Appendix A Riemann Curvature Tensor - Springer978-3-319-01842-3/1.pdf · 244 Appendix A: Riemann...

Appendix ARiemann Curvature Tensor

We will often use the notation (...),μ and (...);μ for a partial and covariant derivativerespectively, and the [anti]-symmetrization brackets defined here by

T[ab] := 1

2(Tab − Tba) and T(ab) := 1

2(Tab + Tba) , (A.1)

for a tensor of second rank Tab, but easily generalized for tensors of arbitrary rank.We can define the Riemannian curvature tensor in coordinate representation by

the action of the commutator of two covariant derivatives on a vector field vα

[∇μ,∇ν] vρ = Rρσμν vσ, (A.2)

with the explicit formula in terms of the symmetric Christoffel symbols �ρμν

Rρσμν = ∂μ�ρσν − ∂ν�

ρσμ + �ραμ�

ασν − �ραν�

ασμ. (A.3)

From this definition it is obvious that Rρσμν possesses the following symmetries

Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ . (A.4)

In addition, there is an identity for the cyclic permutation of the last three indices

Rα[βγδ] = 1

3(Rαβγδ + Rαδβγ + Rαγδβ) = 0. (A.5)

An arbitrary tensor of fourth rank in d dimension has d4 independent components.Since a tensor is built from tensor products, we can think of Rαβγδ as being composedof two (d × d) matrices A1

αβ and A2γδ . From (A.4), it follows that A1 and A2 are

antisymmetric, each having n(n − 1)/2 independent components. Writing Rαβγδ ≡RA1 A2 , where we have collected the index pairs (αβ) → A1 and (γδ) → A2,this corresponds to a matrix, in which each index A1 and A2 labels d(d − 1)/2

C. F. Steinwachs, Non-minimal Higgs Inflation and Frame Dependence 243in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3,© Springer International Publishing Switzerland 2014

244 Appendix A: Riemann Curvature Tensor

components. Taking into account the symmetry under pairwise exchange of (αβ)

and (γδ), we can consider RA1 A2 as a symmetric matrix in A1 and A2, havingA1(A1 + 1)/2 independent components. Altogether, we find

1

2

[1

2d (d − 1)

] {[1

2d (d − 1)

]+ 1

}= 1

8d4 − 1

4d3 + 3

8d2 − 1

4d (A.6)

independent components. We still need to figure out how many components arerelated by the cyclic identity (A.5) in d dimensions. We can artificially write

Rαβγδ = 1

8

[Rαβγδ − Rαβδγ + Rβαδγ − Rβαγδ + (αβ) ↔ (γδ)

](A.7)

and similar expressions for Rαγδβ and Rαδβγ . Inserting these expressions into (A.5)leads to the condition R[αβγδ] = 0. This totally antisymmetric object is identicalzero for two identical indices but gives one additional constraint for each choice offour distinct, orderless indices, reducing the number of independent components byone respectively. In d dimensions, the number of additional constraints correspondsto “choose 4 out of d”. For integers d, n there exists a product formula to calculatethe binomial (

dn

)=

n∏k=1

d − n + k

k. (A.8)

Inserting n = 4, we obtain

(d4

)= 1

24[d (d − 1) (d − 2) (d − 3)] = 1

24d4 − 1

4d3 + 11

24d2 − 1

4d. (A.9)

Thus, the number of independent components IRiem(d) of Rαβγδ is given by

IRiem(d) := 1

4d (d − 1)

[1

2d (d − 1) + 1

]−

(d4

)= 1

12d2 (d2 − 1). (A.10)

We consider IRiem(d) for different dimensions d:

IRiem(1) = 0, IRiem(2) = 1, IRiem(3) = 6, IRiem(4) = 20. (A.11)

This shows that gravity in one dimension is trivial, since there is no dynamical degreeof freedom. It further shows that gravity in two dimensions can be described by theRicci scalar R and in three dimensions by R(3)

μν = R(3)νμ . In general, the d-dimensional

Ricci tensor Rμν := gγδRγμδν has IRic(d) := 12 d (d + 1) independent components.

In particular IRic(4) = 10. Thus, ten components of Rαβγδ not contained in Rμν stillremain. They are contained in the Weyl tensor

Appendix A: Riemann Curvature Tensor 245

Cαβγδ := Rαβγδ − 2

d − 2

(gα[γRδ]β − gβ[γRδ]α

) + 2

(d − 1) (d − 2)gα[γgδ]β R.

(A.12)It follows that the Weyl tensor has (for d > 3)

IWeyl(d) = IRiem(d) − IRic(d) = 1

12d (d + 1) (d + 2) (d − 3) (A.13)

independent components, and in particular IWeyl(4) = 10.

Appendix BVariations and Derivatives

B.1 Covariant Differentiation in General Relativity

The action of the metric compatible covariant derivative ∇μ gαβ = 0 with respect tothe Christoffel symbol �

ρμν on a general tensor is defined as

∇μ T α...γβ...δ = ∂μ T α...γ

β...δ + �αμρ T ρ...γβ...δ + ... + �γμρ T α...ρ

β...δ

− �ρμβ T α...γ

ρ...δ − ... − �ρμδ T α...γ

β...ρ, (B.1)

with the symmetric Christoffel symbol

�αμν(g) := 1

2gαρ

(∂ν gμρ + ∂μ gνρ − ∂ρ gμν

)= �ανμ(g). (B.2)

For scalar functions ∇μϕ = ∂μϕ. For vector fields vμ, a useful formula is

∇μ vμ = 1√g∂μ

(√g vμ

)or

√g∇μ vμ = ∂μ

(√g vμ

). (B.3)

B.2 Functional Derivative

We use again the condensed DeWitt notation for a generalized field φi = φA(x),introduced in Sect. 4.4. We want to emphasize that an object with two DeWitt indiceslike �i j ′ corresponds to a two-point function or a generalized bi-tensor �AB(x, x ′)in conventional notation. The primed indices like j ′ refer to the space-time point x ′.This means we can construct objects which behave as tensors of different rank atdifferent space-time points. A particular case of a generalized bi-tensor is a bi-scalarwhich has no indices A, B, .. at all. A special bi-scalar, in turn, is the Dirac Delta-Distribution δ(x, x ′) which is defined in a 2ω-dimensional curved space-time withmetric gμν(x) for a general test field � A(x) by the equation


http://dx.doi.org/10.1007/978-3-319-01842-3_4

248 Appendix B: Variations and Derivatives

� A(x) =∫

d2ωx ′ |g(x ′)|1/2 δ(x, x ′)� A(x ′), (B.4)

For simplicity, we specify 2ω = 4. In the condensed notation (B.4) takes the form� i = δi

j ′�j ′ with

δij ′ = |g(x ′)|1/2 δA

B δ(x, x ′) = δAB δ(x, x ′) . (B.5)

The quantity δ(x, x ′) := |g(x ′)|1/2δ(x, x ′) is no longer a bi-scalar, since it transformsas a scalar at the point x , but as a scalar density at the point x ′. The expansion ofZ [�] up to linear order in condensed notation is defined by

Z [� + ζ] =: Z [�] + Z, i [�] ζ i , (B.6)

with ζ i := δψi and

Z, i [�] ζ i =∫

d4x ′ δZ [�(x)]δ� A(x ′)

ζ A(x ′) (B.7)

in conventional notation. Therefore, it seems obvious to define the functional deriv-ative Z, i = δZ [�(x)]/δ� A(x ′) by the variation [3]

limε→0

1

ε(Z [� + εζ] − Z [�]) =:

∫d4x ′ δZ [�(x)]

δ� A(x ′)ζ A(x ′). (B.8)

For the special case Z = Id, we obtain from (B.8)

ζ A =∫

d4x ′ δ� A(x)

δ�B(x ′)ζB(x ′) or

δ� A(x)

δ�B(x ′)= δA

B δ(x, x ′). (B.9)

B.3 Lie Derivative

The Lie derivative Lξ of an arbitrary tensor T γ...δα...β in the direction of the vector ξμ is

a measure of the difference between T γ′...δ′α′...β′ (x ′) “dragged” along the integral curve

γξ of ξμ compared to the tensor T γ...δα...β (x ′) at this point. Calculating the infinitesimal

flow of T γ′...δ′α′...β′ (x ′) along γξ back from x ′ to x with the infinitesimal coordinate

transformation matrix ∂xμ′/∂xν = δ

μν + ε ξ

μ, ν yields

T γ′...δ′

α′...β′ (x ′) = T γ...δα...β + ε(ξγ,ρT ρ...δα...β + ... + ξδ,ρT γ...ρα...β − ξ

ρα, T γ...δρ...β − ... − ξ

ρβ,

T γ...δα...ρ

).

(B.10)

Taylor expansion of T γ...δα...β (x ′) around x yields

Appendix B: Variations and Derivatives 249

T γ...δα...β (x + ε ξ) = T γ...δ

α...β (x) + ε T γ...δα...β, ρ ξ

ρ. (B.11)

Since the right-hand sides of (B.10) and (B.11) involve only quantities at x , we cancompare these two objects at x and define the Lie derivative as the difference

(Lξ T)γ...δα...β

(x) := limε→0

T γ...δα...β (x ′) − T γ′...δ′

α′...β′ (x ′)ε

= T γ...δα...β, ρ ξ

ρ − T ρ...δα...β ξ

γ,ρ − ... − T γ...ρ

α...β ξδ,ρ

+ T γ...δρ...β ξ

ρα, + ... + T γ...δ

α...ρ ξρβ,

. (B.12)

A special case is the Lie derivative of the metric tensor. The vanishing of the Liederivative

(Lξ g)μν = 0 in direction ξμ signalizes a symmetry of the space-timemanifold and means that the vector ξμ is a generator of an isometry

(Lξ g)μν = gμν, ρ ξρ + gρν ξ

ρ,μ + gνρ ξ

ρ, ν = ξμ; ν + ξν;μ = 0. (B.13)

The vector fields ξμ obeying (B.13) are called Killing vector fields.

B.4 Variation of Metric Quantities

In the case of pure gravity, � i = gμν(x) and ζ i = hμν(x), we find from (B.9)

δgμν(x)

δgαβ(x ′)= δαβμν δ(x, x ′) with δαβμν := δα(μδ

βν) = 1

2

(δαμδ

βν + δαν δ

βμ

). (B.14)

By using (B.8) with F(�) = F(g) = (gμν)−1 and gμν := (gμν)

−1, it follows

δg gμν = −gμ(α gβ)ν hαβ . (B.15)

By using (B.8) with F(g) = g1/2 and the identity det(g) = exp{

tr[log(gμν)]}, we

find

δg g1/2 = 1

2g1/2 gαβ hαβ . (B.16)

By using (B.4) with an arbitrary test function t (x) �= 0 that does not depend on gμνand differentiating both sides, we find

∫d4x

[δ

δgμν(y)δ(x, x ′)

]t (x ′) = 0 or

δ

δgμν(y)δ(x, x ′) = 0. (B.17)

Combining (B.14) with (B.17) leads to

250 Appendix B: Variations and Derivatives

δ2gμν(x)

δgαβ(y) δgγδ(z)= 0 or δ2

g gμν = δg hμν = 0. (B.18)

With the basic results (B.15), (B.16) and (B.18), we can construct the variation ofmore complicated objects. Using the fact that the operations of variation and partialdifferentiation commute, we calculate the variation of the Christoffel symbol

δg�ρμν = − 1

2gραgδβ

(∂ν gμδ + ∂μ gνδ − ∂δ gμν

)hαβ

+ 1

2gρδ

(∂ν hμδ + ∂μ hνδ − ∂δ hμν

). (B.19)

Using (B.1), we rewrite the ∂μ’s acting on the hμν’s in terms of the ∇μ’s

δg�ρμν = − gρα �βμν hαβ + 1

2gρδ

(∇ν hμδ + ∇μ hνδ − ∇δ hμν

+ �ενμhεδ + �ενδhμε + �εμνhεδ + �εμρhνε − �εμρhεν − �ενρhμε)

= − gρα �βμν hαβ + 1

2gρδ

(∇ν hμδ + ∇μ hνδ − ∇δ hμν + 2�ενμhεδ

)

= 1

2gρδ

(∇ν hμδ + ∇μ hνδ − ∇δ hμν) = 1

2gρδ

(hμδ; ν + hνδ;μ − hμν; δ

).

