Riemann Curvature Tensor

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

1 1 T[ab] := (Tab − Tba) and T( ) := (Tab + Tba) , (A.1) 2 ab 2 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

[∇ , ∇ ] vρ = ρ vσ, μ ν R σμν (A.2)

ρ with the explicit formula in terms of the symmetric Christoffel symbols μν

ρ = ρ − ρ + ρ α − ρ α . R σμν ∂μ σν ∂ν σμ αμ σν αν σμ (A.3)

ρ From this deﬁnition it is obvious that R σμν 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

1 R [ ] = (R + R + R ) = 0. (A.5) α βγδ 3 αβγδ αδβγ αγδβ

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 composed ( × ) 1 2 1 2 of two d d matrices Aαβ and Aγδ.From(A.4), it follows that A and A are antisymmetric, each having n(n − 1)/2 independent components. Writing Rαβγδ ≡ ( ) → 1 ( ) → 2 RA1 A2 , where we have collected the index pairs αβ A and γδ A , 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 243 in 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) ( ) 1 2 and γδ , we can consider RA1 A2 as a symmetric matrix in A and A ,having A1(A1 + 1)/2 independent components. Altogether, we find 1 1 1 1 1 3 1 d (d − 1) d (d − 1) + 1 = d4 − d3 + d2 − d (A.6) 2 2 2 8 4 8 4 independent components. We still need to figure out how many components are related by the cyclic identity (A.5)ind dimensions. We can artificially write

1 R = R − R + R − R + (αβ) ↔ (γδ) (A.7) αβγδ 8 αβγδ αβδγ βαδγ βαγδ 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 identical zero for two identical indices but gives one additional constraint for each choice of four distinct, orderless indices, reducing the number of independent components by one respectively. In d dimensions, the number of additional constraints corresponds to “choose 4 out of d”. For integers d, n there exists a product formula to calculate the binomial n d d − n + k = . (A.8) n k k=1

Inserting n = 4, we obtain d 1 1 1 11 1 = [d (d − 1)(d − 2)(d − 3)] = d4 − d3 + d2 − d. (A.9) 4 24 24 4 24 4

Thus, the number of independent components IRiem(d) of Rαβγδ is given by 1 1 d 1 I (d) := d (d − 1) d (d − 1) + 1 − = d2 (d2 − 1). (A.10) Riem 4 2 4 12

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 degree of freedom. It further shows that gravity in two dimensions can be described by the (3) (3) Ricci scalar R and in three dimensions by Rμν = Rνμ . In general, the d-dimensional := γδ ( ) := 1 ( + ) Ricci tensor Rμν g Rγμδν has IRic d 2 d d 1 independent components. In particular IRic(4) = 10. Thus, ten components of Rαβγδ not contained in Rμν still remain. They are contained in the Weyl tensor Appendix A: Riemann Curvature Tensor 245

2 2 C := R − g [ R ] − g [ R ] + g [ g ] R. αβγδ αβγδ d − 2 α γ δ β β γ δ α (d − 1)(d − 2) α γ δ β (A.12) It follows that the Weyl tensor has (for d > 3)

1 I (d) = I (d) − I (d) = d (d + 1)(d + 2)(d − 3) (A.13) Weyl Riem Ric 12 independent components, and in particular IWeyl (4) = 10. Appendix B Variations and Derivatives

B.1 Covariant Differentiation in General Relativity

The action of the metric compatible covariant derivative ∇μ gαβ = 0 with respect to ρ the Christoffel symbol μν on a general tensor is deﬁned as

∇ α...γ = α...γ + α ρ...γ + ... + γ α...ρ μ T β...δ ∂μ T β...δ μρ T β...δ μρ T β...δ − ρ α...γ − ... − ρ α...γ , μβ T ρ...δ μδ T β...ρ (B.1) with the symmetric Christoffel symbol 1 α (g) := gαρ ∂ g + ∂ g − ∂ g = α (g). (B.2) μν 2 ν μρ μ νρ ρ μν νμ

μ For scalar functions ∇μϕ = ∂μϕ. For vector ﬁelds v , a useful formula is 1 √ √ √ ∇ vμ = √ ∂ g vμ or g ∇ vμ = ∂ g vμ . (B.3) μ g μ μ μ

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 indices like ij 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 at different space-time points. A particular case of a generalized bi-tensor is a bi-scalar which 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 with A metric gμν(x) for a general test field (x) by the equation

C. F. Steinwachs, Non-minimal Higgs Inﬂation and Frame Dependence 247 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 248 Appendix B: Variations and Derivatives / A(x) = d2ω x |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 = i j δ j with i =| ( )|1/2 A ( , ) = A ˜( , ). δ j g x δ B δ x x δ B δ x x (B.5)

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

i Z[ + ζ]=:Z[]+Z, i [] ζ , (B.6) with ζi := δψi and [( )] i 4 δZ x A Z, [] ζ = d x ζ (x ) (B.7) i δ A(x) in conventional notation. Therefore, it seems obvious to deﬁne the functional deriv- A ative Z, i = δZ[(x)]/δ (x ) by the variation [3] Z[(x)] 1 ( [ + ]− []) =: 4 δ A( ). lim Z ζ Z d x ζ x (B.8) →0 δ A(x )

For the special case Z = Id, we obtain from (B.8) A A δ (x) δ (x) ζ A = d4x ζ B(x ) or = δ A δ˜(x, x ). (B.9) δ B(x) δ B(x) B

B.3 Lie Derivative

L γ...δ μ The Lie derivative ξ of an arbitrary tensor Tα...β in the direction of the vector ξ is γ ...δ ( ) a measure of the difference between Tα...β x “dragged” along the integral curve μ γ...δ( ) γξ of ξ compared to the tensor Tα...β x at this point. Calculating the infinitesimal γ ...δ ( ) flow of Tα...β x along γξ back from x to x with the infinitesimal coordinate μ ν μ μ transformation matrix ∂x /∂x = δ ν + ξ, ν yields

γ ...δ ( ) = γ...δ + γ ρ...δ + ... + δ γ...ρ − ρ γ...δ − ... − ρ γ...δ . Tα...β x Tα...β ξ,ρTα...β ξ,ρTα...β ξα, Tρ...β ξβ, Tα...ρ (B.10)

γ...δ( ) Taylor expansion of Tα...β x around x yields Appendix B: Variations and Derivatives 249

γ...δ( + ) = γ...δ( ) + γ...δ ρ. Tα...β x ξ Tα...β x Tα...β, ρ ξ (B.11)

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

γ...δ γ...δ T (x ) − T (x ) L γ...δ ( ) := α...β α ...β ξ T ... x lim α β →0 = γ...δ ρ − ρ...δ γ − ... − γ...ρ δ Tα...β, ρ ξ Tα...β ξ,ρ Tα...β ξ,ρ + γ...δ ρ + ... + γ...δ ρ. Tρ...β ξα, Tα...ρ ξβ, (B.12)

A special case is the Lie derivative of the metric tensor. The vanishing of the Lie L = μ derivative ξ g μν 0 in direction ξ signalizes a symmetry of the space-time manifold and means that the vector ξμ is a generator of an isometry

ρ ρ ρ L = , + + = ; + ; = . ξ g μν gμν ρ ξ gρν ξ, μ gνρ ξ, 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

i i In the case of pure gravity, = gμν(x) and ζ = hμν(x), we ﬁnd from (B.9) δg (x) μν = αβ ˜( , ) αβ := α β = 1 α β + α β . δμν δ x x with δμν δ(μδν) δμ δν δν δμ (B.14) δgαβ(x ) 2

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

μν μ(α β)ν δg g =−g g hαβ. (B.15) 1/2 By using (B.8) with F(g) = g and the identity det(g) = exp tr[log(gμν)] ,we ﬁnd / 1 / δ g1 2 = g1 2 gαβ h . (B.16) g 2 αβ

By using (B.4) with an arbitrary test function t(x) = 0 that does not depend on gμν and differentiating both sides, we ﬁnd δ δ d4x δ˜(x, x ) t(x ) = 0or δ˜(x, x ) = 0. (B.17) δgμν(y) δgμν(y)

Combining (B.14) with (B.17) leads to 250 Appendix B: Variations and Derivatives

δ2g (x) μν = 2 = = . 0orδg gμν δg hμν 0 (B.18) δgαβ(y) δgγδ(z)

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

1 δ ρ =− gραgδβ ∂ g + ∂ g − ∂ g h g μν 2 ν μδ μ νδ δ μν αβ 1 + gρδ ∂ h + ∂ h − ∂ h . (B.19) 2 ν μδ μ νδ δ μν

