World-line construction of a covariant chiral kinetic theory Niklas Mueller with Raju Venugopalan, arXiv:1701.03331, 1702.01233; and with Raju Venugopalan and Yi Yin, in prep. Cold Quantum Coffee Heidelberg, 09.05.2017
Outline of this talk 1. Motivation for a chiral kinetic theory 2. Things that should make you suspicious 3. World line formulation of the effective action 4. Equations of motion and what Berry's phase really is 5. Towards the big picture 1
Motivation - What do heavy ion collisions tell us about fundamental symmetries? P- and CP- odd phenomena in QCD – anomalies and topology The Chiral Magnetic Effect local CP-odd domains + = Electric currents Axial Imbalance Large Magnetic Fields (observable) Kharzeev, McLerran, Warringa 2007 2
Motivation - Topological and anomalous phenomena 2009: initial excitement, charge correlations today: not so sure and signs as predicted (?) STAR; PRL 103 (2009) 251601; PRC 81 (2010) 54908 CMS, arxiv 1610.00263 Experimental verification very hard: - qualitative predictions do exist - quantitative understanding of signal or background do not 3
Motivation - Topological and anomalous phenomena Background: a difficult task - Signal and background have similar centrality dependence Y/<Y> max Prithwish Tribedy (STAR) at QM 2017 N part Paul Soerensen (STAR) at QM 2017 - Local charge conservation (Schlichting et al.) - …. Isobar run coming soon! (Chiral Magnetic Effect Task Force Report, arXiv:1605.01413 [hep-ph] |) But: THEORETICAL understanding needed! 4
Motivation - The limits of what we can calculate, and where the interesting physics lives hydrodynamic CGC over-occupied flux tubes regime kinetic regime colliding nuclei plasma T h e r ma l i z a t i o n ? ? ? S. Schlichting 2016 non-equilibrium anomalous fermion production from coherent fields (Tanji et al. 2016) and sphaleron transitions (Mace et al. 2016) large magnetic fields present ? Anomalous Transport (CME, CSE and CMW) Subsequent interactions in the fire ball, axial transport and relaxation weak to strong coupling classical statistical anomalous chiral kinetic theory simulations + fermions hydrodynamics 5
Motivation - We need a chiral kinetic theory Always a challenge, as natural to formulate in first quantization (point-particles), whereas most of our modern concepts (anomalies etc.) are in second quantization (fields) Berry connection and chiral kinetic theory Son and Yamamoto, Stephanov and Yin (2012) Weyl equation: Effective theory in the adiabatic limit , describing excitations near the fermion surface: Berry phase: Geometric only in the adiabatic limit Berry monopole: Claim: accounts for the dynamics of the anomaly by some continuity arguments (incompressibility of phase space) 6
Chiral Kinetic Theory - a 'hot' topic Big excitement across many communities! Son,Yamamoto, PRL109 (2012), 181602; PRD87 (2013) 085016, Stephanov, Yin, PRL109 (2012) 162001, Chen, Son, Stephanov, Yee, Yin, PRL 113 (2014) 182302, Chen, Son, Stephanov, PRL115 (2015) 021601, Chen,Pu,Wang,Wang, PRL110 (2013) 26230, Gao, Liang, Pu, Wang, Wang, PRL109 (2012) 232301, Stone, Dwivedi,Zhou, PRD91 (2015) 025004, Zahed, PRL109 (2012) 091603; Basar, Kharzeev,Zahed, PRL111 (2013)161601 Stephanov,Yi,Yin,PRD91 (2015) 125014, Manuel, Torres-Rincon, PRD 90, 076007 (2014) … 8+ PRLs in last 5 years! Relation of Berry phase and the anomaly? In Fujikawa's words ... and ... 7
Chiral Kinetic Theory - a closer look Relation of Berry phase and the anomaly? Things to consider ? adiabatic approximation ( absence of level crossin g) vs. level crossing interpretation of the anomaly (Fujikawa 2005) → robustness of the anomaly and approximations ? adiabatic limit and large chemical potential not applicable to ultra-relativistic heavy ion collisions ? 'classical' description of spinning particles → spin connection vs 'where does the anomaly really come from' ? Lorentz covariance? scattering kernels, side jumps etc. 8