(B.20)

Then, the variation of the Riemann tensor yields

δgRρσμν = δg

(∂μ�

ρνσ − ∂ν�

ρμσ + �ρμα�

ανσ − �ρνα�

αμσ

)= (δg�

ρνσ),μ − (δg�

ρμσ), ν + (δg�

ρμα)�

ανσ + (δg�

ανσ)�

ρμα

− (δg�ρνα)�

αμσ − (δg�

αμσ)�

ρνα. (B.21)

Using (B.1), we can again rewrite the partial derivative in terms of a covariant deriv-ative plus terms proportional to the connection

(δg�ρνσ),μ = (δg�

ρνσ);μ − (δg�

ανσ)�

ραμ + (δg�

ρασ)�

ανμ + (δg�

ρνα)�

ασμ. (B.22)

Substituting this expression (and the same term interchanging μ and ν) in (B.21),we find that all terms proportional to (δg�)� cancel identically. Thus, the variationof the Riemann tensor yields

δgRρσμν = (δg�ρνσ);μ − (δg�

ρμσ); ν . (B.23)

For the variation of the Ricci tensor, we contract (B.23) over ρ and μ

Appendix B: Variations and Derivatives 251

δμρ (δgRρσμν) = δg(δμρ Rρσμν) = δgRρσρν = (δg�

ρνσ); ρ − (δg�

ρρσ); ν . (B.24)

Substituting the variation (B.20), we obtain

δgRμν = (δg�ρνμ); ρ − (δg�

ρμρ); ν = 1

2gρδ

(hμρ; νδ + hνδ;μρ − hμν; ρδ − hρδ;μν

).

(B.25)Finally, the variation of the Ricci scalar can ultimately be obtained from (B.15) and(B.25) by the relation

δg R = (δg gμν) Rμν + gμν (δg Rμν). (B.26)

Appendix CYoung Tableaux for SU(3)

The most efficient way to calculate direct products of different representations andtheir dimensionality is to use the technique of Young tableaux. This is an abstractsymbolical notation for projecting out the symmetric and antisymmetric componentsof a tensor. A linear representation ρg of a group element g ∈ G can be written asa multilinear form (a tensor). Each index runs over the dimension of the groupparameters from 1, ..., N . Everything we will work out can easily be generalized toSU (N ), but we are mainly interested in SU (3). We have to specify several conditionsand rules which define a valid diagram and its combination with other diagrams. Thefundamental representation 3 is represented as a single box

3 = . (C.1)

This corresponds to a tensor Ai with one contravariant index (for SU (3) all indicesrun from 1, ..., 3). The conjugated representation 3 corresponds to a tensor Ai withone covariant index. To obtain the corresponding diagram, we have to raise this indexwith the three-dimensional epsilon tensor εi jk . This gives a completely antisymmetriccontravariant tensor A jk of second rank. Anti-symmetrization is diagrammaticallyindicated by vertical boxes, one box for each index and symmetrization by boxes ina row. Thus, we obtain the diagram for 3

3 = . (C.2)

For a valid Young diagram The numbers of boxes in a corresponding row, startingfrom the first row, has to be smaller or equal to the number of boxes in the previousrow. For SU (N ) the number of vertical boxes can never exceed N because otherwisethe procedure of anti-symmetrization of N + 1 indices, each index running from1, ..., N would result in zero. We write tensor product between two fundamentalrepresentations 3 as


254 Appendix C: Young Tableaux for SU (3)

3 ⊗ 3 = ⊗ a = a ⊕ a = 6 ⊕ 3. (C.3)

The algorithm of the calculation can be summarized in different steps:

1. First, we fill all boxes in the first row of the right diagram with a, all boxes inthe second row of the right diagram with b etc., until we have filled all rows ofthe second diagram in that way.

2. Second, we take all boxes filled with a and attach them to the first diagram inall possible ways to construct a valid Young diagram. After having attached theblocks containing the a’s, we do the same procedure successively for the b’s,the c’s etc., until all boxes of the original right diagram are attached to the leftdiagram. This will result in a big number of different diagrams (two diagramsare considered equal only if they have the same shape and the same a’s, b’, etc.in the same boxes).

3. Third, there is one more rule which guarantees the avoidance of double counting.We will write a sequence of letters for each diagram in the following way. Westart with the last row and write down all a’s, b’s etc. in the order they appearreading the row from left to right. We continue writing the sequence generatedby the letters from the last row by attaching a sequence of letters generated fromthe above row in the same way. We do this for all rows ending with the firstrow. We finally have a string of letters for each diagram. We keep only thosediagrams in which for an arbitrary position in the string the number of a’s rightto that position is bigger or equal than the number of b’s right to this position.We require the same for the number of b’s with respect to the number of c’sand so on. We throw away all diagrams which do not satisfy this constraint.The resulting diagrams correspond to the decomposition into a direct sum ofirreducible representations.

We finally have to estimate the dimensionality of the representation correspondingto a diagram. We do this again in several steps.

1. First, we duplicate the corresponding diagram.2. In one diagram we fill the boxes according to the following rule: Begin with the

box of the first row in the right corner. Insert the dimension of the Group (incase of SU(3), a 3). Fill the remaining boxes in the same row by the value ofthe foregoing box and add 1. Fill the first box of the second row with the valueof the box above it and subtract one. Fill the remaining boxes in the same rowaccording to the same rule like for the first row. Proceed in this way until youhave filled all boxes. Calculate the product of the numbers contained in all boxesof the diagram. Take the result as the value of the numerator of the number forthe dimensionality.

3. Fill the copy of the diagram accordingly to the following rule: For each boxcount the number of boxes to the right and the number of boxes below and addone. Fill this value into the box. Fill all boxes in this way and calculate againthe product of the numbers of all boxes. The result is the denominator of thedimensionality.

Appendix C: Young Tableaux for SU (3) 255

4. The fraction of these two numbers gives the dimensionality of the representationcorresponding to each diagram in the direct sum.

The dimensionality for the simple example above can be calculated by

3 4

2 1= 3 · 4

2 · 1= 12

2= 6. (C.4)

Now we calculate the product between the fundamental representation and its con-jugated. It is easier to reverse the order of multiplication because the right-hand sidehas fewer boxes, reducing the amount of calculations.

3 ⊗ 3 = ⊗ a =a

⊕ a = 8 ⊕ 1, (C.5)

where we have calculated the dimensionality by

432

131

= 4 · 3 · 2

3 · 1 · 1= 8,

321

321

= 1. (C.6)

The product of three quarks can be calculated by

3 ⊗ 3 ⊗ 3 = (6 ⊕ 3) ⊗ 3 =(

⊕)

⊗ a

= a ⊕ a ⊕a

⊕ a = 10 ⊕ 8 ⊕ 8 ⊕ 1 (C.7)

Finally, we have to figure out how the product of two gluons transforms. Since thegluon itself transforms in the adjoint representation 8 we have to calculate

8 ⊗ 8 = ⊗a ab =

( a a

⊕a

a ⊕

a

a ⊕a

a ⊕aa

)⊗ b

=a a b

⊕a a

b ⊕

a a

b ⊕a

ab

⊕a

a b

256 Appendix C: Young Tableaux for SU (3)

⊕

aa

b ⊕

a

a

b

⊕

a

ab

⊕

a

ab ⊕

aa

b

⊕a

a b

= 27 ⊕ 10 ⊕ 10 ⊕ 8 ⊕ 8 ⊕ 1 (C.8)

We have calculated all relevant products of representations of SU (3) needed for theapplication in the Standard Model.

Appendix DSynge’s World Function

The notation “world function” dates back to the Irish relativist Synge [7] and describesa non-local extension σ(x, x ′) of the space-time metric gμν(x). It is defined byconnecting the two space-time points {x = z(s1), x ′ = z(s0)} ∈ M, s ∈ R by theunique geodesic γ ≡ z(s) parametrized through s

σ(x, x ′) := 1

2(s1 − s0)

∫ s1

s0

gμν(z(s)) tμtν ds. (D.1)

The restriction to unique geodesics γ excludes conjugate points on γ. This is alwayssatisfied for sufficient small distances between x and x ′ and it is necessary in orderto define the derivative, the normalized tangential vector

tμ := dzμ(s)

ds∈ TM. (D.2)

The world function is a bi-scalar that transforms independently at x and x ′ as ascalar. The tangential vectors are parallelly transported along the geodesic γ

Dtμ

Ds:= d2zμ(s)

ds2 + �μαβ

dzα(s)

ds

dzβ(s)

ds= 0. (D.3)

Thus, the scalar product and in particular the norm are conserved on γ

gμν(z) tμtν := N = const. (D.4)

This allows us to write the world function with �s := (s1 − s0) as

σ(x, x ′) = 1

2(s1 − s0)

2 gμν(z) tμtν = N

2(�s)2. (D.5)


258 Appendix D: Synge’s World Function

The sign of N fixes the causal nature of the geodesic, i.e. N = (−1, +1, 0)

corresponds to γ =(time-like, space-like, light-like). From (D.5) it is clear thatσ(x, x ′) = σ(x ′, x) is symmetric under the exchange x ↔ x ′ For Minkowski space-time (gμν = ημν), geodesics are straight lines and (D.1) reduces to

σ(x, x ′) = 1

2ημν(xμ − x ′μ)(xν − x ′ν). (D.6)

Without loss of generality we can fix one point, say x ′ = 0 and regard σ(x, 0) as anordinary scalar function at x . The light cone structure emerges from

σ(x, 0) = − (x0)2

(√

2)2+ (x1)2

(√

2)2+ (x2)2

(√

2)2+ (x3)2

(√

2)2= 0. (D.7)

D.1 Derivatives of the World Function

By repeated differentiation of the world function, we can generate bi-tensors ofdifferent rank at x and x ′. The covariant derivative of σ(x, x ′) at the point x isdefined as ∇ασ := σ;α = σ,α. Correspondingly, the derivative of σ at the point x ′ isdefined as σ;α′ . The world function and its derivatives transform independently at xand x ′. For example, this means σ;α′β′γ transforms like a tensor of second rank at x ′and like a covariant vector at x . This implies that the order of primed and unprimedindices can be changed arbitrarily for any general bi-tensor A...;β′α = A...;αβ′ . Inparticular, we can freely permute primed and unprimed derivatives of σ as longas the order among the primed and the order among the unprimed indices remainmaintained, e.g.

