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Renormalization and two loop electroweak corrections to lepton anomalous dipole moments in the standard model and beyond (I): heavy fermion contributions

Applying effective Lagrangian method and on-shell scheme, we analyze the electroweak corrections to anomalous dipole moments of lepton from some special two loop diagrams in which a closed heavy fermion loop is attached to the virtual gauge bosons or Higgs fields. As the masses of virtual fermions in inner loop are much heavier than the electroweak scale, we verify the final results satisfying the decoupling theorem explicitly if the interactions among Higgs and heavy fermions do not contain the nondecoupling couplings. At the decoupling limit, we also present the leading corrections to lepton anomalous dipole moments from those two loop diagrams in some popular extensions of the standard model, such as the fourth generation, supersymmetry, universal extra dimension, and the littlest Higgs with T-parity.


I. INTRODUCTION
At both aspects of experiment and theory, the magnetic dipole moments ( For the electroweak corrections, the standard one loop contribution amounts to 19.5 × 10 −10 , and the one loop corrections from new physics sector are generally suppressed by Λ 2 EW /Λ 2 NP . Here Λ EW denotes the electroweak energy scale, and Λ NP denotes the energy scale of new physics. Comparing with the analysis at one loop level, the two loop analysis is more complicated and less advanced. Utilizing the heavy mass expansion approximation (HME) together with the projection operator method, the authors of Ref. [4] have evaluated the two loop standard electroweak corrections to muon MDM. Within the framework of CP conservation, Ref. [5] presents the supersymmetric corrections from some special two-loop diagrams where a closed chargino (neutralino) or scalar fermion loop is inserted into those two-Higgs-doublet one-loop diagrams. Ref. [6] discusses the contributions to muon MDM from the effective vertices H ± W ∓ γ, h 0 (H 0 )γγ which are induced by the scalar quarks of the third generation. Furthermore, the contributions from two loop Bar-Zee-type diagrams to the electric dipole moments (EDMs) of light fermions are discussed extensively in literature [7].
In this paper, we calculate the corrections to the anomalous dipole moments of lepton from some special diagrams in which a closed heavy fermion loop is attached to the virtual electroweak gauge or Higgs fields. The effective Lagrangian method can yield the one loop electroweak corrections to lepton MDMs and EDMs exactly in the SM and beyond, and has been adopted to calculate the two loop supersymmetric corrections for the branching ratio of b → sγ [8], neutron EDM [9] and lepton MDMs and EDMs [10,11]. In concrete calculation, we assume that all external leptons and photon are off-shell, then expand the amplitude of corresponding triangle diagrams according to the external momenta of leptons and photon.
Using loop momentum translating invariance, we formulate the sum of amplitude from those triangle diagrams corresponding to same self energy in the form which explicitly satisfies the Ward identity required by the QED gauge symmetry, then get all dimension 6 operators together with their coefficients. After the equations of motion are applied to external leptons, higher dimensional operators, such as dimension 8 operators, also contribute to the lepton MDMs and EDMs in principle. However, the contributions of dimension 8 operators contain the additional suppression factor m 2 l /Λ 2 EW comparing with that of dimension 6 operators, where m l is the mass of lepton. Setting Λ EW ∼ 100GeV, one obtains easily that this suppression factor is about 10 −6 for the muon lepton. Under current experimental precision, it implies that the contributions of all higher dimension operators (D ≥ 8) can be neglected safely.
We adopt the naive dimensional regularization with the anticommuting γ 5 scheme, where there is no distinction between the first 4 dimensions and the remaining D − 4 dimensions.
Since the bare effective Lagrangian contains the ultraviolet divergence which is induced by divergent subdiagrams, we give the renormalized results in the on-mass-shell scheme [12]. Additional, we adopt the nonlinear R ξ gauge with ξ = 1 for simplification [13]. This special gauge-fixing term guarantees explicit electromagnetic gauge invariance throughout the calculation, not just at the end because the choice of gauge-fixing term eliminates the γW ± G ∓ vertex in the Lagrangian. This paper is composed of the sections as follows. In section II, we introduce the effective Lagrangian method and our notations. We will demonstrate how to obtain the identities among two loop integrals from the loop momentum translating invariance through an example, then obtain the corrections from the relevant diagrams to the lepton MDMs and EDMs. Section III is devoted to the analysis and discussion in some concrete electroweak models. In section IV, we give our conclusion. Some tedious formulae are collected in the appendices.

