The weights are the thermal quantities of the mass states in the plasma, given by. These Boltzmann equations differ from those in the established literature [20], due to the inclusion of soil and tau Yukawa interactions.
Baryons at last
Independent of the nature of the CP-violating source, this collision effect can suppress or reverse the sign of the baryon asymmetry. We now consider an MSSM scenario that illustrates some of the new features discussed in Sec. Here we assume that both squarks are light, with O(100 GeV) masses, since a strong first-order phase transition requires a light top squark.
Although we work within the context of the MSSM, many of our conclusions are much more general. This approach is motivated by the fact that most of the charge transport dynamics occur outside the bubble, in the region of unbroken symmetry.
Lepton-mediated scenario: results
Dashed curve (at zero) is our numerical result obtained by neglecting tau/under Yukawa interactions. As advertised, the total left-handed quark charge is suppressed compared to our calculation that neglects bottom and tau Yukawa interactions (dotted curve). Without tau Yukawa interactions, no lepton charge density is generated; this is indicated by the dotted curve at zero.
However, our numerical (solid) and analytical (dashed) results suggest that tau Yukawa interactions do generate significant lepton charge. The resulting baryon asymmetry crucially depends on whether bottom and tau Yukawa interactions are incorporated in the Boltzmann equations or not.
MSSM parameter exploration
The dashed curves are the corresponding analytical results, in good agreement with our numerical curves. The mass of the right tau-slepton also plays a small role; from Eq. 6.36), we see that the lepton contribution to Lis increased slightly when the right stau is light, i.e., not suppressed by Boltzmann. Our numerical result neglecting bottom interactions and the Yukawa tau is shown by the upper, dotted curve.
In this region, the impact of bottom and tau Yukawa couplings is only a factor-of-two. Furthermore, if we take mA→ ∞ in addition, then bottom and tau Yukawa interactions will be suppressed; one gets the dotted curve back.
Beyond the MSSM
The singlet sector will modify the nature of the phase transition, the properties of the expanding bubbles and perhaps the CP-violating sources. For small tanβ the magnitude of the baryon asymmetry is suppressed because (i) bottom Yukawa interactions have suppressed the quark contribution to nL since (kT −kB) ' 0 and (ii) tau Yukawa interactions are still too small to generate lepton charge. In particular, effects of weak scale supersymmetry (SUSY) — one of the most popular extensions of the Standard Model (SM).
Moreover, measurements of this quantity provide unique probes of deviations from the leptonic universality of the weak charge-current (CC) interaction in the SM, which are induced by loop corrections and possible SM extensions. We show that the pseudoscalar contributions are negligible unless the ratio of the up and down Higgs vacuum expectation values (vevs) is huge (vu/vd ≡ tanβ & 103).
R-parity conserving interactions
Pseudoscalar contributions
Note that the combinationG(0)P S/Gµ×ω`, which introduces Eq. 8.7) is independent of the lepton flavor and will cancel Re/µ. In principle, however, the extraction of Fπ from πµ2 decay could be affected by Higgs exchange with a tree-level charged charge if the correction in Eq. 8.9) is & 0.003 in magnitude, corresponding to a shift comparable to the theoretical SM uncertainty as estimated in Ref. The reason is that in each pair of incoming quarks or outgoing leptons, the two fermions must have opposite chirality to contribute to G(1)P S .
Because CC interactions in the MSSM are purely left-handed, the chirality in each graph must change at least twice, with each flip generating a factor of ². However, if the triscalar SUSY breaking parametersaf are not suppressed by yf as normally assumed, it is possible to have²∼ O(1), potentially leading to significant contributions.
Axial vector contributions
We see that the ∆V + ∆L contributions (thin solid line) vanish for large µ, since in this regime gaugino-Higgsino mixing is suppressed and there is no ∆V + ∆L contribution to ∆RSU SYe/µ. The contributions to ∆RSU SYe/µ disappear ifmeeL =mµeLand is the largest if eithermµeL ¿ meeL ormµeL ÀmeeL. If µ À mZ, then the lack of gaugino-Higgsino mixing suppresses the ∆V + ∆L contributions to ∆RSU SYe/µ.
Finally, we illustrate the interplay between ∆V + ∆Land∆B by considering δRSU SYe/µ as a function of |µ|inmeuL. The solid shaded areas correspond to areas of the |µ|-meuLplane where the maximum value of δRSU SYe/µ is within the indicated ranges.
Contributions from R-parity Violating Processes
The solid shaded areas correspond to the maximum values of δRSUSYe/µ within the indicated ranges. Since ∆0ijk are positive semidefinite quantities, only the contour area in the upper right quadrant is shown. The inside of the dark blue contour corresponds to a fit using the current value of ∆Re/µ/RSMe/µ [67, 68], while the dashed red contour corresponds to a fit using the future expected experimental precision [69], assuming the same central value.
The light green curve shows the future impact of an upcoming measurement of the weak proton charge at Jefferson Lab [86]. On the other hand, the agreement of Re/µ with SM would lead to considerable tightening of the constraints in this scenario, especially in the case of ∆021k, which is currently limited only by Re/µ and the deep inelastic ν(¯ν ) distribution [87].
General Radiative Corrections in the MSSM
In addition, we note that the actual uncertainty associated with RPV effects entering the πµ2 decay rate would affect the value of Fπ to a level about half of the theoretical SM uncertainty as estimated by Ref. We note that the off-diagonal elements mixing gauginos and Higgsinos arise only from weak electric symmetry breaking. There are two classes of off-diagonal elements in M2 that can contribute to sleep mixing: mixing between flavors and mixing between left and right components of a given flavor, both of which arise through SUSY breaking terms.