σ;α′β′γ = σ;α′γβ′ = σ; γα′β′ �= σ;β′α′γ . (D.8)

For an explicit expression of the first derivative of σ at x , we fix the point x ′ and varyx , i.e. δz(s0) := δx ′ = 0 and δz(s1) := δx , see [4]

δσ = σ(x + δx, x ′) − σ(x, x ′) = �s gμν tμ δxν . (D.9)

Comparison with the result of the chain rule δσ = σ,μδxμ yields

σ;μ = �s (gμν tν)|s=s1 or σ;μ = �s tμ, (D.10)

which shows that σ;μ is proportional to the tangential vector tμ with length of thegeodesic distance between x and x ′ and pointing in direction x ′ → x . The derivativeof σ at x ′ can be obtained in an analogue manner

σ;μ′ = −�s (gμ′ν ′ tν′)|s=s0 or σ;μ′ = −�s tμ

′. (D.11)

Appendix D: Synge’s World Function 259

It has the same properties as (D.10) but points in the opposite direction x → x ′. With(D.10) we can derive an important identity for the norm of σ;μ

gμν σ,μσ; ν = gμν (�s)2 tμtν = (�s)2 N = 2σ or σ = 1

2σ;μσ;μ. (D.12)

D.2 Densities and Determinants

For any ordinary tensor Ab1...bna1...an a tensor density Ab1...bn

a1...an of weight w is defined byits transformation behaviour under general coordinate transformations xμ → xμ

Aa1...anb1...bn

= det

(∂xc

∂ xd

)w ∂ xa1

∂xa′1...∂ xan

∂xa′n

∂xb′1

∂ xb1...∂xb′

n

∂ xbnAa′

1...a′n

b′1...b

′n, (D.13)

where g = det(gμν(x)) = ∂(xμ)∂(xν ) is the Jacobian that takes care of a coordinate

transformation from xμ to xν . Since we deal with bi-tensors, we can generalize thisfor the transformation of the two variables (xμ, x ′ρ) that characterize the geodesic γby its endpoints to the variables (σ; ν ′ , x ′

σ) that characterize γ by one endpoint andthe tangent at this point. This Jacobian is given by

∣∣∣∣∂ (σ; ν ′ , x ′σ)

∂ (xμ, x ′ρ)

∣∣∣∣ =∣∣∣∣∂ (σ; ν ′)

∂ (xμ)

∣∣∣∣ = det(σ;μν ′) = det(−Dμν ′). (D.14)

Following [2], we have defined the matrix Dμν ′ := −σ;μν ′ and its determinantas D(x, x ′) := −det(Dμν ′). The convention of the minus sign takes care of theLorentzian signature of space-time. The square root of this determinant can bethought of as the generalize bi-volume factor D1/2(x, x ′) replacing g1/2(x). Dif-ferentiating (D.12) twice with respect to xμ and x ′ν , we obtain the relation

Dμν ′ = σρ; μ Dρ ν ′ + σ

ρ; Dμν ′; ρ. (D.15)

Multiplication with (D−1)μν′, and using Jacobi’s formula yields

D−1 (D σρ; ); ρ = d. (D.16)

with d = 4 in four space-time dimensions. Finally, we introduce the Van-Fleckdeterminate �(x, x ′) by

� := g−1/2 D g′−1/2. (D.17)

Since (g1/2);μ = 0, we obtain relation (D.16) in terms of �

�−1 (�σρ; ); ρ = d. (D.18)


D.3 The Parallel Propagator

We further need the geometrical concept of parallel transport. Following [4], wedefine the parallel propagator as the tensor product of two tetrads. We introduce anorthonormal tetrad eμI (z) which satisfies

gμν eμI eνJ = ηI J , (D.19)

with (I, J, ... = 0, 1, 2, 3) being Lorentz-indices of the frame with the Minkowskimetric ηI J = diag(−1, 1, 1, 1). We define the dual triad

eIμ := η I J gμν eνJ (D.20)

and via the completeness relation

gμν = η I J eμI eνJ it follows eμI eIν = δμν and eμI eJ

μ = δ JI . (D.21)

The tetrad basis eμI (z) and its dual eIμ(z) are parallelly transported along γ = z(s)

DeμIDs

= 0 andDeI

μ

Ds= 0. (D.22)

We can expand each contravariant vector field vν(z) into this orthonormal basis

vμ = v I eμI , (D.23)

whereas the Lorentz components of v can be expanded in the dual basis

v I = vνeIν . (D.24)

If the vector vμ is parallelly transported along γ, then the frame coefficients v I areconstant since eμI is also parallelly transported along γ. Since the v I are constant, wecan express them as v I = vν

′eIν ′ where the vector vν

′and the tetrad eI

ν ′ are evaluatedat x ′. Putting all this together, vμ at x can be expressed by

vμ(x) = (vν′eIν ′) eμI or vμ = g

μμ′ vμ

′, (D.25)

with gμμ′ being the parallel propagator gμμ′ := eμI eIμ′ . It is a bi-vector that transports

the vector vμ′from the point x ′ = z(s1) to the vector vμ at the point x = z(s0) along

the unique geodesic γ = z(s). The inverse parallel transporter from x to x ′ is given

by gμ′μ = eμ

′I eI

μ. It is clear that the parallel transporter gμν ′ satisfies the boundarycondition

limx ′→x

gμν ′ = δμν or lim

x ′→xgμν ′ = gμν . (D.26)


The parallel transport easily generalizes to tensors of arbitrary rank T μ...νρ...σ . Since

a tensor is a multilinear form we need one (inverse) parallel propagator for each(covariant) contravariant index to transport it along γ from x ′ to x .

T μ...νρ...σ = g

μμ′ · · · gνν ′ gρ

′ρ · · · gσ′

σ T μ′...ν ′ρ′...σ′ . (D.27)

Since the tetrads eμI are parallelly propagated along γ this means

DeμI (z(s))

Ds= eμI ; ν

dzν(s)

ds= �s eμI ; ν σ

ν = 0. (D.28)

From (D.28) it follows eμI ; ν σν; = 0 and eI

μ′ ; ν σν; = 0. This, in turn, implies

gμμ′ ; ν σ

ν; = 0 and g

μμ′ ; ν ′ σ ν ′

; = 0. (D.29)

Since σ;μ ∝ −tμ, it is automatically parallelly transported along γ and thus

σ;μ = −gμ′μ σ;μ′ and σ;μ′ = −gμμ′ σ;μ. (D.30)

From the determinants of the tetrad and its dual

det(eμI ) = g−1/2 and det(eIμ) = g1/2, (D.31)

we find for the determinant of the parallel propagator

det(−gμν ′) = det(−gμνgνν ′) = det(−gμνeνI eIν ′)

= det(−gμν) det(eνI ) det(eIν ′) = g1/2g′1/2. (D.32)

D.4 Field Curvature

We assume that we have a set of fields collected in the generalized field φA(x). Eachcomponent of the generalized field consists of some fundamental field present in theaction of the theory together forming the field space C. If we consider C as a manifold,φA(x) corresponds to a coordinate of a point in this manifold. The fields collected inφA(x) do have their own symmetries. In order to define invariant differentiation, weneed to introduce a covariant derivative. We introduce an abbreviation by suppressingthe capital index and write instead φ := φA and a hat for operators O := O B

A . Thelaw of covariant differentiation is not commutative but defines the field (bundle)curvature Rμν

[∇μ, ∇ν]φ =: Rμν φ. (D.33)


To clarify the role of φ, we will consider two examples. We assume φ = φA ≡ qa

consists of a single field qa with an internal gauge symmetry described by the Liegroup G and the group index a. Covariant differentiation of qa means

∇μ qa ≡ ∂μ qa + Aμ qa (D.34)

with the gauge vector field Aμ := AaμGa and the generators Ga of the group G. The

commutator of covariant derivatives then yields

[∇μ, ∇ν]φ ≡ (Aaμ, ν − Aa

ν,μ + cabc Ab

μ Acν)φa = Fa

μν φa (D.35)

with the field strength tensor

Rμν ≡ Faμν = Aa

μ, ν − Aaν,μ + ca

bc Abμ Ac

ν (D.36)

and the structure constants cabc of the group G. If the field φ instead only consists

of a symmetric tensor field φ ≡ qμν = qνμ with ordinary space-time indices, thecovariant derivative is defined with respect to the usual Christoffel symbol ∇μqαβ ≡∇μqαβ = ∂μqαβ − �

γαμqγβ − �

γβμqαγ and we obtain

[∇μ, ∇ν]φ ≡ (∇μ∇ν − ∇ν∇μ) qαβ = −Rγαμνqγβ − Rγβμνqαγ (D.37)

where we have used uα;βγ − uα; γβ = Rδαβγuδ and have written qαβ without lossof generality as the tensor product qαβ := uα ⊗ vβ . This can also be written as

[∇μ, ∇ν]φ ≡ (∇μ∇ν − ∇ν∇μ) qαβ = −2 δ(γ(α Rδ)β)μν qγδ (D.38)

and shows that in this case the field curvature is given by the operator

Rμν ≡ −2 δ(γ(α Rδ)β)μν, (D.39)

acting on qγδ . In general, φ is a collection of many different fields. We can thereforedefine a parallel displacement bi-matrix I (x, x ′), which parallelly transports φ(x) toφ(x ′) along the geodesic γ. It is a matrix in field space and from the general propertiesof a parallel propagator, it follows as in (D.29) and (D.26) that I (x, x ′) must satisfythe relations

σ;μ I;μ = 0 , σ;μ′I;μ′ = 0 and lim

x→x ′ I (x, x ′) = δAB (x) := 1, (D.40)

where 1 is the unit matrix at the point x .


D.5 Coincidence Limits

Since σ(x, x ′) is a bi-scalar and derivatives of σ are bi-tensors, we are interestedin the value of these objects in the coincidence limit x → x ′. For any bi-tensorAν...μ...(x, x ′) we define the limit x → x ′ by the symbolic bracket-notation

[ A......(x, x ′) ] := lim

x ′→xA...

...(x, x ′). (D.41)

The result of this limit is an ordinary tensor at x ′. There is a relation between coin-cidence limits of primed and unprimed indices, denoted Synge’s rule [4]

[A...α′ ] = [A...];α′ − [A...α]. (D.42)

By repeated differentiation of (D.12) we have a recursive algorithm to systematicallycalculate the coincidence limit of higher derivatives [σ;α1,...,αN ].

σ = 1

2σ;λ σ,λ, (D.43)

σ;α = σ λ; α σλ, (D.44)

σ;αβ = σ λ; αβ σ;λ + σ λ

; α σ;λβ, (D.45)

σ;αβγ = σ λ; αβγ σ;λ + σ λ

; αβ σ;λγ + σ λ; αγ σ;λβ + σ λ

; α σ;λβγ, (D.46)

σ;αβγδ = σ λ; αβγδ σ;λ + σ λ

; αβγ σ;λδ + σ λ; αβδ σ;λγ + σ λ

; αβ σ;λγδ,

+ σ λ; αγδ σ;λβ + σ λ

; αγ σ;λβδ + σ λ; αδ σ;λβγ + σ λ

; α σ;λβγδ (D.47)

...

