Minimal gauge invariant couplings at order \ell_p^6 in M-theory
aa r X i v : . [ h e p - t h ] F e b Minimal gauge invariant couplingsat order ℓ p in M-theory Mohammad R. Garousi Department of Physics, Faculty of Science, Ferdowsi University of MashhadP.O. Box 1436, Mashhad, Iran
Abstract
Removing the field redefinitions, the Bianchi identities and the total derivative free-doms from the general form of the gauge invariant couplings at order ℓ p for the bosonicfields of M-theory, we have found that the minimum number of independent couplingsin the structures with even number of the three-form, is 1062. We find that, except twocouplings which have the Ricci scalar, there are schemes in which there is no couplinginvolving R, R µν , ∇ µ F µαβγ . In these schemes, there are sub-schemes in which, exceptone coupling which has the second derivative of F (4) , the couplings can have no term withmore than two derivatives. We find some of the parameters by dimensionally reducingthe couplings on a circle and comparing them with the known couplings of the one-loopeffective action of type IIA superstring theory. In particular, we find the coupling whichhas term with more than two derivatives is zero. [email protected] ontents ℓ p
23 Reduction on a circle 36
M-theory is a consistent quantum theory of gravity which includes all types of superstringtheories at different limits [1]. In particular, the compactification of M-theory on a circleproduces the type IIA superstring theory. A convinent way to study different phenomenain this theory is to use an effective action which is a derivative-expansionin of the theoryin terms of its massless fields [2, 3]. The leading order terms in this expansion is the 11-dimensional supergravity and the next to the leading order terms are at eight-derivative orderor ℓ p -order in which we are interested. There are different techniques to find such couplings[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. One of them is the S-matrix method in which various S-matrixelements are calculated in the M-theory and then they are compared with the correspondingS-matrix elements in the M-theory effective action. To use this method, one needs independentgauge invariant couplings at order ℓ p .To find such independent couplings, one has to impose various Bianchi identities, use fieldredefinitions freedom [15, 16, 17] and remove total derivative terms from the most general gaugeinvariant couplings. Hence, one should first write all gauge invariant couplings at order ℓ p andthen imposes the above freedoms to reduce them to the minimal couplings. The parametersin the gauge invariant action are either ambiguous or unambiguous depending on whether ornot they are changed under these freedoms. Some combinations of the ambiguous parameters,however, remain invariant. This allows one to separate the ambiguous parameters to essentialparameters, and some arbitrary parameters. Depending on which set of parameters are choosingas essential parameters and how to choose the arbitrary parameters, one has different schemes.The minimum number of independent couplings are found in the schemes that all the arbitraryparameters are set to zero. This method has been used in [18] to find 60 independent gaugeinvariant couplings at order α ′ in the bosonic string theory, and in [19] to find 872 independentNS-NS couplings at order α ′ in type II superstring theories. These parameters are then fixedin the tree-level effective action by the T-duality [21, 22]. We are interested in finding suchindependent couplings at order ℓ p in M-theory for bosonic fields. The effective action of M-theory at each order of ℓ p has two sectors: The Chern-Simons sector which has odd number ofthe three-form and another sector which has even number of the three-form. In this paper weare interested in finding the independent couplings in the second sector at order ℓ p .The outline of the paper is as follows: In section 2, we write the most general gauge invariantcouplings involving the 11-dimensional metric g µν and the three-form A (3) at order ℓ p . Thereare 17744 such couplings. Then we add to them the most general total derivative terms and1eld redefinitions with arbitrary parameters. To impose various Bianchi identities, we rewritethem in the local inertial frame, and rewrite the terms which have derivatives of three-fieldstrength F (4) , in terms of potential, i.e., F (4) = dA (3) . We then use the arbitrary parametersin the total derivative terms and in the field redefinitions to show that there are only 1062unambiguous and essential parameters and all other parameters are arbitrary which can be setto zero. We show that there are minimal schemes in which there are 1059 couplings which haveno term with more than two derivatives and no term involving R, R µν , ∇ µ F µαβγ . There is alsoone essential coupling which has two derivatives on F and two unambiguous couplings whichhave R . We write the explicit form of these 1062 couplings in this section. In section 3, webriefly discuss dimensional reduction of the couplings on a circle to find the parameters whenthe three-form is zero, and some of other parameters involving four fields by comparing with theknown couplings in the one-loop effective action of type IIA superstring theory. In particular,this comparison dictates that the coupling which has term with more than two derivatives iszero. ℓ p The bosonic part of the effective action of M-theory has the following derivative-expansion or ℓ p -expansion where ℓ p is the 11-dimensional Plank-length: S eff = 12 κ h Z d x √− g ( L + ℓ p L + · · · ) + Z ( L CS + ℓ p L CS + · · · ) i (1)where we have used the fact that the M-theory has no effective action at four and six derivativeorders. In fact, as we will argue in the next section, the orders of the derivative terms in theM-theory effective action are at ℓ p , ℓ p , ℓ p , ℓ P , ℓ p , · · · .The effective action must be invariant under the coordinate transformations and under the A -field gauge transformations. The metric and A -field must appear in the Lagrangian L n troughtheir field strengths and their covariant derivatives, e.g., the Lagrangian L at the leading orderof ℓ p is L = R − . F µναβ F µναβ (2)The Chern-Simons form at the leading order is not invariant under the gauge transformation, i.e., L CS = − A (3) ∧ F (4) ∧ F (4) (3)However, its corresponding action is invariant. In this paper, we are interested only on thecouplings in L . A systematic method has been used in [18, 19] to find the minimum number ofindependent couplings at order α ′ and α ′ in the effective action of the bosonic string theory.It has been found that there are 60 couplings at order α ′ and 872 couplings at order α ′ . In thissection, we are going to use this method to find similar couplings at order ℓ p in the M-theory.2ollowing [18], one first should write all gauge invariant couplings at eight-derivative orderwhich has even number of the three-form. Using the package ”xAct” [23], one finds there are17744 such couplings in 40 different structures, i.e., L ′ = m ′ F αν δǫ F αβγν F βφεµ F γ ζηφ F δεσθ F ǫζ ισ F µικλ F ηθκλ + · · · (4)where m ′ , · · · , m ′ are some parameters. The above couplings however are not all indepen-dent. Some of them are related by total derivative terms, some of them are related by fieldredefinitions, and some others are related by various Bianchi identities.To remove the total derivative terms from the above couplings, we consider the most generaltotal derivative terms at order ℓ p which have the following structure: ℓ p κ Z d x √− g J = ℓ p κ Z d x √− g ∇ α ( I α ) (5)where the vector I α is all possible covariant and gauge invariant terms at seven-derivative levelwhich has even number of the three-form, i.e., , I α = J F γδǫµ R αβ R βεǫθ ∇ δ F γµεθ + · · · (6)where the coefficients J , · · · , J are 7760 arbitrary parameters. Adding the total deriva-tive terms with arbitrary coefficients to L ′ , one finds the same Lagrangian but with differentparameters m ′′ , m ′′ , · · · . We call the new Lagrangian L ′′ . Hence∆ ′′ − J = 0 (7)where ∆ ′′ = L ′′ − L ′ is the same as L ′ but with coefficients δm ′′ , δm ′′ , · · · where δm ′′ i = m ′′ i − m ′ i .Solving the above equation, one finds some linear relations between only δm ′′ , δm ′′ , · · · whichindicate how the couplings are related among themselves by the total derivative terms. Theabove equation also gives some relation between the coefficients of the total derivative termsand δm ′′ , δm ′′ , · · · in which we are not interested.The couplings in (7), however, are in a fixed field variables. One is free to change the fieldvariables as g µν → g µν + ℓ p δg (6) µν A µνα → A µνα + ℓ p δA (6) µνα (8)where the tensors δg (6) µν and δA (6) µνα are all possible covariant and gauge invariant terms at 6-derivative level. δg (6) µν contains even number of the three-form and δA (6) µνα contains odd numberof the three-form, i.e., , δg (6) αβ = g F { αγδµ F β } γ ǫν F δµεθ F ǫενη F θλκσ F ηλκσ + · · · δA (6) αβµ = e R γδ R δǫε [ α ∇ β F µ ] γ ǫε + · · · (9)3he coefficients g , · · · , g and e , · · · , e are arbitrary parameters. When the field variablesin L are changed according to the above field redefinitions, they produce some couplings atorders ℓ p and higher in which we are not interested in this paper. However, when the fieldvariables in S are changed, up to some total derivative terms, the following couplings at order ℓ p are produced: δS = δS δg αβ δg (3) αβ + δS δA αβµ δA (3) αβµ ≡ ℓ p κ Z d x √− g K = ℓ p κ Z d x √− g h (cid:18) ∇ γ F γαβµ − . ǫ αβµνγλθτκσζ F νγλθ F τκσζ (cid:19) δA (6) αβµ (10) − ( R αβ − F αγδµ F βγδµ ) δg (6) αβ + ( 12 R − F αβγν F αβγν ) δg (6) µµ ) i The second term in the second line produces couplings in the Chern-Simons sector in which weare not interested in this paper, hence, we do not consider the effect of field redefinition δA (6) αβµ on this term. Adding the total derivative terms and field redefinition terms to L ′ , one findsthe same Lagrangian but with different parameters m , m , · · · . We call the new Lagrangian L . Hence ∆ − J − K = 0 (11)where ∆ = L − L ′ is the same as L ′ but with coefficients δm , δm , · · · where δm i = m i − m ′ i .Solving the above equation, one finds some linear relations between only δm , δm , · · · whichindicate how the couplings are related among themselves by the total derivative and fieldredefinition terms. There are also many relations between δm , δm , · · · and the coefficients oftotal derivative terms and field redefinitions in which we are not interested,However, to solve the equation (11) one should write it in terms of independent couplings, i.e., one has to impose the following Bianchi identities as well: R α [ βγδ ] = 0 ∇ [ µ R αβ ] γδ = 0 (12) ∇ [ µ F ναβγ ] = 0[ ∇ , ∇ ] O − R O = 0To impose the Bianchi identities in non-gauge invariant form, one may rewrite the terms in(11) in the local frame in which the first derivative of metric is zero, and rewrite the terms in(11) which have derivatives of F in terms of A-field, i.e., F = dA . In this way, the Bianchiidentities satisfy automatically [18]. In fact, writing the couplings in terms of potential ratherthan field strength, there would be no Bianchi identity at all. This way of imposing the Bianchiidentities is very easy to perform by the