Effective Action in a General Chiral Model: Next to Leading Order Derivative Expansion in the Worldline Method
aa r X i v : . [ h e p - t h ] O c t HD-THEP-07-19
Effective Action in a General Chiral Model: Next to LeadingOrder Derivative Expansion in the Worldline Method
Andres Hernandez (1) , ∗ Thomas Konstandin (2) , † and Michael G. Schmidt (1) ‡ (1) Institut f¨ur Theoretische Physik, Heidelberg University,Philosophenweg 16, D-69120 Heidelberg, Germany and (2)
Department of Theoretical Physics,Royal Institute of Technology (KTH), AlbaNova University Center,Roslagstullsbacken 21, 106 91 Stockholm, Sweden (Dated: October 30, 2018)We present a formalism to determine the imaginary part of a general chiral modelin the derivative expansion. Our formalism is based on the worldline path integralfor the covariant current that can be given in an explicit chiral and gauge covariantform. The effective action is then obtained by integrating the covariant current,taking account of the anomaly.
I. INTRODUCTION
When discussing the influence of fermions on the dynamics of some theory, e.g. of theStandard Particle Model (SM) in a cosmology setting, it is mandatory to integrate out thefermions. This procedure in one-loop order already becomes quite involved if arbitrary chiralcouplings of space-time dependent outer bosonic fields are considered. The extensive workof Salcedo [1, 2] resulted in an effective action to leading order in covariant derivatives ofsuch fields. Basis is a refined calculation in momentum space, handling of anomalies usingthe Wess-Zumino-Witten model (WZW), and last but not least, a very practical shorthandnotation. ∗ [email protected] † [email protected] ‡ [email protected]
Such effective actions are quite important in evaluating some models. For example, inref. [3] Smit discussed a form of ’cold’ electroweak baryogenesis at the end of electroweakscale inflation [4] which could very well work if the rephasing invariant J = s s s c c c sin( δ ) = (3 . ± . × − (1)of the Jarlskog determinant is not accompanied by further suppressions through mass ratios.It was proposed [3] that derivative terms in the effective action that are analytic in the time-dependent masses considered non-perturbatively could be very important in non-equilibrium.Such effects were also observed in ref. [5]. In the work [3] the fourth order derivative resultof ref. [1] turned out not to contain CP violation, but the claim was made that higher ordersof the imaginary part of the effective action will do.Worldline methods in first quantized quantum field theory are ideally adapted for cal-culating effective actions: One considers the propagation of a particle in some space-timedependent background [6], but in x-space path integral formulation [7]. This method [8, 9],also related to the infinite tension limit of String theory [10, 11], was used heavily for thediscussion of various effective actions in one-loop [9, 12, 13, 14] and two-loop [15, 16, 17, 18]order. For example, the high order in the inverse mass calculation of ref. [15] could hardlybe done with other methods.The present paper provides a formalism to determine higher order contributions to theimaginary part of the effective action using the worldline formalism. We are concerned withthe effective action of a multiplet of N Dirac fermions coupled to an arbitrary matrix-valuedset of fields, including a scalar Φ, a pseudoscalar Π, a vector A , a pseudovector B , and anantisymmetric tensor K µν . One peculiar feature of the imaginary part of the effective actionis that it cannot be written in a manifest chiral covariant way, due to the presence of thechiral anomaly. One possibility to arrive at a closed expression for the effective action is toabandon manifest chiral covariance as it was done in ref. [19]. The resulting expression israther complicated and not well suited for higher order calculations. Alternatively, it wasproposed in ref. [1] to determine the covariant current for which a manifestly chiral covariantexpression exists and to take account of the anomaly when integrating the current to yieldthe effective action. Following this idea, we present a worldline path integral formulation ofthe covariant current.Before we do so, we review the worldline formalism by discussing the derivation of thereal part of the effective action. A single Dirac fermion in the presence of both a scalar andpseudoscalar fields in the context of the worldline formalism was first treated in ref. [20],the inclusion of a pseudovector in ref. [21]. In our discussion of the real part of the effectiveaction we will follow the elegant subsequent work of refs. [19, 22].Section II contains the derivation of the real part of the effective action, the derivationof the covariant current and the matching procedure to obtain the imaginary part of theeffective action. In section III, we briefly reproduce the results in lowest order from ref. [1].As a novel result we present the imaginary part of the effective action in two dimensions innext to leading order in section IV. II. EFFECTIVE ACTION
We are concerned with the effective action i W [Φ , Π , A, B, K ] = log Det i [ i ∂ − Φ + i γ Π + A + γ B + i γ µ γ ν K µν ] , (2)and its continuation to Euclidean space. The γ matrices remain unaffected by the contin-uation, but it is useful to introduce the following notation, ( γ E ) j ≡ iγ j , ( γ E ) ≡ γ , and( γ E ) ≡ γ . After Wick-rotation, t → − it , one obtains with this new notation ∂ → i ∂ E , A → i A E , B → i B E , γ µ γ ν K µν → − ( γ E ) µ ( γ E ) ν K Eµν . (3)From now on, the E subscript will be suppressed.The effective action of Eq. (2) now reads − W [Φ , Π , A, B, K ] = log Det [ O ] , (4)with the operator O in momentum space defined by O ≡ 6 p − i Φ( x ) − γ Π( x ) − 6 A ( x ) − γ B ( x ) + γ µ γ ν K µν . (5)As in ref. [2, 22], the real and imaginary parts of the effective action are analyzed separately − W + − i W − = log ( | Det [ O ] | ) + i arg (Det [ O ]) . (6)A perturbative expansion in weak fields [22] shows that graphs with an even number of γ vertices are real, and graphs with an odd number of γ vertices are imaginary. This willprove useful when the behavior of the effective action under complex conjugation is exploredlater on. A. Real Part of the Effective Action
Our intention is to obtain a worldline representation for the effective action with manifestchiral and gauge invariance. This is unproblematic for the real part, but it causes certaindifficulties for the imaginary part due to the chiral anomaly. In order to familiarize the readerwith the worldline method we review the derivation for the real part in four dimensions asit was presented in refs. [19, 22].
