Making the Electroweak Phase Transition (Theoretically) Strong Welcome and Theory Overview April 6, 2017 ACFI Workshop Hiren Patel hhpatel@umass.edu
Workshop Goals 1. How can we get more accurate estimates of physical quantities related to the EWPT in a tractable way for BSM scenarios? a) Effective potential/critical temperatures: V eff , T C b) Bubble nucleation: φ c ( r ), Γ nuc , T N , α , v wall , L wall … c) Sphaleron processes: φ ( r )/A sph ( r ), E sph , Γ sph , … d) Ultimately, Y B , and Ω GW … 2. Can we make the theoretical level of precision comparable to that of experimental/observational cosmology? 3. Can we reliably assign errors to these estimates? 4. Compare perturbation theory to fully non-perturbative lattice simulations. (benchmark models + parameters) Hiren Patel 2
Outline Electroweak Phase Transition Context Motivation Overview of methods Open theoretical problems (partial list) 1. Thermal potential - Spurious imaginary parts - Gauge dependence 2. Bubble nucleation rates - How to calculate them consistently Hiren Patel 3
Cosmic History STRING THEORY ELECTROWEAK BARYOGENSIS � > 3 spatial dimensions Baryon number violation (sphaleron) ν � Curled up? Size scale? � CP violation (e.g. EDM neutron ) ν � Deviations from Newton‘s law � Thermal non-equilibrium H H ν e – q τ q q q H q g ν q τ qq q q q τ n e + ν o q q q ν i t g q q a ν X l f n ν I μ Y ? ν g e + μ e + ? q e – q ν X Y τ τ e + μ ν q q q q e – q W μ ν q q q g q μ ν ν ν q Z ν ν e – τ e – q ν e – qq H τ q H ν ν ν ν BARYOGENESIS (E.G. GUT) PRIMORDIAL NUCLEOSYNTHESIS PRODUCTION OF HEAVY � Baryon number violation ELEMENTS How many neutrons available for � CP violation � Supernova explosions nucleosynthesis � Thermal non-equilibrium � � Nuclear physics in neutron Coupling constants � Lifetime rich stars Big Bang 10 –44 sec 10 –36 sec 10 –10 sec 10 –5 sec 100 sec 400,000 years 10 billion years 13.7 billion years Stephan Paul arXiv:1205.2451 Hiren Patel 4
Electroweak Phase Transition LHC: 2012 Discovery of SM Higgs-like particle measurement of mass m H = 126 GeV ⟹ First principles determination of cosmic history through 10 -10 s. Numerical simulations suggest a crossover K. Kajantie, et al. PRL 77 (1996) 2887, F. Karsch, et al. NPPS 53 (1997) 623, Y . Aoki et al., PRD 56 (1997) 3860 M. Gurtler et al., PRD 56 (1997) 3888 Hiren Patel 5
Electroweak Phase Transition Why do we care: Two big physics motivations - 1. A strong 1st order EWPT satisfies Sakharov’s out-of- equilibrium criteria for baryogenesis 2. A strong 1st order EWPT generates gravity waves, possibly observable in next gen. gravity wave detectors. Also, for intellectual curiosity: 3. Detailed understanding of pattern of EW symmetry breaking in the early universe. Hiren Patel 6
Electroweak Phase Transition The standard model Higgs field alone cannot generate a 1st order EWPT. A strong motivation for BSM Questions for model builders and phenomenologists: 1. Are there fundamental scalars other than the Higgs, and what BSM scenarios can generate a 1st order EWPT? 2. What are the experimental signatures of these scenarios? 3. What are their implications to other theoretical problems (neutrino mass, hierarchy, dark matter, …)? Hiren Patel 7
Electroweak Phase Transition How these questions are answered: Adequate Precision? experiment theory + obs. cosmology Collider pheno: d σ /d Ω T C , T N , Γ sph , α , Particle Physics v wall , L wall Phenomenologist Model Baryon asymmetry, Y B Gravity wave power spectrum Ω GW Issue: Theoretical precision is not competitive with observational cosmology 8 Hiren Patel
Electroweak Phase Transition How these questions are answered: Adequate Precision? experiment theory + obs. cosmology ☺ 😁 Collider pheno: d σ /d Ω 😟 T C , T N , Γ sph , α , N/A Particle Physics v wall , L wall Phenomenologist Model 😁 😲 Baryon asymmetry, Y B Gravity wave 😒 😲 power spectrum Ω GW Issue: Theoretical precision is not competitive with observational cosmology 9 Hiren Patel
Methods of Analysis - The calculation of V eff , T C , φ bubb ( r ), Γ nuc , T N , α , v wall , L wall , E sph , Γ sph , and Y B , Ω GW are notoriously difficult. - Many methods: can be put on a spectrum. Fully dynamical 4D Exact analytic numerical simulation evaluation of Γ eff Analytic V eff + Partial 4D Numerical Analytical numerical Analytic High- T EFT + numerical T C , simulation 3D numerical simulation T N , Γ sph , … Hiren Patel 10
