Third Order Lagrangians, Weyl Invariants and Classical Trace Anomaly in Six Dimensions

Abstract

We have proceeded the analogy (represented in our previous works) of the Einstein tensor and the alternative form of the Einstein field equations for the generic coefficients of the eight terms in the third order of the Lovelock Lagrangian. We have found the constraint between the coefficients into two forms, an independent and a dimensional dependent versions. Each form has three degrees of freedom, and not only the exact coefficients of the third order Lovelock Lagrangian do satisfy the two forms of the constraints, but also the two independent cubic of the Weyl tensor satisfy the independent constraint in six dimensions and yield the dimensional dependent version identically independent of the dimension. Then, we have introduced the most general effective expression for a total third order type Lagrangian with the homogeneity degree number three which includes the previous eight terms plus the new three ones among the all seventeen independent terms. We have proceeded the analogy for this combination, and have achieved the relevant constraint. We have shown that the expressions given in the literature as the third Weyl-invariant combination in six dimensions do satisfy this constraint. Thus, we suggest that these constraint relations to be considered as the necessary consistency conditions on the numerical coefficients that a Weyl-invariant in six dimensions should satisfy. Finally, we have calculated the “classical” trace anomaly (an approach that was presented in our previous works) for the introduced total third order type Lagrangian and have achieved a general expression with four degrees of freedom in more than six dimensions (three degrees in six dimensions). Then, we have demonstrated that the resulted expression contains exactly the relevant coefficient of the Schwinger–DeWitt proper time method (that linked with the relevant heat kernel coefficient) in six dimensions, as a particular case. Of course, this result is a necessary consistency check, nevertheless our approach can be regarded as an alternative (perhaps simpler, and classical) derivation of the trace anomaly which also gives a general expression with the relevant degrees of freedom.

Appendices

Appendix A: Useful Relations & Variation of a Few Actions

In this appendix, we furnish a few useful relations and also supply the metric variation of a few Lagrangian terms.

1.1 A.1 The Useful Relations

The following derivative relations can easily be derived from non-commutativity of the covariant derivatives, and the Bianchi and the contracted Bianchi identities, as [38]

$$ \begin{aligned} &R_{\alpha\mu}{}^{; \mu}{}_{\beta}= R_{; \alpha\beta}/2 , \cr &R_{\mu\alpha\nu\beta}{}^{; \alpha} =R_{\mu\nu; \beta}-R_{\mu\beta; \nu} , \cr &R_{\mu\tau; \nu}{}^{\tau}=R_{; \mu\nu}/2 -R_{\mu\alpha\nu\beta}R^{\alpha\beta}+R_{\mu\alpha} R_{\nu}{}^{\alpha} , \cr &R_{\nu\tau\mu\rho}R^{\tau\mu ; \nu\rho}=K_6/2-K_7/2-K_8-K_{11}/4 \end{aligned} $$

(79)

and also

$$ \bigl(R^{k-1} \bigr)_{; \mu\nu}=(k-1)R^{k-2}R_{; \mu\nu} +(k-1) (k-2)R^{k-3}R_{; \mu}R_{; \nu} . $$

(80)

In the variation process, using the integrating covariantly by parts and the appropriate boundary conditions, one can effectively write

$$ f \delta \bigl(\square ^k R \bigr)\mathrel{ \mathop{=}^{\rm eff}} \bigl(\square ^k f \bigr) \delta R +\sum _{i=0}^{k-1} \bigl(\square ^i f \bigr) \bigl[- \bigl(\square ^{(k-1-i)} R \bigr)_{, \tau} g^{\alpha\rho} \delta \varGamma^\tau {}_{\alpha\rho} + \bigl( \square ^{(k-1-i)} R \bigr)_{; \alpha\rho} \delta g^{\alpha\rho} \bigr], $$

(81)

where the f is any scalar function of the metric and its derivatives.

