m := mu+ md
2 , := mu − md mu+ md
. (2.66)
The quark mass matrix of in eq. (2.65) serves as the starting point of the inves-tigation of the hadronic operators induced by the θ-term in chapter 4.
2.3 Summary
The classical action of QCD is invariant under local SU (2)L×SU(2)R×U(1)V× UA(1) transformation. The fermion measure of the generating functional of QCD which is not invariant under local U (1)A transformations gives rise to an anoma-lous term consisting of gluon fields. Although this anomaanoma-lous term can be ex-pressed as a divergence of another term, it can not be removed by an application of Gauss’ theorem due to topologically non-trivial solutions for the gluon fields.
These topologically nontrivial gluon field configurations give rise to a P and T -violating term in the QCD Lagrangian parametrized by a dimensionless constant θ, which is referred to as the QCD θ-term. The anomalous term generated by the fermion measure of the generating functional of QCD under U (1)A transforma-tions is identical to the QCD θ-term up to a real constant and can be regarded to define the transformation law of the QCD θ-term under U (1)A transformations.
2.3. SUMMARY 21
The QCD θ-term can be removed from the QCD Lagrangian by an U (1)A
rotation at the price of generating a complex phase of the quark mass matrix.
The QCD θ-term expressed as a complex phase of the quark mass matrix is given by
LQCD =· · · +θ¯ 2
(mu+ md)
2 i¯qγ5q + θ¯ 2
(mu− md)
2 i¯qγ5τ3q +· · · , (2.67) where ¯θ = 2φ− θ with a general initial complex phase φ of the quark mass matrix. The QCD θ-term can be removed completely if one of the quark masses vanishes: the quark mass matrix is then equivalent to a real matrix by an axial SU (2)L×SU(2)R rotation as demonstrated in section 4.3 (see eq. (4.148)).
Sources of P and T violation
Extensions of the SM manifest themselves as effective operators to the energy scale Λhad ∼ 1 GeV. The aim of this chapter is to present the complete set of leading non-leptonic P - and T -violating operators at the hadronic energy scale Λhad. The content of this chapter is a brief summary of the results published in refs. [35–38], which have been utilized and extended in the recent publications [39, 40]. It is intended to serve as the starting point of our analysis of the EDMs of light nuclei which are induced by BSM physics.
The SM is an SU (3)C×SU(2)L×U(1)Y (C: color, L: left, Y : weak hypercharge) gauge theory and its Lagrangian for three generations before the spontaneous symmetry breakdown SU (2)L×U(1)Y → U(1)em is given by (see e.g. [39, 64]1)
The quantities qL and lL in eq. (3.1) denote the SU (2)L doublets of left-handed quarks and leptons for all three generations with implied summation over the generation index α: The right-handed quark and lepton singlets in eq. (3.1) are also understood to be vectors in generation space:
uαR= (uR, cR, tR) , dαR= (dR, sR, bR)R, eαR = (eR, µR, τR) . (3.3)
1The notation and conventions of [64] are adopted in this chapter. The brief explanation of the SM Lagrangian follows [64].
22
23
The complex scalar SU (2)L doublet field φ in eq. (3.1) is defined in polar coordinates ~α by proportional to µ2 and to λ in the third line of eq. (3.1) constitute the potential of the scalar φ. For µ2 > 0, the scalar SU (2)L doublet develops a vacuum By the gauge transformation U (~α) as defined in eq. (3.4), the field φ can be re-expressed (in the unitary gauge) as
φ(x)→ φ0(x) = U (~α(x))φ(x) = v
The Higgs boson field h is an excitation of the ground state. All primes denoting fields in the unitary gauge are subsequently omitted for convenience.
The covariant derivative of a general matter field ψ with hypercharge Y is given by:
coupling constant and g and g0 are the SU (2)L and U (1)Y coupling constants, respectively. The term involving gluons is absent from the covariant derivative for leptonic fields and the scalar doublet field φ. The hypercharges Y of the particles in eq. (3.1) are given by: the Yukawa couplings between fermions and the scalar doublet field φ.
The SU (2)L and U (1)Y field-strength tensors in eq (3.1) are given by
Wµνi = ∂µWνi− ∂νWµi + gijkWµjWνk, (3.9)
Bµν = ∂µBν − ∂νBµ. (3.10)
The physical states of the SUL(2) and UY(1) gauge fields are the W± bosons, where θω is the Weinberg-angle. The coupling constants g and g0 obey
g0
g = tan(θω) . (3.15)
Eigenstates of the fermion mass matrices are in general linear combinations of gauge eigenstates, which implies the existence of couplings of fermions from different generations. The quark mass matrix can be diagonalized by a biunitary transformation. For three generations, the resulting diagonal 3× 3 matrix has one overall complex phase. The left-handed SU (2)L quark doublet has therefore to be redefined by
where V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
3.1 P and T violation in the Standard Model
There are two different sources of P and T violation within the SM. The CKM matrix (for three generations) has one P - and T -violating complex phase. As pointed out in [5], any diagram in the SM inducing a P - and T -violating quark-photon coupling has to involve four electroweak vertices and is thus at least a two-loop diagram. Furthermore, the authors of [65] demonstrated that all diagrams potentially contributing to quark EDMs with two loops vanish and the leading non-vanishing contributions emerge only at the three loop level [66, 67]. Due to this significant suppression, the computation of the d-quark EDM, for instance, yielded a numerical estimate of [5, 66] dd≈ 10−34e cm. The only other source of P and T violation within the SM is the above mentioned QCD θ-term which is parametrized by the physical parameter ¯θ. Depending on the size of ¯θ, the θ-term is capable of inducing significantly larger EDMs than the complex phase of the CKM matrix. The current bound on the parameter ¯θ is |¯θ| . 10−10 [45, 48, 51], which has been inferred from the current upper bound on the neutron EDM [14].