Methods of Analysis - The calculation of V eff , T C , φ bubb ( r ), Γ nuc , T N , α , v wall , L wall , E sph , Γ sph , and Y B , Ω GW are notoriously difficult. - Many methods: can be put on a spectrum. - Each method of analysis has some degree of approximation Fully dynamical 4D Exact analytic numerical simulation evaluation of Γ eff Analytic V eff + Partial 4D Numerical Analytical numerical Analytic High- T EFT + numerical T C , simulation 3D numerical simulation T N , Γ sph , … (see T . Parametric Statistical Tenkanen’s talk) O(g 4 ), Parametric, O(m 2 /T 2 ), Lattice spacing… μ /T, (see D. ξ -dep. Weir’s talk) (remainder Level of of talk…) approx. Hiren Patel 11
Equilibrium Effective Potential To ultimately obtain Y B and Ω GW , need dynamical quantities: a) Bubble nucleation: φ bubb (r), Γ nuc , T N , α , v wall , L wall , … b) Sphaleron processes: φ sph (r)/A sph (r), E sph , Γ sph , … For a very rough analytic determination of the strength of phase transition, focus on two quantities: 1. Critical temperature T C , and 2. discontinuity in order parameter φ C These are equilibrium quantities, requiring the calculation of the thermal effective potential V eff . Hiren Patel 12
Effective Potential There are two problems associated with the thermal effective potential. 1. The potential (as naively calculated) has spurious imaginary parts. (see D. Curtin’s talk) 2. The effective potential is gauge dependent. Hiren Patel 13
Effective Potential: Imaginary Part The effective potential is traditionally calculated in perturbation theory: V e ff ( φ , T ) = V tree + V 1-loop + . . . Z ∞ dx x 2 ln(1 ⌥ e − p ✓ m 2 ( φ ) + λ i T 2 ⇢ ◆ � λ i ⇤ 2 ln X m 2 ( φ ) x 2 + m 2 ( φ ) /T 2 ) ⇥ V 1-loop ( φ , T ) = 4(4 π ) 2 2 π 2 µ R 0 i sum over all species, i Zero temperature part Finite temperature part At T=0, if symmetry is broken at tree level (as in SM), Im(V eff ) ≠ 0, near origin (due to logarithm). Re Lee+Weinberg, due to QM instability e ff At very high T, no instabilities should arise, and we must have: Im(V eff )=0 But this is not necessarily satisfied Im e ff Breakdown of PT: requires resummation (see D. Curtin’s talk) 14 Hiren Patel
Effective Potential: Gauge Depedence - Nielsen (1975) showed that extrema - Partition function of system is given by the trace of the density operator. of V eff are gauge independent, although the Higgs condensate Z = Tr[ e − β ˆ H ] is not. Partition Sum over function states - In a gauge theory, only physical states must be - Straightforward extraction of summed over. T C seems to be gauge dependent. - Done by fixing a gauge following method of even though the it only requires Faddeev and Popov (1967). minima of V eff gauge- parameter Z = Tr[ e − β ˆ H ( ξ ) ] - Thermal potential becomes gauge-dependent V e ff ( φ , T ) = − k B T ln Z ( ξ ) Hiren Patel 15
h-bar Expansion H.Patel, M.J. Ramsey-Musolf, JHEP 1107 (2011), 029 A possible resolution: Key is to extremize the potential while maintaining consistency with expansion parameter h-bar . Insert here ~ ~ ( counts # of loops) ~ ~ ~ e ff ~ ~ min Solve consistently order by order e ff min ~ ~ Hiren Patel 16
h-bar Expansion H.Patel, M.J. Ramsey-Musolf, JHEP 1107 (2011), 029 ~ ~ At finite temperature, this gives the thermal energy of the phases of the system as a function of T. Degeneracy condition satisfied at the intersections, yielding the critical temperature. Hiren Patel 17
h-bar Expansion H.Patel, M.J. Ramsey-Musolf, JHEP 1107 (2011), 029 ~ ~ Advantages 1. Can be proven to be strictly gauge-independent (Critical temperature) 2. Numerically straightforward to implement 3. Can be applied to radiatively induced phase transitions but… Hiren Patel 18
h-bar Expansion H.Patel, M.J. Ramsey-Musolf, JHEP 1107 (2011), 029 ~ ~ Open research problems 1. Tends to underestimate T C . The backreaction of thermal bath on condensate is delayed to O(h 2 ) (slow convergence) 2. Incompatible with naive methods of resummation. 3. If there is no solution at zeroth order (solution generated radiatively/thermally), then it will be missed. Nielsen identity does not require h-bar as the power counting scheme. Any consistent power counting scheme would work. - Are there power counting schemes that have better convergence, can capture more solutions? - How do they compare to numerical lattice results? Hiren Patel 19
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