1. Construction of a Positive Operator for the Real Part of the Effective Action
In order to use the worldline formalism, one has to rewrite the effective action in termsof a positive operator, thus obtaining W + = −
12 log Det [ O † O ] . (7)The problem with this operator is that it contains terms linear in the γ matrices, what makesthe transition to a path integral of Grassman fields problematic. One way to avoid thisproblem is by doubling the fermion system and exchanging the operator O for a Hermitianoperator Σ yielding W + = −
12 log Det [ O † O ] = −
14 log Det [Σ ] , Σ ≡ OO † . (8)Since Σ is Hermitian, one can use the Schwinger integral representation of the logarithmwithout any restrictions. One obtains W + = 14 Z ∞ dTT Tr exp( − T Σ ) . (9)At this point, it is natural to introduce six 8 × A matrices. These matricessatisfy { Γ A , Γ B } = 2 δ AB , with A, B = 1 .. µ = γ µ γ µ , Γ = γ γ , Γ = i l − i l . (10)For later use we also introduce the equivalent of γ ,Γ = − i Y A =1 Γ A = l − l , (11)and Γ anticommutes with all other Γ matrices.Expressing Σ in terms of these new matrices yields [19],Σ = Γ µ ( p µ − A µ ) − Γ Φ − Γ Π − i Γ µ Γ Γ B µ − i Γ µ Γ ν Γ K µν . (12)The aim is to turn Eq. (12) into an expression which is manifestly chiral covariant. Thiscan be achieved by changing to a basis in which i Γ Γ is diagonal [22] using the followingtransformation M − i Γ Γ M = l − l , M = l l l l . (13)In this basis, Σ takes the form˜Σ = M − Σ M = γ µ ( p µ − A Lµ ) γ ( − i H + γ µ γ ν K sµν ) − γ ( − i H † + γ µ γ ν K s † µν ) γ µ ( p µ − A Rµ ) , (14)which is manifestly chiral covariant. Here A L = A + B , A R = A − B , H = Φ − i Π, K s = K − i ˜ K and ˜ K µν = ǫ µνρσ K ρσ have been defined.The square of ˜Σ constitutes a positive operator which is suitable for the worldline formal-ism. However, even though this expression contains only even combinations of γ matrices,the coherent state formalism cannot yet be used to transform this expression into a fermionicpath integral. In the coherent state formalism, the γ matrices have to be rewritten as aproduct of the other γ matrices, what would result again in odd combinations. One possi-ble solution of this problem is to enlarge the Clifford space, replacing the γ matrices by Γmatrices γ A → Γ A = γ A ⊗ , A ∈ [1 . . . . (15)The matrix Γ is then independent from the other Γ matrices and the coherent state for-malism with six (instead of four) operators can be used. The doubling of the Clifford spaceinside the trace has to be compensated by a factor , such that Eq. (9) reads W + = 18 Z ∞ dTT Tr exp( − T ˆΣ ) , (16)and the operator ˆΣ is given byˆΣ = ( p − A ) + H + 12 K µν K µν + i µ Γ ν ( F µν + {H , K µν } + i [ K µρ , K ρν ])+ i Γ µ Γ ( D µ H + { p ν − A ν , K µν } ) −
12 Γ µρσ Γ D µ K ρσ −
14 Γ µνρσ K µν K ρσ , (17)with enlarged background fields defined by A µ = A Lµ A Rµ , H = i H − i H † , K µν = i K sµν − i K s † µν . (18)Γ A ...A k ≡ Γ [ A ... Γ A k ] denotes the anti-symmetrized product of k Γ matrices, and the field-strength and the covariant derivative have been defined as F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ , A ν ] , D µ χ = ∂ µ χ − i [ A µ , χ ] . (19)The ˆΣ operator is seen to be manifestly gauge and chiral covariant. It also contains Γmatrices to even powers only, and is well suited for the worldline path integral representation.
2. Worldline Path Integral
With the use of the coherent state formalism [22, 23], one can perform the transitionfrom Γ matrices to a path integral over Grassman fields ψ , with the correspondence Γ A Γ B → ψ A ψ B and Γ A Γ B Γ C Γ D → ψ A ψ B ψ C ψ D , as long as A , B , C , and D are all different. Thefinal form for the real part of the effective action is W + = 18 Z ∞ dTT N Z D x Z AP D ψ tr P e − R T dτ L ( τ ) . (20)Here N denotes a normalization constant coming from a momentum integration and APstands for antiperiodic boundary conditions, which must be fulfilled by the Grassman vari-ables ψ ( T ) = − ψ (0). The Lagrangian is given by L ( τ ) = ˙ x ψ A ˙ ψ A − i ˙ x µ A µ + H + 12 K µν K µν + 2 i ψ µ ψ ( D µ H + i ˙ x ν K µν )+ i ψ µ ψ ν ( F µν + {H , K µν } + i [ K µρ , K ρν ]) − ψ µ ψ ν ψ ρ (2 ψ D µ K µν + ψ σ K µν K ρσ ) . (21)The periodic boundary conditions for the field x ( τ ) suggest to separate the zero modes ofthe free field operator d dτ . The fields x ( τ ) are split into a constant part and a τ dependentpart according to x ( τ ) = x + y ( τ ), with ∂ τ x = 0 and R T dτ y ( τ ) = 0, and the measure inthe integral is changed into D x = D y d D x . The Green function is defined on a subspaceorthogonal to the zero modes. The ψ A fields contain no zero modes, so that the propagatorsfor the y ( τ ) and ψ A ( τ ) fields read h y ( τ ) y ( τ ) i = ( τ − τ ) T − | τ − τ | , h ψ A ( τ ) ψ B ( τ ) i = 12 δ AB sign ( τ − τ ) . (22)This formalism can then be used to determine the real part of the effective action as discussedin ref. [19]. B. Imaginary Part of the Effective Action
As in the case of the real part of the effective action, one requires a positive operator inorder to use the Schwinger trick. Even though this is still possible for the imaginary part,gauge and chiral invariance cannot be manifestly conserved due to the chiral anomaly. Forexample, in ref. [19, 22] a parameter α is introduced, which breaks the chiral invariance, butleads to a positive operator. However the resulting expression is not appropriate for higherorder calculations since the breaking of manifest chiral invariance leads to a large numberof contributions in the perturbative expansion of the path integral.The aim of the present work is to present a worldline representation of the effectivecurrent for which a manifestly chiral covariant expression exists. This current can then beintegrated to obtain the effective action [1, 24, 25]. This integration rather proceeds bymatching: First, a general effective action is proposed, which has the expected chiral andcovariant properties. The functional variation of this action is then matched to the covariantcurrent that is obtained using the worldline formalism. This method has the advantage thatit is both gauge and chiral invariant at each stage of the calculation. The anomaly only leadsto additional complications in the matching procedure of the lowest order contributions aswill be discussed in detail in the next section.Starting point of our analysis is the functional derivative of the imaginary part of theeffective action in Eq. (6) δW − = 12 δ (cid:0) log Det O − log Det O † (cid:1) = 12 Tr (cid:18) δ O O − δ O † O † (cid:19) . (23)This expression can be rewritten in terms of a positive operator which can be used to employthe worldline representation in combination with the heat kernel formula. Incidentally, itcan also be expressed in a manifestly chiral covariant form, what simplifies higher ordercalculations tremendously as compared to the formalism presented in ref. [19].