From (D.5) and (D.10), it follows directly that [σ] = 0 and [σ;μ] = 0. From (D.45)it is clear that [σ;μν] = gμν . The remaining coincidence limits for (D.46) and (D.47)are obtained recursively. By commuting covariant derivatives we generate curvatureexpressions. Using the symmetries of the Riemann tensor, we obtain

[σ;αβγ ] = 0 and [σ;αβγδ ] = 1

3

(Rαγβδ + Rαδβγ

). (D.48)

For further application we will also need two additional coincidence limits. Con-tracting indices in (D.47) and differentiating twice, we obtain after some algebra

[σ

μ ν;μ ν α

]=R;α, (D.49)

[σ

μ ν α;μ ν α

]=8

5R α

; α + 4

15RαβRαβ − 4

15RαβγδRαβγδ. (D.50)


We are ultimately interested in the coincidence limit of the coefficient matrix a2.As shown in Chap. 4, a2 can be calculated by the recursion relation (4.109). Thecoincidence limits can then be again obtained recursively. In order to calculate [ a1 ]and [ a2 ], we must calculate several further coincidence limits. It is easy to see that

[ Dμν ′ ] = gμν, [ D ] = g and [ � ] = 1. (D.51)

For the coincidence limits of higher derivatives of the field parallel propagator matrixI , we differentiate (D.40) and use (D.33) in order to obtain

[I;μ

] = 0,[

I;μν] = −1

2Rμν, (D.52)

[I μ;μ ν

] = 1

3R μμν; ,

[I μ ν;μ ν

] = 1

2RμνRμν . (D.53)

Proceeding in a similar manner, we differentiate (D.18) in order to obtain recursivelythe coincidence limits of the derivatives of the Van-Fleck determinant

[�

1/2;μ

] = 0,[�

1/2;μν

] = −1

6Rμν, (D.54)

[�

−1/2;μν

] = 1

6Rμν,

[�1/2 μ

;μ ν

] = −1

6R; ν (D.55)

[�1/2 μ ν

;μ ν

] = −1

5R α

; α + 1

36R2 − 1

30RαβRαβ + 1

30RαβγδRαβγδ. (D.56)

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Appendix ECorrections to the Initial Values of λ and yt

The pole mass matching scheme used to correct the initial values for λ and yt wasdeveloped in [6, 8]. We have used these results at the one-loop level. The correctionfunction for the Higgs mass is

�H = GF√2

M2Z

16π2 [ζ f1(ζ) + f0(ζ) + ζ−1 f−1(ζ)] (E.1)

with

f1(ζ) = 6 lnM2

t

M2H

+ 3

2ln ζ − 1

2Z

[1

ζ

]− Z

[c2

wζ

]− ln c2

w + 9

2

(25

9− π√

3

), (E.2)

f0(ζ) = − 6 lnM2

t

M2Z

(1 + 2 c2

w − 2M2

t

M2Z

)+ 3 c2

w ζ

ζ − c2w

lnζ

c2w

+ 2 Z

[1

ζ

]

+ 4c2w Z

[c2

wζ

]+

(3 c2

w

s2w

+ 12 c2w

)ln c2

w − 15

2(1 + 2 c2

w)

− 3M2

t

M2Z

(2 Z

[M2

t

M2Z ζ

]+ 4 ln

M2t

M2Z

− 5

), (E.3)

f−1(ζ) = 6 lnM2

t

M2Z

(1 + 2 c2

w − 4M2

t

M2Z

)− 6 Z

[1

ζ

]− 12 c4

w Z

[c2

wζ

]− 12 c4

w ln c2w

+ 8 (1 + 2 c4w) + 24

M4t

M4Z

(ln

M2t

M2Z

− 2 + Z

[M2

t

M2Z ζ

]), (E.4)

the Fermi constant GF and ζ := M2H/M2

Z, s2w := sin2 θW, c2

W := cos2 θW. Thefunction Z [z] is defined with respect to its domains depending on z

Z [z] :={

2 A arctan 1A , (z > 1/4),

A ln 1+A1−A , (z < 1/4),

A := √|1 − 4 z|. (E.5)


266 Appendix E: Corrections to the Initial Values of λ and yt

The correction for the top-quark mass is �t(MH) and reads

�t(MH) = δQCDt + δW

t + δQEDt , δ

QCDt = −4αs(Mt)

3π, δ

QEDt + δW

t = −4α(Mt)

9π.

(E.6)We have neglected two-loop corrections. The numerical values of α and αs can beobtained from [1].

Appendix FTransfer Equations

Not all structures appearing in the divergent part of the one-loop effective actionare independent. Neglecting surface terms and making use of the Bianchi identities,we can convert certain structures into others via an integration by parts. In thisway, we can reduce the number of different structures in (5.102) to a minimum. The“transfer equations” below describe explicitly how the contributions of the dependentstructures are distributed among the minimal set of independent structures:

F �a μ;μ na → − F ′ �4 − F

ϕ(�3 − �4), (F.1)

F �a;μν�

;μνa → F ′ �21 − F(�7 − �17) + F ′

ϕ�15

− 1

2

(F ′

ϕ− F ′′

)�14 + 1

2F ′ �19 − 1

2

F ′

ϕ�12, (F.2)

F �a;μν na�b;μν nb →

(4

F

ϕ2 − 5

2

F ′

ϕ+ 1

2F ′′

)�13 +

(2

F ′

ϕ− 4

F

ϕ2

)�16

+(

3

2F ′ − 3

F

ϕ

)�20 +

(1

2

F ′

ϕ− F

ϕ2

)�14

+ 2F

ϕ�21 − F�8 + F

ϕ�19 + F �18 + F

ϕ2 �15, (F.3)

F �a;μν na �b,μ �

, νb → − 1

2

(F

ϕ− F ′

)�14 +

(F

ϕ− F ′

)�16

− F �21 + 1

2F �19 + 1

2

F

ϕ�12 − F

ϕ�15, (F.4)

F �a;μν�

,μa �

, νb nb → 1

2

(F

ϕ− F ′

)�14 − F

2�19 − 1

2

F

ϕ�12, (F.5)


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268 Appendix F: Transfer Equations

F �a;μν na �

,μb nb �, ν

c nc → 1

2

(3

F

ϕ− F ′

)�13 − F

ϕ�16

− 1

2F �20 − 1

2

F

ϕ�14, (F.6)

F Rμν �a;μν na →

(F

ϕ− F ′

)�8 − 1

2

(F

ϕ− F ′

)�10

− F

ϕ�7 + 1

2F �11 + 1

2

F

ϕ�9, (F.7)

F RμνρσRμνρσ → 4 F �5 − F �6. (F.8)

The last equation is a topological invariant, the Gauss–Bonnet identity. In the singlefield limit, cf. Table 5.1, the other seven transfer equations reduce to the three reduc-tion formulas given in [5]. In principle, even more scalar invariants composed ofdifferent scalar contractions between derivatives ∇μ and field variables (gμν, �a),containing up to four derivatives, can appear at the one-loop level, but they do notoccur in our calculations.

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Appendix GGradient Structures

As already explained in Sect. 4.5.2, the coefficients αi , i = 12, ..., 21 do all corre-spond to gradient structures, symbolically denoted as ∂4��. In the cosmologicalmodel of Higgs inflation, discussed in Chap. 6, these structures are additionally sup-pressed compared to (5.108)–(5.118) and thus less important from the viewpoint ofan effective field theory. For the sake of completeness we list their coefficients in aclosed form.

The coefficient α12 belonging to the structure (�a,μ�

,μa )2:

α12 = s2(

25(U ′)8

48G2ϕ4U4 + 3(U ′)8

16U6 −(U ′)7

4Gϕ3U4 − 3(U ′)7

8ϕU5+ 5G′ (U ′)6

4G2ϕ3U3 − 11(U ′)6

72Gϕ4U3

− 9(U ′)6

8ϕ2U4 − 13G′ (U ′)5

12Gϕ2U3 − G′ (U ′)5

2U4 − 3U ′′ (U ′)5

4Gϕ3U3 + 35(U ′)5

12ϕ3U3

+ 13(G′)2 (

U ′)4

24G2ϕ2U2 − G′ (U ′)4

Gϕ3U2 + 15G′ (U ′)4

8ϕU3 + 3G′′ (U ′)4

8U3 − 695(U ′)4

432ϕ4U2

+(G′)2 (

U ′)3

8GϕU2 − 47G′ (U ′)3

18ϕ2U2 − 3G′′ (U ′)3

4ϕU2 + 3G′U ′′ (U ′)3

4Gϕ2U2 − U ′′ (U ′)3

4ϕ3U2

−(G′)3 (

U ′)2

4G2ϕU+ 139

(G′)2 (

U ′)2

72Gϕ2U+

(G′)2 (

U ′)2

8U2 + 19G′ (U ′)2

36ϕ3U

+ 3G′G′′ (U ′)2

4GϕU+

(G′)3 U ′6GU

−(G′)2 U ′12ϕU

− G′G′′U ′4U

+ G′U ′′U ′4ϕ2U

+(G′)4

48G2 − 7(G′)3

24Gϕ+ 11

(G′)2

24ϕ2 +(G′′)2

8−

(G′)2 G′′

8G+ G′G′′

2ϕ

)

+ s

(− 25

(U ′)6

72G2ϕ4U3 −(U ′)6

8U5+

(U ′)5

6Gϕ3U3 − 37(U ′)5

8ϕU4 + 5G′ (U ′)4

12G2ϕ3U2

− 43(U ′)4

216Gϕ4U2 + 21(U ′)4

4ϕ2U3 + 13G′ (U ′)3

36Gϕ2U2 + 11G′ (U ′)3

12U3 + U ′′ (U ′)3

2Gϕ3U2


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270 Appendix G: Gradient Structures

+ 3U ′′ (U ′)3

2ϕU3 − 107(U ′)3

36ϕ3U2 + 77(G′)2 (

U ′)2

72G2ϕ2U+ 15G′ (U ′)2

4Gϕ3U− 15G′ (U ′)2

8ϕU2

− G′′ (U ′)2

8U2 − 3U ′′ (U ′)2

ϕ2U2 −(U ′)2

12ϕ4U+ 5

(G′)2 U ′

6GϕU+ 13G′U ′

6ϕ2U− G′U ′′U ′

4Gϕ2U

+ U ′′U ′12ϕ3U

−(G′)3

6G2ϕ− 13

(G′)2

12Gϕ2 − G′G′′4Gϕ

+ G′U ′′2ϕU

)+ 25

(U ′)4

432G2ϕ4U2

+(U ′)4

48U4 + 59(U ′)3

36Gϕ3U2 + 19(U ′)3

12ϕU3 − G′ (U ′)2

9G2ϕ3U+ 5

(U ′)2

12Gϕ4U+ 17

(U ′)2

3ϕ2U2

− 4G′U ′3Gϕ2U

− G′U ′4U2 − 5U ′′U ′

12Gϕ3U− U ′′U ′

2ϕU2 − 5U ′3ϕ3U

+ 13GU ′6ϕU2

+ (N − 1)(G′)2

2G2ϕ2 − 5(G′)2

12G2ϕ2 − 7G′6ϕU

+ 5U ′′3ϕ2U

− G2

4U2 (G.1)

The coefficient α13 belonging to the structure (�a,μna�b ,μnb)

2:

α13 = s4(

81(U ′)12

32U 8 − 81U ′′ (U ′)10

4U 7 − 27G ′ (U ′)9

8U 6 + 243(U ′′)2 (

U ′)8

4U 6

+ 81G ′U ′′ (U ′)7

4U 5− 81

(U ′′)3 (

U ′)6

U 5+ 27

(G ′)2 (

U ′)6

16U 4 − 81G ′ (U ′′)2 (U ′)5

2U 4

+ 81(U ′′)4 (

U ′)4

2U 4 − 27(G ′)2

U ′′ (U ′)4

4U 3 − 3(G ′)3 (

U ′)3

8U 2

+ 27G ′ (U ′′)3 (U ′)3

U 3 + 27(G ′)2 (

U ′′)2 (U ′)2

4U 2 + 3(G ′)3

U ′′U ′

4U+

(G ′)4

32

)

+ s3(

− 3(U ′)10

2Gϕ2U 6 + 45(U ′)10

8U 7 − 81(U ′)9

8ϕU 6 − 9G ′ (U ′)8

4GϕU 5+ 19U ′′ (U ′)8

4Gϕ2U 5

+ 27U ′′ (U ′)8

8U 6 + 25(U ′)8

4ϕ2U 5+ G ′ (U ′)7

Gϕ2U 4 − 15G ′ (U ′)7

4U 5+ 81U ′′ (U ′)7

2ϕU 5

+ 5(G ′)2 (

U ′)6

8GU 4 + 155(U ′′)2 (

U ′)6

8Gϕ2U 4 − 351(U ′′)2 (

U ′)6

4U 5+ 39G ′ (U ′)6

8ϕU 4

+ 9G ′′ (U ′)6

8U 4 + 3G ′U ′′ (U ′)6

GϕU 4 − 305U ′′ (U ′)6

12ϕ2U 4 + 3(G ′)2 (

U ′)5

2GϕU 3

− 81(U ′′)2 (

U ′)5

2ϕU 4 − 25G ′ (U ′)5

6ϕ2U 3 + 15G ′U ′′ (U ′)5

4Gϕ2U 3 − 39G ′U ′′ (U ′)5

4U 4

+ 261(U ′′)3 (

U ′)4

2U 4 + 29(G ′)2 (

U ′)4

24U 3 + 93G ′ (U ′′)2 (U ′)4

4GϕU 3 + 803(U ′′)2 (

U ′)4

24ϕ2U 3

+(G ′)2

U ′′ (U ′)4

GU 3 − 8G ′U ′′ (U ′)4

ϕU 3 − 9G ′′U ′′ (U ′)4

2U 3 +(G ′)2 (

U ′)3

8ϕU 2

+ 39G ′ (U ′′)2 (U ′)3

U 3 − 3G ′G ′′ (U ′)3

4U 2 + 9(G ′)2

U ′′ (U ′)3

GϕU 2 + 41G ′U ′′ (U ′)3

4ϕ2U 2

− 27(U ′′)4 (

U ′)2

U 3 + 3(G ′)3 (

U ′)2

4GϕU+ 3

(G ′)2 (

U ′)2

4ϕ2U− 31

(G ′)2 (

U ′′)2 (U ′)2

8GU 2

Appendix G: Gradient Structures 271

+ 13G ′ (U ′′)2 (U ′)2

4ϕU 2 + 9G ′′ (U ′′)2 (U ′)2

2U 2 + 41(G ′)2

U ′′ (U ′)2

24U 2 −(G ′)3

U ′

4U

− 9G ′ (U ′′)3U ′

U 2 − 3(G ′)3

U ′′U ′

2GU+ 3

(G ′)2

U ′′U ′

2ϕU+ 3G ′G ′′U ′′U ′

2U−

(G ′)4

8G+

(G ′)3

8ϕ

− 13(G ′)2 (

U ′′)2

24U+ 1

8

(G ′)2

G ′′)

+ s2(

25(U ′)8

16G2ϕ4U 4 − 19(U ′)8

6Gϕ2U 5− 189

(U ′)8

16U 6

− 5(U ′)7

4Gϕ3U 4 − 123(U ′)7

8ϕU 5+ 15G ′ (U ′)6

4G2ϕ3U 3 + G ′ (U ′)6

4GϕU 4 − U ′′ (U ′)6

6Gϕ2U 4 + 273U ′′ (U ′)6

8U 5

+ 133(U ′)6

24Gϕ4U 3 − 223(U ′)6

72ϕ2U 4 − 25G ′ (U ′)5

24Gϕ2U 3 + 85G ′ (U ′)5

8U 4 − 5G ′′ (U ′)5

4GϕU 3 − 15G ′U ′′ (U ′)5

4G2ϕ2U 3

− 33U ′′ (U ′)5

4Gϕ3U 3 + 87U ′′ (U ′)5

4ϕU 4 + 3U ′′′ (U ′)5

4Gϕ2U 3 − 3U ′′′ (U ′)5

2U 4 − 113(U ′)5

12ϕ3U 3 +(G ′)2 (

U ′)4

G2ϕ2U 2

− 4(G ′)2 (

U ′)4

3GU 3 − 55(U ′′)2 (

U ′)4

6Gϕ2U 3 − 81(U ′′)2 (

U ′)4

8U 4 + 5G ′ (U ′)4

2Gϕ3U 2 + 167G ′ (U ′)4

24ϕU 3

− 7G ′′ (U ′)4

8U 3 + 45G ′U ′′ (U ′)4

4GϕU 3 + 881U ′′ (U ′)4

36ϕ2U 3 + 889(U ′)4

144ϕ4U 2 − 7(G ′)2 (

U ′)3

8GϕU 2

+ 9(U ′′)2 (

U ′)3

2ϕU 3 + 325G ′ (U ′)3

72ϕ2U 2 − 8G ′′ (U ′)3

3ϕU 2 − 9(G ′)2

U ′′ (U ′)3

2G2ϕU 2 − 31G ′U ′′ (U ′)3

2Gϕ2U 2

− 25G ′U ′′ (U ′)3

U 3 − 13G ′′U ′′ (U ′)3

4GϕU 2 − 11U ′′ (U ′)3

4ϕ3U 2 − 3G ′U ′′′ (U ′)3

4GϕU 2 − 6U ′′U ′′′ (U ′)3

U 3

+ U ′′′ (U ′)3

4ϕ2U 2 − 3(G ′)3 (

U ′)2

2G2ϕU− 39

(U ′′)3 (

U ′)2

2U 3 − 8(G ′)2 (

U ′)2

3Gϕ2U− 19

(G ′)2 (

U ′)2

8U 2

− 43G ′ (U ′′)2 (U ′)2

4GϕU 2 − 713(U ′′)2 (

U ′)2

72ϕ2U 2 − 31G ′ (U ′)2

12ϕ3U− 125

(G ′)2

U ′′ (U ′)2

24GU 2

− 2G ′U ′′ (U ′)2

3ϕU 2 + 7G ′′U ′′ (U ′)2

4U 2 − G ′U ′′′ (U ′)2

2U 2 −(G ′)3

U ′

8GU− 23

(G ′)2

U ′

12ϕU

+ 17G ′ (U ′′)2U ′

2U 2 − G ′G ′′U ′

4U+ 3

(G ′)3

U ′′U ′

4G2U− 4G ′U ′′U ′

3ϕ2U+ 3G ′G ′′U ′′U ′

2GU

+ 23G ′′U ′′U ′

12ϕU− G ′U ′′′U ′

4ϕU+ 3

(G ′)4

16G2 + 9(U ′′)4

2U 2 +(G ′)3

8Gϕ+ 5

(G ′)2

8ϕ2 +(G ′′)2

8

+ 13(G ′)2 (

U ′′)2

24GU− 5G ′ (U ′′)2

2ϕU− 3G ′′ (U ′′)2

2U+

(G ′)2

G ′′

8G+ 3G ′G ′′

4ϕ+

(G ′)2

U ′′

2U

)

+ s

(− 25

(U ′)6

24G2ϕ4U 3 − 35(U ′)6

9Gϕ2U 4 + 61(U ′)6

8U 5+ 7G ′ (U ′)5

8G2ϕ2U 3 − 37(U ′)5

6Gϕ3U 3 + 189(U ′)5

8ϕU 4

+ 5(G ′)2 (

U ′)4

8G3ϕ2U 2 − 3G ′ (U ′)4

8G2ϕ3U 2 − 11G ′ (U ′)4

12GϕU 3 − 5G ′′ (U ′)4

8G2ϕ2U 2 + 16U ′′ (U ′)4

9Gϕ2U 3 − 227U ′′ (U ′)4

8U 4

− 79(U ′)4

72Gϕ4U 2 + 521(U ′)4

27ϕ2U 3 − 5(G ′)2 (

U ′)3

2G2ϕU 2 − 56G ′ (U ′)3

9Gϕ2U 2 − 13G ′ (U ′)3

2U 3 − 19G ′′ (U ′)3

12GϕU 2

+ G ′U ′′ (U ′)3

G2ϕ2U 2 + 3U ′′ (U ′)3

2Gϕ3U 2 − 95U ′′ (U ′)3

2ϕU 3 − U ′′′ (U ′)3

2Gϕ2U 2 + 3U ′′′ (U ′)3

U 3 + 101(U ′)3

36ϕ3U 2

+ 3(G ′)3 (

U ′)2

4G3ϕU+ 17

(G ′)2 (

U ′)2

8G2ϕ2U+

(G ′)2 (

U ′)2

4GU 2 + 101(U ′′)2 (

U ′)2

72Gϕ2U 2 + 53(U ′′)2 (

U ′)2

2U 3


+ 11G ′ (U ′)2

8Gϕ3U− 433G ′ (U ′)2

36ϕU 2 − 41G ′′ (U ′)2

24Gϕ2U+ G ′′ (U ′)2

U 2 − 17G ′U ′′ (U ′)2

6GϕU 2 − 21U ′′ (U ′)2

4ϕ2U 2

+ 3U ′′′ (U ′)2

ϕU 2 +(U ′)2

4ϕ4U+ 3

(G ′)3

U ′

8G2U+ 17

(G ′)2

U ′

12GϕU+ 3

(U ′′)2

U ′

2ϕU 2 − 17G ′U ′

12ϕ2U+ G ′G ′′U ′

4GU

− G ′′U ′

2ϕU+ 3

(G ′)2

U ′′U ′

2G2ϕU+ 17G ′U ′′U ′

4Gϕ2U+ 27G ′U ′′U ′

2U 2 + 13G ′′U ′′U ′

12GϕU− 5U ′′U ′

12ϕ3U

+ G ′U ′′′U ′

4GϕU− U ′′U ′′′U ′

U 2 − U ′′′U ′

12ϕ2U−

(G ′)4

8G3 −(G ′)3

8G2ϕ−

(U ′′)3

2U 2 + 23(G ′)2

24U−

(G ′)2

4Gϕ2

−(G ′′)2

4G+ G ′ (U ′′)2

GϕU+

(U ′′)2

6ϕ2U−

(G ′)2

G ′′

8G2 − G ′G ′′

2Gϕ+

(G ′)2

U ′′

2GU+ 25G ′U ′′

12ϕU− G ′U ′′′

U

)

