One-loop β-functions in 4-derivative gauge theory in 6 dimensions

A classically scale-invariant 6d analog of the 4d Yang-Mills theory is the 4-derivative (∇F )2 + F 3 gauge theory with two independent couplings. Motivated by a search for a perturbatively conformal but possibly non-unitary 6d models we compute the one-loop β-functions in this theory. A systematic way of doing this using the back-ground field method requires the (previously unknown) expression for the b6 Seeley-DeWitt coefficient for a generic 4-derivative operator; we derive it here. As an application, we also compute the one-loop β-function in the (1,0) supersymmetric (∇F )2 6d gauge theory con-structed in hep-th/0505082.


Introduction
Like Einstein theory in 4 dimensions, the 6d Yang-Mills theory with the standard F 2 action has dimensional coupling and is not power-counting renormalizable. A 6d analog of the classically scale invariant and renormalizable R 2 + C 2 4d gravity is the 4-derivative (∇F ) 2 + F 3 gauge theory. Such 4-derivative terms are induced as counterterms when considering the standard scalars, fermions or YM vectors coupled to a background gauge field in 6d [1]. While non-unitary, this model may serve as a building block of possible higher-derivative (super)conformal theories in 6 dimensions. 1 Similar 4-derivative 6d gauge theories were discussed, e.g., in [5][6][7][8][9][10][11][12][13].
The aim of the present paper is to compute the one-loop β-functions in the Euclidean 6d theory with the action 2 S = − 1 g 2 d 6 x Tr (∇ m F mn ) 2 + 2γF mn F nk F km (1.1) Here g and γ are the two independent dimensionless coupling parameters. 3 1 In 4 dimensions the F 2 + (∇F ) 2 + F 3 theory was studied in [2] and later in [3]. The result of [2] for the one-loop divergences in this 4d theory was corrected in [4] making it in agreement with that of [3]. 2 We use m, n, k, . . . = 1, . . . , 6 for coordinate indices and flat Euclidean 6d metric so that the position of contracted indices is irrelevant. The gauge group generators are normalized as tr(t a t b ) = −TRδ ab , [t a , t b ] = f abc t c , where TR = 1 2 in the fundamental representation of SU(N ) (we denote the trace in this case as Tr) and TR = C2 = N in the adjoint representation. 3 Two other possible 4-derivative ∇F ∇F invariants are related to the above two by the Bianchi identity, e.g., Fmn∇ 2 Fmn = −2 (∇mFmn) 2 + 4FmnF nk F km + total derivative.

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In general, the UV logarithmically divergent part of the 6d one-loop effective action Γ 1 in a gauge field background may be written as 4 where the 1-loop β-function coefficients β 2 , β 3 depend on the field content of the theory. As we shall find below, their values in the case of the 4-derivative theory (1.1) are given by the following functions of the coupling γ (1-loop coefficients do not depend on the overall g 2 coupling) Somewhat surprisingly, the coefficient β 2A of the (∇F ) 2 divergence turns out to be independent of the coupling γ.
The total values of β 2 , β 3 in a 6d renormalizable model containing the gauge theory (1.1) minimally coupled to the ordinary-derivative "matter" fields -N 0 real scalars, N 1 2 Weyl fermions, N 1 YM vectors and N T self-dual tensors (interacting with A m as in [12]) are then [1,12] (1.4) Note that for the ordinary spin 0, 1/2, 1 fields their contributions to β 3 are proportional to the number of dynamical degrees of freedom. The same is true also for the 4-derivative gauge theory (1.1) with γ = 0: β 3A = 9 is the number of d.o.f. of a 4-derivative gauge vector in 6d. 6 As a consequence one should get β 3 = 0 in a supersymmetric theory; this is consistent with the non-existence of a super-invariant containing tr(F mn F nk F km ). Indeed, for the standard 2-derivative 6d (1,0) SYM theory (N 1 = 1, N 1 2 = 1) and for the scalar (hyper) multiplet (N 0 = 4, N 1 2 = 1) one finds Since ∇ m F mn = 0 on the standard YM equations of motion the (1,0) SYM theory is 1-loop finite on shell. The sum of the contributions of the two multiplets in (1.5) corresponds to the (1,1) SYM theory in 6d (and thus to N = 4 SYM in 4d) which is 1-loop finite even off-shell [1] Here tr is the trace over the matrix indices of a particular representation to which the quantum field belongs; for example, in the gauge theory case it is in the adjoint representation A ab m = f acb A c m , f acd f bcd = C2δ ab . 5 Here all the fields are taken for simplicity in the adjoint representation; in the case of other representations one is to rescale the numbers Ns by TR/C2. We corrected misprints in [1] mentioned in [12]. Note that the vector N1 terms here are formal: they indicate the 6d YM contribution in the absence of higher-derivative terms in (1.1). In the combined F 2 + (∇F ) 2 + F 3 theory discussed below in appendix B the values of β2 and β3 are the same as in the theory (1.1) without the YM F 2 term. 6 While the 2-derivative YM vector in d dimensions has