1. Construction of a Positive Operator for the Imaginary Part of the Effective Action
The expression in Eq. (23) can be transformed using the operator Σ defined in Eq. (8) δW − = 12 Tr δ O− δ O † / O † / O , (24)which, with the introduction of a new matrix χ , can be rewritten as δW − = 12 Tr χδ ΣΣ − , (25)with Σ = OO † , χ = l − l . (26)To produce the positive definite operator Σ in Eq. (25), we multiply and divide by Σ, usingthe cyclic property of the trace and the fact that Σ anticommutes with χ , to obtain δW − = 14 Tr ( χδ ΣΣ + Σ χδ Σ) Σ − = 14 Tr χ [ δ Σ , Σ] Σ − . (27)Since the last factor is a positive operator, it can be reexpressed as an integral, similar tothe expression of the real part of the effective action in Eq. (17), namely δW − = 14 Tr Z ∞ dT χ [ δ Σ , Σ] e − T Σ . (28)As in the case for the real part, the chiral covariance can be made manifest by changing toan appropriate basis. With the help of the matrix M in Eq. (13), one obtains again˜Σ = γ µ ( p µ − A µ ) − γ H − i γ µ γ ν γ K µν . (29)The additional factors χ [ δ Σ , Σ] read M − χM = ˜ χ = γ − γ = χ γ , (30)and for the case δ ˜Σ = − γ µ δ A µ h δ ˜Σ , ˜Σ i = − γ µν { δ A µ , p ν − A ν } − i D µ δ A µ − γ γ µ { δ A µ , H} + i γ γ µ [ δ A ν , K µν ] − i γ γ µλσ { δ A µ , K λσ } . (31)To use the coherent state formalism, it is again necessary to enlarge the Clifford algebraand to replace the γ matrices by Γ matrices. However, taking into account the factor γ in Eq. (30) the imaginary part of the effective action contains only odd combinations of γ matrices. Thus, the replacement γ A → Γ A = γ A ⊗ , A ∈ [1 . . .
5] (32)has to be compensated by a factor − i Γ = l ⊗ . (33)The overall factor Γ changes the boundary condition of the fermionic sector from antiperi-odic to periodic as explained in ref. [22]. This means that the fermionic sector contains zeromodes, which have to be separated in the same way as was done for the bosonic sector.Including the factor in Eq. (33) to compensate for the doubling of the Clifford space, oneobtains δW − = i Z ∞ dT Γ Γ χw ( T ) e − T ˆΣ , (34)where ˆΣ is given in Eq. (17), and the insertion due to the commutator yields w ( T ) = −
12 Γ Γ µν { δ A µ , p ν − A ν } − i Γ D µ δ A µ − Γ µ { δ A µ , H} + i Γ µ [ δ A ν , K µν ] − i µλσ { δ A µ , K λσ } . (35)To transform this expression into a worldline path integral, a similar procedure as for thereal part of the effective action can be followed. Products of Γ matrices can be replaced byGrassman fields, however in this case the Jacobian of the transformation contains additionalcontributions from the zero modes D θ D ¯ θ ≡ dθ dθ dθ d ¯ θ d ¯ θ d ¯ θ D θ ′ D ¯ θ ′ = 1 J dψ dψ dψ dψ dψ dψ D ψ ′ . (36)0The factor J only includes the Jacobian for the zero modes, while the Jacobian for theorthogonal modes is absorbed in the normalization of the correlation functions of the ψ ′ A . J can be calculated from the definition of the Grassman fields ψ in the coherent stateformalism [22] and yields in D dimension J = det (cid:18) ∂θ, ¯ θ∂ψ (cid:19) = ( − i ) ( D +2) / . (37)The final result can be expressed as δW − = 18 tr Z ∞ dT N Z D x Z P D ψ χw ( T ) P e − R T dτ L ( τ ) . (38)The Lagrangian is of the same form as in the real part, Eq. (21), L ( τ ) = ˙ x ψ A ˙ ψ A − i ˙ x µ A µ + H − K µν K µν + 2 i ψ µ ψ ( D µ H + i ˙ x ν K µν )+ i ψ µ ψ ν ( F µν + {H , K µν } ) − ψ µ ψ ν ψ ρ (cid:18) ψ D µ K µν + 12 ψ σ K µν K ρσ (cid:19) . (39)and the trivial integration over ψ can been carried out, so that the insertion yields w ( T ) = − i ψ ψ µ ψ ν δ A µ ˙ x ν − i ψ D µ δ A µ − ψ µ { δ A µ , H} +2 i ψ µ [ δ A ν , K µν ] − i ψ µ ψ λ ψ σ { δ A µ , K λσ } . (40)The normalization N coming from the momentum integration, satisfies N Z D xe − R T dτ ˙ x = (4 πT ) − D/ Z d D x. (41)The Green function for the bosonic field x is the same as for the real part of the effectiveaction, Eq. (22), while the Green function of the Grassman fields ψ A differs due to thepresence of the zero modes. The fermionic fields are split according to ψ A ( τ ) = ψ A + ψ ′ A ( τ ),with ∂ τ ψ A = 0 and R T dτ ψ ′ A ( τ ) = 0 and the measure turns into D ψ = dψ dψ dψ dψ dψ D ψ ′ .The Green function for the ψ ′ A fields, defined on a space orthogonal to the zero modes, reads D ψ ′ A ( τ ) ψ ′ B ( τ ) E = δ AB (cid:18)
12 sign( τ − τ ) − ( τ − τ ) T (cid:19) . (42)These results can be easily generalized to different dimensions. In two dimension, one obtainsan additional overall factor − i from the Jacobian of the zero modes and the fermionicmeasure reads D ψ = dψ dψ dψ D ψ ′ .1
2. The Effective Density
The effective density is obtained by varying with respect to the H field, so that δ ˜Σ = − γ δ H . In comparison to the worldline representation of the covariant current only theinsertion changes into h δ ˜Σ , ˜Σ i = − γ γ µ { δ H , p µ − A µ } + [ δ H , H ] + i γ µ γ ν [ δ H , K µν ] . (43)The corresponding insertion w ( T ) in the path integral reads then w ( T ) = − i ψ µ ˙ x µ δ H + 2 ψ [ δ H , H ] + 2 i ψ µ ψ ν [ δ H , K µν ] . (44)Since δ A carries an index, the effective current is of one order lower than the effective densityand usually results in less terms to calculate. The advantage of the effective density lies inthe matching process, since the factors in the effective density consist of the same type asfound in the effective action. They both combine the same type of object, DH and F , to thesame kind of order, while the effective current combines the terms to a lower order. Besides,there is no distinction between a consistent effective density and a covariant effective density,as there is for the effective current, as will be explained in the next section.