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

ρ =− ρα β + 1 ρδ ∇ +∇ −∇ δg μν g μν hαβ g ν hμδ μ hνδ δ hμν 2 + + + + − − νμhδ νδhμ μν hδ μρhν μρhν νρhμ 1 =−gρα β h + gρδ ∇ h +∇ h −∇ h + 2 h μν αβ 2 ν μδ μ νδ δ μν νμ δ

1 ρδ 1 ρδ = g ∇ h +∇ h −∇ h = g h ; + h ; − h ; . 2 ν μδ μ νδ δ μν 2 μδ ν νδ μ μν δ (B.20)

Then, the variation of the Riemann tensor yields ρ = ρ − ρ + ρ α − ρ α δg R σμν δg ∂μ νσ ∂ν μσ μα νσ να μσ = ( ρ ) − ( ρ ) + ( ρ )α + ( α )ρ δg νσ , μ δg μσ , ν δg μα νσ δg νσ μα − ( ρ )α − ( α )ρ . δg να μσ δg μσ να (B.21)

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

( ρ ) = ( ρ ) − ( α )ρ + ( ρ )α + ( ρ )α . δg νσ , μ δg νσ ; μ δg νσ αμ δg ασ νμ δg να σμ (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 variation of the Riemann tensor yields

ρ = ( ρ ) − ( ρ ) . δg R σμν δg νσ ; μ δg μσ ; ν (B.23)

For the variation of the Ricci tensor, we contract (B.23) over ρ and μ Appendix B: Variations and Derivatives 251

μ( ρ ) = ( μ ρ ) = ρ = ( ρ ) − ( ρ ) . δρ δg R σμν δg δρ R σμν δg R σρν δg νσ ; ρ δg ρσ ; ν (B.24)

Substituting the variation (B.20), we obtain

ρ ρ 1 ρδ δ R = (δ ); − (δ ); = g h ; + h ; − h ; − h ; . g μν g νμ ρ g μρ ν 2 μρ νδ νδ μρ μν ρδ ρδ μν (B.25) Finally, the variation of the Ricci scalar can ultimately be obtained from (B.15) and (B.25) by the relation

μν μν δg R = (δg g ) Rμν + g (δg Rμν). (B.26) Appendix C Young Tableaux for SU(3)

The most efﬁcient way to calculate direct products of different representations and their dimensionality is to use the technique of Young tableaux. This is an abstract symbolical notation for projecting out the symmetric and antisymmetric components of a tensor. A linear representation ρg of a group element g ∈ G can be written as a multilinear form (a tensor). Each index runs over the dimension of the group parameters from 1, ..., N. Everything we will work out can easily be generalized to SU(N), but we are mainly interested in SU(3). We have to specify several conditions and rules which deﬁne a valid diagram and its combination with other diagrams. The fundamental 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 indices ¯ run from 1, ..., 3). The conjugated representation 3 corresponds to a tensor Ai with one covariant index. To obtain the corresponding diagram, we have to raise this index with the three-dimensional epsilon tensor ijk. This gives a completely antisymmetric contravariant tensor A jk of second rank. Anti-symmetrization is diagrammatically indicated by vertical boxes, one box for each index and symmetrization by boxes in a row. Thus, we obtain the diagram for 3¯

3¯ = . (C.2)

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

C. F. Steinwachs, Non-minimal Higgs Inﬂation and Frame Dependence 253 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 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 in the second row of the right diagram with b etc., until we have filled all rows of the second diagram in that way. 2. Second, we take all boxes filled with a and attach them to the first diagram in all possible ways to construct a valid Young diagram. After having attached the blocks 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 left diagram. This will result in a big number of different diagrams (two diagrams are 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. We start with the last row and write down all a’s, b’s etc. in the order they appear reading the row from left to right. We continue writing the sequence generated by the letters from the last row by attaching a sequence of letters generated from the above row in the same way. We do this for all rows ending with the first row. We finally have a string of letters for each diagram. We keep only those diagrams in which for an arbitrary position in the string the number of a’s right to 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’s and so on. We throw away all diagrams which do not satisfy this constraint. The resulting diagrams correspond to the decomposition into a direct sum of irreducible representations. We finally have to estimate the dimensionality of the representation corresponding to 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 (in case of SU(3), a 3). Fill the remaining boxes in the same row by the value of the foregoing box and add 1. Fill the first box of the second row with the value of the box above it and subtract one. Fill the remaining boxes in the same row according to the same rule like for the first row. Proceed in this way until you have filled all boxes. Calculate the product of the numbers contained in all boxes of the diagram. Take the result as the value of the numerator of the number for the dimensionality. 3. Fill the copy of the diagram accordingly to the following rule: For each box count the number of boxes to the right and the number of boxes below and add one. Fill this value into the box. Fill all boxes in this way and calculate again the product of the numbers of all boxes. The result is the denominator of the dimensionality. Appendix C: Young Tableaux for SU(3) 255

4. The fraction of these two numbers gives the dimensionality of the representation corresponding to each diagram in the direct sum. The dimensionality for the simple example above can be calculated by

34 3 · 4 12 = = = 6. (C.4) 21 2 · 1 2

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

a 3¯ ⊗ 3 = ⊗ a = ⊕ a = 8 ⊕ 1, (C.5) where we have calculated the dimensionality by

3 3 4 2 2 4 · 3 · 2 1 = = 8, = 1. (C.6) 3 1 3 · 1 · 1 3 1 2 1

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 the gluon itself transforms in the adjoint representation 8 we have to calculate

a aa aa a a a 8 ⊗ 8 = ⊗ b = ⊕ a ⊕ a ⊕ a ⊕ a ⊗ b

aa aab aa a b a = ⊕ b ⊕ b ⊕ a ⊕ a b 256 Appendix C: Young Tableaux for SU(3)

a a a b a b a b a a a ⊕ b ⊕ a ⊕ a ⊕ 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 the application in the Standard Model. Appendix D Synge’s World Function

The notation “world function” dates back to the Irish relativist Synge [7] and describes a non-local extension σ(x, x ) of the space-time metric gμν(x). It is deﬁned by connecting the two space-time points {x = z(s1), x = z(s0)}∈M, s ∈ R by the unique geodesic γ ≡ z(s) parametrized through s s1 1 μ ν σ(x, x ) := (s1 − s0) gμν(z(s)) t t ds. (D.1) 2 s0

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

dzμ(s) tμ := ∈ TM. (D.2) ds

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

Dtμ d2zμ(s) dzα(s) dzβ(s) := + μ = 0. (D.3) Ds ds2 αβ ds ds Thus, the scalar product and in particular the norm are conserved on γ

μ ν gμν(z) t t := N = const. (D.4)

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

1 N σ(x, x ) = (s − s )2 g (z) tμtν = (s)2. (D.5) 2 1 0 μν 2

C. F. Steinwachs, Non-minimal Higgs Inﬂation and Frame Dependence 257 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 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 that σ(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

1 σ(x, x ) = η (xμ − x μ)(xν − x ν ). (D.6) 2 μν

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

(x0)2 (x1)2 (x2)2 (x3)2 σ(x, 0) =− √ + √ + √ + √ = 0. (D.7) ( 2)2 ( 2)2 ( 2)2 ( 2)2

D.1 Derivatives of the World Function

By repeated differentiation of the world function, we can generate bi-tensors of different rank at x and x. The covariant derivative of σ(x, x) at the point x is deﬁned as ∇ασ := σ; α = σ, α. Correspondingly, the derivative of σ at the point x is deﬁned as σ; α . The world function and its derivatives transform independently at x and x . For example, this means σ; αβγ transforms like a tensor of second rank at x and like a covariant vector at x. This implies that the order of primed and unprimed indices can be changed arbitrarily for any general bi-tensor A...; βα = A...; αβ .In particular, we can freely permute primed and unprimed derivatives of σ as long as the order among the primed and the order among the unprimed indices remain maintained, e.g. σ; αβγ = σ; αγβ = σ; γαβ = σ; βαγ. (D.8)

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

μ ν δσ = σ(x + δx, x ) − σ(x, x ) = s gμν t δx . (D.9)

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

= ( ν)| ; μ = μ, σ; μ s gμν t s=s1 or σ st (D.10) which shows that σ; μ is proportional to the tangential vector tμ with length of the geodesic distance between x and x and pointing in direction x → x. The derivative of σ at x can be obtained in an analogue manner

; =− ( ν )| μ =− μ . σ; μ s gμ ν t s=s0 or σ st (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

μν 2 μ ν 2 1 ; μ g σ, σ; = g (s) t t = (s) N = 2σ or σ = σ; σ . (D.12) μ ν μν 2 μ