The Euler–Lagrange expressions of a few scalar Lagrangian density terms,³² where $\mathcal{L}=F\sqrt{-g}/\kappa^{2}$, are [38]

$$ G^{ (F(R) )}_{\alpha\beta}=F^{ \prime}{} R_{\alpha\beta} -F^{ \prime}{}_{; \alpha\beta}-{1\over2} g_{\alpha\beta} \bigl(F-2 \square F^{ \prime} \bigr), $$

(82)

$$\begin{aligned} G^{ (F(R_{\mu\nu} R^{\mu\nu}) )}_{\alpha\beta} = &2F^{ \prime} R_{\alpha\rho} R_{\beta}{}^\rho - \bigl(F^{ \prime} R_{\alpha\rho} \bigr)_{; \beta}{}^\rho - \bigl(F^{ \prime} R_{\beta\rho} \bigr)_{; \alpha}{}^\rho +\square \bigl(F^{ \prime} R_{\alpha\beta} \bigr) \cr &- {1\over2} g_{\alpha\beta} \bigl[F -2 \bigl(F^{ \prime} R^{\mu\nu} \bigr)_{; \mu\nu} \bigr], \end{aligned}$$

(83)

$$ G^{ (F(R_{\rho\tau\mu\nu} R^{\rho\tau\mu\nu}) )}_{\alpha\beta} =2F^{ \prime} R_{\alpha\rho\mu\nu}R_{\beta}{}^{\rho\mu\nu} +2 \bigl(F^{ \prime} R_{\mu\alpha\nu\beta} \bigr)^{; \mu\nu} +2 \bigl(F^{ \prime} R_{\mu\beta\nu\alpha} \bigr)^{; \mu\nu} -{1\over2} g_{\alpha\beta} F , $$

(84)

$$\begin{aligned} G^{ (F(\sum\limits^p_{k=0} \square ^k R) )}_{\alpha\beta} = & \Biggl(\sum _{k=0}^p\square ^k f_k \Biggr)R_{\alpha\beta} - \Biggl(\sum_{k=0}^p \square ^k f_k \Biggr)_{; \alpha\beta} -\sum _{k=1}^p\sum_{i=0}^{k-1} \bigl(\square ^i f_k \bigr)_{(; \alpha} \bigl(\square ^{(k-1-i)}R \bigr)_{; \beta)} \cr &-{1\over2} g_{\alpha\beta} \Biggl\{F-2 \square \Biggl(\sum_{k=0}^p \square ^k f_k \Biggr) -\sum_{k=1}^p \sum_{i=0}^{k-1} \bigl[ \bigl(\square ^i f_k \bigr) \bigl(\square ^{(k-1-i)}R \bigr)_{;\rho} \bigr]^{;\rho} \Biggr\} \end{aligned}$$

(85)

and

$$\begin{aligned} G^{ (R^N \square ^l R )}_{\alpha\beta} = &\varTheta^{[Nl]}R_{\alpha\beta}- \varTheta^{[Nl]}{}_{; \alpha\beta} -\sum_{i=0}^{l-1} \bigl(\square ^i R^N \bigr)_{(; \alpha} \bigl( \square ^{(l-1-i)}R \bigr)_{; \beta)} \cr {} &-{1\over2} g_{\alpha\beta} \Biggl\{R^N \square ^l R-2 \square \varTheta^{[Nl]}-\sum_{i=0}^{l-1} \bigl[ \bigl(\square ^i R^N \bigr) \bigl(\square ^{(l-1-i)}R \bigr)_{;\rho} \bigr]^{;\rho} \Biggr\}, \end{aligned}$$

(86)

where the prime denotes ordinary derivative with respect to the argument, f _k ≡∂F/∂□^k R and Θ ^[Nl]≡NR ^N−1□^l R+□^l R ^N.

The Weyl conformal tensor, for D≥3 dimensions, satisfies

$$ C_{\alpha\beta\mu\nu} C^{\alpha\beta\mu\nu}= {2\over(D-1)(D-2)}R^2- {4\over(D-2)}R_{\mu\nu}R^{\mu\nu}+ R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu}, $$

(87)

where in four dimensions $C_{\alpha\beta\mu\nu}C^{\alpha\beta\mu\nu}{ |}_{\rm 4-dim.} =R^{2}/3-2R_{\mu\nu}R^{\mu\nu}+R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu}$. And, its associated effective Lagrangian, by considering the Gauss–Bonnet theorem, is −2/3(R ²−3R _μν R ^μν). In six dimensions, it reads $C_{\alpha\beta\mu\nu}C^{\alpha\beta\mu\nu}{ |}_{\rm 6-dim.} =R^{2}/10-R_{\mu\nu}R^{\mu\nu}+R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu}$, also $C_{\alpha\beta\mu\nu}\square C^{\alpha\beta\mu\nu}{ |}_{\rm 6-dim.} =K_{9}/10-K_{10}+K_{11}$ and $C_{\alpha\beta\mu\nu ;\rho}C^{\alpha\beta\mu\nu ;\rho}{ |}_{\rm 6-dim.} =K_{15}/10-K_{16}+K_{17}$. In three dimensions, as the Weyl tensor identically vanishes, from relations (32) and (33) we have, respectively,