3. Distinction between the Consistent and the Covariant Current
With Eq. (38) an expression for the covariant current which is chiral and gauge covariantwas derived. This current cannot be the variation of the effective action, since the effectiveaction contains the chiral anomaly, and in fact the covariant current is not a variation ofany action. The reason for this is that performing the variation does not commute with theregularization procedure we used, namely the Schwinger trick. On the other hand, knowingthe chiral anomaly, one can reproduce the so-called consistent current that denotes the truevariation of the effective action.To explain the relation between the two currents, we define a general variation δ Y = Z dx Y aµ ( x ) δδ A µ ( x ) , (45)so that a gauge variation δ ξ is given by δ ξ = Z dx ( D µ ξ ) ( x ) δδ A µ ( x ) . (46)2Two subsequent variations have then the commutator [ δ Y , δ ξ ] = δ [ Y,ξ ] and in order to findthe transformation properties of the consistent current, one can apply this commutator tothe effective action [ δ Y , δ ξ ] W − [ A µ ] = δ [ Y,ξ ] W − [ A µ ] . (47)Using the anomalous Ward identity [26] δ ξ W − [ A µ ] = Z dx ξ ( x ) G [ A µ ]( x ) , (48)with G [ A µ ] denoting the consistent anomaly, one can evaluate both sides of Eq. (47) toobtain Z dx [ Y µ , ξ ]( x ) δδ A µ ( x ) W − [ A µ ] = δ Y Z dx ξ ( x ) G [ A µ ]( x ) − δ ξ Z dx Y µ ( x ) δδ A µ ( x ) W − [ A µ ] . (49)Defining the consistent current as the variation of the effective action h j µ ( x ) i = δδ A µ ( x ) W − [ A µ ] , (50)one finds Z dx Y µ ( x ) δ ξ h j µ ( x ) i = Z dx Y µ [ h j µ ( x ) i , ξ ]( x ) + Z dx ξ ( x ) δ Y G [ A µ ]( x ) . (51)Since Y was a general variation this leads to δ ξ h j µ ( x ) i = [ h j µ ( x ) i , ξ ] + Z dy ξ b ( y ) δδ A µ ( x ) G [ A µ ]( y ) . (52)This shows that only if the anomaly vanishes, the current transforms covariantly. Thisrelation can be used to determine the connection between the consistent current, i.e. thetrue variation of the action, and the covariant current. The latter is obtained by adding anobject P µ [ A µ ], called the Bardeen-Zumino polynomial [27], to the consistent current so thatthe sum transforms covariantly h ¯ j µ i = h j µ i + h P µ i . (53)This implies the following gauge transformation property for the BZ polynomial δ ξ P µ [ A µ ]( x ) = [ P µ [ A µ ] , ξ ] ( x ) − Z dy ξ ( y ) δδ A µ ( x ) G [ A µ ]( y ) . (54)3It is not obvious that such an object exists, but using P µ [ A µ ] = 148 π ǫ µνλσ tr χ ( A ν F λσ + F λσ A ν + i A ν A λ A σ ) , (55)and the consistent anomaly [26] G [ A µ ] = 124 π ǫ µνλσ tr χ ∂ µ (cid:18) A ν ∂ λ A σ − i A ν A λ A σ (cid:19) , (56)it can be shown that the definition of P µ in Eq. (55) provides a unique polynomial in A µ that satisfies Eq. (54). The corresponding functions in two dimensions are given by P µ = 14 π ǫ µν tr A ν , G [ A µ ] = 14 π ǫ µν tr χ ∂ µ A ν . (57)As stated above, the path integral in Eq. (38) constitutes a worldline representation ofthe covariant current. To obtain the imaginary part of the effective action from the covariantcurrent one can use the following ansatz W − = Γ gW ZW + W − c . (58)Here, Γ gW ZW is an extended gauged Wess-Zumino-Witten action [24, 28, 29], which is chosento reproduce the correct chiral anomaly, and W − c denotes a chiral invariant part. Thevariation of the functional Γ gW ZW , consists of a part that saturates the anomaly, namely theBZ polynomial, and a covariant remainder which has to be added to the variation of W − c toyield the covariant current.
4. The Wess-Zumino-Witten action
When the effective action is separated into two parts, it is required by the non-covariantpart that it reproduces the anomaly. It is well known that the WZW action has this property.The ungauged WZW action in four dimension is e.g. of the formΓ( U ) = i π Z Q d x ǫ abcde tr (cid:20) U − ∂ a UU − ∂ b UU − ∂ c UU − ∂ d UU − ∂ e U (cid:21) , (59)where Q is a five-dimensional space with boundary ∂Q equal to the R flat Euclideanspace. The matrix U is a unitary matrix, and is usually related to the case where themass can be expressed as a constant times that unitary matrix. We are interested in themore general case when the mass matrix is not of this form which is called extended WZW4action. In addition, the presence of the background gauge fields makes a gauging of theaction mandatory. The gauged extended WZW action can be generally expressed as theintegral in five dimensions [24]. Unlike the action itself, the resulting current turns out tobe a total derivative in five dimensions, such that it can be represented by an integral overthe physical four-dimensional space δ Γ gW ZW = 196 π Z d x ǫ µνλσ tr χ (cid:2) δ A µ (cid:0) −H − D ν HH − D λ HH − D σ H + D ν HH − D λ HH − D σ HH − − i (cid:8) H − D ν H − D ν HH − , F λσ (cid:9) + i H (cid:8) H − D ν H , F λσ (cid:9) H − − i H − (cid:8) D ν HH − , F λσ (cid:9) H− {A ν , F λσ } − i A ν A λ A σ )] , (60)or in two dimensions δ Γ gW ZW = 18 π Z d x ǫ µν tr χ (cid:2) δ A µ (cid:0) − i H − D ν H + i D ν HH − − A ν (cid:1)(cid:3) . (61)Notice that in both cases the last term in the current denotes the BZ polynomial. Theremaining chiral covariant terms have to be subtracted from the covariant current before itis matched to the effective action according to the ansatz made in Eq. (58). III. LOWEST ORDER EFFECTIVE ACTIONA. Effective covariant current
