Non-relativistic particles in a thermal bath

Heavy particles are a window to new physics and new phenomena. Since the late eighties they are treated by means of effective field theories that fully exploit the symmetries and power counting typical of non-relativistic systems. More recently these effective field theories have been extended to describe non-relativistic particles propagating in a medium. After introducing some general features common to any non-relativistic effective field theory, we discuss two specific examples: heavy Majorana neutrinos colliding in a hot plasma of Standard Model particles in the early universe and quarkonia produced in heavy-ion collisions dissociating in a quark-gluon plasma.


Real-time formalism
Temperature is introduced via the partition function. Sometimes it is useful to work in the real-time formalism. In real time, the degrees of freedom double ("1" and "2"), however, the advantages are • the framework becomes very close to the one for T = 0 EFTs; • in the heavy-particle sector, the second degrees of freedom, labeled "2", decouple from the physical degrees of freedom, labeled "1".
This usually leads to a simpler treatment with respect to alternative calculations in imaginary time formalism.

Real-time gauge boson propagator
• Gauge boson propagator (in Coulomb gauge): Real-time heavy-particle propagator • The free heavy-particle propagator is proportional to Since [S (0) (p)] 12 = 0, the static quark fields labeled "2" never enter in any physical amplitude, i.e. any amplitude that has the physical fields, labeled "1", as initial and final states.
These properties hold also for interacting heavy particle(s): interactions do not change the nature ("1" or "2") of the interacting fields.
• Brambilla Ghiglieri Vairo Petreczky PRD 78 (2008) 014017 Thermal widths at weak coupling We will consider heavy particles interacting weakly with a weakly coupled plasma: • a heavy Majorana neutrino in the primordial universe that interacts weakly with a plasma of massless SM particles; • a quarkonium formed in heavy-ion collisions of sufficiently high energy that is a Coulombic bound state interacting with a weakly coupled quark-gluon plasma.
We will compute corrections to the width induced by the medium: thermal width, Γ.

A model for neutrino oscillation and thermal leptogenesis
We consider a heavy Majorana neutrino ψ of mass M ≫ M W coupled to the SM only through a Higgs-lepton vertex: • This extension of the SM provides a model for neutrino mass generation through the seesaw mechanism.
• It also provides a model of baryogenesis through thermal leptogenesis.

Baryogenesis
The observed baryons are the remnant of a small matter-antimatter asymmetry n B − nB n B + nB ≈ 10 −10 in the early universe.
Any pre-inflationary asymmetry is diluted by inflation.
There are three necessary Sakharov conditions for a baryon asymmetry to be generated: • baryon number violation, which is allowed in the SM by sphaleron processes (effective at T > M W ); • C and CP violation, which is allowed in the SM by weak interactions and CKM phase, but too small; • nonequilibrium, too small if induced by the expansion of the universe.
The SM cannot account for the observed asymmetry (10 −20 vs 10 −10 ).

Baryogenesis through thermal leptogenesis
• baryon number violation: through sphaleron process, which conserve B − L, hence also induced by lepton number violation due to the Majorana neutrinos; • C and CP violation: besides from the weak interaction from phases in F ; • nonequilibrium: from the Majorana neutrino production, freezeout and decay.
T M M W Production of a net lepton asymmetry starts when the (lightest) sterile neutrino decouples from the plasma.
During the universe expansion, the sterile neutrino continues to decay.
Recombination process is almost absent and a net lepton asymmetry is generated.
Baryon number is protected from washout after sphaleron freezeout.
A key quantity for leptogenesis is the rate at which the plasma of the early universe creates Majorana neutrinos with mass M at a temperature T . This quantity is in turn related to the heavy Majorana neutrino thermal width in the plasma.

Non-relativistic Majorana neutrinos
• We consider the temperature regime • Heavy Majorana neutrinos are non-relativistic, with momentum p µ : In the reference frame where the neutrino is at rest: v µ = (1, 0).

The non-relativistic Majorana neutrino EFT
At an energy scale smaller than M and comparable with T , the low-energy modes of the Majorana neutrino are described by a field N whose effective interactions with the SM particles are encoded in the EFT Lagrangian: Higher-order operators are suppressed by powers of 1/M .