+ 25(U ′)4

144G2ϕ4U 2 + 46(U ′)4

27Gϕ2U 3 + 85(U ′)4

8U 4 −(G ′)3

4G3ϕ− 7G ′ (U ′)3

24G2ϕ2U 2 + 133(U ′)3

36Gϕ3U 2 + 3(U ′)3

4ϕU 3

− 23(G ′)2

24GU−

(G ′)2

2G2ϕ2 − 5(G ′)2 (

U ′)2

24G3ϕ2U− 7G ′ (U ′)2

24G2ϕ3U− 89G ′ (U ′)2

36GϕU 2 −(U ′)2

4Gϕ4U+ 3

(U ′)2

2ϕ2U 2

−(U ′′)2

6Gϕ2U+ 9

(U ′′)2

2U 2 −(G ′)2

U ′

2G2ϕU− 19G ′U ′

12Gϕ2U+ 5

(U ′)2

G ′′

24G2ϕ2U+ U ′G ′′

2GϕU− 9

(U ′)2

U ′′

4Gϕ2U 2

− 3(U ′)2

U ′′

4U 3 + 11G ′U ′′

12GϕU+ G ′U ′U ′′

12G2ϕ2U+ 5U ′U ′′

12Gϕ3U− 6U ′U ′′

ϕU 2 + U ′U ′′′

12Gϕ2U− U ′U ′′′

U 2

+ (N − 1)(G ′)2

8G2ϕ2 + (N − 1)(G ′)3

8G3ϕ+ (N − 1)

(G ′)4

32G4 − (N − 1)G ′G ′′

4G2ϕ

− (N − 1)(G ′)2

G ′′

8G3 + (N − 1)(G ′′)2

8G2 (G.2)


,μa ) (�c

, νnc�d , νnd):

α14 = s3( (

U ′)10

2Gϕ2U6 − 9(U ′)10

8U7 + 9(U ′)9

8ϕU6 + 3G′ (U ′)8

4GϕU5− 19U ′′ (U ′)8

12Gϕ2U5+ 9U ′′ (U ′)8

2U6

+ 29(U ′)8

12ϕ2U5− G′ (U ′)7

3Gϕ2U4 + 9G′ (U ′)7

4U5− 9U ′′ (U ′)7

2ϕU5− 5

(G′)2 (

U ′)6

24GU4

− 155(U ′′)2 (

U ′)6

24Gϕ2U4 − 9(U ′′)2 (

U ′)6

2U5− 19G′ (U ′)6

8ϕU4 − 9G′′ (U ′)6

8U4 − G′U ′′ (U ′)6

GϕU4

− 343U ′′ (U ′)6

36ϕ2U4 −(G′)2 (

U ′)5

2GϕU3 + 9(U ′′)2 (

U ′)5

2ϕU4 − 29G′ (U ′)5

18ϕ2U3 − 5G′U ′′ (U ′)5

4Gϕ2U3

− 15G′U ′′ (U ′)5

2U4 − 95(G′)2 (

U ′)4

72U3 − 31G′ (U ′′)2 (U ′)4

4GϕU3 + 493(U ′′)2 (

U ′)4

72ϕ2U3

−(G′)2 U ′′ (U ′)4

3GU3 + 26G′U ′′ (U ′)4

3ϕU3 + 9G′′U ′′ (U ′)4

2U3 + 29(G′)2 (

U ′)3

24ϕU2

+ 6G′ (U ′′)2 (U ′)3

U3 + 3G′G′′ (U ′)3

4U2 − 3(G′)2 U ′′ (U ′)3

GϕU2 + 31G′U ′′ (U ′)3

12ϕ2U2

−(G′)3 (

U ′)2

4GϕU+

(G′)2 (

U ′)2

4ϕ2U+ 31

(G′)2 (

U ′′)2 (U ′)2

24GU2 − 121G′ (U ′′)2 (U ′)2

12ϕU2


− 9G′′ (U ′′)2 (U ′)2

2U2 + 43(G′)2 U ′′ (U ′)2

18U2 +(G′)3 U ′

4U+

(G′)3 U ′′U ′

2GU

− 7(G′)2 U ′′U ′

2ϕU− 3G′G′′U ′′U ′

2U+

(G′)4

24G− 7

(G′)3

24ϕ− 5

(G′)2 (

U ′′)2

72U

− 1

8

(G′)2 G′′

)+ s2

(− 25

(U ′)8

24G2ϕ4U4 − 13(U ′)8

9Gϕ2U5− 15

(U ′)8

2U6 +(U ′)7

2Gϕ3U4

+ 39(U ′)7

4ϕU5− 5G′ (U ′)6

2G2ϕ3U3 + 7G′ (U ′)6

12GϕU4 − 7U ′′ (U ′)6

9Gϕ2U4 + 141U ′′ (U ′)6

8U5

+ 11(U ′)6

36Gϕ4U3 − 185(U ′)6

54ϕ2U4 + 127G′ (U ′)5

72Gϕ2U3 + 11G′ (U ′)5

8U4 − 5G′′ (U ′)5

12GϕU3

+ 5G′U ′′ (U ′)5

4G2ϕ2U3 + 3U ′′ (U ′)5

2Gϕ3U3 − 111U ′′ (U ′)5

4ϕU4 − 3U ′′′ (U ′)5

4Gϕ2U3 − 35(U ′)5

6ϕ3U3

− 7(G′)2 (

U ′)4

8G2ϕ2U2 + 17(G′)2 (

U ′)4

18GU3 + 55(U ′′)2 (

U ′)4

18Gϕ2U3 − 27(U ′′)2 (

U ′)4

4U4

+ 2G′ (U ′)4

Gϕ3U2 − 47G′ (U ′)4

9ϕU3 + G′′ (U ′)4

U3 − 131G′U ′′ (U ′)4

12GϕU3 + 1249U ′′ (U ′)4

108ϕ2U3

+ 695(U ′)4

216ϕ4U2 −(G′)2 (

U ′)3

4GϕU2 + 39(U ′′)2 (

U ′)3

2ϕU3 + 1307G′ (U ′)3

216ϕ2U2 + 31G′′ (U ′)3

36ϕU2

+ 3(G′)2 U ′′ (U ′)3

2G2ϕU2 − 11G′U ′′ (U ′)3

12Gϕ2U2 + 29G′U ′′ (U ′)3

4U3 − 13G′′U ′′ (U ′)3

12GϕU2

+ U ′′ (U ′)3

2ϕ3U2 + 3G′U ′′′ (U ′)3

4GϕU2 − U ′′′ (U ′)3

4ϕ2U2 + 3(G′)3 (

U ′)2

4G2ϕU− 97

(G′)2 (

U ′)2

24Gϕ2U

+ 2(G′)2 (

U ′)2

3U2 + 25G′ (U ′′)2 (U ′)2

4GϕU2 − 583(U ′′)2 (

U ′)2

216ϕ2U2 − 19G′ (U ′)2

18ϕ3U− 7G′G′′ (U ′)2

4GϕU

+ 143(G′)2 U ′′ (U ′)2

72GU2 − 6G′U ′′ (U ′)2

ϕU2 − 17G′′U ′′ (U ′)2

4U2 − 11(G′)3 U ′

24GU− 2

(G′)2 U ′9ϕU

− 13G′ (U ′′)2 U ′2U2 + G′G′′U ′

2U−

(G′)3 U ′′U ′

4G2U+ 2

(G′)2 U ′′U ′

GϕU+ 5G′U ′′U ′

12ϕ2U+ G′G′′U ′′U ′

2GU

+ 23G′′U ′′U ′36ϕU

+ G′U ′′′U ′4ϕU

−(G′)4

12G2 + 5(G′)3

6Gϕ− 3

(G′)2

4ϕ2 −(G′′)2

4− 49

(G′)2 (

U ′′)2

72GU

+ 49G′ (U ′′)2

18ϕU+ 3G′′ (U ′′)2

2U+

(G′)2 G′′

3G− 11G′G′′

12ϕ− 2

(G′)2 U ′′

3U

)+ s

(25

(U ′)6

36G2ϕ4U3

+ 43(U ′)6

54Gϕ2U4 + 12(U ′)6

U5+ 5G′ (U ′)5

24G2ϕ2U3 −(U ′)5

3Gϕ3U3 − 7(U ′)5

ϕU4 − 5(G′)2 (

U ′)4

24G3ϕ2U2

− 5G′ (U ′)4

24G2ϕ3U2 − 47G′ (U ′)4

36GϕU3 + 5G′′ (U ′)4

8G2ϕ2U2 + 79U ′′ (U ′)4

54Gϕ2U3 − 157U ′′ (U ′)4

8U4 + 43(U ′)4

108Gϕ4U2

− 3281(U ′)4

324ϕ2U3 −(G′)2 (

U ′)3

G2ϕU2 + 137G′ (U ′)3

108Gϕ2U2 − 4G′ (U ′)3

3U3 + 41G′′ (U ′)3

36GϕU2 − 5G′U ′′ (U ′)3

6G2ϕ2U2

− U ′′ (U ′)3

Gϕ3U2 + 23U ′′ (U ′)3

ϕU3 + U ′′′ (U ′)3

2Gϕ2U2 + 3U ′′′ (U ′)3

2U3 + 107(U ′)3

18ϕ3U2 −(G′)3 (

U ′)2

4G3ϕU


− 25(G′)2 (

U ′)2

9G2ϕ2U− 11

(G′)2 (

U ′)2

12GU2 − 65(U ′′)2 (

U ′)2

216Gϕ2U2 + 15(U ′′)2 (

U ′)2

4U3 − 139G′ (U ′)2

24Gϕ3U

+ 1231G′ (U ′)2

216ϕU2 + G′G′′ (U ′)2

2G2ϕU+ 41G′′ (U ′)2

24Gϕ2U− G′′ (U ′)2

24U2 + 35G′U ′′ (U ′)2

6GϕU2

+ 145U ′′ (U ′)2

108ϕ2U2 − 3U ′′′ (U ′)2

ϕU2 +(U ′)2

6ϕ4U−

(G′)3 U ′8G2U

− 79(G′)2 U ′

36GϕU− 11

(U ′′)2 U ′2ϕU2

− 41G′U ′12ϕ2U

+ G′G′′U ′4GU

+ 7G′′U ′6ϕU

−(G′)2 U ′′U ′2G2ϕU

+ 4G′U ′′U ′9Gϕ2U

− 2G′U ′′U ′U2 + 13G′′U ′′U ′

36GϕU

− U ′′U ′6ϕ3U

− G′U ′′′U ′4GϕU

+ U ′′′U ′12ϕ2U

+(G′)4

24G3 + 11(G′)3

24G2ϕ− 19

(G′)2

72U+ 3

(G′)2

2Gϕ2

−(G′′)2

12G− 11G′ (U ′′)2

9GϕU−

(G′)2 G′′24G2 − G′G′′

12Gϕ−

(G′)2 U ′′2GU

+ 7G′U ′′36ϕU

+ 2G′′U ′′3U

+ G′U ′′′2U

)− 25

(U ′)4

216G2ϕ4U2 − 10(U ′)4

81Gϕ2U3 − 25(U ′)4

8U4 − 5G′ (U ′)3

72G2ϕ2U2 − 53(U ′)3

18Gϕ3U2

− 6(U ′)3

ϕU3 + 5(G′)2 (

U ′)2

72G3ϕ2U+ 25G′ (U ′)2

72G2ϕ3U+ 55G′ (U ′)2

108GϕU2 − 5G′′ (U ′)2

24G2ϕ2U; + 125U ′′ (U ′)2

108Gϕ2U2

+ 53U ′′ (U ′)2

12U3 −(U ′)2

6Gϕ4U− 27

(U ′)2

2ϕ2U2 − 3G(U ′)2

U3 +(G′)2 U ′2G2ϕU

+ 35G′U ′12Gϕ2U

+ 15G′U ′4U2 − G′′U ′

2GϕU+ 5G′U ′′U ′

36G2ϕ2U+ U ′′U ′

6Gϕ3U+ 83U ′′U ′

6ϕU2 − U ′′′U ′12Gϕ2U

− U ′′′U ′2U2

− 2GU ′ϕU2 +

(G′)3

12G3ϕ+ 55

(G′)2

72GU+ 3

(G′)2

4G2ϕ2 −(U ′′)2

3U2 + G′ϕU

− G′G′′6G2ϕ

− G′′U

− 37G′U ′′36GϕU

+ 2GU ′′U2 − (N − 1)