In order to reproduce the results from ref. [1], we neglect in this section the antisymmetricfield K µν . The fields A and H are matrices of some internal group, and we only assume that H ( x ) is nowhere singular. With this in mind, we restate our result Eq. (38) from the lastsection in D dimensions δW − = − i D/ Z ∞ dT N Z D x Z P D ψ χw ( T ) P e − R T dτ L ( τ ) , (62)with L ( τ ) = ˙ x ψ A ˙ ψ A − i ˙ x µ A µ + H + 2 i ψ µ ψ D µ H + i ψ µ ψ ν F µν ,w ( T ) = − i ψ ψ µ ψ ν δ A µ ˙ x ν − i ψ D µ δ A µ − ψ µ { δ A µ , H} . (63)Next, the derivative expansion of the heat kernel is used. In the derivative expansion termsare classified by the number of covariant indices that they carry, so that D µ H is of first order,5while F µν is of second order. The worldline formalism is well suited for this expansion, andthere are two major advantages compared to the more traditional methods used e.g. inref. [1]. First, the tedious manipulations using the γ algebra are avoided. Secondly, themomentum integration is omitted and replaced by the rather trivial integration in τ space.The coordinate is split as x ( τ ) = x + y ( τ ), and we work in the Fock-Schwinger gauge [30],in which A ( x ) · y = 0. In this gauge, expressions remain gauge covariant and the field A canbe expressed in terms of the field strength tensor F µν by A µ ( x ) = Z dα α F ρµ ( x + αy ) y ρ . (64)All background fields can then be expanded around the point x in terms of covariantderivatives X ( x + y ( τ )) = exp ( y ( τ ) · D x ) X ( x ) , (65)where D x refers to the covariant derivative in Eq. (19) with respect to x . With theexpansion of the field strength tensor in terms of covariant derivatives and Eq. (64), one canrewrite the field A as A µ ( x ) = 12 y ρ F ρµ ( x ) + 13 y α y ρ D α F ρµ ( x ) + 14 · y α y β y ρ D α D β F ρµ ( x ) + . . . . (66)Since we will not carry out the integration with respect to x we use the following notationin D dimensions h X i D = − (cid:18) i π (cid:19) D/ tr χ Z d D x X. (67)It is important to remember that χ and H anticommute; hence, when the cyclic property ofthe trace is used, a minus sign is generated, for example (cid:10) ǫ µνλσ HF µν H F λσ (cid:11) = − (cid:10) ǫ µνλσ F µν H F λσ H (cid:11) = − (cid:10) ǫ µνλσ H F λσ HF µν (cid:11) = − (cid:10) ǫ µνλσ H F µν HF λσ (cid:11) . (68)After expanding the mass field H ( x ) = H ( x ) + y µ D µ H ( x ) + . . . , the field H ( x ) istreated non-perturbatively. Since all the fields can be matrices of some internal space theresulting expressions normally cannot be expressed in closed form. For this case we use thelabeled operator notation laid down in ref. [2, 31]. The notation works as follows: In anexpression f ( A , B , . . . ) XY . . . , the labels of the operators A , B , . . . denote the position ofthat operator with respect to the remaining operators XY . . . . For instance, for the function6 f ( A, B ) = α ( A ) β ( B ), the expression f ( A , B ) XY represents α ( A ) Xβ ( B ) Y . In the caseat hand, the operator appearing in the functions is always m := H ( x ), such that generalfunctions f can be easily interpreted in the basis where m is diagonal. Using this notation,Eq. (68) can be recast as (cid:10) ǫ µνλσ HF µν H F λσ (cid:11) = (cid:10) ǫ µνλσ m m F µν F λσ (cid:11) = − (cid:10) ǫ µνλσ m m F µν F λσ (cid:11) = − (cid:10) ǫ µνλσ m m F µν F λσ (cid:11) = − (cid:10) ǫ µνλσ H F µν HF λσ (cid:11) . (69)This notation can also be used to simplify the matrix valued derivative. Using the definition( ∇ f )( m , m ) := f ( m ) − f ( m ) m − m , (70)it is possible to prove that D µ f ( m ) = ( ∇ f )( m , m ) D µ H . (71)For example, in the polynomial case f ( m ) = m one obtains D µ f ( m ) = D µ ( H ) = D µ H H + HD µ H H + H D µ H = ( m + m m + m ) D µ H = m − m m − m D µ H = ( ∇ f )( m , m ) D µ H . (72)As mentioned earlier, non-polynomial expressions are hereby interpreted in a basis where m is diagonal, so that for m = diag( d , . . . , d n ) f ( m ) − f ( m ) m − m X = f ( d i ) − f ( d j ) d i − d j X ij . (73)More general, this suggests the following definition for the case with several variables: ∇ k f ( m , . . . , m n ) = f ( m , . . . , ˆ m k +1 , . . . , m n ) − f ( m , . . . , ˆ m k , . . . , m n ) m k − m k +1 , (74)where ˆ m k indicates that the corresponding argument is left out.If all arguments of the functions are of the same type one can further simplify the notationand use subscripts to refer to the argument of the function, e.g. f ( m , m ) =: f and weemploy this notation in the following. Additionally, negative arguments will be denoted byunderlining the corresponding index, f ( − m , m ) =: f . More applications of the labeledoperator notation can be found in refs. [1, 2].7The path ordering in Eq. (62) is defined by P N Y i =1 Z T dτ i ≡ N ! Z T dτ Z τ dτ · · · Z τ N − dτ N = N ! Z T dτ · · · Z T dτ N N − Y i − θ ( τ i − τ i +1 ) . (75)Separating the Lagrangian Eq. (63) into L ( τ ) = L ( τ ) + H ( x ) + L ( τ ), with L ( τ ) = ˙ x ψ A ˙ ψ A , L ( τ ) = − i ˙ x µ A µ ( x ) + 2 i ψ µ ψ D µ H ( x ) + i ψ µ ψ ν F µν ( x ) + y µ D µ H ( x ) + . . . , (76)the terms of the expansion of H ( x ), except the leading term H ( x ), are attributed to L ( τ ),and treated perturbatively. Notice that L commutes with the rest of the Lagrangian, sothat the expansion of the path ordered exponential in Eq. (62) takes the form P e − R T dτ L ( τ ) = e − R T dτ L ( τ ) (cid:18) e − T H ( x ) + Z T dτ e − ( T − τ ) H ( x ) ( −L ( τ )) e − τ H ( x ) + Z T dτ Z τ dτ e − ( T − τ ) H ( x ) ( −L ( τ )) e − ( τ − τ ) H ( x ) ( −L ( τ )) e − τ H ( x ) + . . . (cid:19) . (77)When performing the ψ integrals, the zero modes have to be saturated and at least a factor ψ . . . ψ D ψ is required from the Grassman fields in order to contribute. The first term inEq. (77) lacks the appropriate ψ factor except in two dimensions, where the first term ofthe insertion Eq. (63) already has the appropriate factor. However it contains a factor ˙ x µ which must be contracted with a similar factor to form a Green function, hence it does notcontribute and can be left out. The rest of Eq. (77) can be simplified using the labeledoperator notation. Using the expression m n to denote H ( x ) in n th position, one obtains P e − R T dτ L ( τ ) = e − R T dτ L ( τ ) (cid:18) − Z T dτ e − T m − τ ( m − m ) L ( τ )+ Z T dτ Z τ dτ e − T m − τ ( m − m ) − τ ( m − m ) L ( τ ) L ( τ ) + . . . (cid:19) . (78)The evaluation of the worldline path integral can be summarized as follows: First, all fieldsin Eq. (78) and the insertion are expanded around x . Next, the functional integration overthe y fields is carried out, generating bosonic Green functions. Then, the ψ integrations areperformed saturating the zero modes and generating fermionic Green functions. Finally, the T and τ integrations are performed.Before presenting the actual calculation, we comment on