II. THE CORRECTIONS FROM THE RELATING DIAGRAMS
The lepton MDMs and EDMs can actually be written as the operators L M DM = e 4m l a ll σ µν l F µν , Here, σ µν = i[γ µ , γ ν ]/2, l denotes the lepton fermion which is on-shell, F µν is the electromagnetic field strength, m l is the lepton mass and e represents the electric charge, respectively.
It is convenient to get the corrections from loop diagrams to lepton MDMs and EDMs in terms of the effective Lagrangian method, if the loop diagrams contain the virtual fields which are much heavier than the external lepton, i.e. m V ≫ m l with m V denoting the mass scale of virtual fields. Since / p ′ = / p = m l ≪ m V for on-shell leptons and / k → 0 ≪ m V for photon, we can expand the amplitude of corresponding triangle diagrams according to the external momenta of leptons and photon. The two loop diagrams also contain some virtual light freedoms generally, such as virtual neutrinos, charged leptons or photon, and it is unsuitable to expand the propagators of light freedoms in powers of external momenta obviously. In order to obtain the corrections to lepton MDM and EDM properly, we should firstly match the amplitude of two loop diagrams from full theory to that of corresponding diagrams from effective theory which is composed by the QED Lagrangian and some higher dimension operators of light freedoms, then extract the Wilson coefficients of those high dimension operators which are only depend on the masses of virtual heavy freedoms as well as the possible matching scales. Finally, we strictly analyze the amplitude of corresponding diagrams from effective theory to obtain the contributions from the virtual light freedoms to lepton MDMs and EDMs. As discussed in the section I, it is enough to retain only those dimension 6 operators in later calculations: Certainly, all dimension 6 operators in Eq.(3) induce the effective couplings among photons and leptons. The effective vertices with one external photon are written as If the full theory is invariant under the combined transformation of charge conjugation, parity and time reversal (CPT), the induced effective theory preserves the symmetry after the heavy freedoms are integrated out. The fact implies the Wilson coefficients of the operators O ∓ 2,3,6 satisfying the relations where C ∓ i (i = 1, 2, · · · , 6) represent the Wilson coefficients of the corresponding operators O ∓ i in the effective Lagrangian. After applying the equations of motion to external leptons, we find that the concerned terms in the effective Lagrangian are transformed into Here, ℜ(· · ·) denotes the operation to take the real part of a complex number, and ℑ(· · ·) denotes the operation to take the imaginary part of a complex number. Applying Eq. (2) and Eq. (6), we finally get In other words, the MDM of lepton is proportional to real part of the effective coupling C + 2 + C − * 2 + C + 6 , as well as the EDM of lepton is proportional to imaginary part of the effective coupling C + 2 + C − * 2 + C + 6 . After expanding the amplitude of corresponding triangle diagrams, we extract the Wilson coefficients of operators in Eq.(4) which are formulated in the linear combinations of one and two loop vacuum integrals in momentum space, then obtain the MDMs and EDMs of leptons. Taking those diagrams in which a closed heavy fermion loop is inserted into the propagator of charged gauge boson as an example, we show in detail how to obtain the MDMs and EDMs of leptons through the effective Lagrangian method.
A. The corrections from the diagrams where a closed heavy fermion loop is inserted into the self energy of W ± gauge boson In order to get the amplitude of the diagrams in Fig.1, one can write the renormalizable interaction among the charged electroweak gauge boson W ± and the heavy fermions F α,β in a more universal form as where the concrete expressions of ζ L,R αβ depend on the models employed in our calculation.
The conservation of electric charge requires Q β − Q α = 1, where Q α,β denote the electric charges of the heavy fermions F α,β respectively.
