NGB and their parameters Gradient expansion parameters of
NGB and their parameters · · · Gradient expansion: parameters of the. NGB’s · Dispersion relations for the gluons Masses of the NGB’s The role of the chemical potential for scalar fields: BE condensation 1
Hierarchies of effective lagrangians Integrating out heavy degrees of freedom we have two scales. The gap D and a cutoff, d above which we integrate out. Therefore: two different effective theories, LHDET and LGolds 2
Gradient expansion: NGB’s parameters Recall from HDET that in the CFL phase and in the basis 3
Propagator Coupling to the U(1) NGB: 4
Consider now the case of the. U(1)B NGB. The invariant Lagrangian is: At the lowest order in s generates 3 -linear and 4 -linear couplings 5
Generating functional: 6
At the lowest order: 7
Feynman rules · For each fermionic internal line · For each vertex a termi. Lint · For each internal momentum not constrained by momentum conservation: · Factor 2 x(-1) from Fermi statistics and spin. A factor 1/2 from replica trick. · A statistical factor when needed. 8
+ Goldstone theorem: Expanding in p/D: 9
CFL 10
For the V NGB same result in CFL, whereas in 2 SC With an analogous calculation: 11
Dispersion relation for the NGB’s Different way of computing: Current conservation: 12
Masses of the NGB’s QCD mass term: 13
Calculation of the coefficients from QCD Mass insertion in QCD Effective 4 -fermi Contribution to the vacuum energy 14
Consider: Solving for as HDET in like chemical potential 15
Consider fermions at finite density: as a gauge field. A 0 Invariant under: Define: Invariance under: 16
The same symmetry should hold at the level of the effective theory for the CFL phase (NGB’s), implying that The generic term in the derivative expansion of the NGB effective lagrangian has the form 17
Compare the two contribution to quark masses: kinetic term mass insertion Same order of magnitude for since 18
The role of the chemical potential for scalar fields: Bose-Einstein condensation · A conserved current may be coupled to the a gauge field. · Chemical potential is coupled to a conserved charge. · The chemical potential must enter as the fourth component of a gauge field. 19
Complex scalar field: negative mass term breaks C Mass spectrum: For m < m 20
At m = m, second order phase transition. Formation of a condensate obtained from: Charge density Ground state = Bose-Einstein condensate 21
Mass spectrum At zero momentum 22
At small momentum 23
Back to CFL. From the structure First term from “chemical potential” like kinetic term, the second from mass insertions 24
For large values of ms: and the masses of K+ and K 0 are pushed down. For the critical value masses vanish 25
For larger values of ms these modes become unstable. Signal of condensation. Look for a kaon condensate of the type: (In the CFL vacuum, S = 1) and substitute inside the effective lagrangian negative contribution from the “chemical potential” positive contribution from mass insertion 26
Defining with solution and hypercharge density 27
Mass terms break original SU(3)c+L+R to SU(2)Ix. U(1)Y. Kaon condensation breaks this to U(1) breaking through the doublet as in the SM Only 2 NGB’s from K 0, K+ instead of expected 3 (see Chada & Nielsen 1976) 28
Chada and Nielsen theorem: The number of NGB’s depends on their dispersion relation I. If E is linear in k, one NGB for any broken symmetry II. If E is quadratic in k, one NGB for any two broken generators In relativistic case always of type I, in the nonrelativistic case both possibilities arise, for instance in the ferromagnet there is one NGB of type II, whereas for theantiferromagnet there are two NGB’s of type I 29
Dispersion relations for the gluons The bare Meissner mass The heavy field contribution comes from the term 30
Notice that the first quantized hamiltonian is: Since the zero momentum is the density one gets propagator spin 31
Gluons self-energy Vertices from Consider first 2 SC for the unbroken gluons : 32
· Bare Meissner mass cancels out the constant contribution from the s. e. · All the components of the vacuum polarization have the same wave function renormalization Dielectric constant e = k+1, and magnetic permeability l =1 33
Broken gluons a P 00(0) - Pij(0) 1 -3 0 0 4 -7 3 mg 2/2 8 3 mg 2/3 34
But physical masses depend on the wave function renormalization Rest mass defined as the energy at zero momentum: The expansion in p/D cannot be trusted, but numerically 35
In the CFL case one finds: from bare Meissner mass Recall that from the effective lagrangian we got: implying parameters. and fixing all the 36
We find: Numerically 37
Different quark masses We have seen that for onemassless flavors and a massive one (ms), the condensate may be disrupted for The radii of the Fermi spheres are: As if the two quarks had different chemical potential (ms 2/2 m) 38
Simulate the problem with two massless quarks with different chemical potentials: Can be described by an interactionhamiltonian Lot of attention in normal SC. 39
HI changes the inverse propagator and the gap equation (for spin up and down fermions): This has two solutions: 40
Grand potential: Also: Favored solution 41
Also: First order transition to the normal state at For constant D, Ginzburg -Landau expanding up to D 6 42
- Slides: 42