D.2 Densities and Determinants

...... b1 bn Ab1 bn w For any ordinary tensor Aa1...an a tensor density a1...an of weight is deﬁned by its transformation behaviour under general coordinate transformations xμ →˜xμ

w ã1 ãn b1 bn ... ã1...an ∂xc ∂x ∂x ∂x ∂x a1 an A ... = det ... ... A , (D.13) b1 bn d a a b bn b ...b ∂x˜ ∂x 1 ∂x n ∂x˜ 1 ∂x˜ 1 n

( ˜ ) = ( ( )) = ∂ xμ where g det gμν x ∂(xν ) is the Jacobian that takes care of a coordinate transformation from xμ to x˜ν . Since we deal with bi-tensors, we can generalize this for the transformation of the two variables (xμ, xρ) that characterize the geodesic γ by its endpoints to the variables (σ; ν , xσ) that characterize γ by one endpoint and the tangent at this point. This Jacobian is given by ( , ) ( ) ∂ σ; ν xσ ∂ σ; ν = = det(σ; μν ) = det(−Dμν ). (D.14) ∂ (xμ, xρ) ∂ (xμ)

Following [2], we have deﬁned the matrix Dμν := −σ; μν and its determinant as D(x, x ) := −det(Dμν ). The convention of the minus sign takes care of the Lorentzian signature of space-time. The square root of this determinant can be thought 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 xν , we obtain the relation

ρ ρ = + . Dμν σ; μ Dρν σ; Dμν ; ρ (D.15)

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

−1 ρ D (D σ; ); ρ = d. (D.16) with d = 4 in four space-time dimensions. Finally, we introduce the Van-Fleck determinate (x, x) by − / − / := g 1 2 D g 1 2. (D.17)

1/2 Since (g ); μ = 0, we obtain relation (D.16) in terms of

−1 ρ ( σ; ); ρ = d. (D.18) 260 Appendix D: Synge’s World Function

D.3 The Parallel Propagator

We further need the geometrical concept of parallel transport. Following [4], we define the parallel propagator as the tensor product of two tetrads. We introduce an μ( ) orthonormal tetrad eI z which satisfies μ ν = , gμν eI eJ ηIJ (D.19) with (I, J, ... = 0, 1, 2, 3) being Lorentz-indices of the frame with the Minkowski metric ηIJ = diag(−1, 1, 1, 1). We define the dual triad

I := IJ ν eμ η gμν eJ (D.20) and via the completeness relation

μν = IJ μ ν μ I = μ μ J = J . g η eI eJ it follows eI eν δν and eI eμ δI (D.21)

μ( ) I ( ) = ( ) The tetrad basis eI z and its dual eμ z are parallelly transported along γ z s

Deμ DeI I = 0 and μ = 0. (D.22) Ds Ds

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

vμ = vI μ, eI (D.23) whereas the Lorentz components of v can be expanded in the dual basis

vI = vν I . eν (D.24)

If the vector vμ is parallelly transported along γ, then the frame coefﬁcients vI are μ vI constant since eI is also parallelly transported along γ. Since the are constant, we vI = vν I vν I can express them as eν where the vector and the tetrad eν are evaluated at x. Putting all this together, vμ at x can be expressed by

vμ( ) = (vν I ) μ vμ = μ vμ , x eν eI or g μ (D.25)

μ μ := μ I with g μ being the parallel propagator g μ eI eμ . It is a bi-vector that transports μ μ the vector v from the point x = z(s1) to the vector v at the point x = z(s0) along the unique geodesic γ = z(s). The inverse parallel transporter from x to x is given μ μ = I by g μ eI eμ. It is clear that the parallel transporter gμν satisﬁes the boundary condition μ = μ = . lim g δ ν or lim gμν gμν (D.26) x→x ν x→x Appendix D: Synge’s World Function 261

μ...ν The parallel transport easily generalizes to tensors of arbitrary rank Tρ...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.

μ...ν = μ ··· ν ρ ··· σ μ ...ν . Tρ...σ g μ g ν g ρ g σ Tρ...σ (D.27)

μ Since the tetrads eI are parallelly propagated along γ this means

Deμ(z(s)) dzν (s) I = eμ = seμ σν = 0. (D.28) Ds I ; ν ds I ; ν

μ ν = I ν = From (D.28) it follows eI ; ν σ; 0 and eμ ; ν σ; 0. This, in turn, implies

μ ν = μ ν = . g μ ; ν σ; 0 and g μ ; ν σ; 0 (D.29)

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

μ =− μ =− . σ; μ g μ σ; μ and σ; μ g μ σ; μ (D.30)

From the determinants of the tetrad and its dual

( μ) = −1/2 ( I ) = 1/2, det eI g and det eμ g (D.31) we ﬁnd for the determinant of the parallel propagator

(− ) = (− ν ) = (− ν I ) det gμν det gμνg ν det gμνeI eν = (− ) ( ν ) ( I ) = 1/2 1/2. det gμν det eI det eν g g (D.32)

D.4 Field Curvature

We assume that we have a set of fields collected in the generalized field φA(x). Each component of the generalized field consists of some fundamental field present in the action 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 in φA(x) do have their own symmetries. In order to define invariant differentiation, we need to introduce a covariant derivative. We introduce an abbreviation by suppressing := A ˆ := B the capital index and write instead φ φ and a hat for operators O OA .The law of covariant differentiation is not commutative but defines the field (bundle) ˆ curvature Rμν ˆ ˆ ˆ [∇μ, ∇ν ] φ =: Rμν φ. (D.33) 262 Appendix D: Synge’s World Function

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 Lie group G and the group index a. Covariant differentiation of qa means

ˆ a a a ∇μ q ≡ ∂μ q + Aμ q (D.34)

:= a with the gauge vector ﬁeld Aμ AμGa and the generators Ga of the group G.The commutator of covariant derivatives then yields

[∇ˆ , ∇ˆ ] ≡ ( a − a + a b c ) = a μ ν φ Aμ, ν Aν, μ c bc Aμ Aν φa Fμν φa (D.35) with the ﬁeld strength tensor

Rˆ ≡ a = a − a + a b c μν Fμν Aμ, ν Aν, μ cbc Aμ Aν (D.36)

a and the structure constants cbc of the group G. If the field φ instead only consists of a symmetric tensor field φ ≡ qμν = qνμ with ordinary space-time indices, the ˆ covariant derivative is defined with respect to the usual Christoffel symbol ∇μqαβ ≡ ∇ = − γ − γ μqαβ ∂μqαβ αμqγβ βμqαγ and we obtain

[∇ˆ , ∇ˆ ] ≡ (∇ ∇ −∇ ∇ ) =− γ − γ μ ν φ μ ν ν μ qαβ R αμνqγβ R βμνqαγ (D.37)

− = δ wherewehaveuseduα; βγ uα; γβ R αβγuδ and have written qαβ without loss of generality as the tensor product qαβ := uα ⊗ vβ. This can also be written as

[∇ˆ , ∇ˆ ] ≡ (∇ ∇ −∇ ∇ ) =− (γ δ) μ ν φ μ ν ν μ qαβ 2 δ(α Rβ) μν qγδ (D.38) and shows that in this case the ﬁeld curvature is given by the operator

Rˆ ≡− (γ δ) , μν 2 δ(α Rβ) μν (D.39) acting on qγδ. In general, φ is a collection of many different fields. We can therefore define a parallel displacement bi-matrix Iˆ(x, x), which parallelly transports φ(x) to φ(x) along the geodesic γ. It is a matrix in field space and from the general properties of a parallel propagator, it follows as in (D.29) and (D.26) that Iˆ(x, x) must satisfy the relations

; ; μ ˆ = , μ ˆ = ˆ( , ) = A( ) := ˆ, σ I; μ 0 σ I; μ 0 and lim I x x δB x 1 (D.40) x→x where 1ˆ is the unit matrix at the point x. Appendix D: Synge’s World Function 263

D.5 Coincidence Limits

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

... ... [ A...(x, x ) ]:= lim A...(x, x ). (D.41) x→x

The result of this limit is an ordinary tensor at x. There is a relation between coincidence 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 systematically [ ] calculate the coincidence limit of higher derivatives σ; α1,...,αN .

