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Kinetic Mixing in Field Theory

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Kinetic Mixing in Field Theory Kinetic Mixing in Field Theory DESY Workshop Seminar, Winter Semester 2009 / 2010 Kai Schmitz December 1, 2009 Abstract Models with more than one U(1) gauge symmetry always permit a renormalizable dimension-four operator respecting all symmetries of the Lagrangian that mixes the kinetic terms of the Abelian gauge fields. Though potentially generated at arbitrarily high energies, such kinetic mixing may lead to interesting phenomenology in the ef- fective low-energy theory. Subject of this talk are the field-theoretic origin of kinetic mixing, effects of kinetic mixing below the electroweak scale such as hidden-sector par- ticles carrying mini-electric charges and photon hidden-photon oscillations, as well as kinetic mixing in supersymmetric extensions of the Standard Model. In particular, we show that kinetic mixing may provide soft SUSY-breaking masses that could, if not sufficiently suppressed, pull the weak scale up to the scale of SUSY breaking. We conclude that kinetic mixing may play an important role for both model building and experimental searches for new physics at low energies. Contents 1 Motivation 1 2 Origin of Kinetic Mixing in Field Theory 2 2.1 Renormalization of U(1)gaugecouplings. .. .. .. .. .. .. .. .. 3 2.2 Effective interaction in the low-energy theory . .......... 4 3 Kinetic Mixing at and below the Electroweak Scale 6 3.1 Kinetic mixing with an unbroken U(1)X ..................... 7 3.2 Spontaneous breaking of the hidden U(1)X ................... 7 4 Kinetic mixing in Supersymmetric Theories 9 4.1 Supersymmetrization of kinetic mixing . ........ 9 4.2 Kinetic mixing-mediated SUSY breaking in the MSSM . ........ 10 1 Motivation The three previous talks in this seminar dealt with the axion. As we have learned, heavy axions are already ruled out by observations and the only viable scenario that still persists is that of a very light, weakly coupled axion. In this talk we will now explore a differ- ent example of how new physics that is only weakly coupled to the Standard Model can emerge at low energies: The kinetic mixing between two U(1) gauge groups that establishes a communication channel between some hidden and the visible gauge sector. Hidden sec- tors consist of fields that transform as singlets under the gauge group of the visible sector. They are a common and necessary feature of many theories beyond the Standard Model. 1 For instance, the spontaneous breakdown of supersymmetry (SUSY) in the minimal super- symmetric extension of the Standard Model (MSSM) has to be attributed to processes in a hidden sector. It is not possible to explain spontaneous SUSY breaking only working with the usual MSSM fields and interactions. The communication between the visible and the hidden sector is usually described in the language of an effect field theory. One postulates messenger particles that are charged under the hidden and the visible gauge groups so that they are able to mediate between both sectors. After integrating out these messengers one obtains non-renormalizable interactions that are suppressed by the heavy messenger masses and hence rather weak. Most of the few examples of renormalizable interactions between the visible and the hidden sector are very model-specific as they crucially depend on model assumptions such as charge assignments and coupling strengths. By contrast, one renormalizable coupling is always possible provided that the visible and hidden gauge groups each contain at least one U(1) factor — which is the case in many extensions of the MSSM. In string theory compacti- fications hidden U(1) groups are, for example, omnipresent. In the visible sector we always have the familiar hypercharge U(1)Y or, at lower energies, the U(1)EM of electromagnetism. As we will see in the next section, the exchange of messenger particles at arbitrarily high energies can induce the following dimension-four operator in the effective kinetic Lagrangian at lower energies [1]: χ = Bµν X (1) OKM − 2 · µν where Bµν and Xµν respectively denote the field strengths of the visible and the hidden gauge field and χ is some coefficient that parametrizes the strength of this so-called kinetic mixing operator. Kinetic mixing can only occur between the gauge fields of two U(1) groups. µν µν In the non-Abelian case the field strengths Bi and Xi are not gauge invariant quantities µν by themselves and a term in the Lagrangian proportional to Bi Xjµν would be forbidden. Including kinetic mixing effects, the total kinetic Lagrangian kin. in the