$$ (6 K_1-36 K_2+3 K_3+32 K_4+24 K_5-12 K_6+K_7 ){ |}_{\rm3-dim.} \equiv0 $$

(88)

and

$$ (-7 K_1/2+21 K_2-3 K_3/2-18 K_4-15 K_5+6 K_6+K_8 ){ |}_{\rm3-dim.} \equiv0. $$

(89)

In four dimensions, from relations (32) and (33) and their dependence, one has

$$ \biggl(\frac{8}{9}K_1+18 K_2-3 K_3-20 K_4-24 K_5+18 K_6-K_7+4 K_8 \biggr){ \bigg|}_{\rm4-dim.} \equiv0. $$

(90)

Also, see Ref. [48] and the appendix of Ref. [49]. The scalar action I constructed by the cubic of the Weyl tensor, relations (32) or (33), in a D-dimensional space–time, through the conformal transformation g _μν →Ω ² g _μν , conformally transforms as I→Ω ^D−6 I. Hence, it is a conformal invariant only in six dimensions.

1.2 A.2 The M ⁽³⁾ As A Lagrangian Term

In Sect. 4, we need to know the effect of the M ⁽³⁾, relation (19), as a Lagrangian term. By using the relations

$$\begin{aligned} \displaystyle\square R^2=2 (K_9+K_{15} ),& \end{aligned}$$

(91)

$$\begin{aligned} \displaystyle\square \bigl(R_{\mu\nu}R^{\mu\nu} \bigr)=2 (K_{10}+K_{16} ),& \end{aligned}$$

(92)

$$\begin{aligned} \displaystyle\square \bigl(R_{\mu\nu\rho\tau}R^{\mu\nu\rho\tau} \bigr)=2 (K_{11} +K_{17} ),& \end{aligned}$$

(93)

$$\begin{aligned} \displaystyle\bigl(R^{\mu\nu}R_{; \mu} \bigr)_{; \nu}=K_{12}+{1\over2}K_{15} ,& \end{aligned}$$

(94)

$$\begin{aligned} \displaystyle\bigl(R_{\mu\nu}R^{\mu\rho} \bigr)_{; \rho}{}^\nu=K_4-K_5+K_{12}+K_{13}+{1\over4}K_{15}& \end{aligned}$$

(95)

and

$$ \bigl(R_{\mu\nu; \rho}R^{\rho\mu\nu\lambda} \bigr)_{; \lambda }= {1\over 2}K_6-{1\over2}K_7-K_8- {1\over4}K_{11}+K_{13}-K_{16} , $$

(96)

relation (19) reads as

$$\begin{aligned} M^{(3)} = &- \,{1\over2} \biggl[ (12b_1 + b_2 )R^2 + 2 (2b_2 + b_5 )R_{\mu\nu}R^{\mu\nu} + \biggl(4b_3 + {1\over2}b_6 \biggr)R_{\mu\nu\lambda\tau} R^{\mu\nu\lambda\tau} \biggr]_{;\rho}{}^\rho \cr &-\, (2b_2+b_5 ) \bigl(R^{\mu\nu}R_{; \mu} \bigr)_{; \nu} - (3b_4-2b_5 ) \bigl(R_{\mu\nu}R^{\mu\rho} \bigr)_{; \rho}{}^\nu \cr &+\,2 (b_5+b_6 ) \bigl(R_{\mu\nu; \rho}R^{\rho\mu\nu \lambda} \bigr)_{; \lambda} . \end{aligned}$$

(97)

Obviously all of the terms are complete divergences. Therefore, with careful attention about the appropriate boundary conditions (that need to be applied in the process of the variation when a function is up to the fourth order jet-prolongation of the metric), one can easily show that these terms, and hence the M ⁽³⁾, give no contribution to the variation of the corresponding action.

