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 effective 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 particles 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 sectors 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 supersymmetric 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 implications 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 interaction 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 + ... ; β(1)(f)= q (f)q (f) ab ab · 4π2 ab · 4π2 ab 3 a b 1 ε = 2 (d 4) is a measure for the deviation from d = 4 dimensions andε ˆ is given as −1 −−1 (1) 2 εˆ = ε + γE ln(4π). In the limit ε 0 the self energy contribution Πab (k ,f) diverges and renormalization− becomes necessary.→ In the MS-scheme the divergent piece is split off as follows:

g g 1 µ2ε Π(1)(k2,f) = Πε + Πˆ (k2,f) ; Πε = β(1)(f) a b 0 ab ab ab ab ab · 4π · 2π · εˆ ε Subsequently, Πab is absorbed in counterterms. If we take the external momentum k to be much smaller than the mass mf of the circulating fermion the remaining ﬁnite part 2 Πˆ ab(k ,f) becomes:

g g 1 m2(f) k2 m2 Πˆ (f) β(1)(f) a b ln ≪ f ⇒ ab ≈ ab · 4π · 2π µ2 0

4 Transition to eﬀective ﬁeld theory in a toy model

Let us now study a toy model [1] with a U(1)1 U(1)2 gauge group and a fermion content × µ µ as depicted in Fig. 2. The gauge sector of this theory consists of two gauge fields B A1 µ µ ′ ≡ and X A2 whereas the fermion sector contains two heavy messengers Φ and Φ as well as an arbitrary≡ number of visible and hidden sector fields ϕ and φ. At some high energy Λ the kinetic Lagrangian of the gauge fields has canonical form: 1 1 (Λ) = Bµν B Xµν X Lkin. −4 µν − 4 µν How does this Lagrangian change when we evolve down to a energy scale λ below the mass thresholds µ = m(Φ′) and µ = m(Φ)? At the scale λ the heavy messengers Φ′ and Φ cannot be produced any more as physical particles which is why we would like to remove them from the theory. However, at the scale λ the messengers Φ′ and Φ still appear in loop diagrams as virtual particles. To take care of these diagrams we have to introduce effective interactions in the Lagrangian: χ χ χ (λ)= 1 Bµν B 2 Xµν X Bµν X Lint. − 4 µν − 4 µν − 2 µν These interactions give rise to diagrams which have to be matched with the corresponding 1-loop diagrams from the full high-energy theory [5]:

µ ν µ ν Aa Ab Aa Ab f ! {Φ,Φ′} = P χ k k ab The diagrams on the right-hand side have the following amplitude:

µ ν Aa Ab = i k2gµν kµkν χ − − ab χ k ab The amplitudes of the vacuum polarization diagrams were discussed above. The 1-loop matching conditions then require:

(1) χab = Πab (f) (4) − ′ {ΦX,Φ }

The parameters χ1 χ11 and χ2 χ22 can be absorbed into the gauge fields as it is always possible to perform≡ a rescaling of≡ the fields and their corresponding charges. W.l.o.g. we scale Bµ and Xµ as follows (in order to prevent clutter we do not introduce new symbols for transformed quantities): B (1 + χ )1/2B ; g (1 + χ )−1/2g (5) µ 7→ 1 µ 1 7→ 1 1 X (1 + χ )1/2X ; g (1 + χ )−1/2g (6) µ 7→ 2 µ 2 7→ 2 2 In χ1 and χ2 the explicit dependence on the renormalization scale µ0 does not cancel. We 2 can, however, choose µ0 such that the shifts in Eqs. (5) and (6) well approximate the running of the couplings when evolving down from (Λ) to (λ). χ χ12 = χ21 shifts as follows under this transformation: L L ≡ χ (1 + χ )−1/2(1 + χ )−1/2χ χ 7→ 1 2 ≈ which ensures that the low-energy couplings ga(λ) rather than the high-energy couplings ga(Λ) appear in χ. To leading order χ, however, remains unchanged. According to Eq. (4) we find for χ:

5 g g m2(Φ) m2(Φ′) χ = Π(1)(Φ) Π(1)(Φ′)= β(1)(f) 1 2 ln ln − 12 − 12 − 12 · 8π2 µ2 − µ2 0 0 g g m2(Φ′) = 1 2 q (Φ)q (Φ) ln 12π2 1 2 m2(Φ) From this result we immediately see that one condition for successful kinetic mixing always is some splitting in the messenger mass spectrum. Finally, we arrive at the following Lagrangian:

(λ)= (λ)+ (λ) L Lkin. Lint. 1 1 χ 1 = Bµν B Xµν X Bµν X = T µν (7) −4 µν − 4 µν − 2 µν −4Fµν KF where: 1 χ T = (B ,X ) ; = Fµν µν µν K χ 1 Diagonalization and charge shifts

There are two possible linear transformations T1 and T2 that diagonalize the Lagrangian in Eq. (7) and leave one gauge ﬁeld invariant:

µν −1 1 0 1 χ Ta µν ; Ta Ta = ½ ; T1 = ; T2 = F 7→ F K 7→ K O(χ) χ 1 0 1 Either transformation results in a shift of one of the gauge charges. Before the diagonalization the gauge interaction Lagrangian of the fermions reads: Lf = g q (f)fγ¯ µfB g q (f)fγ¯ µfX Lf − 1 1 µ − 2 2 µ Diagonalizing the kinetic mixing matrix results in: K T : g q (f) g−1g χq (f) fγ¯ µfB g q (f)fγ¯ µfX 1 Lf → − 1 1 − 1 2 2 µ − 2 2 µ T : g q (f)fγ¯ µfB g q (f) g−1g χq (f) fγ¯ µfX 2 Lf → − 1 1 µ − 2 2 − 2 1 1 µ We find that transforming away the kinetic mixing operator leads to U(1)a mini-charges ǫa(f): g2 g1 T1 : ǫ1(f)= χq2(f) ; T2 : ǫ2(f)= χq1(f) −g1 −g2 −1 The minus signs are due to the fact that we have to use the inverse matrices Ta in order to express the initial gauge fields in terms of the transformed fields T . Fµν aFµν 3 Kinetic Mixing at and below the Electroweak Scale

We now identify the two gauge groups U(1)1 and U(1)2 studied in the previous section as the hypercharge gauge group of the Standard Model U(1)Y and some hidden U(1)X . Suppose a hidden Higgs ﬁeld h (1, 0, qX (h)) that carries non-zero U(1)X charge qX (h) acquires a VEV h far below the∼ electroweak scale H 174 GeV [6]: h i h i≈ hHi hhi SU(2) U(1) U(1) U(1) U(1) U(1) ; h H L × Y × X −→ EM × X −→ EM h i ≪ h i What phenomenology can we then expect in low-energy processes? We successively discuss kinetic mixing in the energy regimes H >µ> h and h > µ. h i h i h i

6 3.1 Kinetic mixing with an unbroken U(1)X U(1) U(1) kinetic mixing above the electroweak scale is described by the Lagrangian Y × X in Eq. (7) with Bµν and Xµν identiﬁed as the ﬁeld strengths of the U(1)Y and U(1)X gauge 1 bosons Bµ and Xµ. Once the Standard Model Higgs H (2, 2 , 0) obtains its VEV the Bµ boson is not a mass eigenstate any more and we have to∼ diagonalize the mass matrix of the electroweak gauge bosons. Bµ gets replaced by:

Bµ = cos θ Aµ sin θ Zµ w · − w · After integrating out the heavy weak gauge bosons the kinetic Lagrangian of the gauge fields reads as follows: 1 1 1 = F F µν X Xµν χ cos θ F Xµν (8) Lkin. −4 µν − 4 µν − 2 w µν where F now denotes the field strength of the photon A . Using the relation e g cos θ µν µ ≡ Y w the cosine of the electroweak mixing angle θw can be absorbed into the kinetic mixing parameter, χ cos θw = χe. The Lagrangian in Eq. (8) then obtains the same form as the U(1) U(1) mixing Lagrangian in Eq. (7). Thus, it describes the kinetic mixing between Y × X the visible photon Aµ and a hidden-photon Xµ. As long as the U(1)X remains unbroken we are free to diagonalize kin. at will. Any pair of orthogonal gauge fields will serve as a proper gauge boson basis.L For instance, we can perform the transformation T1 which shifts Xµ by χeAµ and leaves Aµ invariant.