1 ; λ σ = σ σ, , (D.43) 2 λ = λ , σ; α σ; α σλ (D.44) = λ + λ , σ; αβ σ; αβ σ; λ σ; α σ; λβ (D.45) = λ + λ + λ + λ , σ; αβγ σ; αβγ σ; λ σ; αβ σ; λγ σ; αγ σ; λβ σ; α σ; λβγ (D.46) = λ + λ + λ + λ , σ; αβγδ σ; αβγδ σ; λ σ; αβγ σ; λδ σ; αβδ σ; λγ σ; αβ σ; λγδ + λ + λ + λ + λ σ; αγδ σ; λβ σ; αγ σ; λβδ σ; αδ σ; λβγ σ; 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 curvature expressions. Using the symmetries of the Riemann tensor, we obtain

1 [ σ; ]=0 and [ σ; ]= R + R . (D.48) αβγ αβγδ 3 αγβδ αδβγ 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 μν = , σ; μνα R; α (D.49) μνα 8 α 4 αβ 4 αβγδ σ = R; + R R − R R . (D.50) ; μνα 5 α 15 αβ 15 αβγδ 264 Appendix D: Synge’s World Function

We are ultimately interested in the coincidence limit of the coefficient matrix aˆ2. As shown in Chap. 4, aˆ2 can be calculated by the recursion relation (4.109). The coincidence limits can then be again obtained recursively. In order to calculate [â1 ] and [â2 ], 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 matrix Iˆ, we differentiate (D.40) and use (D.33) in order to obtain ˆ ˆ 1 ˆ I; = 0, I; =− R , (D.52) μ μν 2 μν 1 1 Iˆ μ = Rˆ μ, Iˆ μν = Rˆ Rˆ μν. (D.53) ; μν 3 μν; ; μν 2 μν Proceeding in a similar manner, we differentiate (D.18) in order to obtain recursively the coincidence limits of the derivatives of the Van-Fleck determinant

/ / 1 1 2 = 0, 1 2 =− R , (D.54) ; μ ; μν 6 μν −1/2 1 1/2 μ 1 = R , =− R; (D.55) ; μν 6 μν ; μν 6 ν 1/2 μν 1 α 1 2 1 αβ 1 αβγδ =− R; + R − R R + R R . (D.56) ; μν 5 α 36 30 αβ 30 αβγδ Appendix E Corrections to the Initial Values of λ and yt

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

2 GF MZ −1 H = √ [ζ f1(ζ) + f0(ζ) + ζ f−1(ζ)] (E.1) 2 16 π2 with M2 3 1 1 c2 9 25 π f (ζ) = 6ln t + ln ζ − Z − Z w − ln c2 + − √ , (E.2) 1 2 2 2 ζ ζ w 2 9 MH 3 M2 M2 2 ( ) =− t + 2 − t + 3 cw ζ ζ + 1 f0 ζ 6ln 1 2 cw 2 ln 2 Z M2 M2 ζ − c2 c2 ζ Z Z w w 2 2 + 2 cw + 3 cw + 2 2 − 15 ( + 2 ) 4cw Z 12 cw ln cw 1 2 cw ζ s2 2 w M2 M2 M2 − 3 t 2 Z t + 4ln t − 5 , 2 2 2 (E.3) MZ MZ ζ MZ M2 M2 2 ( ) = t + 2 − t − 1 − 4 cw − 4 2 f−1 ζ 6ln 1 2 cw 4 6 Z 12 cw Z 12 cw ln cw M2 M2 ζ ζ Z Z M4 M2 M2 + 8 (1 + 2 c4 ) + 24 t ln t − 2 + Z t , w 4 2 2 (E.4) MZ MZ MZ ζ

:= 2 / 2 2 := 2 2 := 2 the Fermi constant GF and ζ MH MZ, sw sin θW, cW cos θW.The function Z[z] is deﬁned with respect to its domains depending on z 2 A arctan 1 ,(z > 1/4), Z[z]:= A A := |1 − 4 z|. (E.5) 1+A ,(< / ), A ln 1−A z 1 4

C. F. Steinwachs, Non-minimal Higgs Inﬂation and Frame Dependence 265 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 266 Appendix E: Corrections to the Initial Values of λ and yt

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

4 α (M ) 4 α(M ) (M ) = δQCD + δW + δQED, δQCD =− s t , δQED + δW =− t . t H t t t t 3π t t 9 π (E.6) We have neglected two-loop corrections. The numerical values of α and αs can be obtained from [1]. Appendix F Transfer Equations

Not all structures appearing in the divergent part of the one-loop effective action are independent. Neglecting surface terms and making use of the Bianchi identities, we can convert certain structures into others via an integration by parts. In this way, 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 dependent structures are distributed among the minimal set of independent structures:

F F a μ n →−F − ( − ), (F.1) ; μ a 4 ϕ 3 4 F F a ; μν → F − F( − ) + ; μν a 21 7 17 ϕ 15 1 F 1 1 F − − F + F − , (F.2) 2 ϕ 14 2 19 2 ϕ 12 a b; μν F 5 F 1 F F F na nb → 4 − + F + 2 − 4 ; μν ϕ2 2 ϕ 2 13 ϕ ϕ2 16 3 F 1 F F + F − 3 + − 2 ϕ 20 2 ϕ ϕ2 14 F F F + − F + + F + , 2 21 8 19 18 2 15 (F.3) ϕ ϕ ϕ a b, μ , ν 1 F F F na →− − F + − F ; μν b 2 ϕ 14 ϕ 16 1 1 F F − F 21 + F 19 + 12 − 15, (F.4) 2 2 ϕ ϕ , , 1 F F 1 F F a μ ν nb → − F − − , (F.5) ; μν a b 2 ϕ 14 2 19 2 ϕ 12

C. F. Steinwachs, Non-minimal Higgs Inﬂation and Frame Dependence 267 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 268 Appendix F: Transfer Equations a , μ b , ν c 1 F F F na n n → 3 − F − ; μν b c 2 ϕ 13 ϕ 16 1 1 F − F 20 − 14, (F.6) 2 2 ϕ μν a F 1 F FR na → − F − − F ; μν ϕ 8 2 ϕ 10 F 1 1 F − + F + , (F.7) ϕ 7 2 11 2 ϕ 9 μνρσ FR Rμνρσ → 4 F 5 − F 6. (F.8)

The last equation is a topological invariant, the GaussÐBonnet identity. In the single ﬁeld 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 of a different scalar contractions between derivatives ∇μ and ﬁeld variables (gμν, ), containing up to four derivatives, can appear at the one-loop level, but they do not occur in our calculations. Appendix G Gradient Structures

As already explained in Sect. 4.5.2, the coefficients αi , i = 12, ..., 21 do all correspond to gradient structures, symbolically denoted as ∂4. In the cosmological model of Higgs inflation, discussed in Chap. 6, these structures are additionally sup- pressed compared to (5.108)Ð(5.118) and thus less important from the viewpoint of an effective field theory. For the sake of completeness we list their coefficients in a closed form. (a , μ)2 The coefficient α12 belonging to the structure , μ a :

8 8 7 7 6 6 2 25 U 3 U U 3 U 5G U 11 U α12 = s + − − + − 48G2ϕ4U 4 16U 6 4Gϕ3U 4 8ϕU 5 4G2ϕ3U 3 72Gϕ4U 3

9 U 6 13G U 5 G U 5 3U U 5 35 U 5 − − − − + 8ϕ2U 4 12Gϕ2U 3 2U 4 4Gϕ3U 3 12ϕ3U 3

13 G 2 U 4 G U 4 15G U 4 3G U 4 695 U 4 + − + + − 24G2ϕ2U 2 Gϕ3U 2 8ϕU 3 8U 3 432ϕ4U 2

G 2 U 3 47G U 3 3G U 3 3G U U 3 U U 3 + − − + − 8GϕU 2 18ϕ2U 2 4ϕU 2 4Gϕ2U 2 4ϕ3U 2

G 3 U 2 139 G 2 U 2 G 2 U 2 19G U 2 − + + + 4G2ϕU 72Gϕ2U 8U 2 36ϕ3U

3G G U 2 G 3 U G 2 U + + − − G G U + G U U 4GϕU 6GU 12ϕU 4U 4ϕ2U

G 4 7 G 3 11 G 2 G 2 G 2 G + − + + − + G G 48G2 24Gϕ 24ϕ2 8 8G 2ϕ

25 U 6 U 6 U 5 37 U 5 5G U 4 + s − − + − + 72G2ϕ4U 3 8U 5 6Gϕ3U 3 8ϕU 4 12G2ϕ3U 2

43 U 4 21 U 4 13G U 3 11G U 3 U U 3 − + + + + 216Gϕ4U 2 4ϕ2U 3 36Gϕ2U 2 12U 3 2Gϕ3U 2

C. F. Steinwachs, Non-minimal Higgs Inﬂation and Frame Dependence 269 in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, © Springer International Publishing Switzerland 2014 270 Appendix G: Gradient Structures