computer.Using the above steps, one can rewrite the different terms on the left-hand side of (11) interms of independent but non-gauge invariant couplings. The solution to the equation (11)then has two parts. One part is 1062 relations between only δm i ’s, and the other part is some4elations between the coefficients of the total derivative terms, field redefinitions and δm i ’sin which we are not interested. The number of relations in the first part gives the numberof independent couplings in L . In a particular scheme, one may set some of the coefficientsin L ′ to zero, however, after replacing the non-zero terms in (11), the number of relationsbetween only δm i ’s should not be changed, i.e., there must be always 1062 relations. We setthe coefficients of the couplings in L ′ in which each term that has R µν , ∇ µ F µναβ to be zero.After setting these coefficients to zero, there are still 1062 relations between δm i ’s. This meanswe are allowed to remove these terms.We find that there are two couplings which involve the Ricci scalar that their parametersare unambiguous, i.e., L ⊃ m F αβǫε F αβγδ F γδǫκ F ελµν F κλστ F µνστ R + m F αβ ǫε F αβγδ F γδǫε F κλστ F κλµν F µνστ R (13)Obviously there is no total derivative term with the above structure because it does not involvederivative of the Ricci scalar and F . Moreover, the Bianchi identities does not affect theseterms. The field redefinition can not produce the above couplings either. Apart from the abovetwo terms, one can set to zero the coefficients of all other terms that have the Ricci scalar.We then try to set zero the couplings in L ′ which have term with more then two derivatives.Imposing this condition and then solving (11) again, one would find 1061 relations between only δm i ’s. It means that at least one of the independent couplings has terms with more than twoderivatives. We have found this independent coupling to be L ⊃ m F ǫµνσ R αγ ǫε R αβγδ ∇ σ ∇ δ F βεµν (14)The way we have found the above coupling is that we divided the couplings involving morethan two derivatives to two parts. We then set the coefficients of one part to zero. If thecorresponding equations in (11) gives 1062 relations between the remaining δm i ’s then thatchoice is allowed, otherwise the other part is allowed to be zero. Again we divided the non-zeropart to two parts and set half of them to zero. If the corresponding equations in (11) gives1062 relations between the remaining δm i ’s then that choice is allowed, otherwise the otherpart is allowed to be zero. Repeating this strategy one finds the above couplings is one of theindependent couplings. Apart from the above coupling, all other couplings which have termswith more than two derivatives are allowed to be zero. There are still 3304 couplings which haveno term with more than two derivatives and have no terms with structures R, R µν , ∇ µ F µναβ .Hence, there are still many choices for choosing the non-zero coefficients such that they satisfythe 1062 relations δm i = 0. In the particular scheme that we have chosen, there are threecouplings appear in (13) and (14), and the other 1059 couplings appear in the following 9structures.There are 132 couplings with structure F , i.e., L F = m F αǫεµ F αβγδ F β νσλ F γκτω F δϕξζ F ǫνκϕ F εστξ F µλωζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F ετωϕ F κωϕζ F µστ ξ F νλξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F ετωϕ F κϕξζ F µστ ξ F νλωζ +5 F αβ ǫε F αβγδ F γδµν F ǫσλκ F ετωϕ F κϕξζ F µστ ξ F νλωζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F ετωϕ F κωϕζ F µσλξ F ντξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F εστω F κωξζ F µλϕξ F ντϕζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F εστω F κωξζ F µλϕξ F ντϕζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F ετωϕ F κϕξζ F µσλξ F ντωζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F εστω F κτωζ F µλϕξ F νϕξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F ετωϕ F κϕξζ F µσλτ F νωξζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F ετωϕ F κϕξζ F µσλτ F νωξζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F εσλτ F κτξζ F µωϕξ F νωϕζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εσκτ F κτξζ F λωϕζ F ν ωϕξ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εσκτ F κτξζ F λωϕζ F ν ωϕξ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F κτξζ F λωϕζ F µσκτ F ν ωϕξ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F κτξζ F λωϕζ F µσκτ F ν ωϕξ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εκτω F λωξζ F νϕξζ F σκτϕ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εκτω F λωξζ F νϕξζ F σκτϕ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F λωξζ F µκτω F νϕξζ F σκτϕ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F λωξζ F µκτω F νϕξζ F σκτϕ + m F αǫεµ F αβγδ F βǫνσ F γελκ F δτωϕ F µτ ξζ F νλωξ F σκϕζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F ετωϕ F κωϕζ F µν ξζ F σλτξ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F εϕξζ F κωξζ F σλτϕ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ετ ϕξ F κωξζ F σλϕζ + m F αǫεµ F αβγδ F βǫνσ F γελκ F δν τω F κωξζ F µτ ϕξ F σλϕζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµνω F εϕξζ F κτωζ F σλϕξ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F εϕξζ F κτωζ F σλϕξ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ετ ϕξ F κϕξζ F σλωζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµλω F εϕξζ F νκωϕ F στξζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµωϕ F ελξζ F νκωξ F στϕζ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ελϕξ F κωξζ F στϕζ + m F αǫεµ F αβγδ F βǫνσ F γενλ F δκτω F λωξζ F µκϕξ F στϕζ + m F αǫεµ F αβγδ F βǫνσ F γελκ F δν τω F κωξζ F µλϕξ F στϕζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εκτω F λϕξζ F νκϕξ F στωζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εκτω F λϕξζ F νκϕξ F στωζ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F λϕξζ F µκτω F νκϕξ F στωζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F λϕξζ F µκτω F νκϕξ F στωζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµνω F ελϕξ F κτωζ F σϕξζ +6 F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ελϕξ F κτωζ F σϕξζ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ελτ ϕ F κωξζ F σϕξζ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ελτ ϕ F κϕξζ F σωξζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εκτω F λϕξζ F νκτ ϕ F σωξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εκτω F λϕξζ F νκτ ϕ F σωξζ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F λϕξζ F µκτω F νκτ ϕ F σωξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F λϕξζ F µκτω F νκτ ϕ F σωξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F εσλτ F κωξζ F µν ωϕ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εσκτ F λωξζ F νκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εσκτ F λωξζ F νκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F λωξζ F µσκτ F νκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F λωξζ F µσκτ F νκωϕ F τϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F λωξζ F µσκτ F νκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εσκτ F κωξζ F νλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F κωξζ F µσκτ F νλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F κωξζ F µσκτ F νλωϕ F τϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F κωξζ F µσκτ F νλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F εσλτ F µκωϕ F νωξζ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εσκτ F λκωζ F νωϕξ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εσκτ F λκωζ F νωϕξ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F ενκτ F λωξζ F σκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F ενκτ F λωξζ F σκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F λωξζ F µνκτ F σκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F λωξζ F µνκτ F σκωϕ F τϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F λωξζ F µνκτ F σκωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F ενκτ F κωξζ F σλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δεσλ F κωξζ F µν κτ F σλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F κωξζ F µν κτ F σλωϕ F τϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F κωξζ F µν κτ F σλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δεµσ F κωξζ F ν λκτ F σλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεµσ F κωξζ F ν λκτ F σλωϕ F τϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεµσ F κωξζ F ν λκτ F σλωϕ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δµν σ F ελκτ F λκωζ F σωϕξ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δεµσ F λκωζ F νλκτ F σωϕξ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεµσ F λκωζ F νλκτ F σωϕξ F τϕξζ +7 F αβγ ǫ F αβγδ F δεµν F ǫεµσ F λκωζ F νλκτ F σωϕξ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δεµν F κωξζ F σλωϕ F σλκτ F τϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεµν F κωξζ F σλωϕ F σλκτ F τϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεµν F κωξζ F σλωϕ F σλκτ F τϕξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F εστω F λκϕζ F µνϕξ F τωξζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εκτω F λκϕζ F νσϕξ F τωξζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F εστω F µλκϕ F νϕξζ F τωξζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµνω F ελϕξ F σκϕζ F τωξζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµλω F ενϕξ F σκϕζ F τωξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εκτω F ν ϕξζ F σλκϕ F τωξζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εκτω F νκϕξ F σλϕζ F τωξζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµνω F ελκϕ F σϕξζ F τωξζ + m F αβ ǫε F αβγδ F γµνσ F δµλκ F ǫντω F εσϕξ F λκτ ζ F ωϕξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F ετωϕ F λκτ ζ F µνσξ F ωϕξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F εστω F µλτ ϕ F νκξζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F εστω F µλτ ϕ F νκξζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F ετωϕ F µσλτ F νκξζ F ωϕξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F ετωϕ F µσλξ F νκτ ζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εσκτ F λτ ξζ F νκωϕ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F λτ ξζ F µσκτ F νκωϕ F ωϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F λτ ξζ F µσκτ F νκωϕ F ωϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫσλκ F εµστ F κτ ξζ F νλωϕ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F κτ ξζ F µσκτ F νλωϕ F ωϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F κτ ξζ F µσκτ F νλωϕ F ωϕξζ + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εκτω F λκτ ζ F νσϕξ F ωϕξζ + m F αβ ǫε F αβγδ F γǫµν F δσλκ F εστω F µλκϕ F ντ ξζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ F εστω F µλκϕ F ντ ξζ F ωϕξζ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ελτ ϕ F σκξζ F ωϕξζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµνω F ελϕξ F σκτ ζ F ωϕξζ + m F αβ ǫε F αβγδ F γ µνσ F δµλκ F ǫντω F ελϕξ F σκτ ζ F ωϕξζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµλω F ενϕξ F σκτ ζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F ενκτ F λτ ξζ F σκωϕ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F λτ ξζ F µνκτ F σκωϕ F ωϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F λτ ξζ F µνκτ F σκωϕ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εκτω F νκτ ϕ F σλξζ F ωϕξζ +8 F