1.3 A.3 The K ₉ To K ₁₇ As Lagrangian Terms

In Sect. 5, the effect of each linearly independent term of the K ₉ up to K ₁₇, as a third order Lagrangian term, is needed. For this purpose, using the null effect of complete divergences (91)–(96) in the variation of their corresponding action and that, the term K ₁₄ is itself a complete divergence, we obtain³³ the following identities, i.e.

$$\begin{aligned} \displaystyle G^{(K_{11})}_{\alpha\beta}\equiv-4 G^{(K_4)}_{\alpha\beta} +4 G^{(K_5)}_{\alpha\beta}+2 G^{(K_6)}_{\alpha\beta} -2 G^{(K_7)}_{\alpha\beta}-4 G^{(K_8)}_{\alpha\beta} -G^{(K_9)}_{\alpha\beta}+4 G^{(K_{10})}_{\alpha\beta} ,& \end{aligned}$$

(98)

$$\begin{aligned} \displaystyle G^{(K_{12})}_{\alpha\beta}\equiv{1\over 2}G^{(K_9)}_{\alpha\beta},& \end{aligned}$$

(99)

$$\begin{aligned} \displaystyle G^{(K_{13})}_{\alpha\beta}\equiv-G^{(K_4)}_{\alpha\beta} +G^{(K_5)}_{\alpha\beta}-{1\over4}G^{(K_9)}_{\alpha\beta},& \end{aligned}$$

(100)

$$\begin{aligned} \displaystyle G^{(K_{14})}_{\alpha\beta}\equiv0 ,& \end{aligned}$$

(101)

$$\begin{aligned} \displaystyle G^{(K_{15})}_{\alpha\beta}\equiv-G^{(K_9)}_{\alpha\beta},& \end{aligned}$$

(102)

$$\begin{aligned} \displaystyle G^{(K_{16})}_{\alpha\beta}\equiv-G^{(K_{10})}_{\alpha\beta}& \end{aligned}$$

(103)

and

$$ G^{(K_{17})}_{\alpha\beta}\equiv-G^{(K_{11})}_{\alpha\beta} . $$

(104)

Relation (98) has already been used in Ref. [66] (the last relation on its page number 612, after adapting the sign convention and substituting the necessary relations) and Ref. [23] (the first relation of their relation (14)). However, consideration caution has to be exercised in applying identities in general, and identity (98) in particular, for there are associated with them some difficulties, but nonetheless they throw some important effects on the effective actions.

Thus, it appears that among the K ₉ to K ₁₇ Lagrangian terms, it suffices to derive only the metric variation of the K ₉ and K ₁₀ terms; indeed, we mean that

$$ \delta\int\sum^{17}_{i=1}b_i K_i\sqrt{-g} d^D x =\delta\int\sum ^{10}_{i=1}b^{'}_i K_i\sqrt{-g} d^D x , $$

(105)

where

$$ \begin{array}{l@{\qquad}l} b^{'}_1=b_1, &b^{'}_2=b_2,\\ b^{'}_3=b_3, &b^{'}_4=b_4-4b_{11}-b_{13}+4b_{17},\\ b^{'}_5=b_5+4b_{11}+b_{13}-4b_{17}, &b^{'}_6=b_6+2b_{11}-2b_{17},\\ b^{'}_7=b_7-2b_{11}+2b_{17}, &b^{'}_8=b_8-4b_{11}+4b_{17},\\ b^{'}_9=b_9-b_{11}+b_{12}/2-b_{13}/4-b_{15}+b_{17}, &b^{'}_{10}=b_{10}+4b_{11}-b_{16}-4b_{17}. \end{array} $$

(106)

Hence, as it is customary to write a Lagrangian in terms of its effective one, one can write

$$ \sum^{17}_{i=1}b_i K_i\mathrel{\mathop{=}^{\rm eff}}\sum ^{10}_{i=1}b^{'}_i K_i , $$

(107)

where also the word “effective” is often not mentioned. However, as explained in the text, for being able to apply the trace analogy for an effective total third order Lagrangian, we, in practice, need to consider the K ₁ to K ₁₁ terms, and actually neglect identity (98).

