Formulating electroweak pion decays in functional methods and the - - PowerPoint PPT Presentation
Formulating electroweak pion decays in functional methods and the influence of CP-violation. Walid Ahmed Mian Advisor: Axel Maas and Jan M. Pawlowski June 20th, 2017 W. Ah. Mian EW pion decay in FM June 20th, 2017 1 / 31 Outline Motivation
Walid Ahmed Mian Advisor: Axel Maas and Jan M. Pawlowski June 20th, 2017
EW pion decay in FM June 20th, 2017 1 / 31
1
Motivation
2
Quark Propagator with broken Flavor and CP-Violation
3
Bethe-Salpeter Equation of the weak pion decay
4
Conclusion
EW pion decay in FM June 20th, 2017 2 / 31
System of binary neutron stars mergers Source of gravitational waves Possible outcome: neutron star or black hole Depends on neutrino backcoupling, magnetic field etc.
(Y. Sekiguchi et al. PRL 107 (2011), 051102
Foucart et al. arXiv:1510.06398v2 [astro-ph] Rosswog et al. arXiv:0302301v1 [astro-ph] . . .) (http://www.ligo.org/science/GW-Inspiral.php)
EW pion decay in FM June 20th, 2017 3 / 31
System of binary neutron stars mergers Source of gravitational waves Possible outcome: neutron star or black hole Depends on neutrino backcoupling, magnetic field etc.
(Y. Sekiguchi et al. PRL 107 (2011), 051102
Foucart et al. arXiv:1510.06398v2 [astro-ph] Rosswog et al. arXiv:0302301v1 [astro-ph] . . .) (http://www.ligo.org/science/GW-Inspiral.php)
EW pion decay in FM June 20th, 2017 3 / 31
System of binary neutron stars mergers Source of gravitational waves Possible outcome: neutron star or black hole Depends on neutrino backcoupling, magnetic field etc.
(Y. Sekiguchi et al. PRL 107 (2011), 051102
Foucart et al. arXiv:1510.06398v2 [astro-ph] Rosswog et al. arXiv:0302301v1 [astro-ph] . . .) (http://www.ligo.org/science/GW-Inspiral.php)
EW pion decay in FM June 20th, 2017 3 / 31
Micro physics influence gravitational waves Very high neutrino flux Super/Hyper-Kamiokande have good sensitivity Measurement shows the inner structure of the neutron star mergers
(Y. Sekiguchi et al. PRL 107 (2011), 051102
Foucart et al. arXiv:1510.06398v2 [astro-ph] Rosswog et al. arXiv:0302301v1 [astro-ph] . . .) (Foucart et al. arXiv:1510.06398v2 [astro-ph])
EW pion decay in FM June 20th, 2017 4 / 31
Very dense matter ⇒ opaque for neutrinos Reaction inside the core (Foucart et al. arXiv:1510.06398v2 [astro-ph]) νe + n ← → p + e− νe + p ← → n + e+ νe + νe ← → e+ + e− νe + νe ← → γ Electroweak interactions play an important role Consider QCD + electroweak interactions non-perturbative
EW pion decay in FM June 20th, 2017 5 / 31
Full resolution of electroweak interactions is complicated β-decay captures the main features Look at the π±-decay Electroweak interactions approximate by 4-Fermi-interaction Electroweak interactions violates parity No results on non-perturbative backcoupling of C and P violation First: Investigate the effects on the simplest object: Quark propagator Analyse influence through explicit breaking term (A. Maas & W. M., EPJA (2017) 53: 22 ,
arxiv:1611:08130) http://hyperphysics.phy- astr.gsu.edu/hbase/particles/proton.html
EW pion decay in FM June 20th, 2017 6 / 31