Dimension 5 operator
The leading effective low-energy operator describing the decay of the Majorana neutrino into SM particles is the dimension 5 operator It is a neutrino-Higgs vertex, fixed at one loop by the matching condition The Wilson coefficient a develops an imaginary part, which is

The LO Majorana neutrino thermal width
The imaginary part of the coefficient a is the main responsible for the emergence of a heavy neutrino thermal width induced by the interaction with the plasma of SM (Higgs) particles, through the neutrino-Higgs tadpole diagram: The thermal width is • Salvio Lodone Strumia JHEP 1108 (2011) 116 Higher-order corrections • We calculate them at first order in the SM couplings. We consider only Yukawa couplings with the top and neglect Yukawa couplings with other quarks and leptons.
• Thermal corrections are encoded into tadpole diagrams.
• We need to consider only operators with imaginary coefficients (tadpoles do not develop an imaginary part), coupled to bosonic operators with an even number of spatial and time derivatives (the boson propagator in the tadpole is even for space and time reflections) and to fermionic operators with an odd number of derivatives (the massless fermion propagator in the tadpole is odd for spacetime reflections).
• This implies that L (2) does not contribute because it involves either boson fields with one derivative or fermion fields with no derivatives.
Leading momentum dependent operator The Wilson coefficient of this operator is fixed by the relativistic dispersion relation

Colour deconfinement
Transition from hadronic matter to a plasma of deconfined quarks and gluons happening at some critical temperature T c = 154 ± 9 MeV as studied in finite temperature lattice QCD.

Quarkonium as a quark-gluon plasma probe
In 1986, Matsui and Satz suggested quarkonium as an ideal quark-gluon plasma probe.
• Heavy quarkonium formation will be sensitive to the medium.
• The dilepton signal makes the quarkonium a clean experimental probe.

Scales
Quarkonium being a composite system is characterized by several energy scales, these in turn may be sensitive to thermodynamical scales smaller than the temperature: • the scales of a non-relativistic bound state (v is the relative heavy-quark velocity; v ∼ α s for a Coulombic bound state): M (mass), M v (momentum transfer, inverse distance), M v 2 (kinetic energy, binding energy, potential V ), ...
The non-relativistic scales are hierarchically ordered: We assume this to be also the case for the thermodynamical scales: πT ≫ m D

Υ(1S) scales
A weakly coupled quarkonium possibly produced in a weakly coupled plasma is the bottomonium ground state Υ(1S) produced in heavy-ion experiments at the LHC:

Non-relativistic EFTs of QCD
The existence of a hierarchy of energy scales calls for a description of the system (quarkonium at rest in a thermal bath) in terms of a hierarchy of EFTs.  • The Lagrangian is organized as an expansion in 1/M : is the field that annihilates (creates) the (anti)fermion.
• Caswell Lepage PLB 167 (1986) 437 Bodwin Braaten Lepage PRD 51 (1995) 1125 pNRQCD pNRQCD is obtained by integrating out modes associated with the scale M v and possibly with thermal scales larger than M v 2 .
• The degrees of freedom of pNRQCD are quark-antiquark states (color singlet S, color octet O), low energy gluons and light quarks propagating in the medium.
• The Lagrangian is organized as an expansion in 1/M and r: • At leading order in r, the singlet S satisfies a Schrödinger equation.
The explicit form of the potential depends on the version of pNRQCD.

Dissociation mechanisms at LO
A key quantity for describing the observed quarkonium dilepton signal suppression is the quarkonium thermal dissociation width.
Two distinct dissociation mechanisms may be identified at leading order: • gluodissociation, which is the dominant mechanism for M v 2 ≫ m D ; • dissociation by inelastic parton scattering, which is the dominant mechanism for M v 2 ≪ m D .
Beyond leading order the two mechanisms are intertwined and distinguishing between them becomes unphysical, whereas the physical quantity is the total decay width.
Gluodissociation is the dissociation of quarkonium by absorption of a gluon from the medium.
• The exchanged gluon is lightlike or timelike.
• The process happens when the gluon has an energy of order M v 2 .
• Kharzeev Satz PLB 334 (1994) 155 Xu Kharzeev Satz Wang PRC 53 (1996) 3051 From the optical theorem, the gluodissociation width follows from cutting the gluon propagator in the following pNRQCD diagram mv 2 For a quarkonium at rest with respect to the medium, the width has the form Γ nl = q min d 3 q (2π) 3 n B (q) σ nl gluo (q) .
• σ nl gluo is the in-vacuum cross section (QQ) nl + g → Q + Q. • Gluodissociation is also known as singlet-to-octet break up.