T : A A ; X X + χ A 1 µ 7→ µ µ 7→ µ e µ

Deﬁning the physical gauge bosons in this way gives rise to mini-electric charges ǫe(φ) of ﬁelds φ in the hidden sector. This also applies to the hidden Higgs h which now couples to the visible photon via a mini-electric charge ǫe(h): g ǫ (h)= X χ q (h) e − e e X

Note that ǫe(h) is not only suppressed by χe but also depends on the ratio of the gauge couplings g and e. If g e as it is the case in some string scenarios where g is inversely X X ≪ X proportional to the volume of the extra dimensions ǫe(h) can become very small. In any case, if the hidden Higgs h carries a tiny electric charge astrophysical and cosmological bounds on mini-charged particles (MCP) apply and can be translated into bounds on χ:

e ǫmax. χ < e (9) e g q X X

MCP bounds can be derived for astrophysical environments such as white dwarfs and red giants. The theory of Big Bang Nucleosynthesis puts a cosmological bound on the parameter space of MCPs. The seminar talk next week will discuss these and other bounds in detail.

3.2 Spontaneous breaking of the hidden U(1)X

At all scales above the U(1)X breaking scale u the Lagrangian h of the hidden Higgs h reads as follows: L

= (D h)∗ (Dµh) V (h) ; D = ∂ ig q (h)X ; V (h)= µ2 h 2 + λ h 4 Lh µ − µ µ − X X µ − h | | h | | 2 Here Xµ denotes the original hidden-photon before the diagonalization of kin.. With µh > 0 the minimum of the potential V (h) is located at non-zero h: L

u 1 2 2 2 1 2 2 2 µ h = = µ /λh m =2µ ; mass = u g g (h)XµX h i √2 √2 h ⇒ h h L 2 X X q 7 2 iχegX iχ m igX − − e X

ig1 ig2 ∼ φ ∼ φ Aµ Aµ Xµ

µ −i µ J 2 2 J φ k −mX φ

µ Figure 3: Coupling of a fermionic hidden-sector current Jφ to the visible photon Aµ. We only indicate those factors that end up in the coupling prefactors of the respective amplitudes. The suppressed parts of the amplitudes are the same for both diagrams.

This calculation shows that we lose the freedom in deﬁning the physical gauge bosons once the U(1)X gets broken. In the broken phase the masses of the gauge bosons are not degenerate any more and the physical gauge bosons are identiﬁed as the mass eigenstates. In particular, the diagonalization of kin. by performing the transformation T1 results in a gauge boson basis with a non-diagonalL mass matrix: 1 T : m2 (X Xµ 2χ X Aµ) ; m2 = u2g2 q2 (h) (10) 1 Lmass → 2 X µ − e µ X X X

By contrast, the transformation T2 which leaves Xµ invariant diagonalizes kin. and, at the same time, preserves the diagonal form of the gauge boson mass matrix. IfL we stick to our diagonalization by means of T1 we have to deal with two conceptually different sets of gauge bosons: int. T ; mass T ; T = (A ,X ) Aµ ≡ 1Aµ Aµ ≡ 2Aµ Aµ µ µ T rotates the orginial gauge bosons into the interaction basis, that is, those gauge bosons 1 Aµ in terms of which we will formulate all interaction processes. T2 provides us with the mass eigenstates. With regard to the mass eigenstates hidden fermions φ only couple to the hidden-photon. They do not carry mini-electric charges which would imply the interaction with the massless visible photon. In the interaction basis, the hidden fermions, however, do receive small charges ǫe(φ) as we have concluded in our discussion of the unbroken U(1)X . It is easy to see that these two pictures of the hidden-sector photon interactions do not contradict each other. The non-diagonal mass Lagrangian in Eq. (10) leads to photon hidden-photon 2 oscillations with a coupling strength of g(Aµ Xµ) = χmX . These oscillations have to ↔ − µ be taken care of when calculating the coupling of a fermionic hidden-sector current Jφ to the visible photon Aµ. We now have to consider two diagrams, see Fig. 3. We find that the effective coupling geff. g1 + g2 is given as: φ ≡ φ φ