the behavior of the effectiveaction under complex conjugation. As noted earlier, in any contribution to the imaginary8part of the action the field ψ appears an odd number of times. If one attributes a factor i to the operators F and δ A one observes that the remaining expressions in the currentin Eq. (63) are real. Accordingly, all expressions in W − are real as long as a factor i isattributed to the operator F . In addition, notice that the effective action has to be an evenfunction in the masses due to chiral invariance.In order to showcase the method, we present the lowest order calculation in two dimen-sions. The lowest order contribution coming from the first term in the insertion is givenby − i ψ ψ µ ψ ν ˙ y ν ( T ) Z T dτ e − T m − τ ( m − m ) y α ( τ ) D α H δ A µ ( T ) . (79)Performing the y and ψ integrals one obtains i (cid:28) ǫ µν ( m + m ) Z ∞ dTT Z T dτ e − T m − τ ( m − m ) ˙ g B ( T, τ ) D µ H δ A ν (cid:29) = i (cid:10) ǫ µν J ( m + m ) D µ H δ A ν (cid:11) . (80)The second term of the insertion does not contribute at lowest order since it is already ofsecond order in derivatives but lacks the appropriate fermionic factor to saturate the zeromodes. The third term of the insertion leads only to one contribution of the form − ψ µ { δ A µ , H} Z T dτ e − T m − τ ( m − m ) ( − i ψ ν ψ D ν H ) . (81)yielding − i (cid:28) ǫ µν ( m − m ) Z ∞ dTT Z T dτ e − T m − τ ( m − m ) D µ H δ A ν (cid:29) = − i (cid:10) ǫ µν J ( m − m ) D µ H δ A ν (cid:11) . (82)The factor ( m − m ) results from the anticommutator in Eq. (81), and the sign changein the cyclic property of the trace as explained in Eq. (68). The integrals J are given inAppendix A. The total current is hence given by δW − = − i (cid:10) ǫ µν A D µ H δ A ν (cid:11) , (83) A = 1 m − m − m m log ( m /m )( m − m )( m − m ) , (84)where the function A has been defined. This agrees with the results obtained in ref. [1].9 B. Effective Density
The effective density can be obtained similar as the covariant current, utilizing the inser-tion in Eq. (44). Neglecting the antisymmetric tensor K , the insertion is w ( T ) = − i ψ µ ˙ x µ δ H + 2 ψ [ δ H , H ] . (85)The contributions to the effective density are δW − = (cid:28) ǫ µν (cid:18) i (cid:0) J ( m + m ) + J ( m − m ) (cid:1) F µν − (cid:0) J ( m + m ) + J ( m + m ) − J ( m + m ) (cid:1) D µ HD ν H (cid:19) δ H (cid:29) = (cid:28) ǫ µν (cid:18) − i B F µν + B D µ HD ν H (cid:19) δ H (cid:29) . (86)where the functions B and B are given by B = − m + m − m m ( m + m )( m − m ) log (cid:18) m m (cid:19) , (87) B = B R + B L log( m ) + B L log( m ) + B L log( m ) , (88)with B R = 1( m − m )( m − m )( m + m ) , (89) B L = ( m + m m m )( m − m )( m + m )( m − m )( m + m ) , (90)in accordance with ref. [1]. C. Effective Action
We proceed and briefly present the derivation of the imaginary part of the effective actionfollowing ref. [1]. Using the ansatz in Eq. (58), the most general functional for W − c consistentwith chiral and gauge invariance in two dimensions reads W − c = h ǫ µν N D µ HD ν Hi . (91)An additional term proportional to F could be added but it can be removed by partialintegration. Notice that N is a real function according to the comments made in the lastsection.0The function N has some nontrivial restrictions. First of all, the function N is evenin m such that N ( − m , − m ) := N = N . (92)Because of the cyclic property of the trace one obtains N = N = N = N , (93)and due to the Hermiticity of W − N = − N = − N = − N . (94)Varying W − c [ A , H ] with respect to A , one obtains δW − c = − i h ǫ µν ( − m + m ) N ) D µ H δ A ν i . (95)Comparing this to Eq. (83) and adding the covariant contribution in Eq. (61) coming fromΓ gW ZW one has 1 m − m − m m log ( m /m )( m − m )( m − m ) = 12 m − m − m + m ) N , (96)which finally leads to N = 12 m m m − m (cid:18) log( m /m ) m − m − (cid:18) m + 1 m (cid:19)(cid:19) . (97)At higher order, the matching of the effective potential to the current potentially becomesmore intricate. On the other hand, the anomaly only contributes to the leading order,such that the knowledge of the covariant current (that in higher order coincides with theconsistent current) suffices to determine the effective action. D. Four Dimensions
For completeness, we also present the results for the effective action and the effectivecurrent in four dimensions. The matching procedure proceeds the same way as in ref. [1],and we do not repeat it here.The effective current in four dimensions consists out of three terms and reads δW − d =4 = − i (cid:28) ǫ µνλσ (cid:18) − i A F νλ D µ H − i A D µ HF νλ − A D µ HD ν HD λ H (cid:19) δ A σ (cid:29) , (98)1while the effective density can be written as δW − d =4 = (cid:28) ǫ µνλσ (cid:18) B F µν F λσ + i B F λσ D µ HD ν H + i B D µ HF λσ D ν H + i B D µ HD ν HF λσ − B D µ HD ν HD λ HD σ H (cid:19) δ H (cid:29) . (99)The functions A , A , A , B , B , B , B , and B are given in Appendix B. IV. NEXT TO LEADING ORDER EFFECTIVE ACTION IN TWODIMENSIONS
In this section we present as a novel result the imaginary part of the effective action innext to leading order and two dimensions. Even though the results are rather lengthy, theevaluation of the worldline path integral involves only very basic integrals such that it canbe easily implemented using a computer algebra system.In two dimensions and in next to leading order, the imaginary part of the effective actiontakes the form W c = (cid:28) ǫ µν (cid:18) Q D µ D α HD α D ν H + i P F µν D α D α H + e R D α D α HD µ HD ν H + i b R F µν D α HD α H + M D µ HD α HD ν HD α H ) i . (100)At next to leading order the action is chiral invariant and the effective action can hence beimmediately obtained by matching with the covariant current that in this order coincideswith the consistent current. These functions must have the following properties P = − P = P , Q = Q = − Q , (101) e R = − e R = e R , b R = b R = − b R , (102) M = M = − M = M . (103)We have chosen a rather general imaginary effective action at the required order whichpreserves gauge and chiral invariance, but we have included a larger number of terms thannecessary to perform