(G′)2

2G2ϕ2 − (N − 1)(G′)3

4G3ϕ+ (N − 1)G′G′′

2G2ϕ(G.3)

The coefficient α15 belonging to the structure �a,μ�a , ν�

b ,μ�, νb :

α15 = s2(

25(U ′)8

24G2ϕ4U4 + 3(U ′)8

8U6 − 3(U ′)7

ϕU5+ 5G′ (U ′)6

2G2ϕ3U3 + 205(U ′)6

36Gϕ4U3 + 9(U ′)6

ϕ2U4

+ 5G′ (U ′)5

6Gϕ2U3 + G′ (U ′)5

2U4 − 12(U ′)5

ϕ3U3 + 13(G′)2 (

U ′)4

12G2ϕ2U2 + 6G′ (U ′)4

Gϕ3U2 − 3G′ (U ′)4

ϕU3

+ 1681(U ′)4

216ϕ4U2 +(G′)2 (

U ′)3

GϕU2 + 113G′ (U ′)3

18ϕ2U2 −(G′)3 (

U ′)2

2G2ϕU− 77

(G′)2 (

U ′)2

36Gϕ2U

+(G′)2 (

U ′)2

4U2 − 41G′ (U ′)2

18ϕ3U−

(G′)3 U ′6GU

− 2(G′)2 U ′3ϕU

+(G′)4

24G2 +(G′)3

6Gϕ+

(G′)2

6ϕ2

)

+ s

(− 25

(U ′)6

36G2ϕ4U3 + 2(U ′)6

U5− 5

(U ′)5

2Gϕ3U3 +(U ′)5

2ϕU4 − 5G′ (U ′)4

3G2ϕ3U2 − 205(U ′)4

108Gϕ4U2

+ 7(U ′)4

ϕ2U3 − 16G′ (U ′)3

9Gϕ2U2 − G′ (U ′)3

6U3 + 19(U ′)3

6ϕ3U2 − 31(G′)2 (

U ′)2

36G2ϕ2U− 2G′ (U ′)2

Gϕ3U

− 5G′ (U ′)2

2ϕU2 −(G′)2 U ′3GϕU

− 11G′U ′6ϕ2U

+(G′)3

6G2ϕ+

(G′)2

4U+

(G′)2

3Gϕ2

)+ 25

(U ′)4

216G2ϕ4U2


− 17(U ′)4

24U4 + 2(U ′)3

3Gϕ3U2 +(U ′)3

6ϕU3 + G′ (U ′)2

9G2ϕ3U−

(U ′)2

3Gϕ4U− 5

(U ′)2

3ϕ2U2 + G(U ′)2

4U3

+ G′U ′2Gϕ2U

+ U ′′U ′3Gϕ3U

+ 5U ′3ϕ3U

− GU ′6ϕU2 +

(G′)2

6G2ϕ2 + 7G′6ϕU

− 5U ′′3ϕ2U

+ 3G2

2U2 (G.4)

The coefficient α16 belonging to the structure �a,μna�b

,νnb�c,μ�

, νc :

α16 = s3( (

U ′)10

Gϕ2U6 − 9(U ′)10

4U7 + 9(U ′)9

ϕU6 + 3G′ (U ′)8

2GϕU5− 19U ′′ (U ′)8

6Gϕ2U5+ 9U ′′ (U ′)8

U6

− 26(U ′)8

3ϕ2U5− 2G′ (U ′)7

3Gϕ2U4 − 36U ′′ (U ′)7

ϕU5− 5

(G′)2 (

U ′)6

12GU4 − 155(U ′′)2 (

U ′)6

12Gϕ2U4

− 9(U ′′)2 (

U ′)6

U5− 5G′ (U ′)6

2ϕU4 − 2G′U ′′ (U ′)6

GϕU4 + 629U ′′ (U ′)6

18ϕ2U4 −(G′)2 (

U ′)5

GϕU3

+ 36(U ′′)2 (

U ′)5

ϕU4 + 52G′ (U ′)5

9ϕ2U3 − 5G′U ′′ (U ′)5

2Gϕ2U3 + 3G′U ′′ (U ′)5

U4 + 13(G′)2 (

U ′)4

36U3

− 31G′ (U ′′)2 (U ′)4

2GϕU3 − 1451(U ′′)2 (

U ′)4

36ϕ2U3 − 2(G′)2 U ′′ (U ′)4

3GU3 − 2G′U ′′ (U ′)4

3ϕU3

− 4(G′)2 (

U ′)3

3ϕU2 − 6G′ (U ′′)2 (U ′)3

U3 − 6(G′)2 U ′′ (U ′)3

GϕU2 − 77G′U ′′ (U ′)3

6ϕ2U2

−(G′)3 (

U ′)2

2GϕU−

(G′)2 (

U ′)2

ϕ2U+ 31

(G′)2 (

U ′′)2 (U ′)2

12GU2 + 41G′ (U ′′)2 (U ′)2

6ϕU2

− 11(G′)2 U ′′ (U ′)2

9U2 +(G′)3 U ′′U ′

GU+ 2

(G′)2 U ′′U ′ϕU

+(G′)4

12G+

(G′)3

6ϕ− 5

(G′)2 (

U ′′)2

36U

)

+ s2(

− 25(U ′)8

12G2ϕ4U4 + 83(U ′)8

18Gϕ2U5+ 15

(U ′)8

2U6 +(U ′)7

Gϕ3U4 + 9(U ′)7

ϕU5− 5G′ (U ′)6

G2ϕ3U3

− 5G′ (U ′)6

6GϕU4 + 17U ′′ (U ′)6

18Gϕ2U4 − 33U ′′ (U ′)6

2U5− 205

(U ′)6

18Gϕ4U3 − 73(U ′)6

54ϕ2U4 − 17G′ (U ′)5

36Gϕ2U3

− 19G′ (U ′)5

4U4 + 5G′′ (U ′)5

3GϕU3 + 5G′U ′′ (U ′)5

2G2ϕ2U3 + 15U ′′ (U ′)5

2Gϕ3U3 + 6U ′′ (U ′)5

ϕU4 + 73(U ′)5

3ϕ3U3

− 7(G′)2 (

U ′)4

4G2ϕ2U2 + 7(G′)2 (

U ′)4

18GU3 + 55(U ′′)2 (

U ′)4

9Gϕ2U3 − 19G′ (U ′)4

2Gϕ3U2 − 11G′ (U ′)4

18ϕU3

− G′′ (U ′)4

U3 − G′U ′′ (U ′)4

3GϕU3 − 973U ′′ (U ′)4

27ϕ2U3 − 1681(U ′)4

108ϕ4U2 − 24(U ′′)2 (

U ′)3

ϕU3

− 1537G′ (U ′)3

108ϕ2U2 + 23G′′ (U ′)3

9ϕU2 + 3(G′)2 U ′′ (U ′)3

G2ϕU2 + 47G′U ′′ (U ′)3

3Gϕ2U2 + G′U ′′ (U ′)3

U3

+ 13G′′U ′′ (U ′)3

3GϕU2 + 5U ′′ (U ′)3

2ϕ3U2 + 3(G′)3 (

U ′)2

2G2ϕU+ 83

(G′)2 (

U ′)2

12Gϕ2U+

(G′)2 (

U ′)2

3U2

+ 9G′ (U ′′)2 (U ′)2

2GϕU2 + 1361(U ′′)2 (

U ′)2

108ϕ2U2 + 97G′ (U ′)2

18ϕ3U+ G′G′′ (U ′)2

GϕU

+ 29(G′)2 U ′′ (U ′)2

9GU2 + 20G′U ′′ (U ′)2

3ϕU2 + 2G′′U ′′ (U ′)2

U2 + 7(G′)3 U ′12GU

+ 26(G′)2 U ′9ϕU


+ 2G′ (U ′′)2 U ′U2 −

(G′)3 U ′′U ′

2G2U− 2

(G′)2 U ′′U ′

GϕU+ 2G′U ′′U ′

3ϕ2U− 2G′G′′U ′′U ′

GU

− 23G′′U ′′U ′9ϕU

−(G′)4

6G2 − 5(G′)3

6Gϕ−

(G′)2

2ϕ2 + 5(G′)2 (

U ′′)2

36GU− 2G′ (U ′′)2

9ϕU−

(G′)2 G′′

3G

− G′G′′3ϕ

+ 2(G′)2 U ′′

3U

)+ s

(25

(U ′)6

18G2ϕ4U3 + 167(U ′)6

54Gϕ2U4 − 9(U ′)6

U5− 13G′ (U ′)5

12G2ϕ2U3

+ 53(U ′)5

6Gϕ3U3 − 25(U ′)5

2ϕU4 − 5(G′)2 (

U ′)4

12G3ϕ2U2 + 11G′ (U ′)4

6G2ϕ3U2 + 20G′ (U ′)4

9GϕU3 − 175U ′′ (U ′)4

54Gϕ2U3

+ 13U ′′ (U ′)4

2U4 + 151(U ′)4

54Gϕ4U2 − 1735(U ′)4

81ϕ2U3 + 7(G′)2 (

U ′)3

2G2ϕU2 + 172G′ (U ′)3

27Gϕ2U2 + 17G′ (U ′)3

6U3

+ 4G′′ (U ′)3

9GϕU2 − G′U ′′ (U ′)3

6G2ϕ2U2 − U ′′ (U ′)3

Gϕ3U2 + 23U ′′ (U ′)3

ϕU3 − 161(U ′)3

18ϕ3U2 −(G′)3 (

U ′)2

2G3ϕU

+ 4(G′)2 (

U ′)2

9G2ϕ2U+ 2

(G′)2 (

U ′)2

3GU2 − 119(U ′′)2 (

U ′)2

108Gϕ2U2 + 3(U ′′)2 (

U ′)2

U3 + 8G′ (U ′)2

3Gϕ3U

+ 289G′ (U ′)2

27ϕU2 − G′G′′ (U ′)2

2G2ϕU− G′′ (U ′)2

3U2 − 3G′U ′′ (U ′)2

GϕU2 + 373U ′′ (U ′)2

54ϕ2U2 −(U ′)2

3ϕ4U

−(G′)3 U ′4G2U

+ 5(G′)2 U ′

18GϕU+ 4

(U ′′)2 U ′ϕU2 + 9G′U ′

2ϕ2U− G′G′′U ′

2GU− 2G′′U ′

3ϕU−

(G′)2 U ′′U ′

G2ϕU

− 40G′U ′′U ′9Gϕ2U

− 9G′U ′′U ′2U2 − 13G′′U ′′U ′

9GϕU+ U ′′U ′

2ϕ3U+

(G′)4

12G3 −(G′)3

3G2ϕ− 25

(G′)2

36U

−(G′)2

2Gϕ2 +(G′′)2

3G+ 2G′ (U ′′)2

9GϕU−

(U ′′)2

6ϕ2U+

(G′)2 G′′

6G2 + 5G′G′′6Gϕ

− 25G′U ′′9ϕU

− 2G′′U ′′3U

)