Applying Eq.(8), we write firstly the amplitude of those two loop diagrams in Fig.1. For example, the amplitude of Fig.1(a) can be formulated as Here Λ RE denotes the renormalization scale that can take any value in the range from the electroweak scale Λ EW to the new physics scale Λ NP naturally, and we adopt the shortcut notations: c w = cos θ w , s w = sin θ w , with θ w denoting the Weinberg angle. Additionally, p, k are the incoming momenta of lepton and photon fields, ρ is the Lorentz index of photon.
Certainly, the amplitude does not depend on how to mark the momenta of virtual fields for the invariance of loop momentum translation. It can be checked easily that the sum of amplitude for diagrams in Fig.1 satisfies the Ward identity required by the QED gauge where A ww,ρ denotes the sum of amplitudes for the diagrams (a), (b), (c) and (d) in Fig.1, as well as Σ ww denotes the amplitude of corresponding self energy diagram, respectively.
According the external momenta of leptons and photon, we expand the amplitude in Eq.(9) as since we only consider the corrections to lepton MDM and EDM from dimension 6 operators.
After applying those identities derived from loop momentum translating invariance, we formulate the sum of the amplitude for Fig.1(a Since only the operators O ∓ 2,3,6 actually contribute to the MDMs and EDMs of leptons when the equations of motion are applied to the incoming and outgoing leptons separately, the relevant terms in the effective Lagrangian are formulated as: where Q f = −1 represents the charge of leptons, and the expressions of form factors N (i) ww (i = 1, 2, 3, 4) are presented in appendix.
Integrating over loop momenta, one gets the following terms in the effective Lagrangian: where G F = 1.16639 × 10 −5 GeV −2 is the 4-fermion coupling, and α e = e 2 /4π. Note that the above result does not depend on the concrete choice of energy scale Λ, and the concrete The charged gauge boson self energy composed of a closed heavy fermion loop induces the ultraviolet divergence in the Wilson coefficients of effective Lagrangian, the unrenormalized W ± self energy is generally expressed as where the form factors A w 0,1,2 and B w 1,2 only depend on the virtual field masses and renormalization scale. Here, we omit those terms which are strongly suppressed at the limit of heavy virtual fermion masses. The corresponding counter terms are given as The renormalized self energy is given bŷ For on-shell external gauge boson W ± , we have [12] Σ w µν (p, m w )ǫ ν (p) where ǫ(p) is the polarization vector of W ± gauge boson. Inserting Eq. (24) and Eq. (25) into Eq. (27), we derive the counter terms for the W ± self energy in on-shell scheme as We should derive the counter term for the vertex γW + W − here since the corresponding coupling is not zero at tree level. In the nonlinear R ξ gauge with ξ = 1, the counter term where k i (i = 1, 2, 3) denote the incoming momenta of W ± and photon, and µ, ν, ρ denote the corresponding Lorentz indices respectively.
We present the counter term diagrams to cancel the ultraviolet divergence contained by the bare effective Lagrangian in Fig.1(e,f,g), and we can verify that the sum of corresponding amplitude satisfies the Ward identity required by the QED gauge invariance obviously.
Accordingly, the effective Lagrangian from the counter term diagrams is written as Adding the counter terms to bare Lagrangian Eq.(23), we cancel the ultraviolet divergence there. However, the diagrams in Fig.1 include the virtual neutrino which belongs to light freedoms contained by the effective Lagrangian. It is unreasonable obviously in the above analysis that the propagators of virtual neutrino are expanded according to the external momenta. In order to obtain the corrections to lepton MDMs and EDMs from light freedoms properly, we match the sum of amplitude from full theory to that from effective theory [15] at first: which only depend on the masses of virtual fields. It should be clarified that the corrections to lepton EDMs from the diagrams (a,b,c,d) in Fig.1 do not depend on the concrete renormalization scheme adopted here since the relevant terms from bare Lagrangian do not contain the ultraviolet divergence. Using the expansion of Φ(x, y, z) in Eq(.20), we get the asymptotic expressions of T i (x, z, u), (i = 1, · · · , 4) at the limit z, u ≫ x as This implies that the leading contributions contained in the asymptotic form of Eq.32 under the assumption m F = m Fα = m F β ≫ m w can be written as: where ellipses represent those relatively unimportant corrections.