the matching process with the effective current. In fact, the matching2process could be done with solely the functions Q , e R , b R , and M . Instead, we havedecided to include the additional term P , in order to have the option of simplifying theaction by a judicious choice of this extra function. For example, the extra function can beused to ensure that all functions remain finite at the coincidence limit, as will be explainedlater on.The calculation from the worldline formalism leads to the following contributions to thecovariant current δW c = − i ǫ µν (cid:10) I D µ D α D α H δ A ν + i I D α F µα δ A ν + I D α D α HD µ H δ A ν + I D µ HD α D α H δ A ν + I D µ D α HD α H δ A ν + I D α HD µ D α H δ A ν + i I F µα D α H δ A ν + i I D α HF µα δ A ν + I D µ HD α HD α H δ A ν + I D α HD µ HD α H δ A ν + I D α HD α HD µ H δ A ν (cid:11) . (104)The coefficient functions are given in Appendix C. In order to express the current in thisform, partial integration has been used to remove terms of the form D δ A . In addition,indices that are contracted with the ǫ tensor have been moved to the left, such that a termof the form D α D µ yields a sum of terms of the type D µ D α and F µα .The contributions from the variation of Eq. (100) can be grouped in three levels, withthe first level having only contributions from Q and P ; the second level from the previousones and e R and b R ; the last level with all functions. Adding the contributions from theworldline method and the variation of Eq. (100) one obtains for the first level the followingtwo equations P + ( m + m ) Q = I , ( m + m ) P − ( m − m ) Q = I , (105)which have the solution Q = I m − m − P m + m . (106)Besides, there arises the following restriction which is satisfied and can serve as a check forthe corresponding terms in the effective current( m + m ) I = − I , I = − I , I = I . (107)3The matching equations for the next level are − ∇ (( m + m ) Q − P ) + Q + Q + ( m + m )( − e R + e R ) = I , (108) ∇ (( m + m )( Q + Q )) − ( m + m ) e R − Q − Q + b R = I , (109) ∇ (( m + m ) P ) − ( m − m ) ∇ (( m + m ) Q ) − P + 2( m − m ) Q + ( m + m )( Q + 2 Q ) − b R ( m + m ) + ( m − m )( m + m ) e R = I , (110)and their complex conjugates.The first Eq. (108) is of the form e R − e R = − ˜ I m + m , (111)and a set of solutions to Eqs. (108) and (109) is hence given by e R = − e I m + m ! − e I m + m ! − e I m + m ! (112) b R = b I − ( m − m ) e R . (113)The functions e I and b I are hereby defined as e I = I + ∇ (( m + m ) Q − P ) − Q − Q , (114) b I = I − ∇ (( m + m )( Q + Q )) + 2 Q + 2 Q . (115)The last Eq. (110) leads to a constraint on the I functions that is given in Appendix C.The function e R possesses the required symmetries, and it reproduces the effective currentcorrectly, but it is not necessarily finite in the coincidence limit, m → − m . One wayof solving this problem is to choose the function P appropriately which up to this pointremained undetermined. Such a choice is e.g. given by P = I m + m , Q = 0 , (116)which leaves e I as e I = I − I ( m − m )( m − m ) + I ( m − m )( m − m ) . (117)With this choice, e R is finite in the coincidence limit, as can be checked explicitly, and since b I is also finite, so is b R .4For the last level, the following three equations hold ∇ b R − ( ∇ + ∇ ) (cid:16) ( m + m ) e R (cid:17) + 2 e R − m + m ) M = I , (118)( ∇ − ∇ ) (cid:16) e R ( m + m ) (cid:17) + ∇ b R − e R + 2 e R − m + m ) M = I , (119)( ∇ + ∇ ) (cid:16) ( m + m ) e R (cid:17) + ∇ b R − e R +2( m + m ) M ) = I . (120)One of these equations can be used to determine M , while the other two lead again toconstraints on the I functions. the sum of the three equations has the especially simple form − m + m ) M = − ( ∇ + ∇ + ∇ ) b R + I + I + I . (121)Since all previous functions in the effective action have been chosen finite in the coinci-dence limit, so is M . Eqs. (118) and (120) show that M is finite in the limit m → m ,while Eq. (119) shows that M is finite in the limit m → − m . This concludes the dis-cussion of the next to leading order contributions in two dimensions. V. CONCLUSIONS
We presented a worldline formalism to calculate the imaginary part of the covariant cur-rent of a general chiral model in the derivative expansion and our results are best summarizedby Eqs. (62) and (63).The resulting covariant current can be used to reproduce the imaginary part of theeffective action by integration or matching. The advantage of this approach, compared toexplicit formulas of the effective action, as given e.g. in ref. [19], consists in the manifest chiralcovariance. Chiral covariance reduces the number of possible contributions to the currentenormously and makes even next to leading order calculations manageable as demonstratedin section IV in the case of two dimensions. Besides the chiral covariance, the use of theworldline formalism is essential in our approach. The evaluation of the worldline pathintegral requires neither performing Dirac algebra nor integrating over momentum space, incontrast to the more traditional methods used in ref. [1].5In principle, it is possible to use the presented formalism to determine the effective CPviolation resulting from integrating out the fermions of the Standard model. For example,in next to leading order in four dimensions, an operator of the form
DHDHF F could arisefrom the CP violation in the CKM matrix. Since the mass terms of the fermions are treatednon-perturbatively in the derivative expansion, the resulting effective CP-violating operatoris not necessarily proportional to the Jarlskog determinant. The discussion of this questionis postponed to a forthcoming publication.
Acknowledgments
T.K. is supported by the Swedish Research Council (Vetenskapsr˚adet), Contract No. 621-2001-1611. A.H. is supported by CONACYT/DAAD, Contract No. A/05/12566.
APPENDIX A: INTEGRALS USED IN THE CALCULATION
In this section, the function g denotes the bosonic Green function g ( T, τ ) = h y ( T ) y ( τ ) i , (A1)and ˙ g ( T, τ ) = h ˙ y ( T ) y ( τ ) i = − h ψ A ( T ) ψ A ( τ ) i , (A2)where the last expression does not contain a summation over the index A .
