Quark mixing

The quarks of the other families are treated in exactly the same manner, with the SU 2)i doublets being given by 

The prime for the down, the strange and the bottom quark is a tribute to the strangeness changing decays, that are observed experimentally, and 

The bottom quark is therefore only weakly coupled to the down and the strange quark. 

Returning now to the weak interactions, it would be bizarre if in the quark sector the Cabibbo mixing only took place between d and s. So we generalize the Cabibbo scheme by assuming that the coupling in the weak interactions is to three linearly independent combinations of d, s and b  

The matrix V with elements where i = u,c,t and j = d,s,b must be unitary as discussed in Section 1.1. 

It is then assumed that the weak interactions couple to the left-handed weak isospin doublets 

L fj /L T /L In this way we have a complete parallelism between the three lepton doublets and the three quark doublets. The transformations under SU(2) are identical, but because of the fractional charges one needs 

Just as for the leptons, we require right-handed parts for u,d, c,s, t and b in order to have the correct electromagnetic current, and these are taken to be invariant under the 517(2) gauge transformations. To achieve a gauge invariant theory they must transform imder the 17(1) transformations as follows  

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In our case, nearly equal volume fractions of the two quark phases are likely to form alternating layers (slabs) of matter. The energy cost per unit volume to produce such layers scales as a2/3(r 2SC — niN )2/3 where a is the surface tension [25], Therefore, the quark mixed phase is a favorable phase of matter only if the surface tension is not too large. Our simple estimates show that max < 20 MeV/fm2. However, even for slightly larger values, 20 < a < 50 MeV/fm2, the mixed phase is still possible, but its first appearance would occur at larger densities, 3po < Pn < 5po. The value of the maximum surface tension obtained here is comparable to the estimate in the case of the hadronic-CFL mixed phase obtained in Ref. [26], The thickness of the layers scales as a1 /3(r/i2 SY -) — niN ) 2/3 [25], and its typical value is of order 10 fm in the quark mixed phase. This is similar to the estimates in various hadron-quark and hadron-hadron mixed phases [25, 26], While the actual value of the surface tension in quark matter is not known, in this study we assume that it is... 

We provide a much expanded disc sion of quark mixing (the... 

Figure 9. Mass-radius relation for pure strange quark matter stars (left) and hybrid stars (right). GO - G4 models of hybrid stars corresponding to different parameters of the model. H pure hadron star, QC star has a quark core, MC star has a mixed core, from Thoma et al. (2003).

From the intersection of the two surfaces representing the hadron and the quark phase one can calculate the equilibrium chemical potentials of the mixed phase,... 

The EOS resulting from this procedure is shown in Fig. 10(b), where the pure hadron, mixed, and pure quark matter portions are indicated. The mixed phase begins actually at a quite low density around po- Clearly the outcome of the mixed phase construction might be substantially changed, if surface and Coulomb energies were taken into account [36], For the time being these are, however, unknown and have been neglected. 

The analysis of Ref. [42] as well as the NJL-type model investigation of Ref. [43] are based on a comparison of homogeneous phases. The neutrality conditions can, however, also be fulfilled giving up the requirement of separately neutral phases and to consider mixed phases in chemical equilibrium which are only neutral in total. This procedure has been pushed forward by Glendenning in the context of the quark-hadron phase transition in neutron stars where a similar problem related to electrical neutrality occurs [44], For the case of electrically and color neutral quark matter the phase boundaries are... 

Table 2. Composition of electrically and color neutral mixed phases, corresponding quark number chemical potentials and average baryon number densities pB = n/3 in unities of nuclear matter saturation density po = 0.17/fm3. The various components are defined in Tab. 1.

Figure f. Pressure as a function of //, for homogeneous neutral quark matter in the CFL phase (solid), 2SC (dashed), and the normal quark matter phase (dotted). Also shown is the pressure of the mixed phase solution (dash-dotted). 

Abstract The ground state of dense up and down quark matter under local and global charge neutrality conditions with / -equilibrium has at least four possibilities normal, regular 2SC, gapless 2SC phases, and mixed phase composed of 2SC phase and normal components. The discussion is focused on the unusual properties of gapless 2SC phase at zero as well as at finite temperature. 

Figure 8. At r = 0.75, pressure as a function of gi = gis /3 and /re for the normal and color superconducting quark phases (the same as in Figure 7, but from a different viewpoint). The dark solid line represents the mixed phase of negatively charged normal quark matter and positively charged 2SC matter.

Different components of the mixed phase occupy different volumes of space. To describe this quantitatively, we introduce the volume fraction of normal quark matter as follows X sc = Vnq/V (notation Xb means volume fraction of phase A in a mixture with phase B). Then, the volume fraction of the 2SC phase is given by X2nq = (1 — X.2sc)- From the definition, it is clear that 0[Pg.236]

Under the assumptions that the effect of Coulomb forces and the surface tension is small, the mixed phase of normal and 2SC quark matter is the most favorable neutral phase of matter in the model at hand with r/ = 0.75. This should be clear from observing the pressure surfaces in Figs. 7 and 8. For a given value of the baryon chemical potential /i = fin/3, the mixed phase is more favorable than the gapless 2SC phase, while the gapless 2SC phase is more favorable than the neutral normal quark phase. 

Under global charge neutrality condition, assuming that the effect of Coulomb forces and the surface tension is small, one can construct a mixed phase composed of positive charged 2SC phase and negative charged normal quark matter. 

In bulk matter the quark-hadron mixed phase begins at the static transition point defined according to the Gibbs criterion for phase equilibrium... 

Figure 1. Chemical potentials of the three phases of matter (H, Q, and Q ), as defined by Eq. (2) as a function of the total pressure (left panel) and energy density of the H- and Q-phase as a function of the baryon number density (right panel). The hadronic phase is described with the GM3 model whereas for the Q and Q phases is employed the MIT-like bag model with ms = 150 MeV, B = 152.45 MeV/fm3 and as = 0. The vertical lines arrows on the right panel indicate the beginning and the end of the mixed hadron-quark phase defined according to the Gibbs criterion for phase equilibrium. On the left panel P0 denotes the static transition point.

B > B1. These high values of the bag constant do not allow the quark deconfinement to occur in the maximum mass hadronic star either. Here B1 denotes the value of the bag constant for which the central density of the maximum mass hadronic star is equal to the critical density for the beginning of the mixed quark-hadron phase. For these values of B, all compact stars are pure hadronic stars. 

Figure 6. Solutions of the gap equations and the charge neutrality condition (solid black line) in the /// vs //, plane. Two branches are shown states with diquark condensation on the upper right (A > 0) and normal quark matter states (A = 0) on the lower left. The plateau in between corresponds to a mixed phase. The lines for the /3-equilibium condition are also shown (solid and dashed straight lines) for different values of the (anti-)neutrino chemical potential. Matter under stellar conditions should fulfill both conditions and therefore for //,( = 0 a 2SC-normal quark matter mixed phase is preferable.

A quark-hadron mixed phase between 5 x 1014g/cm3 and 1015g/cm3 3.5po where both quark and hadrons are present. In this phase transition region, the EoS substantially softens with T 1 — 1.5. 

In strong and electromagnetic interactions, hadronic flavor is conserved, i.e. the conversion of a quark of one flavor (d, u, s, c, 6, t) into a quark of another flavor is forbidden. In the Standard Model, the weak interactions violate these conservation laws in a manner described by the Cabibbo-Kobayashi-Maskawa mixing (see the section Cabibbo-Kobayashi-Maskawa Mixing Matrix ). The way in which these conservation laws are violated is tested as follows ... 

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