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In the (1, 0) supersymmetric 4-derivative gauge theory with the action given by the superextension [5] of tr (∇ m F mn ) 2 (containing also interacting / ∇ 3 Weyl fermion and three ∇ 2 scalars) we will find below that This result is in agreement (modulo notation change) with the one given in the recently revised version of [5]. This theory is non-unitary and is also formally inconsistent having a chiral anomaly [6] (the same as in the (1,0) 6d SYM theory containing Weyl fermion). One may still hope to cancel all of its anomalies by adding some higher derivative 6d "matter" multiplets (cf. [14][15][16]). The calculation of the β-functions (1.3) is most straightforward in the background field method and using the heat kernel expansion to extract the log divergences of the determinants. This requires the knowledge of the corresponding b 6 Seeley-DeWitt coefficient for the 4-derivative operator ∆ 4 = ∇ 4 + . . . in a gauge field background. While b 6 is available for the 2-derivative ∆ 2 operators [17], its expression for ∆ 4 was not known so far. The main new technical result of this paper is the computation of b 6 (∆ 4 ). We shall use the same strategy as employed previously in [2] to obtain b 4 (∆ 4 ) from the known expression for b 4 (∆ 2 ) by considering special factorized cases of the operator ∆ 4 .
It would be interesting to extend the computation of the b 6 coefficient for the 4-derivative operators to the case of a curved metric background (finding the analog of the corresponding expression for b 4 in [2]). This would allow, in particular, to compute the one-loop UV divergences in d = 6 conformal supergravity and verify the expectation [7,8] that the higher derivative (2,0) 6d conformal supergravity coupled to exactly 26 (2,0) tensor multiplets has vanishing conformal anomaly. 7 Another important step would be to extend the background field approach to the computation of UV divergences in 4-derivative gauge theories to the two-loop level generalizing the methods of [20][21][22].
The rest of the paper is organized as follows. In section 2 we present the general form of the one-loop effective action of the theory (1.1). In section 3 the result for the heat kernel coefficient b 6 that controls the logarithmic divergence of the determinant of a generic 4-derivative operator is given. In section 4 this expression is applied to compute the oneloop divergences in the bosonic gauge theory (1.1) and its (1,0) supersymmetric extension (with γ = 0). Details of the derivation of b 6 (∆ 4 ) are described in appendix A. In appendix B we discuss divergences of the combined 2-and 4-derivative 1 g 2 [κ 2 F 2 +(∇F ) 2 +γF 3 ] gauge theory and its (1,0) supersymmetric version: adding F 2 does not change the βfunctions (1.3) for g and γ but leads to the γ-dependent β-function for κ.

One-loop effective action
The derivation of the one-loop effective action in the 4-derivative theory (1.1) in 6d follows the same steps as in the 4d case discussed in appendix C of [2] (for a review, see also [4]).

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Expanding the invariants in (1.1) near a classical background A a m → A a m +Ã a m we get where F mn and ∇ m depend on the background A m and a, b are indices in the adjoint representation. Then the quadratic part of the fluctuation Lagrangian in (1.1) may be written as The second term here can be cancelled by adding a gauge-fixing (∇ mÃm = f (x)) term averaged with the operator −∇ 2 . The 4-derivative operator ∆ 4A acting onÃ a m can be written in the following "symmetric" form The operator that appears in the effective action after path-integral is performed (i.e. ∆ 4A in (2.3)) should be self-adjoint and this is so for (2.4) with (2.5). 8 The 1-loop effective action is then given by where ∆ gh = −∇ 2 is the ghost operator and H = −∇ 2 is the gauge-condition averaging operator required to cancel the last term in (2.3). Using the proper-time cutoff, the log divergent part of a determinant can be expressed (in general dimension d) in terms of the corresponding 8 Note that (2.4) is a completely general form for a fourth-order elliptic differential operator without the three-derivative term. The self-adjointness can be imposed via the following additional conditions on the coefficientsV † mn =Vmn,N † m = −Nm,Û † =Û where † is transposition if the field is real, and hermitian conjugation if the field is complex. 9 Here we ignore boundary terms. Note also that in the dimensional regularization one is to replace