Similarly, up-type squarks, down-type squarks and sneutrinos have mixing matrices ZU, ZD and Zν respectively, which are defined identically to ZL - except for the fact that there are no right-handed sneutrinos in the MSSM and thus there are only three sneutrinos mass eigenstates. There are three types of contributions to ∆RSU SYe/µ in the MSSM: external leg, vertex,.
Conclusions
Under these conditions, the size of ∆RSU SYe/µ can fall within the sensitivity of the new Re/µ measurements. The agreement between the results of the new Re/µ and RSMe/µ measurements could provide important new constraints on these possibilities. At low energies, recent evidence of a potentially significant deviation of the muon anomalous magnetic moment (gµ−2) from SM expectations provides at least a tantalizing hint of low-energy SUSY in the largetanβ regime [79, 92 ].
In this paper we study the implications of SUSY for the weak decay of muons, neutrons and nuclei. The magnitude of the effects in µ decay is below the current sensitivity of decay correlation studies.
Weak Decay Correlations: General Features
However, the presence of SUSY-induced scalar interaction may modify the derivation of the Fermi constant (Gµ) from the next generation of τµ measurements at PSI. In Section 9.3 we discuss the computation of the corresponding SUSY corrections and give analytical expressions for the resulting operators. In the limit of vanishing leptonic masses, weak non-QED SM electro-electric radiative corrections to the tree-level amplitude are absorbed in the definition of Gµ.
These corrections may affect the uniformity tests of the first row of the CKM matrix, as they must be subtracted from aVLL when determining Vud from β-decay half-lives. The conserved vector flux property of the SM CC interaction means that gV(0) = 1 in the limit of exact isospin symmetry.
SUSY-Induced Scalar and Tensor Interactions
The amplitude relative to the diagram shown in (a) includes left-right sleepon mixing, while those in (a) do not disappear if lepton flavor mixing is present in the sleepon sector. Note that in the latter case, a given eigenstate of the virtual slepton mass will couple to both the first and second generation of charged leptons, requiring the presence of non-zero flavor mixing. Indeed, studies of decay correlation parameters may provide a means of testing this alignment hypothesis.
The corresponding spermion mass eigenstates F˜j are given as a linear combination of the flavor eigenstates f˜Ias. In general, charginos (χ˜+j ) and neutralinos entering the loop graphs are mixtures of the electroweak gauginos and Higgsinos.
Phenomenological Constraints and Implications
Lepton Flavor Mixing Contributions
The contributions to gSRR of δμ(b.1,2) depend on the products of stermion mixing matrices Zν1m∗Zν2m andZL5iZL4i∗, which alone do not vanish in the presence of flavor mixing between the first two generations of sneutrinos and RH-charged roosts, respectively . It is also instructive to demonstrate the sensitivity of gSRR to the degree of flavor mixing and to demonstrate the corresponding impact of the LFV searches. To this end, we quantify the amount of lepton flavor mixing with a parameter δLFV, defined as.
Since the amplitudes δ˜(b.1,2) μ depend on the flavor mixing between both LH and RH sleepons, they only contribute if both δL and δR are small. In contrast, Br(µ → eγ) survives in the presence of flavor mixing between LH or RH sleep.) Obviously, the contribution of flavor mixing to gRRS is greatest when δLF V is smallest, as shown in Figs.
Left-Right Mixing Contributions
The possible existence of almost straight lines in the MSSM potential leads to additional limits on the size of scalar trilinear couplings from the condition of avoiding charge and color discontinuity minima. The Higgs sector does not enter loops that contribute to any of the quantities at stake here, and the magnitude of the mA therefore does not affect our results. 9.4, (a), the values of |∆τµ|/τµSM (left axis) and |gRRS |(right axis) obtained in our scan as a function of the supersymmetric contribution to the muon anomalous magnetic moment, δaµ.
Comparison of these quantities can provide a test of SM (or MSSM) at the level of electroweak radiation corrections via the relation [125]. In the case of β-decay, we show an analogue of the picture described above for muon decay, in fig.
Discussion
Second, the interaction fields ϕ are related to their Heisenberg counterparts by the time evolution operator U. The operator U obeys the usual relations:. Reading from right to left, Eq. A.8) corresponds to the start with the state "in" |n−i, then the time evolution from −∞ to +∞, the operation with the field operators at times x0 and y0 along this path, and finally the time evolution from +∞back to −∞, return to the "in" state. This time contour, labeled C, is a "closed time path"; it is closed in the sense that the contour starts and ends at t = −∞, connecting states "v" to states "v".
The path order prescription is to order the (+) fields in time, to place the (–) fields in anti-time order (T†), and finally to place all (–) fields to the left of the (+). fields. The functions of these Greens are the free or complete propagators of fields in the interaction or Heisenberg pictures, respectively.
Spectral functions
This is the "constraint equation"; it defines the spectra of degrees of freedom of φ. Let us consider how ρ(k) is modified in the presence of interactions by studying a simple example:. For the purpose of estimating the particle collision rate in the plasma, the dominant corrections to the spectral function are temperature-limited masses.
In the next section, we show how to solve for g(k, X) by solving the Boltzmann equation. The Boltzmann equation will allow us to solve for the dynamic evolution of the distribution function g(k, X) in the presence of collisions and a CP-violating source.