3U U 3 107 U 3 77 G 2 U 2 15G U 2 15G U 2 + − + + − 2ϕU 3 36ϕ3U 2 72G2ϕ2U 4Gϕ3U 8ϕU 2

G U 2 3U U 2 U 2 5 G 2 U − − − + + 13G U − G U U 8U 2 ϕ2U 2 12ϕ4U 6GϕU 6ϕ2U 4Gϕ2U

G 3 13 G 2 25 U 4 + U U − − − G G + G U + 12ϕ3U 6G2ϕ 12Gϕ2 4Gϕ 2ϕU 432G2ϕ4U 2

U 4 59 U 3 19 U 3 G U 2 5 U 2 17 U 2 + + + − + + 48U 4 36Gϕ3U 2 12ϕU 3 9G2ϕ3U 12Gϕ4U 3ϕ2U 2 − 4G U − G U − 5U U − U U − 5U + 13GU 3Gϕ2U 4U 2 12Gϕ3U 2ϕU 2 3ϕ3U 6ϕU 2

2 2 (N − 1) G 5 G 7G 5U G2 + − − + − (G.1) 2G2ϕ2 12G2ϕ2 6ϕU 3ϕ2U 4U 2

(a b , μ )2 The coefﬁcient α13 belonging to the structure , μna nb :

81 U 12 81U U 10 27G U 9 243 U 2 U 8 α = s4 − − + 13 32U 8 4U 7 8U 6 4U 6

81GU U 7 81 U 3 U 6 27 G 2 U 6 81G U 2 U 5 + − + − 4U 5 U 5 16U 4 2U 4

81 U 4 U 4 27 G 2 U U 4 3 G 3 U 3 + − − 2U 4 4U 3 8U 2

27G U 3 U 3 27 G 2 U 2 U 2 3 G 3 U U G 4 + + + + U 3 4U 2 4U 32 3 U 10 45 U 10 81 U 9 9G U 8 19U U 8 + s3 − + − − + 2Gϕ2U 6 8U 7 8ϕU 6 4GϕU 5 4Gϕ2U 5

27U U 8 25 U 8 G U 7 15G U 7 81U U 7 + + + − + 8U 6 4ϕ2U 5 Gϕ2U 4 4U 5 2ϕU 5

5 G 2 U 6 155 U 2 U 6 351 U 2 U 6 39G U 6 + + − + 8GU4 8Gϕ2U 4 4U 5 8ϕU 4

9G U 6 3GU U 6 305U U 6 3 G 2 U 5 + + − + 8U 4 GϕU 4 12ϕ2U 4 2GϕU 3

81 U 2 U 5 25G U 5 15GU U 5 39GU U 5 − − + − 2ϕU 4 6ϕ2U 3 4Gϕ2U 3 4U 4

261 U 3 U 4 29 G 2 U 4 93G U 2 U 4 803 U 2 U 4 + + + + 2U 4 24U 3 4GϕU 3 24ϕ2U 3

G 2 U U 4 8GU U 4 9GU U 4 G 2 U 3 + − − + GU3 ϕU 3 2U 3 8ϕU 2

39G U 2 U 3 3GG U 3 9 G 2 U U 3 41GU U 3 + − + + U 3 4U 2 GϕU 2 4ϕ2U 2

27 U 4 U 2 3 G 3 U 2 3 G 2 U 2 31 G 2 U 2 U 2 − + + − U 3 4GϕU 4ϕ2U 8GU2 Appendix G: Gradient Structures 271

13G U 2 U 2 9G U 2 U 2 41 G 2 U U 2 G 3 U + + + − 4ϕU 2 2U 2 24U 2 4U

9G U 3 U 3 G 3 U U 3 G 2 U U G 4 G 3 − − + + 3G G U U − + U 2 2GU 2ϕU 2U 8G 8ϕ

2 2 8 8 8 13 G U 1 25 U 19 U 189 U − + G 2 G + s2 − − 24U 8 16G2ϕ4U 4 6Gϕ2U 5 16U 6

5 U 7 123 U 7 15G U 6 G U 6 U U 6 273U U 6 − − + + − + 4Gϕ3U 4 8ϕU 5 4G2ϕ3U 3 4GϕU 4 6Gϕ2U 4 8U 5

133 U 6 223 U 6 25G U 5 85G U 5 5G U 5 15GU U 5 + − − + − − 24Gϕ4U 3 72ϕ2U 4 24Gϕ2U 3 8U 4 4GϕU 3 4G2ϕ2U 3

33U U 5 87U U 5 3U U 5 3U U 5 113 U 5 G 2 U 4 − + + − − + 4Gϕ3U 3 4ϕU 4 4Gϕ2U 3 2U 4 12ϕ3U 3 G2ϕ2U 2

4 G 2 U 4 55 U 2 U 4 81 U 2 U 4 5G U 4 167G U 4 − − − + + 3GU3 6Gϕ2U 3 8U 4 2Gϕ3U 2 24ϕU 3

7G U 4 45GU U 4 881U U 4 889 U 4 7 G 2 U 3 − + + + − 8U 3 4GϕU 3 36ϕ2U 3 144ϕ4U 2 8GϕU 2

9 U 2 U 3 325G U 3 8G U 3 9 G 2 U U 3 31GU U 3 + + − − − 2ϕU 3 72ϕ2U 2 3ϕU 2 2G2ϕU 2 2Gϕ2U 2

25GU U 3 13GU U 3 11U U 3 3GU U 3 6U U U 3 − − − − − U 3 4GϕU 2 4ϕ3U 2 4GϕU 2 U 3

U U 3 3 G 3 U 2 39 U 3 U 2 8 G 2 U 2 19 G 2 U 2 + − − − − 4ϕ2U 2 2G2ϕU 2U 3 3Gϕ2U 8U 2

43G U 2 U 2 713 U 2 U 2 31G U 2 125 G 2 U U 2 − − − − 4GϕU 2 72ϕ2U 2 12ϕ3U 24GU2

2GU U 2 7GU U 2 GU U 2 G 3 U 23 G 2 U − + − − − 3ϕU 2 4U 2 2U 2 8GU 12ϕU

17G U 2 U 3 G 3 U U + − G G U + − 4G U U + 3G G U U 2U 2 4U 4G2U 3ϕ2U 2GU

3 G 4 9 U 4 G 3 5 G 2 G 2 + 23G U U − G U U + + + + + 12ϕU 4ϕU 16G2 2U 2 8Gϕ 8ϕ2 8

13 G 2 U 2 5G U 2 3G U 2 G 2 G G 2 U + − − + + 3G G + 24GU 2ϕU 2U 8G 4ϕ 2U 25 U 6 35 U 6 61 U 6 7G U 5 37 U 5 189 U 5 + s − − + + − + 24G2ϕ4U 3 9Gϕ2U 4 8U 5 8G2ϕ2U 3 6Gϕ3U 3 8ϕU 4

5 G 2 U 4 3G U 4 11G U 4 5G U 4 16U U 4 227U U 4 + − − − + − 8G3ϕ2U 2 8G2ϕ3U 2 12GϕU 3 8G2ϕ2U 2 9Gϕ2U 3 8U 4

79 U 4 521 U 4 5 G 2 U 3 56G U 3 13G U 3 19G U 3 − + − − − − 72Gϕ4U 2 27ϕ2U 3 2G2ϕU 2 9Gϕ2U 2 2U 3 12GϕU 2

GU U 3 3U U 3 95U U 3 U U 3 3U U 3 101 U 3 + + − − + + G2ϕ2U 2 2Gϕ3U 2 2ϕU 3 2Gϕ2U 2 U 3 36ϕ3U 2

3 G 3 U 2 17 G 2 U 2 G 2 U 2 101 U 2 U 2 53 U 2 U 2 + + + + + 4G3ϕU 8G2ϕ2U 4GU2 72Gϕ2U 2 2U 3 272 Appendix G: Gradient Structures