αβ ǫε F αβγδ F γǫµν F δµσλ F εκτω F νκϕξ F σλτ ζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εκτω F νκϕξ F σλτ ζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F κτ ξζ F µν κτ F σλωϕ F ωϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F κτ ξζ F µν κτ F σλωϕ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεµσ F κτ ξζ F ν λκτ F σλωϕ F ωϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεµσ F κτ ξζ F ν λκτ F σλωϕ F ωϕξζ + m F αβ ǫε F αβγδ F γµνσ F δλκτ F ǫµν ω F ελκϕ F στ ξζ F ωϕξζ + m F αβ ǫε F αβγδ F γ µνσ F δλκτ F ǫµλω F ενκϕ F στ ξζ F ωϕξζ + m F αβ ǫε F αβγδ F γδµν F ǫεµν F κτ ξζ F σλωϕ F σλκτ F ωϕξζ + m F αβγ ǫ F αβγδ F δεµν F ǫεµν F κτ ξζ F σλωϕ F σλκτ F ωϕξζ + m F αβ ǫε F αβγδ F γµνσ F δµλκ F ǫνστ F ελκτ F F + m F αβ ǫε F αβγδ F γµνσ F δλκτ F ǫµνλ F εσκτ F F + m F αβ ǫε F αβγδ F γǫµν F δσλκ F εσλτ F µνκτ F F + m F αǫεµ F αβγδ F βǫνσ F γελκ F δνλτ F µσκτ F F + m F αβ ǫε F αβγδ F γǫµν F δµσλ F εσκτ F νλκτ F F + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F εσκτ F νλκτ F F + m F αβ ǫε F αβγδ F γǫµν F δεσλ F µσκτ F νλκτ F F + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F µσκτ F νλκτ F F + m F αβ ǫε F αβγδ F γδµν F ǫµσλ F ενκτ F σλκτ F F + m F αβ ǫε F αβγδ F γδµν F ǫεσλ F µν κτ F σλκτ F F + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ F µν κτ F σλκτ F F + m F αβ ǫε F αβγδ F γǫµν F δεµν F F + m F αβ ǫε F αβγδ F γδµν F ǫεµν F F + m F F (15)where F F = F µναβ F µναβ . Note that number of all contractions of F without imposing thefield redefinition is 174. The are 104 couplings in (15) that their coefficients are unambiguous.The coefficients of the couplings which have F F or F µαβγ F ναβγ are essential parameters.There are 47 couplings with structure of one Riemann curvature and six F , i.e., L F R = m F αǫεµ F βνσλ F γǫκτ F δν ωϕ F εσκω F µλτϕ R αβγδ + m F αǫεµ F βνσλ F γǫνκ F δτωϕ F εσκτ F µλωϕ R αβγδ + m F αǫεµ F βǫεν F γσλκ F δτωϕ F µσλτ F νκωϕ R αβγδ + m F αβǫε F γǫµν F δσλκ F ετωϕ F µσλτ F νκωϕ R αβγδ + m F αβǫε F γδµν F ǫσλκ F ετωϕ F µσλτ F νκωϕ R αβγδ + m F αβǫε F γ µνσ F δλκτ F ǫµωϕ F ελωϕ F νσκτ R αβγδ + m F αǫεµ F βǫνσ F γελκ F δτωϕ F µλτω F νσκϕ R αβγδ + m F αǫεµ F βǫνσ F γελκ F δλτω F µτωϕ F νσκϕ R αβγδ +9 F αǫεµ F βνσλ F γǫεκ F δτωϕ F µτωϕ F νσλκ R αβγδ + m F αǫεµ F βǫνσ F γελκ F δτωϕ F µτωϕ F νσλκ R αβγδ + m F αǫεµ F βǫνσ F γλκτ F δλωϕ F εµκτ F νσωϕ R αβγδ + m F αǫεµ F βǫνσ F γελκ F δτωϕ F µλκτ F νσωϕ R αβγδ + m F αǫεµ F βǫεν F γσλκ F δσλτ F µκωϕ F ντωϕ R αβγδ + m F αβǫε F γǫµν F δσλκ F εσλτ F µκωϕ F ντωϕ R αβγδ + m F αβǫε F γδµν F ǫσλκ F εσλτ F µκωϕ F ντωϕ R αβγδ + m F αǫεµ F βǫεν F γσλκ F δστω F µλκϕ F ντωϕ R αβγδ + m F αβǫε F γǫµν F δσλκ F εστω F µλκϕ F ντωϕ R αβγδ + m F αβǫε F γδµν F ǫσλκ F εστω F µλκϕ F ντωϕ R αβγδ + m F αǫεµ F βǫεν F γσλκ F δτωϕ F µσλκ F ντωϕ R αβγδ + m F αβǫε F γǫµν F δσλκ F ετωϕ F µσλκ F ντωϕ R αβγδ + m F αβǫε F γδµν F ǫσλκ F ετωϕ F µσλκ F ντωϕ R αβγδ + m F αβǫε F γ µνσ F δλκτ F ǫµν ω F ελωϕ F σκτϕ R αβγδ + m F αβǫε F γ µνσ F δλκτ F ǫµλω F ενωϕ F σκτϕ R αβγδ + m F αβǫε F γ µνσ F δλκτ F ǫεωϕ F µνλω F σκτϕ R αβγδ + m F αβǫε F γ µνσ F δµλκ F ǫντω F ελτ ϕ F σκωϕ R αβγδ + m F αǫεµ F βνσλ F γǫκτ F δν ωϕ F εµωϕ F σλκτ R αβγδ + m F αβǫε F γ µνσ F δλκτ F ǫµωϕ F ενωϕ F σλκτ R αβγδ + m F αβǫε F γ µνσ F δµλκ F ǫντω F ετωϕ F σλκϕ R αβγδ + m F αǫεµ F βνσλ F γǫεκ F δτωϕ F µντω F σλκϕ R αβγδ + m F αǫεµ F βǫνσ F γελκ F δτωϕ F µντω F σλκϕ R αβγδ + m F αβǫε F γǫµν F δσλκ F ετωϕ F µντω F σλκϕ R αβγδ + m F αǫεµ F βνσλ F γǫνκ F δετω F µτωϕ F σλκϕ R αβγδ + m F αǫεµ F βνσλ F γǫεκ F δν τω F µτωϕ F σλκϕ R αβγδ + m F αǫεµ F βǫνσ F γελκ F δν τω F µτωϕ F σλκϕ R αβγδ + m F αβǫε F γǫµν F δσλκ F εµτω F ντωϕ F σλκϕ R αβγδ + m F αǫεµ F βνσλ F γǫκτ F δν ωϕ F εµκω F σλτϕ R αβγδ + m F αβǫε F γ µνσ F δµν λ F ǫκτω F εκτ ϕ F σλωϕ R αβγδ + m F αǫεµ F βνσλ F γǫκτ F δν ωϕ F εµκτ F σλωϕ R αβγδ + m F αǫεµ F βνσλ F γǫνκ F δτωϕ F εµκτ F σλωϕ R αβγδ + m F αǫεµ F βνσλ F γǫεν F δκτω F µκτ ϕ F σλωϕ R αβγδ + m F αǫεµ F βνσλ F γǫεκ F δτωϕ F µνκτ F σλωϕ R αβγδ + m F αǫεµ F βǫεν F γµσλ F δκτω F νκτ ϕ F σλωϕ R αβγδ + m F αβǫε F γǫµν F δµσλ F εκτω F νκτ ϕ F σλωϕ R αβγδ +10 F αβǫε F γδµν F ǫµσλ F εκτω F νκτ ϕ F σλωϕ R αβγδ + m F αβǫε F γδµν F ǫεσλ F µκτω F νκτ ϕ F σλωϕ R αβγδ + m F αβǫε F γδµν F ǫεµσ F κτωϕ F ν λκτ F σλωϕ R αβγδ + m F αβǫε F γδµν F ǫεµν F κτωϕ F σλωϕ F σλκτ R αβγδ (16)There are 63 couplings with structure of two Riemann curvatures and four F , i.e., L F R = m F ǫεσλ F ǫεµν F µσκτ F νλκτ R αγβδ R αβγδ + m F ǫεσλ F ǫεµν F µν κτ F σλκτ R αγβδ R αβγδ + m F ǫεµσ F ǫεµν F νλκτ F σλκτ R αγβδ R αβγδ + m F βµνσ F δµνσ F ǫλκτ F ελκτ R αǫγε R αβγδ + m F βµνσ F δµν λ F ǫσκτ F ελκτ R αǫγε R αβγδ + m F βµνσ F δµλκ F ǫνλτ F εσκτ R αǫγε R αβγδ + m F βεµν F δǫσλ F µσκτ F νλκτ R αǫγε R αβγδ + m F βǫµν F δεσλ F µσκτ F νλκτ R αǫγε R αβγδ + m F βδµν F ǫεσλ F µσκτ F νλκτ R αǫγε R αβγδ + m F βεµν F δǫσλ F µν κτ F σλκτ R αǫγε R αβγδ + m F βǫµν F δεσλ F µν κτ F σλκτ R αǫγε R αβγδ + m F βδµν F ǫεσλ F µν κτ F σλκτ R αǫγε R αβγδ + m F βδǫε F µν κτ F µνσλ F σλκτ R αǫγε R αβγδ + m F βεµν F δǫµσ F ν λκτ F σλκτ R αǫγε R αβγδ + m F βδµν F ǫεµσ F ν λκτ F σλκτ R αǫγε R αβγδ + m F βγνσ F δν λκ F ǫεστ F µλκτ R αǫεµ R αβγδ + m F βγνσ F δενσ F ǫλκτ F µλκτ R αǫεµ R αβγδ + m F βγνσ F δεν λ F ǫσκτ F µλκτ R αǫεµ R αβγδ + m F βενσ F γǫνλ F δµκτ F σλκτ R αǫεµ R αβγδ + m F βγǫν F δεν σ F µλκτ F σλκτ R αǫεµ R αβγδ + m F βγεν F δǫσλ F µν κτ F σλκτ R αǫεµ R αβγδ + m F βγǫν F δεσλ F µν κτ F σλκτ R αǫεµ R αβγδ + m F βγǫε F δνσλ F µνκτ F σλκτ R αǫεµ R αβγδ + m F βγǫν F δεµσ F νλκτ F σλκτ R αǫεµ R αβγδ + m F αǫσλ F βµσκ F γελτ F δνκτ R αβγδ R ǫεµν + m F αǫσλ F βεσλ F γµκτ F δνκτ R αβγδ R ǫεµν + m F αǫσλ F βεσκ F γµλτ F δνκτ R αβγδ R ǫεµν + m F αǫσλ F βεκτ F γµσλ F δνκτ R αβγδ R ǫεµν + m F αβ σλ F γǫσλ F δεκτ F µνκτ R αβγδ R ǫεµν +11 F αβ σλ F γǫσκ F δελτ F µνκτ R αβγδ R ǫεµν + m F αβ σλ F γδσλ F ǫεκτ F µνκτ R αβγδ R ǫεµν + m F αβ σλ F γδσκ F ǫελτ F µνκτ R αβγδ R ǫεµν + m F αβ σλ F γδκτ F ǫεσλ F µνκτ R αβγδ R ǫεµν + m F αβ σλ F γǫσκ F δεκτ F µνλτ R αβγδ R ǫεµν + m F αβ σλ F γδκτ F ǫεσκ F µνλτ R αβγδ R ǫεµν + m F αβ σλ F γǫκτ F δεκτ F µνσλ R αβγδ R ǫεµν + m F αβ σλ F γǫµσ F δεκτ F νλκτ R αβγδ R ǫεµν + m F αβ σλ F γǫµκ F δεστ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γεµσ F δλκτ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γεσλ F δµκτ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γελκ F δµστ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γεµλ F δσκτ F νλκτ R αβγδ R ǫεµν + m F αβǫε F γµσλ F δσκτ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γδµσ F ελκτ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γδσλ F εµκτ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γδλκ F εµστ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γδµλ F εσκτ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γδελ F µσκτ F νλκτ R αβγδ R ǫεµν + m F αβǫε F γδσλ F µσκτ F νλκτ R αβγδ R ǫεµν + m F αβǫσ F γεµλ F δλκτ F νσκτ R αβγδ R ǫεµν + m F αβǫσ F γελκ F δµλτ F νσκτ R αβγδ R ǫεµν + m F αβǫσ F γδλκ F εµλτ F νσκτ R αβγδ R ǫεµν + m F αβ σλ F γǫµκ F δεκτ F νσλτ R αβγδ R ǫεµν + m F αβǫσ F γεµλ F δν κτ F σλκτ R αβγδ R ǫεµν + m F αβǫε F γµσλ F δν κτ F σλκτ R αβγδ R ǫεµν + m F αβǫσ F γδελ F µν κτ F σλκτ R αβγδ R ǫεµν + m F αβǫε F γδσλ F µν κτ F σλκτ R αβγδ R ǫεµν + m F αβǫε F γδµσ F ν λκτ F σλκτ R αβγδ R ǫεµν + m F αβǫε F γδµν F F R αβγδ R ǫεµν + m F βεµν F δǫµν F F R αǫγε R αβγδ + m F βδµν F ǫεµν F F R αǫγ ε R αβγδ + m F βγǫν F δεµν F F R αǫεµ R αβγδ + m F F R αγβδ R αβγδ (17)There are 24 couplings with structure of two Riemann curvatures and four F , i.e., L F R = m F δν σλ F ǫµσλ R αǫγ ε R αβγδ R βµεν + m F δµσλ F ǫνσλ R αǫγε R αβγδ R βµεν +12 F δǫσλ F µνσλ R αǫγ ε R αβγδ R βµεν + m F ενσλ F µνσλ R αβ ǫε R αβγδ R γǫδµ + m F δν σλ F εµσλ R αβǫε R αβγδ R γ µǫν + m F δµσλ F ενσλ R αβ ǫε R αβγδ R γµǫν + m F δεσλ F µνσλ R αβǫε R αβγδ R γ µǫν + m F βµν λ F δǫσλ R αǫεµ R αβγδ R γ νεσ + m F βδµλ F ǫνσλ R αǫεµ R αβγδ R γ νεσ + m F βδǫλ F µνσλ R αǫεµ R αβγδ R γ νεσ + m F βǫεν F δµσλ R αǫεµ R αβγδ R γ νσλ + m F βδǫν F εµσλ R αǫεµ R αβγδ R γ νσλ + m F ενσλ F µνσλ R αγβ ǫ R αβγδ R δεǫµ + m F ǫεσλ F µνσλ R αγβ ǫ R αβγδ R δεµν + m F βµσλ F δνσλ R αγ ǫε R αβγδ R ǫµεν + m F βδσλ F µνσλ R αγ ǫε R αβγδ R ǫµεν + m F βδµλ F ενσλ R αγ ǫε R αβγδ R ǫµνσ + m F ǫεσλ F µνσλ R αγβδ R αβγδ R ǫεµν + m F δεµλ F ǫνσλ R αγβ ǫ R αβγδ R εµνσ + m F βεµν F δǫσλ R αǫγ ε R αβγδ R µνσλ + m F βǫµν F δεσλ R αǫγε R αβγδ R µνσλ + m F βδµν F ǫεσλ R αǫγε R αβγδ R µνσλ + m F F R αǫγ ε R αβγδ R βεδǫ + m F F R αβǫε R αβγδ R γǫδε (18)There are 7 couplings with structure of four Riemann curvatures, i.e., L R = m R αβ ǫε R αβγδ R γµǫν R δµεν + m R αǫγε R αβγδ R βµǫν R δνεµ + m R αβ ǫε R αβγδ R γµǫν R δνεµ + m R αβǫε R αβγδ R γ µδν R ǫµεν + m R αγβ ǫ R αβγδ R δεµν R ǫµεν + m R αǫγε R αβγδ R β µδν R ǫνεµ + m R αγβδ R αβγδ R ǫµεν R ǫεµν (19)There are 530 couplings with structure of four F and two ∇ F , i.e., L F ( ∂F ) = m F αβǫε F αβγδ F γµνσ F µνλκ ∇ ǫ F δστω ∇ κ F ελτω + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ ǫ F δν τω ∇ κ F σλτω + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ ε F δǫτω ∇ κ F σλτω + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ ε F δν τω ∇ κ F σλτω + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ κ F εστω ∇ λ F δǫτω + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ κ F γǫσω ∇ λ F δετω + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ κ F γενσ ∇ λ F δµτω + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ κ F γεσω ∇ λ F δµτω + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ κ F ǫστω ∇ λ F δν τω + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ κ F εστω ∇ λ F δν τω + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ κ F µστω ∇ λ F δν τω + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ κ F µντω ∇ λ F ǫστω + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ κ F µντω ∇ λ F εστω + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ κ F µντω ∇ λ F εστω + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ ε F γδνσ ∇ λ F µκτω + m F αβγ ǫ F αβγδ F εµν λ F εµνσ ∇ ǫ F δκτω ∇ λ F σκτω +13 F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ ε F δκτω ∇ λ F σκτω + m F αǫεµ F αβγδ F βǫνσ F γενλ ∇ λ F σκτω ∇ µ F δκτω + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ κ F γενσ ∇ µ F δλτω + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ κ F σλτω ∇ µ F δν τω + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ κ F νλτω ∇ µ F ǫστω + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ κ F νλτω ∇ µ F εστω + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ κ F νλτω ∇ µ F εστω + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ κ F σλτω ∇ ν F δετω + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ κ F εστω ∇ ν F δλτω + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ κ F ελτω ∇ ν F δστω + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ κ F σλτω ∇ ν F ǫετω + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ λ F εκτω ∇ σ F γδǫω + m F αǫεµ F αβγδ F β νσλ F γκτω ∇ λ F µκτω ∇ σ F δǫεν + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ κ F ελτω ∇ σ F δǫτω + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ κ F νλτω ∇ σ F δετω + m F αβγ ǫ F αβγδ F εµν λ F εµνσ ∇ λ F ǫκτω ∇ σ F δκτω + m F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ λ F εκτω ∇ σ F δκτω + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ κ F ǫλτω ∇ σ F δν τω + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ κ F ελτω ∇ σ F δν τω + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ κ F ελτω ∇ σ F δν τω + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ κ F µλτω ∇ σ F δν τω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ κ F δνλω ∇ σ F ǫετω + m F αβγ ǫ F αβγδ F εµνσ F λκτω ∇ λ F δεµν ∇ σ F ǫκτω + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ λ F νκτω ∇ σ F ǫκτω + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ κ F νλτω ∇ σ F ǫµτω + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ κ F ελτω ∇ σ F ǫντω + m F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ λ F δκτω ∇ σ F εκτω + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ λ F δν τω ∇ σ F εκτω + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ λ F ǫντω ∇ σ F εκτω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ν F δǫλω ∇ σ F εκτω + m F αβǫε F αβγδ F γǫµν F δµσλ ∇ λ F νκτω ∇ σ F εκτω + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ κ F νλτω ∇ σ F εµτω + m F αβγ ǫ F αβγδ F δεµν F ǫσλκ ∇ κ F νλτω ∇ σ F εµτω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ κ F δǫλω ∇ σ F εντω + m F αβǫε F αβγδ F γµνσ F λκτω ∇ κ F δǫµλ ∇ σ F εντω +14 F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ κ F δµλω ∇ σ F εντω + m F αβǫε F αβγδ F γδǫµ F νσλκ ∇ κ F µλτω ∇ σ F εντω + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ λ F δν τω ∇ σ F µκτω + m F αβǫε F αβγδ F γǫµν F δεσλ ∇ λ F νκτω ∇ σ F µκτω + m F αβǫε F αβγδ F γδµν F ǫεσλ ∇ λ F νκτω ∇ σ F µκτω + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ ∇ λ F νκτω ∇ σ F µκτω + m F αǫεµ F αβγδ F βǫνσ F λκτω ∇ ε F γδλκ ∇ σ F µντω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ κ F δελω ∇ σ F µντω + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ λ F µνσω ∇ τ F γδεκ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ λ F εµσω ∇ τ F γδνκ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ λ F γǫσω ∇ τ F δεκω + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ κ F γǫσω ∇ τ F δελω + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ µ F γεσω ∇ τ F δλκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F ǫενω ∇ τ F δλκω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ σ F εµνω ∇ τ F δλκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ σ F εµνω ∇ τ F δλκω + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ κ F γεσω ∇ τ F δµλω + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ λ F γδσω ∇ τ F ǫεκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ λ F δνσω ∇ τ F ǫεκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F δνλω ∇ τ F ǫεκω + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ κ F γδσω ∇ τ F ǫελω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ κ F δνλω ∇ τ F ǫεσω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ν F δλκω ∇ τ F ǫεσω + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ σ F δµν ω ∇ τ F ǫλκω + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ λ F δµν ω ∇ τ F ǫσκω + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ ν F δµλω ∇ τ F ǫσκω + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ ǫ F γδσω ∇ τ F ελκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ǫ F δνσω ∇ τ F ελκω + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ν F δµσω ∇ τ F ελκω + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ σ F γδǫω ∇ τ F ελκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F δǫν ω ∇ τ F ελκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ σ F δµν ω ∇ τ F ελκω + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ σ F δµν ω ∇ τ F ελκω + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ σ F ǫµνω ∇ τ F ελκω + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ κ F ǫσλω ∇ τ F εµνω +15 F αβǫε F αβγδ F γǫµν F σλκτ ∇ λ F δµσω ∇ τ F ενκω + m F αβǫε F αβγδ F γδµν F σλκτ ∇ λ F ǫµσω ∇ τ F ενκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ǫ F δλκω ∇ τ F ενσω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ κ F δµλω ∇ τ F ενσω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ µ F δλκω ∇ τ F ενσω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ǫ F δνλω ∇ τ F εσκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ λ F δǫνω ∇ τ F εσκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ λ F δµν ω ∇ τ F εσκω + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ λ F ǫµνω ∇ τ F εσκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ν F δǫλω ∇ τ F εσκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ν F δµλω ∇ τ F εσκω + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ ν F ǫµλω ∇ τ F εσκω + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ σ F γδεω ∇ τ F µλκω + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ε F ǫσλω ∇ τ F µνκω + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ λ F δεσω ∇ τ F µνκω + m F αβǫε F αβγδ F γδµν F σλκτ ∇ λ F ǫεσω ∇ τ F µνκω + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ λ F ǫεσω ∇ τ F µνκω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ε F δλκω ∇ τ F µνσω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ κ F δελω ∇ τ F µνσω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ν F δελω ∇ τ F µσκω + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ ε F γδκω ∇ τ F µσλω + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ κ F γδεω ∇ τ F µσλω + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ µ F δεσω ∇ τ F νλκω + m F αβǫε F αβγδ F γδµν F σλκτ ∇ µ F ǫεσω ∇ τ F νλκω + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ µ F ǫεσω ∇ τ F νλκω + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ σ F δεµω ∇ τ F νλκω + m F αβǫε F αβγδ F γδµν F σλκτ ∇ σ F ǫεµω ∇ τ F νλκω + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ σ F ǫεµω ∇ τ F νλκω + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ ǫ F δµλω ∇ τ F νσκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ε F δǫλω ∇ τ F νσκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ε F δµλω ∇ τ F νσκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ λ F δǫεω ∇ τ F νσκω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ λ F δεµω ∇ τ F νσκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ λ F δεµω ∇ τ F νσκω + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ λ F ǫεµω ∇ τ F νσκω +16 F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ µ F δελω ∇ τ F νσκω + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ ǫ F δµν ω ∇ τ F σλκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ε F δǫν ω ∇ τ F σλκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ε F δµν ω ∇ τ F σλκω + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ µ F γδεω ∇ τ F σλκω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ µ F δεν ω ∇ τ F σλκω + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ν F δǫεω ∇ τ F σλκω + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ν F δεµω ∇ τ F σλκω + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ν F δεµω ∇ τ F σλκω + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ ν F ǫεµω ∇ τ F σλκω + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ λ F εµνσ ∇ ω F γδκτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ λ F γǫνσ ∇ ω F δεκτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ κ F γǫνσ ∇ ω F δελτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ σ F γǫνκ ∇ ω F δελτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ µ F γενσ ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ σ F ǫεµν ∇ ω F δλκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ λ F γενσ ∇ ω F δµκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ κ F γενσ ∇ ω F δµλτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ σ F γενκ ∇ ω F δµλτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ λ F γδνσ ∇ ω F ǫεκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ λ F δµνσ ∇ ω F ǫεκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ λ F δνσω ∇ ω F ǫεκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ σ F δµνλ ∇ ω F ǫεκτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ κ F γδνσ ∇ ω F ǫελτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ σ F γδνκ ∇ ω F ǫελτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ µ F δλκτ ∇ ω F ǫενσ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ τ F δµλκ ∇ ω F ǫενσ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ κ F δµνλ ∇ ω F ǫεστ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ κ F δνλω ∇ ω F ǫεστ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ν F δµλκ ∇ ω F ǫεστ + m F αβγ ǫ F αβγδ F εµνσ F λκτω ∇ σ F δεµν ∇ ω F ǫλκτ + m F αβγ ǫ F αβγδ F εµνσ F λκτω ∇ κ F δεµλ ∇ ω F ǫνστ + m F αβγ ǫ F αβγδ F εµνσ F λκτω ∇ λ F δεµν ∇ ω F ǫσκτ + m F αβγ ǫ F αβγδ F εµνσ F λκτω ∇ ν F δεµλ ∇ ω F ǫσκτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ ǫ F γδνσ ∇ ω F ελκτ +17 F αβǫε F αβγδ F γµνσ F λκτω ∇ ǫ F δµνσ ∇ ω F ελκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ǫ F δνσω ∇ ω F ελκτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ ǫ F δστω ∇ ω F ελκτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ σ F γδǫν ∇ ω F ελκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ σ F δǫµν ∇ ω F ελκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F δǫν ω ∇ ω F ελκτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ σ F δǫτω ∇ ω F ελκτ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ σ F δµν ω ∇ ω F ελκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ λ F γδνσ ∇ ω F εµκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ κ F γδνσ ∇ ω F εµλτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ σ F γδνκ ∇ ω F εµλτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ǫ F δλκτ ∇ ω F εµνσ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ τ F δǫλκ ∇ ω F εµνσ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ ν F γδκτ ∇ ω F εµσλ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ τ F γδνκ ∇ ω F εµσλ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ǫ F δµλκ ∇ ω F ενστ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ κ F δǫλω ∇ ω F ενστ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ κ F δǫµλ ∇ ω F ενστ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ µ F δǫλκ ∇ ω F ενστ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ǫ F δµνλ ∇ ω F εσκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ λ F δǫµν ∇ ω F εσκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ λ F δǫνω ∇ ω F εσκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ν F δǫλω ∇ ω F εσκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ν F δǫµλ ∇ ω F εσκτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ δ F γǫνκ ∇ ω F εσλτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ ǫ F γδνκ ∇ ω F εσλτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ κ F γδǫν ∇ ω F εσλτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ ε F γδνσ ∇ ω F µλκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ σ F γδεν ∇ ω F µλκτ + m F αǫεµ F αβγδ F β νσλ F γκτω ∇ σ F δǫεν ∇ ω F µλκτ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ λ F δεσω ∇ ω F µνκτ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ λ F ǫεσω ∇ ω F µνκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ ε F γδκτ ∇ ω F µνσλ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ τ F γδεκ ∇ ω F µνσλ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ε F δǫλκ ∇ ω F µνστ +18 F αǫεµ F αβγδ F βǫνσ F λκτω ∇ κ F γδελ ∇ ω F µνστ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ κ F δǫελ ∇ ω F µνστ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ κ F δελω ∇ ω F µνστ + m F αǫεµ F αβγδ F βǫνσ F λκτω ∇ ν F γδελ ∇ ω F