Comparing the result in Eq.(32), the contributions from the corresponding diagrams contain the additional suppressed factor m 2 l /Λ 2 EW when both of virtual charged gauge bosons in Fig.1(a,b, Similarly, the renormalizable interaction among the electroweak charged Goldstone/Higgs G ± (H ± ) and the heavy fermions F α,β can be expressed in a more universal form as among the charged Goldstone/Higgs and leptons are written as where the parameter B c also depends on the concrete models adopted in our analysis. In full theory, the couplings in Eq.(35) induce the corrections to lepton MDMs and EDMs through the diagrams in Fig.3, and the corresponding diagram of effective theory is same as that presented in Fig.2.
After the steps taken in W W sector, the corresponding corrections from the diagrams in Fig.3 to the lepton MDMs and EDMs are formulated as The expressions of form factors F i (x, y, z, u) (i = 1, · · · , 4) can be found in appendix.
Using the asymptotic formulae of form factors F i , (i = 1, · · · , 4) under the condition z, u ≫ x, y, we simplify the expressions of Eq.(37) in the limit m F = m Fα = m F β ≫ m w as: The results indicate that the corrections to a l , d l from the diagrams in Fig.3  In order to get the amplitude of the diagram in Fig.4, we write the renormalizable interaction among the electroweak neutral gauge boson Z and the heavy fermions F α,β in a more universal form as where the expressions of ξ L,R αβ depend on the concrete models employed in our calculation, and the CPT symmetry requires ξ L αβ = ξ L * βα , ξ R αβ = ξ R * βα . It is easy to check that the divergent amplitude of Fig.4(a) satisfies the Ward identity: where p, k are the incoming momenta of lepton and photon fields, ρ is the Lorentz index of photon, A (a) zz,ρ denotes the sum of amplitude for the diagrams in Fig.5(a), and Σ zz denotes the amplitude of corresponding self energy diagram, respectively. After a tedious calculation, the bare Lagrangian from the diagram Fig.4(a) can be written as where T Z f = −1/2 represents the isospin of charged leptons. Generally, the unrenormalized self energy of the weak gauge boson Z can be written as Here, we omit those terms which are strongly suppressed at the limit of the large virtual fermion masses, and those form factors are actually decided by the masses of virtual fields and the renormalization scale. Correspondingly, the counter terms are given as The renormalized self energy is expressed aŝ For on-shell external gauge boson Z, we have [12] Σ z µν (p, m z )ǫ ν (p) where ǫ(p) is the polarization vector of neutral gauge boson. From Eq.(46), we get the counter terms at electroweak scale in on-shell scheme: Accordingly, the effective Lagrangian from the counter term diagrams is written as As a result of the preparation mentioned above, we use the contributions from the counter term diagram in Fig.4(b) to cancel the ultraviolet divergence in Eq.(42). After matching the amplitude from Fig.4(a, b) to that from the diagram of effective theory (Fig.5(a)) and analyzing the corrections from light freedoms properly, we get the theoretical prediction on the lepton MDMs as In fact, the four-fermion interaction does not induce the corrections to lepton MDMs and EDMs in Fig.5(a). Using the asymptotic expression of T 5 (x, z, u) at the limit z, u ≫ x: where ellipsis represents those relatively unimportant corrections.
In a similar way, we can verify the identity zz,ρ denotes the sum of amplitude for the diagrams in Fig.4(c,d). After the steps adopted above, the corrections from Fig.4(c,d) to lepton MDMs and EDMs are formulated as As a closed heavy fermion loop is inserted into neutral or charged gauge boson propagator, the counter terms to self energy diagrams of Z or W gauge bosons induce the renormalization for Weinberg angle where the dots indicate again nonleading contributions. Furthermore, ∆ρ F denotes the 1loop corrections from the heavy fermions to the ρ parameter, which appears in the ratio of weak neutral to charged current amplitudes and comparisons of the Z and W ± masses.