11G U 2 433G U 2 41G U 2 G U 2 17GU U 2 21U U 2 + − − + − − 8Gϕ3U 36ϕU 2 24Gϕ2U U 2 6GϕU 2 4ϕ2U 2

3U U 2 U 2 3 G 3 U 17 G 2 U 3 U 2 U + + + + + − 17G U + G G U ϕU 2 4ϕ4U 8G2U 12GϕU 2ϕU 2 12ϕ2U 4GU

3 G 2 U U − G U + + 17G U U + 27G U U + 13G U U − 5U U 2ϕU 2G2ϕU 4Gϕ2U 2U 2 12GϕU 12ϕ3U

G 4 G 3 U 3 23 G 2 G 2 + G U U − U U U − U U − − − + − 4GϕU U 2 12ϕ2U 8G3 8G2ϕ 2U 2 24U 4Gϕ2

G 2 G U 2 U 2 G 2 G G 2 U − + + − − G G + + 25G U − G U 4G GϕU 6ϕ2U 8G2 2Gϕ 2GU 12ϕU U

25 U 4 46 U 4 85 U 4 G 3 7G U 3 133 U 3 3 U 3 + + + − − + + 144G2ϕ4U 2 27Gϕ2U 3 8U 4 4G3ϕ 24G2ϕ2U 2 36Gϕ3U 2 4ϕU 3

23 G 2 G 2 5 G 2 U 2 7G U 2 89G U 2 U 2 3 U 2 − − − − − − + 24GU 2G2ϕ2 24G3ϕ2U 24G2ϕ3U 36GϕU 2 4Gϕ4U 2ϕ2U 2

U 2 9 U 2 G 2 U 5 U 2 G 9 U 2 U − + − − 19G U + + U G − 6Gϕ2U 2U 2 2G2ϕU 12Gϕ2U 24G2ϕ2U 2GϕU 4Gϕ2U 2

3 U 2 U − + 11G U + G U U + 5U U − 6U U + U U − U U 4U 3 12GϕU 12G2ϕ2U 12Gϕ3U ϕU 2 12Gϕ2U U 2

(N − 1) G 2 (N − 1) G 3 (N − 1) G 4 ( − ) + + + − N 1 G G 8G2ϕ2 8G3ϕ 32G4 4G2ϕ

(N − 1) G 2 G (N − 1) G 2 − + (G.2) 8G3 8G2

(a , μ)(c d , ν ) The coefﬁcient α14 belonging to the structure , μ a , ν nc nd :

10 10 9 8 8 8 3 U 9 U 9 U 3G U 19U U 9U U α14 = s − + + − + 2Gϕ2U 6 8U 7 8ϕU 6 4GϕU 5 12Gϕ2U 5 2U 6

29 U 8 G U 7 9G U 7 9U U 7 5 G 2 U 6 + − + − − 12ϕ2U 5 3Gϕ2U 4 4U 5 2ϕU 5 24GU4

155 U 2 U 6 9 U 2 U 6 19G U 6 9G U 6 G U U 6 − − − − − 24Gϕ2U 4 2U 5 8ϕU 4 8U 4 GϕU 4

343U U 6 G 2 U 5 9 U 2 U 5 29G U 5 5G U U 5 − − + − − 36ϕ2U 4 2GϕU 3 2ϕU 4 18ϕ2U 3 4Gϕ2U 3

15G U U 5 95 G 2 U 4 31G U 2 U 4 493 U 2 U 4 − − − + 2U 4 72U 3 4GϕU 3 72ϕ2U 3

G 2 U U 4 26G U U 4 9G U U 4 29 G 2 U 3 − + + + 3GU3 3ϕU 3 2U 3 24ϕU 2

6G U 2 U 3 3G G U 3 3 G 2 U U 3 31G U U 3 + + − + U 3 4U 2 GϕU 2 12ϕ2U 2

G 3 U 2 G 2 U 2 31 G 2 U 2 U 2 121G U 2 U 2 − + + − 4GϕU 4ϕ2U 24GU2 12ϕU 2 Appendix G: Gradient Structures 273

9G U 2 U 2 43 G 2 U U 2 G 3 U G 3 U U − + + + 2U 2 18U 2 4U 2GU

7 G 2 U U G 4 7 G 3 5 G 2 U 2 − − 3G G U U + − − 2ϕU 2U 24G 24ϕ 72U

8 8 8 7 1 25 U 13 U 15 U U − G 2 G + s2 − − − + 8 24G2ϕ4U 4 9Gϕ2U 5 2U 6 2Gϕ3U 4

39 U 7 5G U 6 7G U 6 7U U 6 141U U 6 + − + − + 4ϕU 5 2G2ϕ3U 3 12GϕU 4 9Gϕ2U 4 8U 5

11 U 6 185 U 6 127G U 5 11G U 5 5G U 5 + − + + − 36Gϕ4U 3 54ϕ2U 4 72Gϕ2U 3 8U 4 12GϕU 3

5G U U 5 3U U 5 111U U 5 3U U 5 35 U 5 + + − − − 4G2ϕ2U 3 2Gϕ3U 3 4ϕU 4 4Gϕ2U 3 6ϕ3U 3

7 G 2 U 4 17 G 2 U 4 55 U 2 U 4 27 U 2 U 4 − + + − 8G2ϕ2U 2 18GU3 18Gϕ2U 3 4U 4

2G U 4 47G U 4 G U 4 131G U U 4 1249U U 4 + − + − + Gϕ3U 2 9ϕU 3 U 3 12GϕU 3 108ϕ2U 3

695 U 4 G 2 U 3 39 U 2 U 3 1307G U 3 31G U 3 + − + + + 216ϕ4U 2 4GϕU 2 2ϕU 3 216ϕ2U 2 36ϕU 2

3 G 2 U U 3 11G U U 3 29G U U 3 13G U U 3 + − + − 2G2ϕU 2 12Gϕ2U 2 4U 3 12GϕU 2

U U 3 3G U U 3 U U 3 3 G 3 U 2 97 G 2 U 2 + + − + − 2ϕ3U 2 4GϕU 2 4ϕ2U 2 4G2ϕU 24Gϕ2U

2 G 2 U 2 25G U 2 U 2 583 U 2 U 2 19G U 2 7G G U 2 + + − − − 3U 2 4GϕU 2 216ϕ2U 2 18ϕ3U 4GϕU

143 G 2 U U 2 6G U U 2 17G U U 2 11 G 3 U 2 G 2 U + − − − − 72GU2 ϕU 2 4U 2 24GU 9ϕU

13G U 2 U G 3 U U 2 G 2 U U − + G G U − + + 5G U U + G G U U 2U 2 2U 4G2U GϕU 12ϕ2U 2GU

G 4 5 G 3 3 G 2 G 2 49 G 2 U 2 + 23G U U + G U U − + − − − 36ϕU 4ϕU 12G2 6Gϕ 4ϕ2 4 72GU

2 2 2 2 6 49G U 3G U G G 11G G 2 G U 25 U + + + − − + s 18ϕU 2U 3G 12ϕ 3U 36G2ϕ4U 3

43 U 6 12 U 6 5G U 5 U 5 7 U 5 5 G 2 U 4 + + + − − − 54Gϕ2U 4 U 5 24G2ϕ2U 3 3Gϕ3U 3 ϕU 4 24G3ϕ2U 2

5G U 4 47G U 4 5G U 4 79U U 4 157U U 4 43 U 4 − − + + − + 24G2ϕ3U 2 36GϕU 3 8G2ϕ2U 2 54Gϕ2U 3 8U 4 108Gϕ4U 2

3281 U 4 G 2 U 3 137G U 3 4G U 3 41G U 3 5G U U 3 − − + − + − 324ϕ2U 3 G2ϕU 2 108Gϕ2U 2 3U 3 36GϕU 2 6G2ϕ2U 2

U U 3 23U U 3 U U 3 3U U 3 107 U 3 G 3 U 2 − + + + + − Gϕ3U 2 ϕU 3 2Gϕ2U 2 2U 3 18ϕ3U 2 4G3ϕU 274 Appendix G: Gradient Structures

25 G 2 U 2 11 G 2 U 2 65 U 2 U 2 15 U 2 U 2 139G U 2 − − − + − 9G2ϕ2U 12GU2 216Gϕ2U 2 4U 3 24Gϕ3U

1231G U 2 G G U 2 41G U 2 G U 2 35G U U 2 + + + − + 216ϕU 2 2G2ϕU 24Gϕ2U 24U 2 6GϕU 2

145U U 2 3U U 2 U 2 G 3 U 79 G 2 U 11 U 2 U + − + − − − 108ϕ2U 2 ϕU 2 6ϕ4U 8G2U 36GϕU 2ϕU 2

G 2 U U − 41G U + G G U + 7G U − + 4G U U − 2G U U + 13G U U 12ϕ2U 4GU 6ϕU 2G2ϕU 9Gϕ2U U 2 36GϕU

G 4 11 G 3 19 G 2 3 G 2 − U U − G U U + U U + + − + 6ϕ3U 4GϕU 12ϕ2U 24G3 24G2ϕ 72U 2Gϕ2