µσκτ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ν F δελω ∇ ω F µσκτ + m F αǫεµ F αβγδ F β νσλ F γκτω ∇ δ F ǫενκ ∇ ω F µσλτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ ε F γδνκ ∇ ω F µσλτ + m F αǫεµ F αβγδ F β νσλ F γκτω ∇ ε F δǫνκ ∇ ω F µσλτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ κ F γδεν ∇ ω F µσλτ + m F αǫεµ F αβγδ F β νσλ F γκτω ∇ κ F δǫεν ∇ ω F µσλτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ ν F γδεκ ∇ ω F µσλτ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ µ F δεσω ∇ ω F νλκτ + m F αβǫε F αβγδ F γδµν F σλκτ ∇ µ F ǫεσω ∇ ω F νλκτ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ µ F ǫεσω ∇ ω F νλκτ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ σ F δεµω ∇ ω F νλκτ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ σ F δετω ∇ ω F νλκτ + m F αβǫε F αβγδ F γδµν F σλκτ ∇ σ F ǫεµω ∇ ω F νλκτ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ σ F ǫεµω ∇ ω F νλκτ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ σ F ǫµτω ∇ ω F νλκτ + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ σ F εµτω ∇ ω F νλκτ + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ σ F εµτω ∇ ω F νλκτ + m F αβγ ǫ F αβγδ F εµνσ F λκτω ∇ ǫ F δεµλ ∇ ω F νσκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ε F δǫλω ∇ ω F νσκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ε F δǫµλ ∇ ω F νσκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ λ F δǫεµ ∇ ω F νσκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ λ F δǫεω ∇ ω F νσκτ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ λ F δεµω ∇ ω F νσκτ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ λ F δεµω ∇ ω F νσκτ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ λ F ǫεµω ∇ ω F νσκτ + m F αǫεµ F αβγδ F βǫνσ F λκτω ∇ µ F γδελ ∇ ω F νσκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ µ F δǫελ ∇ ω F νσκτ + m F αβγ ǫ F αβγδ F εµνσ F λκτω ∇ ǫ F δεµν ∇ ω F σλκτ + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ ǫ F δµν ω ∇ ω F σλκτ + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ ǫ F δν τω ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ε F δǫµν ∇ ω F σλκτ +19 F αβǫε F αβγδ F γµνσ F µλκτ ∇ ε F δǫν ω ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ε F δµν ω ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ ε F δν τω ∇ ω F σλκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ µ F γδεν ∇ ω F σλκτ + m F αǫεµ F αβγδ F βǫνσ F λκτω ∇ µ F γδεν ∇ ω F σλκτ + m F αǫεµ F αβγδ F β νσλ F γκτω ∇ µ F δǫεν ∇ ω F σλκτ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ µ F δεν ω ∇ ω F σλκτ + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ µ F δν τω ∇ ω F σλκτ + m F αβǫε F αβγδ F γδǫµ F νσλκ ∇ µ F εντω ∇ ω F σλκτ + m F αβǫε F αβγδ F µκτω F µνσλ ∇ ν F γδǫε ∇ ω F σλκτ + m F αǫεµ F αβγδ F β νσλ F ǫκτω ∇ ν F γδεµ ∇ ω F σλκτ + m F αǫεµ F αβγδ F β νσλ F γκτω ∇ ν F δǫεµ ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F λκτω ∇ ν F δǫεµ ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ν F δǫεω ∇ ω F σλκτ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ν F δεµω ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ν F δεµω ∇ ω F σλκτ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ν F δεµω ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ ν F δετω ∇ ω F σλκτ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ ν F δετω ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ ν F ǫεµω ∇ ω F σλκτ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ν F ǫεµω ∇ ω F σλκτ + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ ν F ǫετω ∇ ω F σλκτ + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ ν F ǫκτω ∇ ω F σλκτ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ ν F ǫµτω ∇ ω F σλκτ + m F αβǫε F αβγδ F γǫµν F δµσλ ∇ ν F εκτω ∇ ω F σλκτ + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ ν F εµτω ∇ ω F σλκτ + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ ν F εµτω ∇ ω F σλκτ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ ω F σλκτ ∇ ω F γδǫε + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ λ F εκτω ∇ ω F γδǫσ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ τ F ελκω ∇ ω F γδǫσ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ ω F ελκτ ∇ ω F γδǫσ + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ λ F µστω ∇ ω F γδεκ + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ τ F µσλω ∇ ω F γδεκ + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ ω F µσλτ ∇ ω F γδεκ + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ ω F σλκτ ∇ ω F γδεµ +20 F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ τ F µλκω ∇ ω F γδεσ + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ ω F µλκτ ∇ ω F γδεσ + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ ω F εµσλ ∇ ω F γδκτ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ τ F ǫελω ∇ ω F γδσκ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ ω F ǫελτ ∇ ω F γδσκ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ ω F ǫεκτ ∇ ω F γδσλ + m F αβǫε F αβγδ F µν κτ F µνσλ ∇ ω F δελτ ∇ ω F γǫσκ + m F αǫεµ F αβγδ F β νσλ F ǫνκτ ∇ ω F δµλτ ∇ ω F γεσκ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ τ F νσκω ∇ ω F δǫελ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F νσκτ ∇ ω F δǫελ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ τ F σλκω ∇ ω F δǫεν + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F σλκτ ∇ ω F δǫεν + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ κ F σλτω ∇ ω F δǫετ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ τ F σλκω ∇ ω F δǫετ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ ω F σλκτ ∇ ω F δǫετ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F εντω ∇ ω F δǫλκ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ τ F ενσω ∇ ω F δǫλκ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F ενστ ∇ ω F δǫλκ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ κ F εστω ∇ ω F δǫλτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ σ F εκτω ∇ ω F δǫλτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ τ F εσκω ∇ ω F δǫλτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ ω F εσκτ ∇ ω F δǫλτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F εκτω ∇ ω F δǫνλ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ τ F εσκω ∇ ω F δǫνλ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F εσκτ ∇ ω F δǫνλ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ τ F ελκω ∇ ω F δǫνσ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F ελκτ ∇ ω F δǫνσ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ κ F ελτω ∇ ω F δǫστ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ τ F ελκω ∇ ω F δǫστ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ ω F ελκτ ∇ ω F δǫστ + m F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ τ F σλκω ∇ ω F δεκτ + m F αβǫε F αβγδ F γǫµν F µνσλ ∇ τ F σλκω ∇ ω F δεκτ + m F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ ω F σλκτ ∇ ω F δεκτ + m F αβǫε F αβγδ F γǫµν F µνσλ ∇ ω F σλκτ ∇ ω F δεκτ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ σ F µντω ∇ ω F δελκ +21 F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ τ F µνσω ∇ ω F δελκ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ω F µνστ ∇ ω F δελκ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ τ F νσκω ∇ ω F δεµλ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ω F νσκτ ∇ ω F δεµλ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ω F νσκτ ∇ ω F δεµλ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ τ F σλκω ∇ ω F δεµν + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ τ F σλκω ∇ ω F δεµν + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ ω F σλκτ ∇ ω F δεµν + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ω F σλκτ ∇ ω F δεµν + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ω F σλκτ ∇ ω F δεµν + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ τ F νλκω ∇ ω F δεµσ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ω F νλκτ ∇ ω F δεµσ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ σ F µκτω ∇ ω F δενλ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ κ F σλτω ∇ ω F δεντ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ τ F σλκω ∇ ω F δεντ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ τ F σλκω ∇ ω F δεντ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ ω F σλκτ ∇ ω F δεντ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ ω F σλκτ ∇ ω F δεντ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ τ F µνκω ∇ ω F δεσλ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ω F µνκτ ∇ ω F δεσλ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ κ F νλτω ∇ ω F δεστ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ τ F νλκω ∇ ω F δεστ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ ω F νλκτ ∇ ω F δεστ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F ǫενω ∇ ω F δλκτ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ σ F εµνω ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ σ F εµνω ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F ǫενσ ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ω F εµνσ ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ σ F ǫετω ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ σ F εντω ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ τ F ǫεσω ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ τ F ενσω ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F µνλκ ∇ ω F ǫεστ ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ ω F ενστ ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ τ F εσκω ∇ ω F δλκτ +22 F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ ω F εσκτ ∇ ω F δλκτ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ σ F εντω ∇ ω F δµλκ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ τ F ενσω ∇ ω F δµλκ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ω F ενστ ∇ ω F δµλκ + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ σ F ǫκτω ∇ ω F δµνλ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ σ F εκτω ∇ ω F δµνλ + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ τ F ǫσκω ∇ ω F δµνλ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ τ F εσκω ∇ ω F δµνλ + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ ω F ǫσκτ ∇ ω F δµνλ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ω F εσκτ ∇ ω F δµνλ + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ τ F ǫλκω ∇ ω F δµνσ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ τ F ελκω ∇ ω F δµνσ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ τ F ελκω ∇ ω F δµνσ + m F αβγ ǫ F αβγδ F ελκτ F εµνσ ∇ ω F ǫλκτ ∇ ω F δµνσ + m F αβǫε F αβγδ F γµνσ F ǫλκτ ∇ ω F ελκτ ∇ ω F δµνσ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ω F ελκτ ∇ ω F δµνσ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ν F εκτω ∇ ω F δµσλ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ τ F ενκω ∇ ω F δµσλ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ω F ενκτ ∇ ω F δµσλ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ σ F ǫετω ∇ ω F δνλκ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ τ F ǫεσω ∇ ω F δνλκ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F ǫεστ ∇ ω F δνλκ + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ κ F ǫστω ∇ ω F δνλτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ κ F εστω ∇ ω F δνλτ + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ κ F µστω ∇ ω F δνλτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ σ F εκτω ∇ ω F δνλτ + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ τ F ǫσκω ∇ ω F δνλτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ τ F εσκω ∇ ω F δνλτ + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ τ F µσκω ∇ ω F δνλτ + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ ω F ǫσκτ ∇ ω F δνλτ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ ω F εσκτ ∇ ω F δνλτ + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ ω F µσκτ ∇ ω F δνλτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ τ F ǫεκω ∇ ω F δνσλ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ ω F ǫεκτ ∇ ω F δνσλ + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ κ F ǫλτω ∇ ω F δνστ +23 F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ κ F ελτω ∇ ω F δνστ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ κ F ελτω ∇ ω F δνστ + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ κ F µλτω ∇ ω F δνστ + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ τ F ǫλκω ∇ ω F δνστ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ τ F ελκω ∇ ω F δνστ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ τ F ελκω ∇ ω F δνστ + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ τ F µλκω ∇ ω F δνστ + m F αβγ ǫ F αβγδ F εµλκ F εµνσ ∇ ω F ǫλκτ ∇ ω F δνστ + m F αβǫε F αβγδ F γµνσ F ǫµλκ ∇ ω F ελκτ ∇ ω F δνστ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ ω F ελκτ ∇ ω F δνστ + m F αǫεµ F αβγδ F βǫνσ F γελκ ∇ ω F µλκτ ∇ ω F δνστ + m F αβǫε F αβγδ F γǫµν F µνσλ ∇ λ F εκτω ∇ ω F δσκτ + m F αβγ ǫ F αβγδ F εµν λ F εµνσ ∇ τ F ǫλκω ∇ ω F δσκτ + m F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ τ F ελκω ∇ ω F δσκτ + m F αβǫε F αβγδ F γǫµν F µνσλ ∇ τ F ελκω ∇ ω F δσκτ + m F αβγ ǫ F αβγδ F εµν λ F εµνσ ∇ ω F ǫλκτ ∇ ω F δσκτ + m F αβǫε F αβγδ F γµνσ F ǫµνλ ∇ ω F ελκτ ∇ ω F δσκτ + m F αβǫε F αβγδ F γǫµν F µνσλ ∇ ω F ελκτ ∇ ω F δσκτ + m F αβǫε F αβγδ F γǫµν F σλκτ ∇ ε F µντω ∇ ω F δσλκ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ ε F νκτω ∇ ω F δσλτ + m F αβǫε F αβγδ F γǫµν F µσλκ ∇ ν F εκτω ∇ ω F δσλτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ δ F νστω ∇ ω F ǫελκ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ ω F νσκτ ∇ ω F ǫεµλ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ τ F σλκω ∇ ω F ǫεµν + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ ω F σλκτ ∇ ω F ǫεµν + m F αβǫε F αβγδ F γδµν F σλκτ ∇ ω F σλκτ ∇ ω F ǫεµν + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ω F σλκτ ∇ ω F ǫεµν + m F αβǫε F αβγδ F γδµν F σλκτ ∇ τ F νλκω ∇ ω F ǫεµσ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ τ F νλκω ∇ ω F ǫεµσ + m F αβǫε F αβγδ F γδµν F σλκτ ∇ ω F νλκτ ∇ ω F ǫεµσ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ω F νλκτ ∇ ω F ǫεµσ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ δ F σκτω ∇ ω F ǫενλ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ δ F λκτω ∇ ω F ǫενσ + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ κ F σλτω ∇ ω F ǫεντ + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ τ F σλκω ∇ ω F ǫεντ +24 F αβǫε F αβγδ F γµνσ F δµλκ ∇ ω F σλκτ ∇ ω F ǫεντ + m F αβǫε F αβγδ F γδµν F σλκτ ∇ τ F µνκω ∇ ω F ǫεσλ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ τ F µνκω ∇ ω F ǫεσλ + m F αβǫε F αβγδ F γδµν F σλκτ ∇ ω F µνκτ ∇ ω F ǫεσλ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ω F µνκτ ∇ ω F ǫεσλ + m F αβγ ǫ F αβγδ F δεµν F εµνσ ∇ τ F σλκω ∇ ω F ǫλκτ + m F αβγ ǫ F αβγδ F δεµν F εµνσ ∇ ω F σλκτ ∇ ω F ǫλκτ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ σ F εκτω ∇ ω F ǫµνλ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ τ F εσκω ∇ ω F ǫµνλ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ ω F εσκτ ∇ ω F ǫµνλ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ τ F ελκω ∇ ω F ǫµνσ + m F αβǫε F αβγδ F γµνσ F δλκτ ∇ ω F ελκτ ∇ ω F ǫµνσ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ τ F σλκω ∇ ω F ǫµντ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ ω F σλκτ ∇ ω F ǫµντ + m F αβǫε F αβγδ F γδµν F σλκτ ∇ τ F ενκω ∇ ω F ǫµσλ + m F αβǫε F αβγδ F γδµν F σλκτ ∇ ω F ενκτ ∇ ω F ǫµσλ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ κ F νλτω ∇ ω F ǫµστ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ τ F νλκω ∇ ω F ǫµστ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ ω F νλκτ ∇ ω F ǫµστ + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ τ F σλκω ∇ ω F ǫνκτ + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ ω F σλκτ ∇ ω F ǫνκτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ δ F εστω ∇ ω F ǫνλκ + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ κ F εστω ∇ ω F ǫνλτ + m F αβǫε F αβγδ F γµνσ F µλκτ ∇ δ F εκτω ∇ ω F ǫνσλ + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ κ F ελτω ∇ ω F ǫνστ + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ τ F ελκω ∇ ω F ǫνστ + m F αβǫε F αβγδ F γµνσ F δµλκ ∇ ω F ελκτ ∇ ω F ǫνστ + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ λ F νκτω ∇ ω F ǫσκτ + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ τ F νλκω ∇ ω F ǫσκτ + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ ω F νλκτ ∇ ω F ǫσκτ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ τ F εµνω ∇ ω F ǫσλκ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ω F εµντ ∇ ω F ǫσλκ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ κ F µντω ∇ ω F ǫσλτ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ τ F µνκω ∇ ω F ǫσλτ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ ω F µνκτ ∇ ω F ǫσλτ +25 F αβ ǫε F αβγδ F γǫµν F δµν σ ∇ τ F σλκω ∇ ω F ελκτ + m F αβ ǫε F αβγδ F γǫµν F δµν σ ∇ ω F σλκτ ∇ ω F ελκτ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ δ F νστω ∇ ω F εµλκ + m F αǫεµ F αβγδ F βǫνσ F γ λκτ ∇ δ F σκτω ∇ ω F εµνλ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ǫ F λκτω ∇ ω F εµνσ + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ τ F σλκω ∇ ω F εµντ + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ τ F σλκω ∇ ω F εµντ + m F αβγ ǫ F αβγδ F δεµν F ǫσλκ ∇ τ F σλκω ∇ ω F εµντ + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ ω F σλκτ ∇ ω F εµντ + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ ω F σλκτ ∇ ω F εµντ + m F αβγ ǫ F αβγδ F δεµν F ǫσλκ ∇ ω F σλκτ ∇ ω F εµντ + m F αβγ ǫ F αβγδ F δεµν F σλκτ ∇ ǫ F νκτω ∇ ω F εµσλ + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ κ F νλτω ∇ ω F εµστ + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ κ F νλτω ∇ ω F εµστ + m F αβγ ǫ F αβγδ F δεµν F ǫσλκ ∇ κ F νλτω ∇ ω F εµστ + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ τ F νλκω ∇ ω F εµστ + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ τ F νλκω ∇ ω F εµστ + m F αβγ ǫ F αβγδ F δεµν F ǫσλκ ∇ τ F νλκω ∇ ω F εµστ + m F αβǫε F αβγδ F γǫµν F δσλκ ∇ ω F νλκτ ∇ ω F εµστ + m F αβǫε F αβγδ F γδµν F ǫσλκ ∇ ω F νλκτ ∇ ω F εµστ + m F αβγ ǫ F αβγδ F δεµν F ǫσλκ ∇ ω F νλκτ ∇ ω F εµστ + m F αβ ǫε F αβγδ F γǫµν F δµσλ ∇ τ F σλκω ∇ ω F ενκτ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ ∇ τ F σλκω ∇ ω F ενκτ + m F αβ ǫε F αβγδ F γǫµν F δµσλ ∇ ω F σλκτ ∇ ω F ενκτ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ ∇ ω F σλκτ ∇ ω F ενκτ + m F αβ ǫε F αβγδ F γδǫµ F νσλκ ∇ κ F µλτω ∇ ω F ενστ + m F αβ ǫε F αβγδ F γδǫµ F νσλκ ∇ τ F µλκω ∇ ω F ενστ + m F αβ ǫε F αβγδ F γδǫµ F νσλκ ∇ ω F µλκτ ∇ ω F ενστ + m F αβ ǫε F αβγδ F γǫµν F δµσλ ∇ λ F νκτω ∇ ω F εσκτ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ ∇ λ F νκτω ∇ ω F εσκτ + m F αβ ǫε F αβγδ F γǫµν F δµσλ ∇ τ F νλκω ∇ ω F εσκτ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ ∇ τ F νλκω ∇ ω F εσκτ + m F αβ ǫε F αβγδ F γǫµν F δµσλ ∇ ω F νλκτ ∇ ω F εσκτ + m F αβ ǫε F αβγδ F γδµν F ǫµσλ ∇ ω F νλκτ ∇ ω F εσκτ + m F αβ ǫε F αβγδ F γǫµν F δσλκ ∇ κ F µντω ∇ ω F εσλτ +26 F αβ ǫε F αβγδ F γδµν F ǫσλκ ∇ κ F µντω ∇ ω F εσλτ + m F αβ ǫε F αβγδ F γǫµν F δσλκ ∇ τ F µνκω ∇ ω F εσλτ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ ∇ τ F µνκω ∇ ω F εσλτ + m F αβ ǫε F αβγδ F γǫµν F δσλκ ∇ ω F µνκτ ∇ ω F εσλτ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ ∇ ω F µνκτ ∇ ω F εσλτ + m F αβ ǫε F αβγδ F γǫµν F δεσλ ∇ τ F σλκω ∇ ω F µν κτ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ ∇ τ F σλκω ∇ ω F µν κτ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ ∇ τ F σλκω ∇ ω F µν κτ + m F αβ ǫε F αβγδ F γδǫµ F ενσλ ∇ τ F σλκω ∇ ω F µν κτ + m F αβ ǫε F αβγδ F γδǫε F µνσλ ∇ τ F σλκω ∇ ω F µν κτ + m F αβ ǫε F αβγδ F γǫµν F δεσλ ∇ ω F σλκτ ∇ ω F µν κτ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ ∇ ω F σλκτ ∇ ω F µν κτ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ ∇ ω F σλκτ ∇ ω F µν κτ + m F αβ ǫε F αβγδ F γδǫµ F ενσλ ∇ ω F σλκτ ∇ ω F µν κτ + m F αβ ǫε F αβγδ F γδǫε F µνσλ ∇ ω F σλκτ ∇ ω F µν κτ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ ǫ F λκτω ∇ ω F µνστ + m F αβ ǫε F αβγδ F γǫµν F δσλκ ∇ ε F λκτω ∇ ω F µνστ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ ∇ ε F λκτω ∇ ω F µνστ + m F αβ ǫε F αβγδ F γǫµν F δεσλ ∇ τ F νλκω ∇ ω F µσκτ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ ∇ τ F νλκω ∇ ω F µσκτ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ ∇ τ F νλκω ∇ ω F µσκτ + m F αβ ǫε F αβγδ F γǫµν F δεσλ ∇ ω F νλκτ ∇ ω F µσκτ + m F αβ ǫε F αβγδ F γδµν F ǫεσλ ∇ ω F νλκτ ∇ ω F µσκτ + m F αβγ ǫ F αβγδ F δεµν F ǫεσλ ∇ ω F νλκτ ∇ ω F µσκτ + m F αβγ ǫ F αβγδ F δεµν F εσλκ ∇ ǫ F νκτω ∇ ω F µσλτ + m F αβ ǫε F αβγδ F γǫµν F δσλκ ∇ ε F νκτω ∇ ω F µσλτ + m F αβ ǫε F αβγδ F γδµν F ǫσλκ ∇ ε F νκτω ∇ ω F µσλτ + m F αβ ǫε F αβγδ F γǫµν F δεµσ ∇ τ F σλκω ∇ ω F ν λκτ + m F αβ ǫε F αβγδ F γδµν F ǫεµσ ∇ τ F σλκω ∇ ω F ν λκτ + m F αβγ ǫ F αβγδ F δεµν F ǫεµσ ∇ τ F σλκω ∇ ω F ν λκτ + m F αβ ǫε F αβγδ F γǫµν F δεµσ ∇ ω F σλκτ ∇ ω F ν λκτ + m F αβ ǫε F αβγδ F γδµν F ǫεµσ ∇ ω F σλκτ ∇ ω F ν λκτ + m F αβγ ǫ F αβγδ F δεµν F ǫεµσ ∇ ω F σλκτ ∇ ω F ν λκτ + m F αβγ ǫ F αβγδ F δεµν F εµσλ ∇ ǫ F λκτω ∇ ω F νσκτ + m F αβ ǫε F αβγδ F γǫµν F δµσλ ∇ ε F λκτω ∇ ω F νσκτ +27 F αβ ǫε F αβγδ F γδǫµ F ενσλ ∇ µ F λκτω ∇ ω F νσκτ + m F αβ ǫε F αβγδ F γǫµν F δεµν ∇ ω F σλκτ ∇ ω F σλκτ + m F αβ ǫε F αβγδ F γδµν F ǫεµν ∇ ω F σλκτ ∇ ω F σλκτ + m F αβγ ǫ F αβγδ F δεµν F ǫεµν ∇ ω F σλκτ ∇ ω F σλκτ + m F ǫσλκ F ǫεµν F F ∇ κ F νλτω ∇ σ F εµτω + m F ǫεσλ F ǫεµν F F ∇ λ F νκτω ∇ σ F µκτω + m F ǫεµν F σλκτ F F ∇ λ F ǫεσω ∇ τ F µνκω + m F ǫεµν F σλκτ F F ∇ σ F ǫεµω ∇ τ F νλκω + m F ǫεµν F σλκτ F F ∇ ω F σλκτ ∇ ω F ǫεµν + m F ǫεµν F σλκτ F F ∇ τ F νλκω ∇ ω F ǫεµσ + m F ǫεµν F σλκτ F F ∇ ω F νλκτ ∇ ω F ǫεµσ + m F ǫεµν F σλκτ F F ∇ ω F µνκτ ∇ ω F ǫεσλ + m F ǫσλκ F ǫεµν F F ∇ τ F σλκω ∇ ω F εµντ + m F ǫσλκ F ǫεµν F F ∇ ω F σλκτ ∇ ω F εµντ + m F ǫσλκ F ǫεµν F F ∇ κ F νλτω ∇ ω F εµστ + m F ǫσλκ F ǫεµν F F ∇ τ F νλκω ∇ ω F εµστ + m F ǫσλκ F ǫεµν F F ∇ ω F νλκτ ∇ ω F εµστ + m F ǫεσλ F ǫεµν F F ∇ τ F σλκω ∇ ω F µν κτ + m F ǫεσλ F ǫεµν F F ∇ ω F σλκτ ∇ ω F µν κτ + m F ǫεσλ F ǫεµν F F ∇ τ F νλκω ∇ ω F µσκτ + m F ǫεσλ F ǫεµν F F ∇ ω F νλκτ ∇ ω F µσκτ + m F ǫεµσ F ǫεµν F F ∇ τ F σλκω ∇ ω F ν λκτ + m F ǫεµσ F ǫεµν F F ∇ ω F σλκτ ∇ ω F ν λκτ + m F F ∇ ω F σλκτ ∇ ω F σλκτ (20)There are 217 couplings with structure of one Riemann curvature, two F and two ∇ F , i.e., L RF ( ∂F ) = m F ǫσλκ F ǫεµν R αβγδ ∇ γ F αεµσ ∇ δ F βνλκ + m F ǫεσλ F ǫεµν R αβγδ ∇ γ F αµσκ ∇ δ F βνλκ + m F ǫσλκ F ǫεµν R αβγδ ∇ γ F αεµν ∇ δ F βσλκ + m F ǫεσλ F ǫεµν R αβγδ ∇ γ F αµν κ ∇ δ F βσλκ + m F ǫεµσ F ǫεµν R αβγδ ∇ γ F αν λκ ∇ δ F βσλκ + m F αǫεµ F νσλκ R αβγδ ∇ ǫ F βνσλ ∇ κ F γδεµ + m F αǫεµ F νσλκ R αβγδ ∇ ε F βǫνσ ∇ κ F γδµλ + m F αǫεµ F νσλκ R αβγδ ∇ γ F βνσλ ∇ κ F δǫεµ + m F αǫεµ F νσλκ R αβγδ ∇ γ F βǫνσ ∇ κ F δεµλ +28 F αǫεµ F νσλκ R αβγδ ∇ ǫ F βγνσ ∇ κ F δεµλ + m F ǫσλκ F ǫεµν R αβγδ ∇ γ F αβεσ ∇ κ F δµνλ + m F αǫεµ F