Full resolution of electroweak interactions is complicated β-decay captures the main features Look at the π±-decay Electroweak interactions approximate by 4-Fermi-interaction Electroweak interactions violates parity No results on non-perturbative backcoupling of C and P violation First: Investigate the effects on the simplest object: Quark propagator Analyse influence through explicit breaking term (A. Maas & W. M., EPJA (2017) 53: 22 ,
arxiv:1611:08130)
EW pion decay in FM June 20th, 2017 6 / 31
Full resolution of electroweak interactions is complicated β-decay captures the main features Look at the π±-decay Electroweak interactions approximate by 4-Fermi-interaction Electroweak interactions violates parity No results on non-perturbative backcoupling of C and P violation First: Investigate the effects on the simplest object: Quark propagator Analyse influence through explicit breaking term (A. Maas & W. M., EPJA (2017) 53: 22 ,
arxiv:1611:08130)
↓
EW pion decay in FM June 20th, 2017 6 / 31
Symmetry breaking ⇒ More involved tensor structure Pure QCD: P(p2) = ˜ A(p2) i / p + ˜ B(p2)1 1 Parity violation: P(p2) = ˜ A(p2) i / p + ˜ B(p2)1 1 + ˜ C(p2) i / pγ5 + ˜ D(p2)γ5 Flavor and parity violation: PAB(p2) = ˜ AAB(p2) i / p + ˜ BAB(p2)1 1 + ˜ CAB(p2) i / pγ5 + ˜ DAB(p2)γ5
EW pion decay in FM June 20th, 2017 7 / 31
Pure QCD: P−1(p2) = −A(p2) i / p + B(p2)1 1 P(p2) = ˜ A(p2) i / p + ˜ B(p2)1 1 ˜ A(p2) = A(p2) A2(p2)p2 + B2(p2) = Z(p2) p2 + M2(p2) ˜ B(p2) = B(p2) A2(p2)p2 + B2(p2) = M(p2) p2 + M2(p2) Wavefunctionrenormalization and Massfunction: Z(p2) = 1 A(p2) M(p2) = B(p2) A(p2)
EW pion decay in FM June 20th, 2017 8 / 31
Flavor and parity violation: P−1
AB(p2) = −AAB(p2) i /
p + BAB(p2)1 1 + CAB(p2) i / pγ5 + DAB(p2)γ5 PAB(p2) = ˜ AAB(p2) i / p + ˜ BAB(p2)1 1 + ˜ CAB(p2) i / pγ5 + ˜ DAB(p2)γ5 Complicated and very lengthy relation: ˜ AAB = ˜ AAB(ACD, BCD, CCD, DCD)
EW pion decay in FM June 20th, 2017 9 / 31
P0,uu(p2) = 1 N(p2)
d + (1 − 2g2 w)p2) i /
p + mu(m2
d + p2)1
1 + 2g2
wp2 i /
pγ5 P0,ud(p2) = gw N(p2)
p − (mu + md)p21 1 − (mumd + p2) i / pγ5 − gw(mu − md)p2 N(p2) γ5 N(p2) =m2
dm2 u + (m2 u + m2 d)p2 + (1 − 4g2 w)p4
Pseudo scalar channel of the mixed propagator (tree-level) is proportional to mass splitting
EW pion decay in FM June 20th, 2017 10 / 31
DSEs: Equation of motion for the correlation functions Pure QCD:
−1 = −1 +
P−1(p2) = P−1 +
(2π)4 gγνSνµ(q − p)P(q2)Γµ(p, q) Rainbow-Truncation gSνµ(q − p)Γµ(p, q) ∝ α((p − q)2)S0,νµ(q − p)γµ
EW pion decay in FM June 20th, 2017 11 / 31
DSEs: Equation of motion for the correlation functions Pure QCD:
−1 = −1 +
P−1(p2) = P−1 +
(2π)4 gγνSνµ(q − p)P(q2)Γµ(p, q) Rainbow-Truncation gSνµ(q − p)Γµ(p, q) ∝ α((p − q)2)S0,νµ(q − p)γµ
EW pion decay in FM June 20th, 2017 11 / 31
α(q2) = π ω6 Dq4 e− q2
ω2 +
2πγm[1 − exp (− q2
m2
t )]
ln[e2 −1 + (1 +
q2 Λ2
QCD )2]
ΛQCD =0.234 GeV ω =0.4 GeV D =0.93 GeV mt =1.0 GeV γm = 12 11Nc − 2Nf = 12 11 · 3 − 2 · 2
2 4 6 8 10 12 1e-06 1e-04 1e-02 1 100 10000 1e+06 1e+08 p2 [GeV2]
( P. Maris and P. C. Tandy, PRC 60, 055214 (1999))
EW pion decay in FM June 20th, 2017 12 / 31
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1e-06 0.0001 0.01 1 100 10000 1e+06 chiral up down strange charm bottom top