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The values of b p for 2-derivative Laplacian ∆ 2 (in general curved space and gauge field background) are known up to p = 10 (see, e.g., [17,[23][24][25][26]) while for the 4-derivative operator ∆ 4 only b 2 and b 4 were found so far [2,[27][28][29]. Thus to compute the divergent part of (2.6) we need first to determine the coefficient b 6 for ∆ 4 in (2.4). This will be the subject of the next section and appendix A.
In general, given an elliptic differential operator ∆ of an even order in d dimensions one has where tr is the trace over internal indices of the operator. The heat kernel has an asymptotic expansion for t → 0 so that (see, e.g., [25,28,29]) The Seeley-DeWitt coefficients b p are local invariant expressions of dimension p constructed out of the background metric and gauge field (below we shall consider them up to total derivative terms). Using the proper-time cutoff ε = Λ − we obtain for the divergent part of (3.1) 10 The renormalization scale µ in log will be sometimes left implicit below. For example, for the 2-derivative operator defined on a vector bundle with the covariant derivative ∇ m and the curvature F mn = [∇ m , ∇ n ] one has 11 To find b 6 (∆ 4 ) for the operator in (2.4) we will use the same idea as in [2] and consider several special cases of factorized operators ∆ 4 for which 10 Note that the form of (3.3) is universal for any order of the differential operator -that is the reason for the above normalization of the Seeley-DeWitt coefficients. 11 Here we will somewhat abuse the notation and adopt the same labels for the connection, covariant derivative and its curvature of the vector bundle as in the gauge theory (Am, ∇m, Fmn) with an implicit understanding that the connection in the differential operators ∆ may take more general values that in a particular representation of a gauge group.

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The 4-derivative operator that we are interested in is given in (2.4). As explained in appendix A, a general expression for its b 6 coefficient is In contrast to what happens in the case of ∆ 2 in (3.5), some of the coefficients in (3.7) in general depend on the number of dimensions d. In the case of d = 6 we are interested in here one findŝ (3.8)

Divergences of 4-derivative 6d gauge theories
Let us now apply the above general expression (3.7), (3.8) for b 6 (∆ 4 ) to the gauge theories of interest.

Bosonic theory
Starting with the explicit form of the coefficient functions (2.4), (2.5) in the operator ∆ 4A and applying (3.7), (3.8) as well as (3.5), we can compute the coefficient b 6 in the divergent part of the effective action (2.6), (2.7) of the 4-derivative bosonic 6d gauge theory (1.1) 12 Thus finally out to be independent of the parameter γ: various terms in b 6 in (3.7) generically do give γ-dependent (∇F ) 2 contributions and they cancel out only when combined together weighted with thek i coefficients in (3.8).
As the sign of the F 3 term in (1.1) is not a priori constrained by the requirement of positivity of the Euclidean action we formally define a second coupling h 2 = γ −1 g 2 that may assume positive as well as negative values. Then near the fixed points h 2 also goes to zero in the UV, i.e. like g 2 the second coupling is also asymptotically free.
In appendix B we shall present also the one-loop β-functions for the combined YM plus 4-derivative gauge theory with L = 1 g 2 κ 2 F 2 + (∇F ) 2 + γF 3 .
In this case γ = 0 since, in general, there is no supersymmetric extension of the F 3 term. 13 The field content includes the 4-derivative gauge field A m , the 3-derivative 6d Weyl spinor Ψ, and the three 2-derivative real scalars Φ I (I = 1, 2, 3). 14 In total, one has 9 + 3 bosonic and 3 × 4 fermionic on-shell degrees of freedom (for each value of the internal index).
Using an off-shell harmonic superspace formulation ref. [5] found the following (1,0) supersymmetric 6d action 15 (4.7) We suppressed interactions that are more than second order in the scalars and fermions, as they will not contribute to the one-loop divergences in a gauge-field background. Note 13 This can be easily understood using, e.g., the standard N = 1 4d superspace formulation: the YM field strength Fmn is part of the spinor superfield strength Wα and thus constructing an invariant cubic in Wα is not possible. 14 In the case of the standard (1,0) SYM theory (corresponding to N = 2 SYM theory in 4d) the latter correspond to the auxiliary scalars. 15 Our notation differ significantly from that of [5] (where, e.g., the scalar kinetic term is defined using ij to raise the indices and thus implicitly is negative definite). Here, the Dirac matrices Γm are 8 × 8 hermitian complex matrices satisfying Γ (m Γ n) = 1 2 {Γm, Γn} = δmn and Γmn ≡ Γ [m Γ n] .