G 2 11G U 2 G 2 G G 2 U − − − − G G − + 7G U + 2G U 12G 9GϕU 24G2 12Gϕ 2GU 36ϕU 3U

25 U 4 10 U 4 25 U 4 5G U 3 53 U 3 + G U − − − − − 2U 216G2ϕ4U 2 81Gϕ2U 3 8U 4 72G2ϕ2U 2 18Gϕ3U 2

6 U 3 5 G 2 U 2 25G U 2 55G U 2 5G U 2 125U U 2 − + + + − ;+ ϕU 3 72G3ϕ2U 72G2ϕ3U 108GϕU 2 24G2ϕ2U 108Gϕ2U 2

53U U 2 U 2 27 U 2 3G U 2 G 2 U + − − − + + 35G U 12U 3 6Gϕ4U 2ϕ2U 2 U 3 2G2ϕU 12Gϕ2U + 15G U − G U + 5G U U + U U + 83U U − U U − U U 4U 2 2GϕU 36G2ϕ2U 6Gϕ3U 6ϕU 2 12Gϕ2U 2U 2

G 3 55 G 2 3 G 2 U 2 − 2GU + + + − + G − G G − G ϕU 2 12G3ϕ 72GU 4G2ϕ2 3U 2 ϕU 6G2ϕ U

2 3 37G U 2GU (N − 1) G (N − 1) G (N − 1)G G − + − − + (G.3) 36GϕU U 2 2G2ϕ2 4G3ϕ 2G2ϕ

a b , μ, ν The coefﬁcient α15 belonging to the structure , μ a , ν b :

8 8 7 6 6 6 2 25 U 3 U 3 U 5G U 205 U 9 U α15 = s + − + + + 24G2ϕ4U 4 8U 6 ϕU 5 2G2ϕ3U 3 36Gϕ4U 3 ϕ2U 4

5G U 5 G U 5 12 U 5 13 G 2 U 4 6G U 4 3G U 4 + + − + + − 6Gϕ2U 3 2U 4 ϕ3U 3 12G2ϕ2U 2 Gϕ3U 2 ϕU 3

1681 U 4 G 2 U 3 113G U 3 G 3 U 2 77 G 2 U 2 + + + − − 216ϕ4U 2 GϕU 2 18ϕ2U 2 2G2ϕU 36Gϕ2U

G 2 U 2 41G U 2 G 3 U 2 G 2 U G 4 G 3 G 2 + − − − + + + 4U 2 18ϕ3U 6GU 3ϕU 24G2 6Gϕ 6ϕ2

25 U 6 2 U 6 5 U 5 U 5 5G U 4 205 U 4 + s − + − + − − 36G2ϕ4U 3 U 5 2Gϕ3U 3 2ϕU 4 3G2ϕ3U 2 108Gϕ4U 2

7 U 4 16G U 3 G U 3 19 U 3 31 G 2 U 2 2G U 2 + − − + − − ϕ2U 3 9Gϕ2U 2 6U 3 6ϕ3U 2 36G2ϕ2U Gϕ3U

5G U 2 G 2 U G 3 G 2 G 2 25 U 4 − − − 11G U + + + + 2ϕU 2 3GϕU 6ϕ2U 6G2ϕ 4U 3Gϕ2 216G2ϕ4U 2 Appendix G: Gradient Structures 275

17 U 4 2 U 3 U 3 G U 2 U 2 5 U 2 G U 2 − + + + − − + 24U 4 3Gϕ3U 2 6ϕU 3 9G2ϕ3U 3Gϕ4U 3ϕ2U 2 4U 3

2 G U U U 5U GU G 7G 5U 3G2 + + + − + + − + (G.4) 2Gϕ2U 3Gϕ3U 3ϕ3U 6ϕU 2 6G2ϕ2 6ϕU 3ϕ2U 2U 2

a b c, μ, ν The coefﬁcient α16 belonging to the structure ,μna ,ν nb c :

10 10 9 8 8 8 3 U 9 U 9 U 3G U 19U U 9U U α16 = s − + + − + Gϕ2U 6 4U 7 ϕU 6 2GϕU 5 6Gϕ2U 5 U 6

26 U 8 2G U 7 36U U 7 5 G 2 U 6 155 U 2 U 6 − − − − − 3ϕ2U 5 3Gϕ2U 4 ϕU 5 12GU4 12Gϕ2U 4

9 U 2 U 6 5G U 6 2G U U 6 629U U 6 G 2 U 5 − − − + − U 5 2ϕU 4 GϕU 4 18ϕ2U 4 GϕU 3

36 U 2 U 5 52G U 5 5G U U 5 3G U U 5 13 G 2 U 4 + + − + + ϕU 4 9ϕ2U 3 2Gϕ2U 3 U 4 36U 3

31G U 2 U 4 1451 U 2 U 4 2 G 2 U U 4 2G U U 4 − − − − 2GϕU 3 36ϕ2U 3 3GU3 3ϕU 3

4 G 2 U 3 6G U 2 U 3 6 G 2 U U 3 77G U U 3 − − − − 3ϕU 2 U 3 GϕU 2 6ϕ2U 2

G 3 U 2 G 2 U 2 31 G 2 U 2 U 2 41G U 2 U 2 − − + + 2GϕU ϕ2U 12GU2 6ϕU 2

11 G 2 U U 2 G 3 U U 2 G 2 U U G 4 G 3 5 G 2 U 2 − + + + + − 9U 2 GU ϕU 12G 6ϕ 36U

25 U 8 83 U 8 15 U 8 U 7 9 U 7 5G U 6 + s2 − + + + + − 12G2ϕ4U 4 18Gϕ2U 5 2U 6 Gϕ3U 4 ϕU 5 G2ϕ3U 3

5G U 6 17U U 6 33U U 6 205 U 6 73 U 6 17G U 5 − + − − − − 6GϕU 4 18Gϕ2U 4 2U 5 18Gϕ4U 3 54ϕ2U 4 36Gϕ2U 3

19G U 5 5G U 5 5G U U 5 15U U 5 6U U 5 73 U 5 − + + + + + 4U 4 3GϕU 3 2G2ϕ2U 3 2Gϕ3U 3 ϕU 4 3ϕ3U 3

7 G 2 U 4 7 G 2 U 4 55 U 2 U 4 19G U 4 11G U 4 − + + − − 4G2ϕ2U 2 18GU3 9Gϕ2U 3 2Gϕ3U 2 18ϕU 3

G U 4 G U U 4 973U U 4 1681 U 4 24 U 2 U 3 − − − − − U 3 3GϕU 3 27ϕ2U 3 108ϕ4U 2 ϕU 3

1537G U 3 23G U 3 3 G 2 U U 3 47G U U 3 G U U 3 − + + + + 108ϕ2U 2 9ϕU 2 G2ϕU 2 3Gϕ2U 2 U 3

13G U U 3 5U U 3 3 G 3 U 2 83 G 2 U 2 G 2 U 2 + + + + + 3GϕU 2 2ϕ3U 2 2G2ϕU 12Gϕ2U 3U 2

9G U 2 U 2 1361 U 2 U 2 97G U 2 G G U 2 + + + + 2GϕU 2 108ϕ2U 2 18ϕ3U GϕU

29 G 2 U U 2 20G U U 2 2G U U 2 7 G 3 U 26 G 2 U + + + + + 9GU2 3ϕU 2 U 2 12GU 9ϕU 276 Appendix G: Gradient Structures

2G U 2 U G 3 U U 2 G 2 U U + − − + 2G U U − 2G G U U U 2 2G2U GϕU 3ϕ2U GU

G 4 5 G 3 G 2 5 G 2 U 2 2G U 2 G 2 G − 23G U U − − − + − − 9ϕU 6G2 6Gϕ 2ϕ2 36GU 9ϕU 3G

2 6 6 6 5 GG 2 G U 25 U 167 U 9 U 13G U − + + s + − − 3ϕ 3U 18G2ϕ4U 3 54Gϕ2U 4 U 5 12G2ϕ2U 3

53 U 5 25 U 5 5 G 2 U 4 11G U 4 20G U 4 175U U 4 + − − + + − 6Gϕ3U 3 2ϕU 4 12G3ϕ2U 2 6G2ϕ3U 2 9GϕU 3 54Gϕ2U 3

13U U 4 151 U 4 1735 U 4 7 G 2 U 3 172G U 3 17G U 3 + + − + + + 2U 4 54Gϕ4U 2 81ϕ2U 3 2G2ϕU 2 27Gϕ2U 2 6U 3

4G U 3 G U U 3 U U 3 23U U 3 161 U 3 G 3 U 2 + − − + − − 9GϕU 2 6G2ϕ2U 2 Gϕ3U 2 ϕU 3 18ϕ3U 2 2G3ϕU