p2 [GeV2] Z(p2)
EW pion decay in FM June 20th, 2017 13 / 31
1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 1e-06 0.0001 0.01 1 100 10000 1e+06 chiral up down strange
p2 [GeV2] M(p2) [GeV]
EW pion decay in FM June 20th, 2017 14 / 31
−1
=
−1
+
Weak interaction: Non-vanishing diagonal elements By inversion: Quark propagators of different flavor influence each
EW pion decay in FM June 20th, 2017 15 / 31
0.2 10-6 10-4 10-2 1 102 104 106 108 C ~(p2) [1/GeV2] p2 [GeV2] chiral gw=0 gw=0.01 gw=0.1 gw=0.2 gw=0.3 gw=0.4
EW pion decay in FM June 20th, 2017 16 / 31
Positive in UV Negative in IR Existence of a transition scale
1x10-10 10-6 10-4 10-2 1 102 104 106 108 C ~(p2) [1/GeV2] p2 [GeV2] mu=2.3MeV and md=4.8MeV, gw=10-5 up down
EW pion decay in FM June 20th, 2017 17 / 31
Same behavior as in the chiral limit
1x10-8 2x10-8 10-6 10-4 10-2 1 102 104 106 108 C ~(p2) [1/GeV2] p2 [GeV2] mu=2.3MeV and md=4.8MeV, gw=5x10-5 up down
EW pion decay in FM June 20th, 2017 18 / 31
Existence of a threshold strength: Qualitative change
2x10-9 10-6 10-4 10-2 1 102 104 106 108 C ~(p2) [1/GeV2] p2 [GeV2] mu=md=2.3MeV up, gw=10-5 down, gw=10-5 up, gw=5x10-5 down, gw=5x10-5
EW pion decay in FM June 20th, 2017 19 / 31
Change of behavior due to mass splitting
2x10-15 10-6 10-4 10-2 1 102 104 106 108 C ~(p2) [1/GeV2] p2 [GeV2] mt=160GeV and mb=4.18GeV, gw=10-6 top bottom
5x10-18 1x10-17 1.5x10-17 10-6 10-4 10-2 1 102 104 106 108 C ~(p2) [1/GeV2] p2 [GeV2] mt=mb=160GeV, gw=10-6 top bottom
Threshold value for gw is decreased for bigger mass splitting Transition scale is shifted to higher value
EW pion decay in FM June 20th, 2017 20 / 31
Contributions from quark propagators to left ˜ L and right handed ˜ R: ˜ LAB = ˜ AAB − ˜ CAB
1 − γ5)
RAB = ˜ AAB + ˜ CAB
1 + γ5)
r: relative ratio for left handed to right handed contributions ˜ rAB(p2) = ˜ LAB(p2) − ˜ RAB(p2) ˜ LAB(p2) + ˜ RAB(p2) = − ˜ CAB(p2) ˜ AAB(p2) More left handed or right handed contributions related to sign of ˜ C for pure flavor quark propagators ( ˜ A always positive)
EW pion decay in FM June 20th, 2017 21 / 31
2x10-9 4x10-9 6x10-9 8x10-9 1x10-8 1.2x10-8 1.4x10-8 10-6 10-4 10-2 1 102 104 106 108 r ~(p2) p2 [GeV2] mu=2.3MeV and md=4.8MeV up, gw=10-5 down, gw=10-5 up, gw=5x10-5 down, gw=5x10-5
1x10-9 2x10-9 3x10-9 4x10-9 5x10-9 10-6 10-4 10-2 1 102 104 106 108 r ~(p2) p2 [GeV2] mu=md=2.3MeV up, gw=10-5 down, gw=10-5 up, gw=5x10-5 down, gw=5x10-5
Below the threshold strength
1
Dominantly right handed in the UV
2
Dominantly left handed in the IR
Above the threshold strength
1
No qualitative change for up quark
2
Change for down quark: Dominantly right handed in UV and IR
Absolute value for the ratio is increased due to mass splitting
EW pion decay in FM June 20th, 2017 22 / 31
10-10 10-8 10-6 10-4 10-2 1 10-6 10-4 10-2 1 102 104 106 108 |r ~(p2)| p2 [GeV2] up gw=10-5 gw=10-4 gw=10-3 gw=10-2 gw=0.1 gw=0.2 gw=0.3 10-10 10-8 10-6 10-4 10-2 1 10-6 10-4 10-2 1 102 104 106 108 |r ~(p2)| p2 [GeV2] down gw=10-5 gw=10-4 gw=10-3 gw=10-2 gw=0.1 gw=0.2 gw=0.3
Absolute value is increased by two order of magnitude, when gw is increased by one order of magnitude