1. Integrals in Two Dimensions
In the calculation in two dimensions the following integrals have been used J = Z ∞ dTT Z T dτ e − T m − τ ( m − m ) = log( m /m ) m − m ,J = Z ∞ dTT Z T dt e − T m − τ ( m − m ) ˙ g ( T, τ )= − m − m + ( m + m )( m − m ) log (cid:18) m m (cid:19) ,J = Z ∞ dTT Z T dt e − T m − τ ( m − m ) ˙ g ( T, τ ) g ( T, τ )= − m + m ( m − m ) + ( m + 4 m m + m )( m − m ) log (cid:18) m m (cid:19) . (A3)6The remaining occurring integrals can be expressed as J = Z ∞ dTT Z T dτ e − T m − τ ( m − m ) g ( T, τ ) = J m − m ,J = Z ∞ dTT Z T dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) = − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ ) = − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ ) = − J . (A4)
2. Integrals in Four Dimensions
In four dimensions the following integrals with three indices are used J = Z ∞ dTT Z T dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) = m log( m )( m − m )( m − m ) − m ( m − m ) log( m )( m − m )( m − m )( m − m )+ m log( m )( m − m )( m − m ) ,J = Z ∞ dTT Z T dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − m ( m − m )( m − m ) + m ( m − m m ) log( m )( m − m ) ( m − m ) − m m log( m )( m − m ) ( m − m ) + m m log( m )( m − m ) ( m − m ) ,J = Z ∞ dTT Z T dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ ) = − J . (A5)7The integrals with four indices can be expressed as J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) = − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − J . (A6)Finally, the integrals with five indices read J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) = − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − ∇ J m + m ,J = Z ∞ dTT Z T dτ Z τ dτ Z τ dτ Z τ dτ e − T m − P i =1 τ i ( m i +1 − m i ) ˙ g ( T, τ )= − ∇ J m + m . (A7)8 APPENDIX B: RESULTS IN FOUR DIMENSIONS
In this section, we give the coefficient functions for the effective current and the effectivedensity in four dimensions introduced in section III D.The functions of the covariant current are given by A = m m − m m − m m ( m + m )( m − m )( m − m )+ m ( m ( m − m ) − m m ) log[ m /m ]( m + m )( m − m )( m − m )( m − m )+ m ( m ( m − m ) + 2 m m ) log[ m /m ]( m + m )( m − m )( m − m )( m − m ) , (B1) A = − A , (B2)and A = A R + A L log[ m ] + A L log[ m ] + A L log[ m ] + A L log[ m ] , (B3)with A R = m m m − m m m + m m m − m m m ( m − m )( m + m )( m − m )( m − m )( m + m )( m − m ) ,A L = m ( − m − m m m + m m m − m m m + 2 m m m )( m − m )( m + m )( m − m )( m − m )( m − m )( m − m ) . (B4)The explicit functions occurring in the effective density are rather lengthy and hence wedisplay them in terms of the integrals presented in the last section. B = J ( m + m ) − J ( m − m ) − J ( m − m ) ,B = J ( m + m ) − J ( m − m ) − J ( m + m ) + J ( m + m ) ,B = J ( m + m ) − J ( m + m ) − J ( m − m ) + J ( m + m ) ,B = B ,B = J ( m + m ) − J ( m + m ) + J ( m + m ) − J ( m + m ) + J ( m + m ) . (B5)9 APPENDIX C: COVARIANT CURRENT IN NEXT TO LEADING ORDER
In this section we summarize the contributions to the covariant current in next to leadingorder. For the first two levels, they are given by I = I m − m , I = − m − m ) J + 4( m + m ) J ,I = m − m + 2 m m − m ∇ ( I ) + 8 m + m m − m ∇ ( m m J )+16 m m J m − m − I ( m − m )( m − m ) ,I = 2 ∇ ( I ) m − m + 3 I ( m − m )( m − m ) − I − I m − m ,I = ( m + m ) (cid:18) ∇ I m + m (cid:19) − ∇ I m + m ! + 3 m − m m + m ∇ I +4( m + m )( m ( m − m − m ) + m m ) (cid:18) ∇ J m + m (cid:19) − ∇ J m + m ! − m + m ) (cid:18) ∇ ( m m J ) m + m (cid:19) − ∇ ( m m J ) m + m ! +4( m − m ) (cid:18) ∇ (( m − m m + m ) J ) m + m − ∇ (cid:0) ( m − m ) J (cid:1)(cid:19) , (C1)and the relations I = − I , I = − I , I = I . (C2)0The contributions to the last level read I = − I ,I = − m + m m − m I + 12 (cid:18) ∇ + m + m m − m ( ∇ ) (cid:19) I + (cid:18) ∇ + ∇ + m + m m − m ( ∇ ) (cid:19) m + m m − m ( I − ( ∇ I ) )+ 12 (cid:18) ∇ + ∇ + m + m m − m ( −∇ + ∇ + ∇ ) (cid:19) ( I − ∇ I )+ 12 (cid:18) −∇ + ∇ + m + m m − m ( ∇ ) (cid:19) ( I − ( ∇ I ) ) − m + m m − m )( m − m ) ( I − ( ∇ I ) ) + 2 m + m ( I − ( ∇ I ) ) − m − m ) ( I − ( ∇ I ) ) ,I = ∇ I − I m − m − m − m ) J ( m − m )( m − m ) − m + m ) J m − m − J m − m − m + m m − m (cid:18) ∇ − m + m m + m ∇ (cid:19) (cid:18) ∇ (( m − m )( I − m m J ))( m − m )( m + m ) (cid:19) + ( m + m ) ( ∇ ( I − m m J )) ( m − m )( m − m ) + 4( m + m )( m − m ) m + m )( m − m ) ∇ J − m + m ) m − m (cid:18) −∇ + m + m m + m ∇ (cid:19) ( m + m ) J − ( m − m ) J m − m − m + m )( m − m )( m + m ) m + m ∇ ( ∇ J ) m − m − m + m ) m + m ∇ J − m + m )( m + m ) m − m ∇ J . (C3)All functions I are finite in the coincidence limit and must fulfill the following constraintsdue to the behavior of the terms in the effective action under cyclic permutation and complex1conjugation:( m + m ) I m − m + I + I ( m − m )( m − m ) + I ( m − m )( m − m ) − ( m + m ) I ( m − m )( m − m ) − I ( m − m )( m − m ) = 0 ,I + I − ( m + m ) I m − m + I + I + I m − m + I ( m − m )( m − m ) − I ( m − m )( m − m ) = 0 , ( m + m ) I + ( m + m ) I − I + 2 I m − m + (3 m − m + 2 m ) I ( m − m )( m + m ) = 0 , (C4)and I = I = 0 , I = I = 0 , I = − I = 0 . (C5) [1] L.L. Salcedo. ”Derivative expansion for the effective action of chiral gauge fermions. Theabnormal parity component”. Eur.Phys.J. , C20:161, 2001. (hep-th/0012174).[2] L.L. Salcedo. ”Derivative expansion for the effective action of chiral gauge fermions. Thenormal parity component”.
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