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that with our definition of the coupling constant g (i.e. the choice of the overall sign of the action) the gauge field term in (4.7) is positive definite (cf. (1.1)) but the scalar term is not, and this is one indication of the non-unitarity of the theory. 16 The 4-derivative operator for the fluctuations of the gauge field is given by (2.4), (2.5) with γ = 0, i.e. it is ∆ (0) 4A ≡ ∆ 4A γ=0 , while the 3-derivative fermion and the 2-derivative scalar operators in gauge field background may be written as 17 (4.8) Here i / ∇ 3 is the cube of the Dirac operator ∆ 1Ψ = −i / ∇ = −iΓ m ∇ m whose square is As a result, the one-loop effective action of the supersymmetric theory (4.7) is the following generalization of the bosonic case (2.6) Here the contributions of the ghost and gauge-averaging operators in (2.6) got canceled against the contribution of the three scalars Φ I . We also used that det ∆ Ψ is defined for the Dirac 6d spinors so that the factor 1 2 accounts for the fact that the fermion Ψ is a Weyl spinor. As a result, the coefficient of the log divergent part of the effective action (2.7) is given by (cf.  To compute the fermionic contribution, let us first construct a 4-derivative operator by taking the product of ∆ 3Ψ in (4.8) with the standard Dirac operator (4.14) 16 In [5] the opposite overall sign was chosen so that their coupling is related to ours by g 2 → −g 2 . This translates into the opposite sign of the β-function for g in (4.17). Note that here there is thus no "preferred" choice of the sign of the action (redefining the scalars ΦI → iΦI leads to imaginary Φ 3 interaction, i.e. to non-hermiticity of the action). For a review of related issues in higher-derivative theories see [30]. 17 In the first form of ∆3Ψ the derivative in the second term acts all the way to the right while in the third term it acts only on Fmn. 18 Notice that this operator is not self-adjoint, i.e. the symmetry requirements in footnote 8 are not satisfied.

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Applying the general expression for b 6 (∆ 4 ) that we found in (3.7), (3.8) (where now the connection and its curvature are understood to include also the internal spinor indices, see footnote 11) and also using that squaring ∆ 1Ψ one obtains (4.9), for which b 6 can then found from (3.5), we end up with Combining the bosonic (4.12) and the fermionic (4.15) contributions to (4.11) we conclude that the F 3 terms cancel as expected and finally corresponding to an asymptotically free behaviour. This agrees with the (recently revised) result of [5] (cf. footnote 16). Note that the computation of the β-function in [5] was done in the scalar field Φ I background while here we used the gauge field background, thus providing an independent check of the result. For comparison, let us recall the result [1] of a similar computation in the ordinaryderivative (1,0) 6d SYM theory where Ψ is a Weyl spinor, Φ I are 3 auxiliary fields (cf. (4.7)) and κ is a mass scale. The analog of the one-loop effective action in a gauge field background (4.10) here is Using (3.5) we get As a result, the one-loop logarithmic divergence is given by (2.7) with

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Once again, the F 3 divergence cancels, and (4.21) implies the value of β 2 = −20 in (1.2), (1.5). Since here ∇ m F mn = 0 is an equation of motion, the divergence (4.21) vanishes onshell, i.e. the (1,0) 6d SYM theory is finite on-shell 19 though is not renormalizable off-shell. The (1,1) 6d SYM found by combining the (1,0) SYM with a scalar multiplet (cf. (1.5)) is one-loop finite even off-shell [1] (cf. also [32]). Let us also note that it is easy to check the cancellation of F 3 divergences in the (1,0) supersymmetric gauge theory (4.7) by restricting the background to satisfy ∇ m F mn = 0 (which is a special on-shell background also in this theory). Then ∆ 3Ψ in (4.8) becomes simply (∆ 1Ψ ) 3 = i / ∇ 3 and also the vector field operator in (2.4), (2.5) (with γ = 0) becomes a square of the standard YM operator in (4.19), i.e. ∆ 4A = (∆ 2A ) 2 . As a result, the effective action (4.10) reduces to i.e. equal to the sum of twice the effective action of the standard (1,0) SYM in (4.19) with the effective action of the scalar (hyper) multiplet (containing 4 real scalars and one Weyl fermion). Each of these do not contribute to the F 3 divergent terms according to (1.5).

