefeld, 06.05.2010 22 / 22