4 G 2 U 2 2 G 2 U 2 119 U 2 U 2 3 U 2 U 2 8G U 2 + + − + + 9G2ϕ2U 3GU2 108Gϕ2U 2 U 3 3Gϕ3U

289G U 2 G G U 2 G U 2 3G U U 2 373U U 2 U 2 + − − − + − 27ϕU 2 2G2ϕU 3U 2 GϕU 2 54ϕ2U 2 3ϕ4U

G 3 U 5 G 2 U 4 U 2 U G 2 U U − + + + 9G U − G G U − 2G U − 4G2U 18GϕU ϕU 2 2ϕ2U 2GU 3ϕU G2ϕU

G 4 G 3 25 G 2 − 40G U U − 9G U U − 13G U U + U U + − − 9Gϕ2U 2U 2 9GϕU 2ϕ3U 12G3 3G2ϕ 36U

G 2 G 2 2G U 2 U 2 G 2 G − + + − + + 5G G − 25G U − 2G U 2Gϕ2 3G 9GϕU 6ϕ2U 6G2 6Gϕ 9ϕU 3U

25 U 4 128 U 4 9 U 4 13G U 3 55 U 3 7 U 3 − − + + − + 108G2ϕ4U 2 81Gϕ2U 3 4U 4 36G2ϕ2U 2 18Gϕ3U 2 2ϕU 3

5 G 2 U 2 G U 2 53G U 2 59U U 2 2U U 2 U 2 + − + + − + 36G3ϕ2U 18G2ϕ3U 27GϕU 2 54Gϕ2U 2 3U 3 3Gϕ4U

8 U 2 19G U 2 + + − G U − 13G U − 2G U U − U U − 22U U ϕ2U 2 2U 3 2Gϕ2U 2U 2 9G2ϕ2U 2Gϕ3U 3ϕU 2

3 2 2 2 G 7 G U 2 U G GG G GU + + + − − + + + (G.5) 6G3ϕ 36GU 6Gϕ2U 3U 2 ϕU 6G2ϕ U 9GϕU

a μ ν The coefﬁcient α17 belonging to the structure ; μ a; ν : 4 2 2 4 U GU G U 3 U α = s − − + − − 17 4GU2ϕ2 2U 4G 12Uϕ2 4U 3

2 2 U U U + − + (G.6) 12GUϕ2 Uϕ 4U 2

(c μ )2 The coefﬁcient α18 belonging to the structure ; μ nc : Appendix G: Gradient Structures 277 9G U 3 G 2 81 U 6 U 4 G 2 U 2 α = s2 + + + s − + 18 4U 2 8 8U 4 4GU2ϕ2 4G 12Uϕ2 27 U 4 U 2 13 U 2 (N − 1) G 2 − − + U + − U + 4U 3 12GUϕ2 Uϕ 2U 2 U 8G2 (G.7)

(a , μ)(b ν ) The coefﬁcient α19 belonging to the structure , μ a ; ν nb :

7 7 6 5 5 5 2 U 27 U 27 U 3G U 3U U U α19 = s − − + − − − 4Gϕ2U 4 8U 5 4ϕU 4 4GϕU 3 4Gϕ2U 3 12ϕ2U 3

15G U 4 3 G 2 U 3 7G U 3 9G U 3 3G U U 3 + + − − + 8U 3 8GU2 4ϕU 2 4U 2 4GϕU 2

U U 3 3 G 2 U 2 G 3 G 2 − − + G U U + − − G G 4ϕ2U 2 4GϕU 4ϕU 8G 2ϕ 4

U 5 15 U 5 5G U 4 2 U 4 3G U 3 U U 3 + s + + − + + 6Gϕ2U 3 8U 4 8G2ϕ2U 2 ϕU 3 4GϕU 2 2Gϕ2U 2

3U U 3 37 U 3 3 G 2 U 2 41G U 2 3G U 2 + + + + − 2U 3 36ϕ2U 2 4G2ϕU 24Gϕ2U 2U 2

3U U 2 G 2 U G 3 − + + G U + G U − G U U + U U − ϕU 2 4GU ϕU 2U 4GϕU 12ϕ2U 8G2

53 U 3 U 3 5G U 2 14 U 2 + G U + − − + − G U 2U 36Gϕ2U 2 4U 3 24G2ϕ2U ϕU 2 2GϕU

G 2 (N − 1) G 2 − U U − U U + U + 2GU − − G − U + 12Gϕ2U 2U 2 ϕ2U U 2 4G2ϕ 2U ϕU 2G2ϕ (G.8)

(a b , μ )(c ν ) The coefﬁcient α20 belonging to the structure , μna nb ; ν nc :

9 7 6 2 5 4 3 81 U 81U U 45G U 81 U U 9G U U α20 = s − − + + 8U 6 2U 5 8U 4 2U 4 U 3

3 G 2 U 3 9G U 2 U 2 3 G 2 U U G 3 3 U 7 + + + + + s2 8U 2 2U 2 2U 8 4Gϕ2U 4

45 U 7 45 U 6 G U 5 9U U 5 135U U 5 U 5 − − − + + + 8U 5 4ϕU 4 2GϕU 3 2Gϕ2U 3 4U 4 4ϕ2U 3

5G U 4 21G U 4 9U U 4 3 G 2 U 3 63 U 2 U 3 + − + − − 4Gϕ2U 2 8U 3 ϕU 3 8GU2 2U 3

4G U 3 9G U 3 13G U U 3 3U U 3 5G U 2 + + − + + 3ϕU 2 4U 2 4GϕU 2 2ϕ2U 2 12ϕ2U 278 Appendix G: Gradient Structures

15G U U 2 G 2 U 3 G 2 U U G 3 3 G 2 − − + + 23G U U + + 4U 2 2U 2GU 12ϕU 8G 4ϕ

2 5 5 4 4 3G U GG 7 U 15 U 11G U U − + + s + − − 2U 4 4Gϕ2U 3 8U 4 8G2ϕ2U 2 Gϕ3U 2

3 U 4 25G U 3 U U 3 12U U 3 7 U 3 19G U 2 + − − − − − ϕU 3 12GϕU 2 Gϕ2U 2 U 3 3ϕ2U 2 8Gϕ2U

3G U 2 3U U 2 U 2 G 2 U 15 U 2 U + − − + + − G U + 13G U U U 2 ϕU 2 3ϕ3U 2GU 2U 2 4ϕU 12GϕU

G 3 G 2 13 U 3 29 U 3 11G U 2 + U U − − − G G − − + 6ϕ2U 8G2 2Gϕ 2G 6Gϕ2U 2 4U 3 24G2ϕ2U

U 2 33 U 2 + − + G U − 3U − U U + 33U U + 3U − U 3Gϕ3U 2ϕU 2 4GϕU ϕ2U 6Gϕ2U 2U 2 ϕU U

2 3 (N − 1) G (N − 1) G (N − 1)G G − − + (G.9) 4G2ϕ 8G3 4G2

a μ, ν b The coefﬁcient α21 belonging to the structure ; μ a , ν nb:

7 7 6 5 5 5 2 U 9 U 9 U 5G U 15U U 9U U α21 = s − − + + − + 2Gϕ2U 4 4U 5 2ϕU 4 4GϕU 3 4Gϕ2U 3 2U 4

U 5 5G U 4 3G U 4 9U U 4 5G U 3 5G U U 3 − − − − + + 6ϕ2U 3 4Gϕ2U 2 4U 3 ϕU 3 12ϕU 2 2GϕU 2

5U U 3 3 G 2 U 2 5G U 2 3G U U 2 G 2 U − + − + + 4ϕ2U 2 4GϕU 12ϕ2U U 2 4U

2 3 2 5 5 3 G U U 13G U U G G 23 U 3 U − − − − + s − + 2GU 6ϕU 4G 4ϕ 12Gϕ2U 3 4U 4

3G U 4 U 4 U 4 4G U 3 U U 3 3U U 3 + + − + + + 4G2ϕ2U 2 Gϕ3U 2 ϕU 3 3GϕU 2 2Gϕ2U 2 2U 3

47 U 3 3 G 2 U 2 2G U 2 G U 2 6U U 2 U 2 + − + − + + 36ϕ2U 2 4G2ϕU 3Gϕ2U 2U 2 ϕU 2 3ϕ3U

3 G 2 U G 3 G 2 − − 3G U − G U − 5G U U − U U + + 4GU 4ϕU 2U 6GϕU 4ϕ2U 4G2 2Gϕ

25 U 3 G U 2 U 2 5 U 2 + G G + − − + + G U 2G 36Gϕ2U 2 4G2ϕ2U 3Gϕ3U 2ϕU 2 4GϕU

2 U U U U 2U 13GU G G 2U + − + − + + − (G.10) 4Gϕ2U U 2 ϕ2U 2U 2 4G2ϕ U ϕU Appendix G: Gradient Structures 279

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