2 2 2 eff. χegX mX k 0 for k =0 g = χegX = χegX = φ − − k2 m2 − k2 m2 χ g for k2 m2 − X − X (− e X ≫ X 2 In the low-momentum regime when the hidden-photon mass mX is non-negligible the two diagrams in Fig. 3 exactly cancel. This is the anticipated result: In the broken phase the coupling of the hidden sector fields to the visible photon is swichted off. If the momentum 2 2 transfer k is, however, large compared to mX the U(1)X symmetry is essentially restored and our previous considerations for an unbroken U(1)X apply. The pratical conclusion from this calculation is that we can expect hidden sector fields to behave like MCPs even if the U(1)X symmetry is broken provided that the typical momentum transfer exceeds the mass of the hidden-photon. In other words: In astrophysical

8 environments with temperatures T k m2 hidden sector particles carry mini-electric ∼ | | ≫ X charges. In Eq. (9) we stated how one can constrain the possible χe values from bounds on MCPs. Now we see for what mass regions these bounds apply.

4 Kinetic mixing in Supersymmetric Theories

Mini-charged hidden-sector particles are an interesting phenomenological consequence of kinetic mixing below the electroweak scale. From the theoretical point of view kinetic mixing, however, also has dangereous implications such as the generation of large SUSY- breaking soft masses. We now supersymmetrize our description of kinetic mixing and discuss the resulting bounds on the mixing parameter χ.

4.1 Supersymmetrization of kinetic mixing Let us go back above the electroweak scale and again study the U(1) U(1) mixing. In Y × X the context of SUSY the existence of a hidden U(1)X that could mix the U(1)Y is a natural assumption. As mentioned earlier the MSSM requires the introduction of a hidden sector which could well contain a U(1)X gauge factor. In the supersymmetric case, the messenger fermions that we identified as being respon- sible for kinetic mixing fall into chiral supermultiplets. Thus not only fermionic messenger loops but also the corresponding scalar loops generate kinetic mixing. This, however, only results in a rescaling of all quantities that are derived from the gauge field wavefunction renormalization diagrams by a factor of 3/4: The messenger fermions are now represented by left-handed Weyl spinors. Their contribution to the gauge field vacuum polarization are thus half as large as in the non-supersymmetric case where the messenger fermions are described by four-component Dirac spinors. The contributions from the scalar messenger loops are in turn half as large as the fermion contributions. In total, we find for a toy model with two heavy messenger multiplets: 1 1 g g m2(Φ′) Π (k2) 1+ Π (k2) χ Y X q (Φ)q (Φ) ln ab → 2 2 ab ⇒ → 16π2 Y X m2(Φ) Before we continue let us estimate the magnitude of this mixing parameter for typical values of the couplings, charges and masses. We take both charges qY (Φ) and qX (Φ) to be ′ 2 2 1, set m(Φ )/m(Φ) to 10 and vary the couplings αY = gY /4π and αX = gX /4π between 1/60 α (M ) and 1/25 α (GUT). We find that: ≈ Y Z ≈ Y 10−3 . χ . 10−2 (11)

Later on we will have to compare this estimate with the theoretical bounds that one obtains from assuming particular SUSY-breaking mechanisms. As discussed in the previous section the hidden U(1)X might get broken. Now we diagonalize such that the resulting orthogonal ﬁelds will represent the mass basis once the Lkin. U(1)X is sponateneously broken. That is, we perform the transformation T2 and work with the (would-be) mass eigenstates. Consequently, not the hidden-sector but the MSSM supermultiplets ϕ receive charge shifts. After the diagonalization of they carry a mini-charge Lkin. ǫX (ϕ). Note that now not only the MSSM fermions, but also their scalar superparners couple to the hidden-photon. Moreover, to guarantee SUSY invariance of the total Lagrangian tot. gauge interactions with the U(1)X gaugino, the hidden-photino X˜µ [7], and D-term interactionsL must be introduced. contains the following terms [8]: Ltot. √2g q (f) f˜∗fX˜ + c.c. + g q (f)f˜∗fD˜ (12) Ltot. ⊃ − X X µ X X X {Xϕ,φ} h i Fig. 4 presents all possible interactions between the visible and the hidden sector.