Using the above equation, one can express the corresponding counter terms for the vertices Zll as Inserting the counter terms into one loop standard model diagrams, we finally obtain In principle, we should consider the corrections from the counter terms for W +ν l l: However, the corrections can be absorbed in the one-loop result as we parametrize the final result in terms of G F determined from the muon's lifetime.
The contributions from the corresponding diagrams contain the additional suppressed factor m 2 l /Λ 2 EW when both of virtual neutral gauge bosons in Fig.4 are replaced with the neutral Goldstone G 0 . Nevertheless, we should consider the corrections from those two loop diagrams in which one of virtual neutral gauge bosons is replaced with the neutral Goldstone  Similarly, the renormalizable interaction among the electroweak neutral Goldstone/Higgs G 0 (h 0 , A 0 ) and the heavy fermions F α,β can be expressed in a more universal form as where the concrete expressions of G N αβ , H E αβ and H O αβ depend on the models employed in our calculation, and the conservation of electric charge requires Q β = Q α . Generally, the couplings among the neutral Goldstone/Higgs and leptons are written as and the parameters B E , B O depend on the concrete models adopted in our analysis. In full theory, the couplings in Eq.(58) induce the corrections to lepton MDMs and EDMs from the diagrams in Fig.6. Accordingly, the corresponding diagram of effective theory is presented in Fig.5(b). In principle, we should also consider the corrections from those two loop diagrams in which one of virtual Z gauge bosons is replaced with the neutral Goldstone/Higgs Fig.4(a). However, the sum of amplitude from those two loop diagrams does not contribute to the lepton MDMs and EDMs when one of virtual neutral gauge bosons in Fig.4(a) is replaced with the neutral Goldstone/Higgs G 0 (h 0 , A 0 ) [11].
After the steps taken in W W sector, we formulate the corresponding corrections to the lepton MDMs and EDMs from the diagrams in Fig.6 as The expressions of form factors F i (x, y, z, u) (i = 5, 6) can be found in appendix. In the heavy mass limit m F = m Fα = m F β ≫ m h , m A , m w , we have E. The corrections from the diagrams where a closed heavy fermion loop is attached to the virtual γ and Z bosons When a closed fermion loop is attached to the virtual γ and Z gauge bosons (Fig.7), the corresponding diagrams of effective theory are presented in Fig.8. Taking the steps above, one can get the tedious correction to the effective Lagrangian. If we ignore the terms which are proportional to the suppression factor 1 − 4s 2 w , the correction from this sector to the lepton MDMs from this sector is drastically simplified as and the correction to the lepton EDMs is zero. In the limit m F = m Fα ≫ m z , we approximate the correction to the lepton MDMs from this sector as As a closed fermion loop is attached to the virtual neutral Higgs and photon fields, a real photon can be emitted from either the virtual lepton or the virtual charginos in the self energy diagram. When a real photon is emitted from the virtual charginos, the corresponding "triangle" diagrams belong to the typical two-loop Bar-Zee-type diagrams [16]. Within the framework of CP violating MSSM, the contributions from two-loop Bar-Zee-type diagrams to the EDMs of those light fermions are discussed extensively in literature [7]. When a real photon is attached to the internal standard fermion, the correction from corresponding triangle diagram to the effective Lagrangian is zero because of the Furry theorem, this point is also verified through a strict analysis. Replacing the virtual neutral gauge boson with photon in Fig.6, one obtains the relevant diagrams in full theory. Meanwhile, the diagram of effective theory is same as that presented in Fig.8(a). After the steps adopted above, the corresponding corrections to lepton MDMs and EDMs from this sector are expressed as Note here that the corrections from this sector to the MDM of lepton depend on real parts of the effective couplings H E αα and H O αα , and the corrections from this sector to the EDM of lepton depend on imaginary parts of the effective couplings H E αα and H O αα . In the limit m F = m Fα ≫ m h , m A , the above expressions are simplified as Similarly, the corrections to the lepton MDMs and EDMs from the γG sector are: The corrections from this sector to the MDM of lepton are proportional to real parts of the effective couplings G N αα , and the corrections from this sector to the EDM of lepton are proportional to imaginary parts of the effective couplings G N αα , separately. In the limit m F = m Fα ≫ m z , we have which are suppressed by the masses of heavy fermions.  (Fig.9). Adopting the same steps in W W sector, we formulate the corrections from Fig.9 to lepton MDM and EDM respectively as: which coincide with the well known results in Ref. [24]. For masses m Fα ≥ 100GeV, these corrections from this sector to the muon MDM a µ are below 10 −13 and hence negligible. In this section, we will present the leading terms from two loop corrections to a l within some electroweak models discussed extensively in literature.