νσλκ R αβγδ ∇ γ F βǫεν ∇ κ F δµσλ + m F αǫεµ F νσλκ R αβγδ ∇ ε F βγǫν ∇ κ F δµσλ + m F ǫσλκ F ǫεµν R αβγδ ∇ γ F αβεµ ∇ κ F δνσλ + m F ǫεσλ F ǫεµν R αβγδ ∇ γ F αβµκ ∇ κ F δνσλ + m F αǫεµ F νσλκ R αβγδ ∇ γ F βǫεµ ∇ κ F δνσλ + m F αǫεµ F ǫνσλ R αβγδ ∇ γ F βεµκ ∇ κ F δνσλ + m F αǫεµ F ǫενσ R αβγδ ∇ γ F βµλκ ∇ κ F δνσλ + m F αǫεµ F νσλκ R αβγδ ∇ δ F βγνσ ∇ κ F ǫεµλ + m F αǫεµ F νσλκ R αβγδ ∇ δ F βγǫν ∇ κ F εµσλ + m F αǫεµ F ǫνσλ R αβγδ ∇ δ F βγνκ ∇ κ F εµσλ + m F αβ ǫε F µνσλ R αβγδ ∇ ǫ F γδµκ ∇ κ F ενσλ + m F αǫεµ F νσλκ R αβγδ ∇ δ F βγǫε ∇ κ F µνσλ + m F αǫεµ F ǫνσλ R αβγδ ∇ δ F βγεκ ∇ κ F µνσλ + m F αǫεµ F ǫενσ R αβγδ ∇ δ F βγ λκ ∇ κ F µνσλ + m F αǫεµ F γ νσλ R αβγδ ∇ δ F βǫεκ ∇ κ F µνσλ + m F αǫεµ F γǫνσ R αβγδ ∇ δ F βελκ ∇ κ F µνσλ + m F αǫεµ F β νσλ R αβγδ ∇ ε F γδǫκ ∇ κ F µνσλ + m F αβ ǫε F µνσλ R αβγδ ∇ ε F γδǫκ ∇ κ F µνσλ + m F αǫεµ F βǫνσ R αβγδ ∇ ε F γδλκ ∇ κ F µνσλ + m F αβ ǫε F ǫµνσ R αβγδ ∇ ε F γδλκ ∇ κ F µνσλ + m F αβ ǫε F γµνσ R αβγδ ∇ ε F δǫλκ ∇ κ F µνσλ + m F αβ ǫε F γǫµν R αβγδ ∇ ε F δσλκ ∇ κ F µνσλ + m F ǫεσλ F ǫεµν R αβγδ ∇ κ F βδσλ ∇ κ F αγµν + m F ǫεσλ F ǫεµν R αβγδ ∇ κ F βδνλ ∇ κ F αγµσ + m F ǫεµσ F ǫεµν R αβγδ ∇ κ F βδσλ ∇ κ F αγν λ + m F αǫεµ F γ νσλ R αβγδ ∇ κ F δεµλ ∇ κ F βǫνσ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F γδσλ ∇ κ F βεµν + m F αǫεµ F γǫνσ R αβγδ ∇ κ F δµσλ ∇ κ F βενλ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F γδµλ ∇ κ F βενσ + m F αǫεµ F ǫενσ R αβγδ ∇ κ F γδσλ ∇ κ F βµν λ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F γδεµ ∇ κ F βνσλ + m F αǫεµ F γ νσλ R αβγδ ∇ κ F δǫεµ ∇ κ F βνσλ + m F αǫεµ F ǫενσ R αβγδ ∇ κ F γδµλ ∇ κ F βνσλ +29 F αǫεµ F γǫνσ R αβγδ ∇ κ F δεµλ ∇ κ F βνσλ + m F αǫεµ F ǫεµν R αβγδ ∇ κ F γδσλ ∇ κ F βνσλ + m F αǫεµ F γǫεν R αβγδ ∇ κ F δµσλ ∇ κ F βνσλ + m F αǫεµ F γǫεµ R αβγδ ∇ κ F δνσλ ∇ κ F βνσλ + m F αǫεµ F β νσλ R αβγδ ∇ κ F µνσλ ∇ κ F γδǫε + m F αβ ǫε F µνσλ R αβγδ ∇ κ F µνσλ ∇ κ F γδǫε + m F αβ ǫε F µνσλ R αβγδ ∇ κ F ενσλ ∇ κ F γδǫµ + m F αǫεµ F β νσλ R αβγδ ∇ κ F εµσλ ∇ κ F γδǫν + m F αǫεµ F βǫνσ R αβγδ ∇ κ F µνσλ ∇ κ F γδελ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F µνσλ ∇ κ F γδελ + m F αǫεµ F ǫνσλ R αβγδ ∇ β F νσλκ ∇ κ F γδεµ + m F αǫεµ F ǫνσλ R αβγδ ∇ β F µσλκ ∇ κ F γδεν + m F αǫεµ F ǫενσ R αβγδ ∇ β F νσλκ ∇ κ F γδµλ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F ενσλ ∇ κ F γδµλ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F ǫεσλ ∇ κ F γδµν + m F αǫεµ F ǫενσ R αβγδ ∇ β F µσλκ ∇ κ F γδν λ + m F αǫεµ F ǫνσλ R αβγδ ∇ β F εµλκ ∇ κ F γδνσ + m F αǫεµ F ǫεµν R αβγδ ∇ β F νσλκ ∇ κ F γδσλ + m F αǫεµ F βǫεν R αβγδ ∇ κ F µνσλ ∇ κ F γδσλ + m F αβ ǫε F ǫεµν R αβγδ ∇ κ F µνσλ ∇ κ F γδσλ + m F αǫεµ F β νσλ R αβγδ ∇ κ F δνσλ ∇ κ F γǫεµ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F δνσλ ∇ κ F γǫεµ + m F αǫεµ F β νσλ R αβγδ ∇ κ F δµσλ ∇ κ F γǫεν + m F αβ ǫε F µνσλ R αβγδ ∇ ε F δσλκ ∇ κ F γǫµν + m F αβ ǫε F µνσλ R αβγδ ∇ κ F δεσλ ∇ κ F γǫµν + m F αǫεµ F βǫνσ R αβγδ ∇ κ F δνσλ ∇ κ F γεµλ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F δνσλ ∇ κ F γεµλ + m F αǫεµ F ǫνσλ R αβγδ ∇ β F δσλκ ∇ κ F γεµν + m F αǫεµ F ǫνσλ R αβγδ ∇ β F δµλκ ∇ κ F γενσ + m F αǫεµ F ǫενσ R αβγδ ∇ β F δσλκ ∇ κ F γµν λ + m F αβ ǫε F ǫµνσ R αβγδ ∇ δ F εσλκ ∇ κ F γµν λ + m F αβ ǫε F ǫµνσ R αβγδ ∇ ε F δσλκ ∇ κ F γµν λ + m F αβ ǫε F µνσλ R αβγδ ∇ δ F ǫελκ ∇ κ F γµνσ + m F αǫεµ F βǫεν R αβγδ ∇ κ F δνσλ ∇ κ F γµσλ + m F αβ ǫε F ǫεµν R αβγδ ∇ κ F δνσλ ∇ κ F γµσλ +30 F αǫεµ F ǫεµν R αβγδ ∇ β F δσλκ ∇ κ F γν σλ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F µνσλ ∇ κ F δǫελ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F ενσλ ∇ κ F δǫµλ + m F αβ ǫε F γǫµν R αβγδ ∇ κ F µνσλ ∇ κ F δεσλ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F ǫεσλ ∇ κ F δµν λ + m F αβ ǫε F γǫµν R αβγδ ∇ κ F ενσλ ∇ κ F δµσλ + m F αβ ǫε F γǫεµ R αβγδ ∇ κ F µνσλ ∇ κ F δνσλ + m F αβ ǫε F γµνσ R αβγδ ∇ δ F νσλκ ∇ κ F ǫεµλ + m F αβ ǫε F γδµν R αβγδ ∇ κ F µνσλ ∇ κ F ǫεσλ + m F αβ ǫε F γµνσ R αβγδ ∇ δ F εσλκ ∇ κ F ǫµν λ + m F αβ ǫε F γδµν R αβγδ ∇ κ F ενσλ ∇ κ F ǫµσλ + m F αβ ǫε F γǫµν R αβγδ ∇ δ F νσλκ ∇ κ F εµσλ + m F αβ ǫε F γδǫµ R αβγδ ∇ κ F µνσλ ∇ κ F ενσλ + m F αβ ǫε F γǫεµ R αβγδ ∇ δ F νσλκ ∇ κ F µνσλ + m F αβ ǫε F γδǫε R αβγδ ∇ κ F µνσλ ∇ κ F µνσλ + m F αǫεµ F νσλκ R αβγδ ∇ κ F δǫεµ ∇ λ F βγνσ + m F ǫεσλ F ǫεµν R αβγδ ∇ κ F αγµσ ∇ λ F βδνκ + m F ǫεσλ F ǫεµν R αβγδ ∇ κ F αγµν ∇ λ F βδσκ + m F ǫεµσ F ǫεµν R αβγδ ∇ κ F αγν λ ∇ λ F βδσκ + m F αǫεµ F ǫνσλ R αβγδ ∇ ε F βνσκ ∇ λ F γδµκ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F βενσ ∇ λ F γδµκ + m F αǫεµ F ǫενσ R αβγδ ∇ κ F βνσλ ∇ λ F γδµκ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F βεµν ∇ λ F γδσκ + m F αǫεµ F ǫενσ R αβγδ ∇ κ F βµν λ ∇ λ F γδσκ + m F αǫεµ F ǫεµν R αβγδ ∇ κ F βνσλ ∇ λ F γδσκ + m F αǫεµ F ǫνσλ R αβγδ ∇ γ F βνσκ ∇ λ F δεµκ + m F αǫεµ F γ νσλ R αβγδ ∇ ǫ F βνσκ ∇ λ F δεµκ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F βγνσ ∇ λ F δεµκ + m F αǫεµ F γ νσλ R αβγδ ∇ κ F βǫνσ ∇ λ F δεµκ + m F αǫεµ F γǫνσ R αβγδ ∇ κ F βνσλ ∇ λ F δεµκ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F γǫµν ∇ λ F δεσκ + m F αǫεµ F ǫνσλ R αβγδ ∇ γ F βενκ ∇ λ F δµσκ + m F αǫεµ F ǫνσλ R αβγδ ∇ ε F βγν κ ∇ λ F δµσκ + m F αǫεµ F γ νσλ R αβγδ ∇ ε F βǫνκ ∇ λ F δµσκ + m F αǫεµ F β νσλ R αβγδ ∇ ε F γǫνκ ∇ λ F δµσκ +31 F αǫεµ F ǫνσλ R αβγδ ∇ κ F βγεν ∇ λ F δµσκ + m F αǫεµ F γǫνσ R αβγδ ∇ κ F βενλ ∇ λ F δµσκ + m F αǫεµ F γǫεν R αβγδ ∇ κ F βν σλ ∇ λ F δµσκ + m F αǫεµ F β νσλ R αβγδ ∇ κ F γǫεν ∇ λ F δµσκ + m F ǫεσλ F ǫεµν R αβγδ ∇ γ F αβµκ ∇ λ F δνσκ + m F αǫεµ F ǫνσλ R αβγδ ∇ γ F βεµκ ∇ λ F δνσκ + m F αǫεµ F γǫεµ R αβγδ ∇ κ F βνσλ ∇ λ F δνσκ + m F αǫεµ F β νσλ R αβγδ ∇ κ F γǫεµ ∇ λ F δνσκ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F γǫεµ ∇ λ F δνσκ + m F αǫεµ F βǫνσ R αβγδ ∇ κ F γεµλ ∇ λ F δνσκ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F γεµλ ∇ λ F δνσκ + m F αǫεµ F βǫεν R αβγδ ∇ κ F γµσλ ∇ λ F δνσκ + m F αβ ǫε F ǫεµν R αβγδ ∇ κ F γµσλ ∇ λ F δνσκ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F γδµν ∇ λ F ǫεσκ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F δµν λ ∇ λ F ǫεσκ + m F αǫεµ F ǫνσλ R αβγδ ∇ δ F βγνκ ∇ λ F εµσκ + m F αǫεµ F γ νσλ R αβγδ ∇ δ F βǫνκ ∇ λ F εµσκ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F γδǫµ ∇ λ F ενσκ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F γδµλ ∇ λ F ενσκ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F δǫµλ ∇ λ F ενσκ + m F αβ ǫε F γǫµν R αβγδ ∇ κ F δµσλ ∇ λ F ενσκ + m F αβ ǫε F γδµν R αβγδ ∇ κ F ǫµσλ ∇ λ F ενσκ + m F αǫεµ F ǫνσλ R αβγδ ∇ δ F βγεκ ∇ λ F µνσκ + m F αǫεµ F γ νσλ R αβγδ ∇ δ F βǫεκ ∇ λ F µνσκ + m F αǫεµ F β νσλ R αβγδ ∇ κ F γδǫε ∇ λ F µνσκ + m F αǫεµ F βǫνσ R αβγδ ∇ κ F γδελ ∇ λ F µνσκ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F γδελ ∇ λ F µνσκ + m F αǫεµ F βǫεν R αβγδ ∇ κ F γδσλ ∇ λ F µνσκ + m F αβ ǫε F ǫεµν R αβγδ ∇ κ F γδσλ ∇ λ F µνσκ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F δǫελ ∇ λ F µνσκ + m F αβ ǫε F γǫµν R αβγδ ∇ κ F δεσλ ∇ λ F µνσκ + m F αβ ǫε F γδµν R αβγδ ∇ κ F ǫεσλ ∇ λ F µνσκ + m F αβ ǫε F γδǫµ R αβγδ ∇ κ F ενσλ ∇ λ F µνσκ + m F ǫσλκ F ǫεµν R αβγδ ∇ κ F γδνλ ∇ µ F αβεσ + m F αǫεµ F νσλκ R αβγδ ∇ κ F δνσλ ∇ µ F βγǫε +32 F αǫεµ F ǫνσλ R αβγδ ∇ λ F δνσκ ∇ µ F βγεκ + m F αǫεµ F γ νσλ R αβγδ ∇ λ F δνσκ ∇ µ F βǫεκ + m F αǫεµ F νσλκ R αβγδ ∇ κ F γδσλ ∇ µ F βǫεν + m F αǫεµ F ǫνσλ R αβγδ ∇ λ F γδσκ ∇ µ F βενκ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F ενσλ ∇ µ F γδǫκ + m F αβ ǫε F µνσλ R αβγδ ∇ λ F ενσκ ∇ µ F γδǫκ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F βνσλ ∇ µ F γδεκ + m F αǫεµ F ǫνσλ R αβγδ ∇ κ F βενσ ∇ µ F γδλκ + m F αǫεµ F ǫενσ R αβγδ ∇ κ F βνσλ ∇ µ F γδλκ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F ενσλ ∇ µ F γδλκ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F δνσλ ∇ µ F γǫεκ + m F αǫεµ F β νσλ R αβγδ ∇ λ F δνσκ ∇ µ F γǫεκ + m F αβ ǫε F µνσλ R αβγδ ∇ λ F δνσκ ∇ µ F γǫεκ + m F αǫεµ F γ νσλ R αβγδ ∇ κ F βνσλ ∇ µ F δǫεκ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F ενσλ ∇ µ F δǫλκ + m F αǫεµ F νσλκ R αβγδ ∇ γ F βǫνσ ∇ µ F δελκ + m F αǫεµ F γ νσλ R αβγδ ∇ κ F βǫνσ ∇ µ F δελκ + m F αǫεµ F γǫνσ R αβγδ ∇ κ F βνσλ ∇ µ F δελκ + m F αǫεµ F νσλκ R αβγδ ∇ γ F βǫεν ∇ µ F δσλκ + m F αǫεµ F ǫνσλ R αβγδ ∇ γ F βενκ ∇ µ F δσλκ + m F αǫεµ F ǫενσ R αβγδ ∇ γ F βνλκ ∇ µ F δσλκ + m F αǫεµ F γǫεν R αβγδ ∇ κ F βν σλ ∇ µ F δσλκ + m F αǫεµ F β νσλ R αβγδ ∇ κ F γǫεν ∇ µ F δσλκ + m F ǫσλκ F ǫεµν R αβγδ ∇ κ F γδσλ ∇ ν F αβεµ + m F ǫεσλ F ǫεµν R αβγδ ∇ λ F γδσκ ∇ ν F αβµκ + m F αǫεµ F νσλκ R αβγδ ∇ κ F δµσλ ∇ ν F βγǫε + m F αǫεµ F ǫνσλ R αβγδ ∇ λ F δµσκ ∇ ν F βγεκ + m F αǫεµ F β νσλ R αβγδ ∇ κ F εµσλ ∇ ν F γδǫκ + m F αǫεµ F β νσλ R αβγδ ∇ λ F εµσκ ∇ ν F γδǫκ + m F αβ ǫε F µνσλ R αβγδ ∇ κ F ǫεσλ ∇ ν F γδµκ + m F αβ ǫε F µνσλ R αβγδ ∇ λ F ǫεσκ ∇ ν F γδµκ + m F αǫεµ F β νσλ R αβγδ ∇ λ F δµσκ ∇ ν F γǫεκ + m F αβ ǫε F µνσλ R αβγδ ∇ λ F δεσκ ∇ ν F γǫµκ + m F ǫεσλ F ǫεµν R αβγδ ∇ γ F αβµκ ∇ ν F δσλκ + m F αǫεµ F ǫεµν R αβγδ ∇ γ F β σλκ ∇ ν F δσλκ +33 F αǫεµ F βǫεν R αβγδ ∇ κ F γµσλ ∇ ν F δσλκ + m F αβ ǫε F ǫεµν R αβγδ ∇ κ F γµσλ ∇ ν F δσλκ + m F αǫεµ F βǫεν R αβγδ ∇ µ F γ σλκ ∇ ν F δσλκ + m F αβ ǫε F ǫεµν R αβγδ ∇ µ F γ σλκ ∇ ν F δσλκ + m F αβ ǫε F γǫµν R αβγδ ∇ κ F δµσλ ∇ ν F εσλκ + m F αǫεµ F γǫεν R αβγδ ∇ δ F βσλκ ∇ ν F µσλκ + m F ǫσλκ F ǫεµν R αβγδ ∇ κ F γδνλ ∇ σ F αβεµ + m F αǫεµ F νσλκ R αβγδ ∇ κ F δεµλ ∇ σ F βγǫν + m F ǫεµσ F ǫεµν R αβγδ ∇ κ F αγν λ ∇ σ F βδλκ + m F αǫεµ F ǫενσ R αβγδ ∇ κ F βµν λ ∇ σ F γδλκ + m F αǫεµ F ǫενσ R αβγδ ∇ µ F βνλκ ∇ σ F γδλκ + m F ǫεµσ F ǫεµν R αβγδ ∇ ν F αβλκ ∇ σ F γδλκ + m F αǫεµ F ǫενσ R αβγδ ∇ γ F βνλκ ∇ σ F δµλκ + m F αǫεµ F γǫνσ R αβγδ ∇ ε F βνλκ ∇ σ F δµλκ + m F αǫεµ F ǫενσ R αβγδ ∇ κ F βγν λ ∇ σ F δµλκ + m F αǫεµ F γǫνσ R αβγδ ∇ κ F βενλ ∇ σ F δµλκ + m F αǫεµ F βǫνσ R αβγδ ∇ κ F γεν λ ∇ σ F δµλκ + m F ǫεµσ F ǫεµν R αβγδ ∇ γ F αβ λκ ∇ σ F δνλκ + m F αǫεµ F ǫενσ R αβγδ ∇ γ F βµλκ ∇ σ F δνλκ + m F αǫεµ F βǫνσ R αβγδ ∇ κ F γεµλ ∇ σ F δνλκ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F γεµλ ∇ σ F δνλκ + m F αǫεµ F ǫενσ R αβγδ ∇ µ F βγλκ ∇ σ F δνλκ + m F αǫεµ F βǫνσ R αβγδ ∇ µ F γελκ ∇ σ F δνλκ + m F αβ ǫε F ǫµνσ R αβγδ ∇ µ F γελκ ∇ σ F δνλκ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F δµν λ ∇ σ F ǫελκ + m F αβ ǫε F ǫµνσ R αβγδ ∇ κ F γδµλ ∇ σ F ενλκ + m F αβ ǫε F γµνσ R αβγδ ∇ κ F δǫµλ ∇ σ F ενλκ + m F αǫεµ F ǫενσ R αβγδ ∇ δ F βγ λκ ∇ σ F µνλκ + m F αǫεµ F γǫνσ R αβγδ ∇ δ F βελκ ∇ σ F µνλκ + m F αǫεµ F βǫνσ R αβγδ ∇ κ F γδελ ∇ σ F µνλκ + m F F R αβγδ ∇ γ F ασλκ ∇ δ F βσλκ + m F F R αβγδ ∇ κ F βδσλ ∇ κ F αγ σλ + m F F R αβγδ ∇ κ F αγ σλ ∇ λ F βδσκ (21)34here are 24 couplings with structure of two Riemann curvatures and two ∇ F , i.e., L R ( ∂F ) = m R αǫγε R αβγδ ∇ ǫ F β µνσ ∇ ε F δµνσ + m R αβ ǫε R αβγδ ∇ ǫ F γµνσ ∇ ε F δµνσ + m R αǫγε R αβγδ ∇ δ F β µνσ ∇ ε F ǫµνσ + m R αǫεµ R αβγδ ∇ γ F βενσ ∇ µ F δǫνσ + m R αǫεµ R αβγδ ∇ γ F βǫνσ ∇ µ F δενσ + m R αǫεµ R αβγδ ∇ ǫ F βγνσ ∇ µ F δενσ + m R αǫεµ R αβγδ ∇ δ F βγ νσ ∇ µ F ǫενσ + m R αβγδ R ǫεµν ∇ µ F αβǫσ ∇ ν F γδεσ + m R αβγδ R ǫεµν ∇ ε F αβǫσ ∇ ν F γδµσ + m R αβγδ R ǫεµν ∇ γ F αβǫσ ∇ ν F δεµσ + m R αβγδ R ǫεµν ∇ σ F βδεν ∇ σ F αγǫµ + m R αǫεµ R αβγδ ∇ µ F δǫνσ ∇ σ F βγεν + m R αǫεµ R αβγδ ∇ ǫ F γδνσ ∇ σ F βεµν + m R αǫεµ R αβγδ ∇ ν F γδǫσ ∇ σ F βεµν + m R αǫεµ R αβγδ ∇ σ F γδǫν ∇ σ F βεµν + m R αǫγ ε R αβγδ ∇ ν F δǫµσ ∇ σ F βεµν + m R αǫγε R αβγδ ∇ σ F δǫµν ∇ σ F βεµν + m R αβ ǫε R αβγδ ∇ ν F ǫεµσ ∇ σ F γδµν + m R αβ ǫε R αβγδ ∇ σ F ǫεµν ∇ σ F γδµν + m R αβ ǫε R αβγδ ∇ ν F δεµσ ∇ σ F γǫµν + m R αβ ǫε R αβγδ ∇ σ F δεµν ∇ σ F γǫµν + m R αγβ ǫ R αβγδ ∇ ν F ǫεµσ ∇ σ F δεµν + m R αγβ ǫ R αβγδ ∇ σ F ǫεµν ∇ σ F δεµν + m R αγβδ R αβγδ ∇ σ F ǫεµν ∇ σ F ǫεµν (22)Finally, there are 15 couplings with structure of four ∇ F , i.e., L ( ∂F ) = m ∇ ǫ F αβ εµ ∇ ǫ F αβγδ ∇ λ F εµνσ ∇ λ F γδνσ + m ∇ ǫ F αβ εµ ∇ ǫ F αβγδ ∇ λ F δµνσ ∇ λ F γενσ + m ∇ ǫ F αβγ ε ∇ ǫ F αβγδ ∇ λ F εµνσ ∇ λ F δµνσ + m ∇ ǫ F αβγδ ∇ ε F αβγδ ∇ λ F εµνσ ∇ λ F ǫµνσ + m ∇ δ F µνσλ ∇ ǫ F αβγδ ∇ ε F αβγǫ ∇ λ F εµνσ + m ∇ ǫ F αβγδ ∇ ǫ F αβγδ ∇ λ F εµνσ ∇ λ F εµνσ + m ∇ ǫ F αβγδ ∇ λ F εµνσ ∇ λ F δǫνσ ∇ µ F αβγ ε + m ∇ ǫ F αβγδ ∇ λ F ǫµνσ ∇ λ F δενσ ∇ µ F αβγ ε + m ∇ ǫ F αβγδ ∇ ε F ǫνσλ ∇ λ F δµνσ ∇ µ F αβγ ε + m ∇ ǫ F αβγδ ∇ λ F ǫενσ ∇ λ F δµνσ ∇ µ F αβγ ε + m ∇ ǫ F δνσλ ∇ ǫ F αβγδ ∇ µ F ενσλ ∇ µ F αβγ ε + m ∇ ǫ F αβγδ ∇ λ F ǫεµσ ∇ λ F γδν σ ∇ ν F αβ εµ + m ∇ ǫ F αβγδ ∇ λ F γǫνσ ∇ µ F δεσλ ∇ ν F αβ εµ + m ∇ ǫ F αβγδ ∇ λ F γǫεσ ∇ ν F αβεµ ∇ σ F δµνλ + m ∇ ǫ F αβγδ ∇ λ F δµνσ ∇ µ F αβγ ε ∇ σ F ǫενλ (23)Even though the total number of minimal gauge invariant couplings at order ℓ p are fixed, i.e., L F areinvariant under field redefinition, Bianchi identities and total derivative terms. They are schemeindependent. All other couplings dependent on the scheme that one uses for the couplings. Thevalues of the 1062 parameters may be fixed by various techniques in M-theory.They may be fixed by reducing the couplings on a circle to produce the type IIA couplingsat one-loop. Then one may find the 1062 parameters by calculating various S-matrix elementsin the resulting type IIA effective field theory and comparing them with the corresponding S-matrix elements in the type IIA superstring theory which has no arbitrary parameters. In thismethod one has to calculate in the string theory various S-matrix elements which produces 1062independent contact terms. In the next section we briefly discuss the dimensional reduction ofthe couplings on a circle to fix some of the parameters.35 Reduction on a circle
To reduce the 11-dimensional couplings to the 10-dimensional couplings, one first uses theKaluza-Kelin (KK) reduction of the metric, i.e., g µν = e − / (cid:18) G ab + e C a C b e C a e C b e (cid:19) ; g µν = e / (cid:18) G ab − C a − C b e − + C a C a (cid:19) (24)where G ab is the inverse of the 10-dimensional metric which raises the index of the the R-Rvector C a , and the following reductions for the three-form: A abc = C abc ; A aby = B ab (25)where C (3) is the R-R three-form and B is the antisymmetric B -field of the type IIA superstringtheory. Using these reduction one can calculate the reduction of different 11-dimensional cou-plings to the 10 dimensions, e.g., the reduction of the overall factor √− g and the Ricci scalarin S are √− g = e − / √− GR = e / (cid:18) R − ∇ a Φ ∇ a Φ + 143 ∇ a ∇ a Φ − . e F ab F ab (cid:19) (26)where F ac if field strength of the R-R one-form. Up to a total derivative term they produce thestandard reduction, i.e., √− gR = e − √− G (cid:18) R + 4 ∇ a Φ ∇ a Φ − . e F ab F ab (cid:19) (27)The reduction of the coupling in the action S involving the field strength of the three-form is − . √− gF µναβ F µναβ = e − √− G (cid:18) − . H abc H abc − . e ¯ F abcd ¯ F abcd (cid:19) (28)where the R-R field strength ¯ F (4) is ¯ F abcd = F abcd + H [ abc C d ] . Note that the dilaton factorindicates that the reduction of S correspond to the sphere-level effective action of type IIA.There are stringy corrections to the sphere-level effective action of type IIA which are relatedto the non-zero modes of the KK mass spectrum [5].Using the relation between type IIA coupling g s , the string length ℓ s and the 11-dimensionalPlank length ℓ p , i.e., ℓ p = g / s ℓ s , and the fact that the dilaton factor in the n h -loop effectiveaction of type IIA superstring theory is given by e − (2 − n h )Φ , one finds the relation 6 n h = n between the derivative couplings in the M-theory at the level ℓ np , and the n h -loop couplings in thetype IIA theory. Then the allowed couplings in the ℓ p -expansion are at n = 0 , , , , , · · · .They are correspond to the loop-level couplings in type IIA theory with n h = 0 , , , , , · · · ,respectively. In other words, the couplings at each loop-level has no higher-loop corrections.However, there are stringy corrections at each loop-level which are related to the non-zeromodes of the KK mass spectrum. 36he reduction of the couplings in S which have no three-form is ℓ p κ Z d x √− g L ( R )6 = 2 πR ℓ p κ Z d x √− G h m R αβ ǫε R αβγδ R γµǫν R δµεν (29)+ m R αβ ǫε R αβγδ R γµǫν R δνεµ + m R αβ ǫε R αβγδ R γ µδν R ǫµεν + m R αγβ ǫ R αβγδ R δεµν R ǫµεν + m R αǫγε R αβγδ R β µδν R ǫνεµ + m R αγβδ R αβγδ R ǫµεν R ǫεµν + m R αǫγ ε R αβγδ R βµǫν R δνεµ + · · · i where R = ℓ s g s is the radious of the circle and dots represent the R-R one-form and thedilaton couplings in the effective action of the type IIA theory. Note that as expected thereis no overall dilaton factor which indicates that the above couplings correspond to the tours-level effective action of type IIA. On the other hand, the one-loop gravity couplings in typeIIA theory are given in a scheme which includes the Ricci curvature and the Ricci scalar, as[24, 25, 26] S ( G ) = ℓ s g κ a . Z d x √− G ( t t − ǫ ǫ ) R (30)where a is a constant number, κ = 2 πℓ s g s κ and the tensors ǫ ǫ and t are defined as ǫ µ ··· µ ǫ ν ··· ν = 12 ǫ µ ··· µ αβ ǫ ν ··· ν αβ (31) t µ ··· µ M µ µ M µ µ M µ µ M µ µ = 8Tr( M M M M ) + 8Tr( M M M M )+8Tr( M M M M ) − M M )Tr( M M ) − M M )Tr( M M ) − M M )Tr( M M )where M , · · · , M are four arbitrary antisymmetric matrices. The Ricci curvature and theRicci scalar in above couplings can be removed by a field redefinition. The Riemann curvaturecouplings can then be compared with the couplings in (29). One finds the following parametersfor the couplings in (19): m = 0 ; m = m = − m = 14 m = − m = − m = − a (32)The complete one-loop effective action of type IIA for other NS-NS or R-R fields are not known.Hence, the other parameters in the M-theory effective action can not be fixed completely inthis way.However, the couplings involving four NS-NS fields are known to be given by (30) in whichthe Riemann curvature is replaced by the following expression [29]: R µναβ = R µναβ + 12 ∂ β H µνα − ∂ α H µνβ (33)where R µναβ is the linearised Riemann curvature. The last term in (30) has no effect in four-point functions. One can compare the four-point functions resulting from the first term in3714) with the corresponding four-point functions in the dimensional reduction of the couplingsin (14), (22) and (23). This S-matrix constraint fixes the parameter in (14) to be zero, i.e., m = 0, and fixes the following relations between the couplings in (22): m = − a/ m / , m = a/ − m , m = − a + 2 m , m = a − m ,m = − a + m , m = 2 m , m = 4 a − m − m − m , m = − a/ ,m = − a/ − m / m − m , m = 3 m − m − m ,m = − a/ − m / m , m = a/ m / − m / ,m = m − m − m , m = − m / , m = a − m + m ,m = − a/ m , m = a/ − m /
16 (34)and the following relations between the couplings in (23): m = a/ − m / − m / , m = − a/
48 + 2 m / − m / ,m = a/ − m / m / m , m = a/ − m / m / m ,m = − a/
48 + 2 m / m , m = − a/
48 + m + 2 m / ,m = a/
288 + m / − m /
36 + 4 m / m / ,m = 5 a/
144 + m − m / − m / m / m / ,m = a/ − m / − m /
48 (35)It is extremely difficult to fix all 1062 parameters by the S-matrix method. One may usesymmetries of the effective action to fix them all.The sphere-level gravity couplings in type II theory at order α ′ is known to be Z d x √− Ge − ( t t + 14 ǫ ǫ ) R (36)In this case the reduction of the classical theory on a circle has a O (1 , Z -subgroupof this symmetry has been used in [21] to fix all 872 parameters of the NS-NS couplings.There is no such bosonic symmetry in the one-loop effective action. The sypersymmetry ofthe effective actions, however, exsists in the classical and loop levels. It has been shown in[4, 12] that the R couplings and the Chern-Simons couplings C ∧ R ∧ R ∧ R ∧ R transforminto each other under the supersymmetry transformations. It would be interesting to imposethe supersymmetry constraint to fix all 1062 parameters in L and also the parameters in theChern-Simons sector L CS . 38 eferences [1] J. H. Schwarz, Nucl. Phys. B Proc. Suppl. , 1-32 (1997) doi:10.1016/S0920-5632(97)00070-4 [arXiv:hep-th/9607201 [hep-th]].[2] P. S. Howe and P. C. West, Nucl. Phys. B , 181 (1984). doi:10.1016/0550-3213(84)90472-3[3] E. Witten, Nucl. Phys. B , 85 (1995) doi:10.1016/0550-3213(95)00158-O[hep-th/9503124].[4] M. B. Green and P. Vanhove, Phys. Lett. B , 122-134 (1997) doi:10.1016/S0370-2693(97)00785-5 [arXiv:hep-th/9704145 [hep-th]].[5] M. B. Green, M. Gutperle and P. Vanhove, Phys. Lett. B , 177-184 (1997)doi:10.1016/S0370-2693(97)00931-3 [arXiv:hep-th/9706175 [hep-th]].[6] L. Anguelova, P. A. Grassi and P. Vanhove, Nucl. Phys. B , 269-306 (2004)doi:10.1016/j.nuclphysb.2004.09.024 [arXiv:hep-th/0408171 [hep-th]].[7] M. Cederwall, U. Gran, M. Nielsen and B. E. W. Nilsson, JHEP , 041 (2000)doi:10.1088/1126-6708/2000/10/041 [arXiv:hep-th/0007035 [hep-th]].[8] M. Cederwall, U. Gran, B. E. W. Nilsson and D. Tsimpis, JHEP , 052 (2005)doi:10.1088/1126-6708/2005/05/052 [arXiv:hep-th/0409107 [hep-th]].[9] S. de Haro, A. Sinkovics and K. Skenderis, Phys. Rev. D , 084010 (2003)doi:10.1103/PhysRevD.67.084010 [arXiv:hep-th/0210080 [hep-th]].[10] P. S. Howe and D. Tsimpis, JHEP , 038 (2003) doi:10.1088/1126-6708/2003/09/038[arXiv:hep-th/0305129 [hep-th]].[11] A. Rajaraman, Phys. Rev. D , 085018 (2006) doi:10.1103/PhysRevD.72.125008[arXiv:hep-th/0512333 [hep-th]].[12] Y. Hyakutake, Prog. Theor. Phys. , 109 (2007) doi:10.1143/PTP.118.109[arXiv:hep-th/0703154 [hep-th]].[13] K. Peeters, J. Plefka and S. Stern, JHEP , 095 (2005) doi:10.1088/1126-6708/2005/08/095 [arXiv:hep-th/0507178 [hep-th]].[14] H. R. Bakhtiarizadeh, Eur. Phys. J. C , no.8, 686 (2018) doi:10.1140/epjc/s10052-018-6152-y [arXiv:1711.11313 [hep-th]].[15] D. J. Gross and E. Witten, Nucl. Phys. B , 1 (1986).[16] A. A. Tseytlin, Nucl. Phys. B (1986) 391 Erratum: [Nucl. Phys. B (1987) 876].3917] S. Deser and A. N. Redlich, Phys. Lett. B (1986) 350 Erratum: [Phys. Lett. B (1987) 461].[18] M. R. Garousi and H. Razaghian, Phys. Rev. D , no.10, 106007 (2019)doi:10.1103/PhysRevD.100.106007 [arXiv:1905.10800 [hep-th]].[19] M. R. Garousi, Eur. Phys. J. C , no.11, 1086 doi:10.1140/epjc/s10052-020-08662-9[arXiv:2006.09193 [hep-th]].[20] M. R. Garousi, Eur. Phys. J. C , no.10, 827 (2019) doi:10.1140/epjc/s10052-019-7357-4[arXiv:1907.06500 [hep-th]].[21] M. R. Garousi, [arXiv:2011.02753 [hep-th]].[22] M. R. Garousi, [arXiv:2012.15091 [hep-th]].[23] T. Nutma, Comput. Phys. Commun. , 1719 (2014) doi:10.1016/j.cpc.2014.02.006[arXiv:1308.3493 [cs.SC]].[24] N. Sakai and Y. Tanii, Nucl. Phys. B , 457 (1987) doi:10.1016/0550-3213(87)90114-3[25] I. Antoniadis, S. Ferrara, R. Minasian and K. S. Narain, Nucl. Phys. B , 571-588 (1997)doi:10.1016/S0550-3213(97)00572-5 [arXiv:hep-th/9707013 [hep-th]].[26] E. Kiritsis and B. Pioline, Nucl. Phys. B , 509-534 (1997) doi:10.1016/S0550-3213(97)00645-7 [arXiv:hep-th/9707018 [hep-th]].[27] A. Sen, Phys. Lett. B , 295-300 (1991) doi:10.1016/0370-2693(91)90090-D[28] O. Hohm, A. Sen and B. Zwiebach, JHEP , 079 (2015) doi:10.1007/JHEP02(2015)079[arXiv:1411.5696 [hep-th]].[29] D. J. Gross and J. H. Sloan, Nucl. Phys. B291