9 Xµ X˜µ X˜µ

ϕ ϕ ϕ ϕ˜ ϕ˜ ϕ

Xµ

φ˜ φ˜

ϕ˜ ϕ˜ ϕ˜ ϕ˜

Figure 4: Supersymmetrized gauge interactions of a mini-charged MSSM supermultiplet (ϕ, ϕ˜) with the hidden-photon Xµ, the hidden-photino X˜µ and hidden scalar ﬁelds φ˜.

4.2 Kinetic mixing-mediated SUSY breaking in the MSSM

If the U(1)X is broken by the VEV φ˜ of some hidden-sector scalar ﬁeld φ˜ the D-term interactions in Eq. (12) generate soft SUSY-breaking masses [m(˜ϕ)] for the MSSM sfermions KM ϕ˜ [9]. We will now calculate [m(˜ϕ)]KM and compare it with the D-term contributions to the soft scalar masses [m(˜ϕ)]EWSB from electroweak symmetry breaking. We take the non-zero VEVs φ˜ to be of the usual order of SUSY breaking. That is why, in contrast to our discussion in the previous section, the hidden U(1)X now is broken above and not below the electroweak scale. First of all, we need an explicit expression for the hidden auxiliary ﬁeld DX . The classical equation of motion for DX has the following solution:

D = g q (f)f˜∗f˜ = g q (φ)φ˜∗φ˜ g ǫ (ϕ)˜ϕ∗ϕ˜ X − X X − X X − X X ϕ {Xϕ,φ} Xφ X

With the usual expression for ǫX (ϕ) we find that the sum over the MSSM fieldsϕ ˜ is proportional to the auxiliary D-field in the hypercharge gauge supermultiplet (Bµ, B˜µ,DY ): g g ǫ (ϕ)˜ϕ∗ϕ˜ = g Y χ g (ϕ)˜ϕ∗ϕ˜ = χg q (f)f˜∗f˜ = χ D − X X − X −g Y Y Y − · Y ϕ X ϕ X X {Xϕ,φ} The VEV of DX is determined by the hidden scalar fields which acquire non-zero VEVs: 2 D = g q (φ) φ˜ h X i − X X φ˜ 6=0 hXi After breaking of the hidden U(1)X we therefore obtain: D = D χ D + ... X h X i− · Y where the dots represent terms involving the hidden scalar fields that parametrize the fluc- tuations around the corresponding vacuum values. In the scalar potential V f,˜ f˜∗ this form of the hidden D-field now induces the follwing terms: 1 1 V f,˜ f˜∗ D D ( 2χ D D )= χ D g q (ϕ)˜ϕ∗ϕ˜ ⊃ 2 X X ⊃ 2 − h X i Y h X i Y Y ϕ X 10 We finally arrive at the anticipated result: Kinetic mixing in combination with the breaking 2 of the hidden U(1)X is able to generate scalar squared masses m (˜ϕ) KM which as such break the SUSY invariance of the Lagrangian: V f,˜ f˜∗ m2(˜ϕ) ϕ˜∗ϕ˜ ; m2(˜ϕ) = χg q (ϕ) D ⊃ KM KM Y Y h X i ϕ X In general kinetic mixing is, of course, not the only source for scalar squared masses. The usual mechanisms that are presumed to communicate SUSY breaking to the visible sector — gauge-mediated (GM) and supergravity-mediated (SUGRA) SUSY-breaking — also generate masses for the MSSM sfermions. Besides that, also the breaking of the electroweak symmetry results in soft masses.