A. the SM
Within the framework of SM, only top quark belongs to the fermion which mass is heavier than that of weak gauge bosons. The couplings in Eq. (8) and Eq.(40) are respectively expressed as Here, the 1-loop corrections to ρ-parameter from the heavy fermions of 4th generation can be written as In this extension of SM, the additional possible heavy fermions include two charginos χ ± 1,2 and four neutralinos χ 0 i (i = 1, · · · , 4) [17]. The couplings among weak gauge bosons and heavy fermions are given as Here, the 4×4 matrix N denotes the mixing matrix of neutralinos, two 2×2 matrices U L , U R denote the left-and right-handed mixing matrices of charginos, respectively. In the limit of heavy masses, the mass spectra of charginos and neutralinos are respectively approached as Here, µ H represents the µ parameter in superpotential, and m 2 , m 1 denote the soft breaking masses of SU(2) × U(1) gauginos,respectively.
Applying Eq.(74) and Eq.(75), we get the two loop corrections to lepton MDMs in the the 1-loop corrections to ρ-parameter from the heavy supersymmetric fermions can be written as [18] ∆ρ SU SY = α e 16πs 2 The abbreviation symbols c β = cos β, s β = sin β with tan β = υ 2 /υ 1 denoting the ratio between the absolute values of two vacuum expectations: υ 1 , υ 2 . As 1 ≤ tan β ≤ 60 and m F ∼ 1TeV, the contributions from this sector to muon MDMs is well below 10 −11 which can be ignored safely.
At large tan β, the dominant two-loop supersymmetric corrections to lepton anomalous dipole moments are originated from those Bar-Zee type diagrams which are analyzed extensively. To obtain the corrections from those sectors, we formulate the relevant couplings as with Z E is the mixing matrix of two CP even Higgs. Assuming |µ H | = |m 2 | = |m 1 | = m F and θ 1 = θ 2 = θ µ , we expand the effective couplings in Eq.39 and Eq.65 in powers of m w /m F and get where θ 1,2 = arg(m 1,2 ), θ µ = arg(µ H ) are the corresponding CP violating phases. Applying the above equations and B E = B O = B C = tan β, we find Here, the form factors A, B, C, D depend on the masses of higgs and the mixing matrix of neutral CP-even higgs.
The two loop corrections to lepton MDMs and EDMs are proportional to 1/m 2 F in large tan β limit, which are consistent with the result presented in Ref. [5] qualitatively. Using HME approximation and projection operator method, Ref. [5] approaches the corrections to the muon MDM from two loop diagrams in which a closed chargino/neutralino loop is inserted into those two Higgs doublet one loop diagrams as under the assumption µ H = m 2 = (3c 2 w /5s 2 w )m 1 = m A = M SUSY and large tan β limit, where m A denotes the mass of neutral CP-odd Higgs. To compare our result with that presented in Ref. [5] numerically, we take the same assumption µ H = m 2 = (3c 2 w /5s 2 w )m 1 = m A = M SUSY on the parameter space of supersymmetry. In addition, the existence of a CP-even SM like Higgs with mass above 115 GeV sets a strong constraint on the parameter space of the employed model. To address this problem, we include all loop corrected effects in the Higgs potential [25], and choose the Yukawa couplings of the 3rd generation sfermions as A t = A b = A τ = M SUSY . Considering those points above, we present the numerical results in Fig.10. For large tan β case, our numerical results agree with the approximation presented in Eq.(81) very well. It implies that the equation Eq.(81) fits the exact result perfectly for large tan β and µ H = m 2 = (3c 2 w /5s 2 w )m 1 = m A = M SUSY .