low-energy theory turns out to be of the following form: L µν 1 χ1 χ B kin. = (Bµν Xµν ) µν (2) L −4 χ χ2 X The parameters χ1, χ2 and χ can be related to vacuum polarization diagrams that take care of the wavefunction renormalization of the gauge fields. As evident from Eq. (2) kinetic mixing is a direct analog of mass mixing. Instead of a mass mixing matrix we now have to deal with a kinetic mixing matrix. In order to obtain the physical gauge fields, that is, those fields for which kin. acquires its canonical form, this kinetic mixing matrix has to be normalized and diagonalized:L χ1 χ 1 0 χ χ2 −→ 0 1 In this talk we will first show how kinetic mixing is generated and then discuss what impli- cations this transformation to canonical kinetic terms may have on the phenomenology of the model. 2 Origin of Kinetic Mixing in Field Theory Kinetic mixing is rooted in vacuum polarization diagrams with virtual messenger particles circulating in the loops, see Fig. 1. These diagrams constitute off-diagonal contributions to the wavefunction renormalizations in the space of U(1) gauge fields. In the full high-energy theory such diagrams create small charge shifts through the evolution of the renormalization group equations (RGEs). In the effective low-energy theory they lead to an explicit interac- tion in the Lagrangian as given in Eq. (1). We start by discussing charge renormalization in a model with several U(1) factors and then turn to the effective theory that one obtains after integrating out the heavy messengers. First of all, we consider the case of one U(1). 2 φ (hidden sector fermion) Xµ (hidden sector gauge boson) Heavy Φ messengers Bµ (visible sector gauge boson) ϕ (visible sector fermion) Figure 1: Effective interaction between the visible and the hidden sector due to loop diagrams involving heavy messenger fields. 2.1 Renormalization of U(1) gauge couplings One U(1) gauge factor By virtue of the Ward-Takahashi identity the renormalization of a U(1) gauge coupling g is solely determined by the wavefunction renormalization of the corresponding Abelian gauge field Aµ [2]. Effects from the fermion wavefunction and vertex renormalization exactly cancel to all orders in perturbation theory. The bare coupling g0 is related to the observable 0 coupling g by Z, the rescaling factor of the bare gauge field wavefunction Aµ: A0 = A Z1/2 ; g = g0 Z1/2 ; Z (1 Π(0))−1 (3) µ µ · · ≡ − Z is defined in terms of the self energy Π(k2) of the gauge field evaluated at zero momentum transfer (k2 = 0). The self energy in turn follows from all 1-particle-irreducible insertions into the gauge field propagator which gives Πµν (k). The tensor structure of Πµν (k) is fixed µν by gauge invariance which requires that the contraction kµΠ (k) vanishes: Πµν (k)= k2gµν kµkν Π(k2) − Several U(1) gauge factors Consider now a gauge group U(1)N G where N 2 and with G being a semi-simple group. × ≥ µ A fermion charged under U(1)a then couples to the U(1)b gauge boson Ab with a strength gab. We may assume that at some high scale the different U(1) interactions become diagonal and gab = gaδab. Evolving down in energy vacuum polarization diagrams such as in Fig. 1 will, however, lead to deviations from this relation. Eq. (3) now turns into a matrix equation [3]: g = g0 Z1/2 ; Z (1 Π (0))−1 ab ac · cb cb ≡ − cb Diagramatically we can write: µ ν Ac Ab iΠµν (k)= i k2gµν kµkν Π (k2) cb − cb ≡ 1 PI k The renormalization constants Zcb yield the β-functions βcb which we need in order to construct the RGEs: d d 1 β (ln Z) ; g = g β (RGEs) ; t = ln µ cb ≡ dt cb dt ab 2 ac cb Given this form of the RGEs it is comprehensible that the β-functions βc6=b generate non-zero off-diagonal values in gab when running down to lower energy scales µ. 3 Figure 2: Toy model for the generation of kinetic mixing. The gauge charges of the fermions are all non-zero if not stated explicitly. 2.2 Effective interaction in the low-energy theory In the rest of this talk we will investigate the leading order effects of kinetic mixing in the effective low-energy theory. From now on we will neglect all effects of (χ2). To get started 2 O we need an explicit expression for the leading contribution to Πab(k ) which follows from the evaluation of the 1-loop diagrams. Making use of the procedure of dimensional regularization (1) 2 we obtain the contribution Πab (k ,f) to the vacuum polarization from a fermion f [4]: g g 1 µ2ε 1 m2(f) k2x(1 x) Π(1)(k2,f)= β(1)(f) a b 0 +6 dx x(1 x) ln − − ab ab · 4π · 2π εˆ − µ2 Z0 0 (1) Here βab (f) denotes the first coefficient in the expansion of the β-function βab: g g g g 2 2 β = β(1)(f) a b + β(2)(f) a b + ..
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