2 2 2 2 2 m (˜ϕ) tot. = m (˜ϕ) GM + m (˜ϕ) SUGRA + m (˜ϕ) KM + m (˜ϕ) EWSB

The EWSB contributions originate from the non-zero VEVs of the auxiliary D-ﬁelds DY and DW 3 that are developed when the MSSM Higgs doublets Hu and Hd acquire their VEVs vu and vd:

1 2 2 H = v ; H = v D = D 3 = g v v h ui u h di d ⇒ h Y i − h W i 2 Y d − u These VEVs lead to soft masses in the scalar potential:

∗ 1 1 2 ∗ V f,˜ f˜ D D + D 3 D 3 m (˜ϕ) ϕ˜ ϕ˜ ⊃ 2 Y Y 2 W W ⊃ EWSB ϕ X where: m2(˜ϕ) = M 2 cos(2β) e(ϕ)cos2 θ q (ϕ) EWSB Z w − Y The joint contribution from kinetic mixing and electroweak symmetry breaking can then be written as: η m2(˜ϕ) + m2(˜ϕ) = M 2 cos(2β) e(ϕ)cos2 θ q (ϕ) 1 KM EWSB Z w − Y − cos(2β) Here we have introduced the parameter η as a measure for the relevance of the soft masses from kinetic mixing [10]: DX η gY χh 2 i ≡ MZ Let us briefly discuss the case of large η 1: In this scenario kinetic mixing is the dominant effect for the generation of soft SUSY-breaking≫ masses. The scalar masses are pulled up above MZ towards the hidden-sector SUSY-breaking scale DX . This destabilizes the gauge hierarchy. Furthermore, fields with negative hyperchargeh obtaini negative masses and thus experience tachyonic instabilities. They acquire non-zero VEVs and thereby break all gauge symmetries under which they are charged — a desaster! We conclude that U(1)Y U(1)X mixing with η 1 is not possible in the MSSM. Instead η must be kept small. The condition× η . 1 allows us≫ to constrain the mixing parameter χ:

M 2 χ . Z (13) g D Y h X i To get numerical bounds for χ we estimate D for models in which SUSY is either h X i broken by means of SUGRA or gauge mediation. The typical soft masses msoft that are generated by these two mechanisms can be estimated as follows:

D α2 2 D SUGRA: m h X i ; GM: m Y h X i soft ∼ M soft ∼ 4π M P Z

11 In order to avoid large loop corrections to scalar squared masses the soft masses msoft should be comparable to M . The SUSY-breaking scale D can thus be estimated to be: Z h X i α2 −2 SUGRA: D M M ; GM: D Y M 2 h X i∼ Z P h X i∼ 4π Z We plug these relations into Eq. (13) and ﬁnd:

M α2 2 1 SUGRA: χ . Z 10−16 ; GM: χ . Y 10−6 g M ∼ 4π g ∼ Y P Y Comparing these bounds with our naive estimate in Eq. (11) we ﬁnd that χ is apparently required to take on a value that is much smaller than one would expect. This constitutes a naturalness problem comparable to the strong CP problem that asks why the QCD vacuum angle θ¯ is extremely small in nature. One possible solution to this “χ problem” can be found in string theory. In some string scenarios extremely small χ values can be obtained quite naturally. The seminar talk in two weeks will tell us more about that. Another way out is to argue that χ is actually zero. This can happen in a variety of scenarios. For instance, if the hidden U(1)X were part of a larger non-Abelian gauge group G, the ﬁeld strength Xµν would not be a gauge-invariant quantity and the kinetic mixing operator could not be part of the Lagrangian.

U(1) G χ =0 X ⊂ ⇒

Also, if U(1)X originated from some non-Abelian gauge group G˜ that gets spontaneously ′ broken into U(1)X and some other non-Abelian group G˜ , the mixing parameter χ could, under circumstances, turn out to be zero at tree-level. The U(1)X generator would be a linear combination of the traceless G generators and hence be traceless itself. If the mass degeneracy of the G multiplets still persists after symmetry breaking, this tracelessness would lead to vanishing χ:

G˜ U(1) G˜′ + mass degeneracy χ = 0 (at tree-level) → X × ⇒ However, loop eﬀects would spoil the mass degeneracy and lead, after all, to non-zero χ. A more robust method to prevent kinetic mixing would be to introduce a U(1)X charge parity. Such a new symmetry would forbid us to use χ qX qY as a parameter in the Lagrangian from the beginning. Moreover, it would imply the∼ existence of conjugate fermion pairs with degenerate masses but diﬀerent charge signs which would also imply χ = 0.

References

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