World line formulation of the effective action some rather old stuff (Schwinger, Feynman) … re-discovered many times (Strassler) … and some pioneers of the modern interpretation from Heidelberg (Schmidt, Schubert)! ... quantum mechanical one-loop effective actions particle, quantized on closed loop (world line) 1 st ingredient: integral- representation of the logarithm Will show you: - anomalies from the - natural interpretation of path integral measure pseudo-classical approximation - Witten & Alvarez-Gaume (quasi-particles) (1983): fermionic determinant - Lorentz-covariant and anomalies - Gauge covariant → path integral representation of the index of the Dirac 9 operator
World line formulation WARNING! EQUATIONS! of the effective action Matter + Gauge fields: anomalies live Fermions (do not expect a complete derivation here!) here real part: 2 nd ingredient: representation of the trace over Gamma matrices via 10 fermionic coherent states
Equations of motion After many manipulations the real part of the effective action is written as a QM path integral: fermionic = no variables approximations! world-line (QFT!!!) The quasi-particle limit is very illustrative in this formulation … → the saddle point approximation gives: These are the covariant form of the famous Bargman-Michel-Telegdi and Wong's equations (when written down for QCD) ! So where is the anomaly? And where would Berry's phase be? 11
The origin of Berry's phase ... Adiabatic (and non-relativistic) approximation of the world-line path integral gives Berry's phase It is pretty clear what the role of Berry's phase is: → spin transport along the world-line of the particle → related to defining the 'moving frame' for the spin variables Most importantly: It is not robust! (Part of the dynamics away from adiabatic approx.) … we got it from the real part of the effective action … … the anomaly stems from the imaginary part (more soon) … what Berry's phase has to do with the anomaly: … not much, really ... 12
So where does the anomaly then come from? Reminder: Alvarez-Gaume and Witten 80's (Euclidean): - Imaginary part of the effective action = phase of the fermion determinant anomalies live is ill defined!!! → ORIGIN of the anomaly here - a naïve heat-kernel / world-line representation of the imaginary part of the effective action is possible – ONLY if we break chiral symmetry explicitly: Where did this come from? → path integral representation for imaginary part: - remember quantization on closed loop! (c.f. Schmidt, Schubert, D'Hoker, Gagne) - for the imaginary part: periodic boundary conditions for fermionic variables → zero modes = source of the anomaly (Polyakov) 13
A small recap' We have shown: - The identification of the topology of Berry's phase with that of the anomaly in current literature (8+ PRLs) is a misconception. - The origin of the anomaly can be understood using world-lines to arise from fermionic zero modes in the path integral in first quantization - This allows to derive a kinetic theory (first quantization, particles!) from first principles → consistent derivation of anomalies → Lorentz covariant → gauge covariant The world-line technique is extremely powerful! (Fun fact: Bern and Kosower have derived it from string theory in the infinite string tension limit.) 14
The big picture - and other cool stuff hydrodynamic CGC over-occupied flux tubes regime kinetic regime colliding nuclei plasma T h e r ma l i z a t i o n ? ? ? S. Schlichting 2016 non-equilibrium anomalous fermion production from coherent fields (Tanji et al. 2016) and sphaleron transitions (Mace et al. 2016) large magnetic fields present ? Anomalous Transport (CME, CSE and CMW) Subsequent interactions in the fire ball, axial transport and relaxation weak to strong coupling classical statistical chiral kinetic theory anomalous 15 simulations + fermions hydrodynamics
The big picture - and other cool stuff World-lines: a powerful tool to derive kinetic theories from first principles! Some work ahead, but straightforward: 16
The big picture - and other cool stuff Quantitative understanding of the Chiral Magnetic Effect. Neutron stars and supernovae: understanding helicity transport, neutrinos and large helical magnetic fields (see Yamamoto, Kaplan, etc.) Spin structure of the nucleon. one bosonic 'field' for every dimension (Lorentz-covariant) → Sound interesting, cold atom friends? also note: can go beyond one-loop. And yes, it is what you think it is: scatterings of world- lines. (ask M. Schmidt for more details!) 17
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