Acknowledgments
We are grateful to E. Ivanov and A. Smilga for discussions related to the value of the β-function in [5]. LC wishes to thank T. Bertolini for useful discussions. AAT acknowledges K.-W. Huang and R. Roiban for discussions related to [12]. LC is supported by the International Max Planck Research School for Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory. AAT was supported by the STFC grant ST/P000762/1.
Note added. After this paper was submitted to the arXiv we learned about the earlier work [33] (see also [34]) in which a diagrammatic computation of the two-loop β-functions in the 6d gauge theory (1.1) coupled to standard fermions was performed. 20 After correcting a mistake in the original version of this paper we found that our result (1.3), (1.4) for the β-functions of the theory (1.1) coupled to fermions is in full agreement with the one-loop β-functions in [33]. 21 the action in [33] contained (∇ k Fmn) 2 with the two invariants related as in footnote 3. As a result, the couplings g1 and g2 in [33] are related to ours as g1 = g, g2 = 3g(1 + γ) (using also that g2 → −g2 due to apparent sign difference in notation for Fmn).

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A Derivation of the expression for b 6 (∆ 4 ) The operator that we shall consider is which is the most general fourth-order elliptic differential operator without 3-derivative term. It is related to the "symmetrized" operator in (2.4) by The general expression for its coefficient b 6 including only independent invariants may be written as where the trace is over internal indices and k i are real coefficients. 22 Their values in d = 6 found below are To determine k i we shall exploit the factorization property (3.6), i.e. b 6 (∆ 4 ) = b 6 (∆ 2 ) + b 6 (∆ 2 ) , where b 6 (∆ 2 ) is given by (3.5). One needs to identify enough special cases and consistency conditions to fix all k i . When comparing the two sides of the b 6 -relation in (A.5) it is important to take into account (i) that they are defined up to total derivatives (which we drop in discussing UV divergences), (ii) that the terms can be cyclically permuted because they appear under an overall trace, and (iii) relations between the invariants (implied, e.g., by the Bianchi identity). Considering ∆ 2 = −∇ 2 + X and ∆ 2 = −∇ 2 + X their product is given by (A.1) with The relations between the ki andki in (3.8) are, using (A.2),k6 = k6 + 1 2 k15 − 1 4 k17,k8 = k8 + 1 2 k16, k10 = k10 − 1 2 k15 + 1 4 k17,k11 = k11 + 1 2 k14,k15 = k15 − k17,k16 = k16 + k18 withki = ki otherwise.

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Using ( Next, let us assume that Here ∇ m K n = ∂ m K n + [A m , K n ] (K m is in the adjoint representation of the gauge group). The coefficient functions in the corresponding operator (A.10) Using (3.5) and the relations one can compute b 6 (∆ ± ) and then compare to b 6 (∆ 4 ) in (A.5).

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Their coefficients can then be compared to get ((K m K n K p ) 2 and (K m K p K m ) 2 give the same equation) 3. General unconstrained K n , comparing the terms with one K m or two of them contracted together. A basis of such tensors contains In this case we obtain (the two KKF F terms give the same equation) 2k 11 +k 14 = 0 , 8k 12 +2k 13 = 0 , 2k 11 −2k 13 +2k 14 = 0 , We also checked some of the coefficients k i by explicit diagrammatic calculations of the corresponding UV divergences.
B One-loop divergences in F 2 + (∇F ) 2 + F 3 theory It is straightforward to generalize the expression for the effective action (2.6) to the case when one adds to the action (1.1) the standard YM term, i.e. the first term in (4.18) Here ∆ 2A is given in (4.19). The quadratic and logarithmic divergences of (B.1) are determined by the total b 4 and b 6 coefficients (cf.
Let us note also that on ∇ m F mn = 0 background (B.16) becomes the following generalization of (4. The coefficient of the field independent κ 4 quadratic divergence is proportional to the number of degrees of freedom and thus vanishes in supersymmetric cases.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.