EW pion decay in FM June 20th, 2017 23 / 31
0.5 1 10-6 10-4 10-2 1 102 104 106 108 r ~
ud(p2)
p2 [GeV2] up-down gw=10-4 gw=10-3 gw=10-2 gw=0.1 gw=0.2 gw=0.3 gw=0.4
0.5 1 10-6 10-4 10-2 1 102 104 106 108 r ~
ud(p2)
p2 [GeV2] up-down, degenerate masses gw=10-4 gw=10-3 gw=10-2 gw=0.1 gw=0.2 gw=0.3 gw=0.4
Propagator from up to down
1
UV: Dominantly right handed
2
IR: Dominantly left handed
No qualitative change due to mass splitting Quantitative shifting towards right handed for mass splitting
EW pion decay in FM June 20th, 2017 24 / 31
0.5 1 10-6 10-4 10-2 1 102 104 106 108 r ~
ud(p2)
p2 [GeV2] top-bottom gw=10-4 gw=10-3 gw=10-2 gw=0.1 gw=0.2 gw=0.3 gw=0.4
0.5 1 10-6 10-4 10-2 1 102 104 106 108 r ~
ud(p2)
p2 [GeV2] top-bottom, degenerate masses gw=10-4 gw=10-3 gw=10-2 gw=0.1 gw=0.2 gw=0.3 gw=0.4
UV and IR dominant contribution from right handed quarks Qualitative change due to mass splitting (from left handed to right handed in the IR)
EW pion decay in FM June 20th, 2017 25 / 31
To obtain masses: Quark propagator needed in Minkowski space, but too complicated Other possibility: Schwinger function (Alkofer et. al., Phys. Rev. D70, 014014 (2004)) ∆AB(t) = 1 π ∞ d p4 cos(tp4)σAB(p2
4).
Real pole m ∆(t) ∼ e−mt Complex conjugate poles m = a ± i b ∆(t) ∼ e−at cos(bt + δ).
EW pion decay in FM June 20th, 2017 26 / 31
10-8 10-6 10-4 10-2 1 5 10 15 20 25 30 |∆uu(t)| t [1/GeV] chiral 0.01 0.1 0.2 0.3 0.4 10-12 10-10 10-8 10-6 10-4 10-2 1 5 10 15 20 25 30 |∆ud(t)| t [1/GeV] chiral 10-6 10-5 10-4 10-3 10-2 0.1 0.2 0.3 0.4
Complex conjugate poles Shifts closer to the origin
EW pion decay in FM June 20th, 2017 27 / 31
BSEs: Bound state equations derived from DSEs and evaluated
Total momenta P = p1 − p2 At the pole MPole Γ(4) ∝ N ΨΨ P2 + M2
Pole
Ψ: Bethe-Salpeter-Amplitude Ψ = Γ(3)
EW pion decay in FM June 20th, 2017 28 / 31
u u u u u u u u u u d d d d d d d d d d Γ(4) Γ(4) Γ(4) Γ(4) = + + +
EW pion decay in FM June 20th, 2017 29 / 31
π d u = π d u u d + π d u ν e π e ν = π e ν u d Ψ = KΨ Solve the system self-consistent P2 = −M2
Pole: K eigenvalue 1
Resonances: Pole in the 2nd Riemann sheet (Haag, Local Quantum Physics Fields,
Particles, Algebras)
EW pion decay in FM June 20th, 2017 30 / 31
Consider QCD and electroweak interactions non-perturbative Goal: β-decay in neutron stars First step: Dynamical weak Pion decay Influence of broken C, P and flavor symmetry at the level of the quark propagator can be studied through explicit breaking term
1
Non-perturbative: Amplification of the backcoupling
2
Caution with perturbative extrapolation
Self-consistent backcoupling at the level of the Pion
EW pion decay in FM June 20th, 2017 31 / 31
Consider QCD and electroweak interactions non-perturbative Goal: β-decay in neutron stars First step: Dynamical weak Pion decay Influence of broken C, P and flavor symmetry at the level of the quark propagator can be studied through explicit breaking term
1
Non-perturbative: Amplification of the backcoupling
2
Caution with perturbative extrapolation
Self-consistent backcoupling at the level of the Pion
EW pion decay in FM June 20th, 2017 31 / 31