− 25(U ′)4

108G2ϕ4U2 − 128(U ′)4

81Gϕ2U3 + 9(U ′)4

4U4 + 13G′ (U ′)3

36G2ϕ2U2 − 55(U ′)3

18Gϕ3U2 + 7(U ′)3

2ϕU3

+ 5(G′)2 (

U ′)2

36G3ϕ2U− G′ (U ′)2

18G2ϕ3U+ 53G′ (U ′)2

27GϕU2 + 59U ′′ (U ′)2

54Gϕ2U2 − 2U ′′ (U ′)2

3U3 +(U ′)2

3Gϕ4U

+ 8(U ′)2

ϕ2U2 + 19G(U ′)2

2U3 − G′U ′2Gϕ2U

− 13G′U ′2U2 − 2G′U ′′U ′

9G2ϕ2U− U ′′U ′

2Gϕ3U− 22U ′′U ′

3ϕU2

+(G′)3

6G3ϕ+ 7

(G′)2

36GU+

(U ′′)2

6Gϕ2U− 2

(U ′′)2

3U2 − G′ϕU

+ G′G′′6G2ϕ

+ G′′U

+ G′U ′′9GϕU

(G.5)

The coefficient α17 belonging to the structure �a μ;μ � ν

a; ν :

α17 = s

(−

(U ′)4

4GU 2ϕ2 − G ′U ′

2U+

(G ′)2

4G−

(U ′)2

12Uϕ2 − 3(U ′)4

4U 3

)

+(U ′)2

12GUϕ2 − U ′

Uϕ+

(U ′)2

4U 2 (G.6)

The coefficient α18 belonging to the structure (�c μ;μ nc)

2:


α18 = s2

(9G ′ (U ′)3

4U 2 +(G ′)2

8+ 81

(U ′)6

8U 4

)+ s

( (U ′)4

4GU 2ϕ2 −(G ′)2

4G+

(U ′)2

12Uϕ2

−27(U ′)4

4U 3

)−

(U ′)2

12GUϕ2 + U ′

Uϕ+ 13

(U ′)2

2U 2 − U ′′

U+ (N − 1)

(G ′)2

8G2

(G.7)


,μa )(�b ν

; ν nb):

α19 = s2(

−(U ′)7

4Gϕ2U4 − 27(U ′)7

8U5+ 27

(U ′)6

4ϕU4 − 3G′ (U ′)5

4GϕU3 − 3U ′′ (U ′)5

4Gϕ2U3 −(U ′)5

12ϕ2U3

+ 15G′ (U ′)4

8U3 + 3(G′)2 (

U ′)3

8GU2 − 7G′ (U ′)3

4ϕU2 − 9G′′ (U ′)3

4U2 + 3G′U ′′ (U ′)3

4GϕU2

− U ′′ (U ′)3

4ϕ2U2 − 3(G′)2 (

U ′)2

4GϕU+ G′U ′′U ′

4ϕU+

(G′)3

8G−

(G′)2

2ϕ− G′G′′

4

)

+ s

( (U ′)5

6Gϕ2U3 + 15(U ′)5

8U4 + 5G′ (U ′)4

8G2ϕ2U2 − 2(U ′)4

ϕU3 + 3G′ (U ′)3

4GϕU2 + U ′′ (U ′)3

2Gϕ2U2

+ 3U ′′ (U ′)3

2U3 + 37(U ′)3

36ϕ2U2 + 3(G′)2 (

U ′)2

4G2ϕU+ 41G′ (U ′)2

24Gϕ2U− 3G′ (U ′)2

2U2

− 3U ′′ (U ′)2

ϕU2 +(G′)2 U ′4GU

+ G′U ′ϕU

+ G′′U ′2U

− G′U ′′U ′4GϕU

+ U ′′U ′12ϕ2U

−(G′)3

8G2

+ G′U ′′2U

)+ 53

(U ′)3

36Gϕ2U2 −(U ′)3

4U3 − 5G′ (U ′)2

24G2ϕ2U+ 14

(U ′)2

ϕU2 − G′U ′2GϕU

− U ′′U ′12Gϕ2U

− U ′′U ′2U2 + U ′

ϕ2U+ 2GU ′

U2 −(G′)2

4G2ϕ− G′

2U− U ′′ϕU

+ (N − 1)(G′)2

2G2ϕ(G.8)

The coefficient α20 belonging to the structure (�a,μna�b ,μnb)(�

c ν; ν nc):

α20 = s3(

81(U ′)9

8U6 − 81U ′′ (U ′)7

2U5− 45G′ (U ′)6

8U4 + 81(U ′′)2 (

U ′)5

2U4 + 9G′U ′′ (U ′)4

U3

+ 3(G′)2 (

U ′)3

8U2 + 9G′ (U ′′)2 (U ′)2

2U2 + 3(G′)2 U ′′U ′

2U+

(G′)3

8

)+ s2

(3

(U ′)7

4Gϕ2U4

− 45(U ′)7

8U5− 45

(U ′)6

4ϕU4 − G′ (U ′)5

2GϕU3 + 9U ′′ (U ′)5

2Gϕ2U3 + 135U ′′ (U ′)5

4U4 +(U ′)5

4ϕ2U3

+ 5G′ (U ′)4

4Gϕ2U2 − 21G′ (U ′)4

8U3 + 9U ′′ (U ′)4

ϕU3 − 3(G′)2 (

U ′)3

8GU2 − 63(U ′′)2 (

U ′)3

2U3

+ 4G′ (U ′)3

3ϕU2 + 9G′′ (U ′)3

4U2 − 13G′U ′′ (U ′)3

4GϕU2 + 3U ′′ (U ′)3

2ϕ2U2 + 5G′ (U ′)2

12ϕ2U


− 15G′U ′′ (U ′)2

4U2 −(G′)2 U ′

2U+ 3

(G′)2 U ′′U ′

2GU+ 23G′U ′′U ′

12ϕU+

(G′)3

8G+ 3

(G′)2

4ϕ

− 3G′ (U ′′)2

2U+ G′G′′

4

)+ s

(7

(U ′)5

4Gϕ2U3 + 15(U ′)5

8U4 − 11G′ (U ′)4

8G2ϕ2U2 −(U ′)4

Gϕ3U2

+ 3(U ′)4

ϕU3 − 25G′ (U ′)3

12GϕU2 − U ′′ (U ′)3

Gϕ2U2 − 12U ′′ (U ′)3

U3 − 7(U ′)3

3ϕ2U2 − 19G′ (U ′)2

8Gϕ2U

+ 3G′ (U ′)2

U2 − 3U ′′ (U ′)2

ϕU2 −(U ′)2

3ϕ3U+

(G′)2 U ′2GU

+ 15(U ′′)2 U ′2U2 − G′U ′

4ϕU+ 13G′U ′′U ′

12GϕU

+ U ′′U ′6ϕ2U

−(G′)3

8G2 −(G′)2

2Gϕ− G′G′′

2G

)− 13

(U ′)3

6Gϕ2U2 − 29(U ′)3

4U3 + 11G′ (U ′)2

24G2ϕ2U

+(U ′)2

3Gϕ3U− 33

(U ′)2

2ϕU2 + G′U ′4GϕU

− 3U ′ϕ2U

− U ′U ′′6Gϕ2U

+ 33U ′U ′′2U2 + 3U ′′

ϕU− U ′′′

U

− (N − 1)(G′)2

4G2ϕ− (N − 1)

(G′)3

8G3 + (N − 1)G′G′′4G2 (G.9)

The coefficient α21 belonging to the structure �a μ;μ �

, νa �b

, νnb:

α21 = s2(

−(U ′)7

2Gϕ2U4 − 9(U ′)7

4U5+ 9

(U ′)6

2ϕU4 + 5G′ (U ′)5

4GϕU3 − 15U ′′ (U ′)5

4Gϕ2U3 + 9U ′′ (U ′)5

2U4

−(U ′)5

6ϕ2U3 − 5G′ (U ′)4

4Gϕ2U2 − 3G′ (U ′)4

4U3 − 9U ′′ (U ′)4

ϕU3 + 5G′ (U ′)3

12ϕU2 + 5G′U ′′ (U ′)3

2GϕU2

− 5U ′′ (U ′)3

4ϕ2U2 + 3(G′)2 (

U ′)2

4GϕU− 5G′ (U ′)2

12ϕ2U+ 3G′U ′′ (U ′)2

U2 +(G′)2 U ′

4U

− 3(G′)2 U ′′U ′

2GU− 13G′U ′′U ′

6ϕU−

(G′)3

4G−

(G′)2

4ϕ

)+ s

(− 23

(U ′)5

12Gϕ2U3 + 3(U ′)5

4U4

+ 3G′ (U ′)4

4G2ϕ2U2 +(U ′)4

Gϕ3U2 −(U ′)4

ϕU3 + 4G′ (U ′)3

3GϕU2 + U ′′ (U ′)3

2Gϕ2U2 + 3U ′′ (U ′)3

2U3

+ 47(U ′)3

36ϕ2U2 − 3(G′)2 (

U ′)2

4G2ϕU+ 2G′ (U ′)2

3Gϕ2U− G′ (U ′)2

2U2 + 6U ′′ (U ′)2

ϕU2 +(U ′)2

3ϕ3U

− 3(G′)2 U ′4GU

− 3G′U ′4ϕU

− G′′U ′2U

− 5G′U ′′U ′6GϕU

− U ′′U ′4ϕ2U

+(G′)3

4G2 +(G′)2

2Gϕ

+ G′G′′2G

)+ 25

(U ′)3

36Gϕ2U2 − G′ (U ′)2

4G2ϕ2U−

(U ′)2

3Gϕ3U+ 5

(U ′)2

2ϕU2 + G′U ′4GϕU

+ U ′′U ′4Gϕ2U

− U ′′U ′U2 + 2U ′

ϕ2U− 13GU ′

2U2 +(G′)2

4G2ϕ+ G′

U− 2U ′′

ϕU(G.10)


References

1. Amsler, C. et al. (Particle Data Group): Review of particle physics. Phys. Lett. B 667, 1 (2008)2. DeWitt, B.S.: Dynamical Theory of Groups and Fields. Blackie & Son, London (1965)3. Parker, L.E., Toms, D.J.: Quantum Field Theory in Curved Spactime. Cambridge University

Press, Cambridge (2009)4. Poisson, E., Pound, A., Vega, I.: The motion of point particles in curved spacetime. Living Rev.

Rel. 7, 6 (2004). Cited on 19 December 2011. http://relativity.livingreviews.org/Articles/lrr-2011-7

5. Shapiro, I.L., Takata, H.: One-loop renormalization of the four-dimensional theory for quantumdilaton gravity. Phys. Rev. D 52, 2162 (1995)

6. Sirlin, A., Zucchini, R.: Dependence of the Higgs coupling hMS(M) on m(H) and the possibleonset of new physics. Nucl. Phys. B 266, 389 (1986)

7. Synge, J.L.: Relativity: The General Theory. North-Holland, Amsterdam (1960)8. Tarrach, R.: The pole mass in perturbative QCD. Nucl. Phys. B 183, 384 (1981)

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Appendix A Riemann Curvature Tensor - Springer978-3-319-01842-3/1.pdf · 244 Appendix A: Riemann...

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