The K ₉, K ₁₀ and K ₁₁ Lagrangian terms give

$$ G^{(K_9)}_{\alpha\beta}= \bigl[2 (\square R ) R_{\alpha\beta}-2 (\square R )_{; \alpha\beta} -R_{; \alpha}R_{; \beta} \bigr]-{1\over2} g_{\alpha\beta} (-4 K_{14}-K_{15}), $$

(108)

$$ G^{(K_{10})}_{\alpha\beta}=R^{(K_{10})}_{\alpha\beta}- {1\over2} g_{\alpha\beta} (-2 K_6+2 K_7+4 K_8+2 K_{10}+K_{11} -4 K_{13}-K_{14}+5 K_{16} ) $$

(109)

and

$$ G^{(K_{11})}_{\alpha\beta}=R^{(K_{11})}_{\alpha\beta}- {1\over2} g_{\alpha\beta} (-K_{17} ), $$

(110)

where³⁴

$$ {\rm \operatorname{Trace}}\, R^{(K_9)}_{\alpha\beta}= {1\over3} (2 K_9-2 K_{14}-K_{15}), $$

(111)

$$\begin{aligned} {\rm \operatorname{Trace}}\, R^{(K_{10})}_{\alpha\beta}= {1\over3} \biggl( 2 K_4-2 K_5-2 K_6+2 K_7+4 K_8+4 K_{10}+K_{11} -2 K_{13}-{1\over2} K_{15}+5 K_{16} \biggr) \end{aligned}$$

(112)

and

$$ {\rm \operatorname{Trace}} R^{(K_{11})}_{\alpha\beta}= {1\over3} (8 K_6-8 K_7 -16 K_8-4 K_{10}+16 K_{13}+2 K_{14}-20 K_{16}+K_{17} ). $$

(113)

Appendix B: Dimensional Dependent Constraints For $L^{(3)}_{\rm total}$

In this appendix, we perform the analogy for the appearance of the $G^{(3b)}_{({\rm total})\alpha\beta}$ in accord with part (b), where

$$ G^{(3b)}_{({\rm total})\alpha\beta}=G^{(3b)}_{({\rm generic})\alpha\beta}+b_9 G^{(K_9)}_{\alpha\beta} +b_{10} G^{(K_{10})}_{\alpha\beta}+b_{11} G^{(K_{11})}_{\alpha\beta} $$

(114)

and similar expressions for $R^{(3b)}_{({\rm total})\alpha\beta}$ and $R^{(3b)}_{\rm total}$. The $G^{(3b)}_{({\rm generic})\alpha\beta}$ is given by relation (18) and the Euler–Lagrange expressions of the K ₉ to K ₁₁ Lagrangian terms (that are given by relations (108)–(110)) must be rewritten in accord with part (b). By imposing the condition ${\sl \operatorname{Trace}}\, R^{(3b)}_{({\rm total})\alpha\beta}=R^{(3b)}_{\rm total}$, and performing the bulk calculations, we find that the trace relation, for non-zero coefficients, holds if and only if

$$ \begin{aligned} b_5={}& \bigl[-12(D-1)^2b_1-\bigl(3D^2+3D-4 \bigr)b_2/3-4(3D-5)b_3/3-b_4 \bigr]/(D-1), \cr b_6={}& \bigl[6(D-1)^2b_1- \bigl(3D^2-19D+14\bigr)b_2/6-2(D+1)b_3/3 \cr &{}-\bigl(3D^2-7D+2\bigr)b_4/4 \bigr]/(D-1), \cr b_7={}&(D-2) \bigl[3(D-1)^2b_1+ \bigl(5D^2+3D-6\bigr)b_2/12+\bigl(3D^2-7D+6 \bigr)b_3/3 \cr &{}+\bigl(D^2+3D-2\bigr)b_4/8 \bigr]/\bigl[2(D-1)\bigr], \cr b_8={}&4(D-1)^2b_1+(D-1) (D+2)b_2/3+4(D-1)b_3/3+(D-2)b_4/2, \cr b_9={}& (b_2/6-2b_3/3+b_4/4 )/(D-1), \cr b_{10}={}&-2(D-1)b_9, \cr b_{11}={}&(D-1) (D-2)b_9/2=-(D-2)b_{10}/4. \end{aligned} $$

(115)

If one sets one of the b ₉ or b ₁₀ or b ₁₁ zero, then constraint (115) will reduce to constraint (40). As mentioned in the text, the authors of Ref. [23] have also derived almost the same relations through a proportionality parameter, say u. Indeed, if we set the b ₁₃=0 and substitute for their proportional parameter in terms of the other coefficients, namely u=[6(D−1)² b ₁−(D ²−8D+6)b ₂/3+4D(D−2)b ₃/3−D(D−2)b ₄/2]/(D−1), in their relation (B15), it will reduce to constraint (115).