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

TL;DR

This paper computes the one-loop beta-functions for a class of 6-dimensional 4-derivative gauge theories, including a supersymmetric case, using the background field method and deriving the necessary Seeley-DeWitt coefficient.

Contribution

It introduces a systematic method to compute the $b_6$ Seeley-DeWitt coefficient for 4-derivative operators and applies it to determine beta-functions in 6d gauge theories.

Findings

Derived the $b_6$ Seeley-DeWitt coefficient for generic 4-derivative operators.

Calculated the one-loop beta-functions for a classically scale-invariant 6d gauge theory.

Computed the beta-function for a supersymmetric (1,0) 6d gauge theory.

Abstract

A classically scale-invariant 6d analog of the 4d Yang-Mills theory is the 4-derivative $ (\nabla 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 $\beta$-functions in this theory. A systematic way of doing this using the background field method requires the expression for the $b_6$ Seeley-DeWitt coefficient for a generic 4-derivative operator. It was previously unknown and we derive it here. As an application, we also compute the one-loop $\beta$-function in the (1,0) supersymmetric $ (\nabla F)^2$ 6d gauge theory constructed in hep-th/0505082.

Equations

$$\begin{split}S&=-\frac{1}{g^{2}}\int d^{6}x\,\operatorname{Tr}\Big{[}\left({\mathcal{\nabla}}_{m}{F}_{mn}\right)^{2}+2\gamma F_{mn}F_{nk}F_{km}\Big{]}\\ &=\frac{1}{2g^{2}}\int d^{6}x\,\Big{[}\left({\mathcal{\nabla}}_{m}F^{a}_{mn}\right)^{2}+\gamma f^{abc}F^{a}_{mn}F^{b}_{nk}F^{c}_{km}\Big{]}\ .\end{split}$$

$$\Gamma_{1\infty}=-\frac{\log\Lambda}{(4\pi)^{3}}\;\int d^{6}x\ \operatorname{tr}\Big{[}-\frac{1}{60}\beta_{2}\left(\nabla_{m}F_{mn}\right)^{2}+\frac{1}{90}\beta_{3}F_{mn}F_{nk}F_{km}\Big{]}\ ,$$

$${\beta }_{2A}$$

$${\beta }_2$$

$${\beta }_3$$

$${\beta }_{2{}_{(1,0)\text{SYM}}}=-20,{\beta }_{2\mathrm{scal}}=20,{\beta }_{3{}_{(1,0)\text{SYM}}}={\beta }_{3\mathrm{scal}}=0.$$

$${\beta }_{2{}_{(1,1)\text{SYM}}}={\beta }_{3{}_{(1,1)\text{SYM}}}=0.$$

$${\beta }_{2(1,0)}=220,{\beta }_{3(1,0)}=0.$$

$$\mathrm{Tr}{\left({\nabla }_m{F}_{mn}\right)}^2$$

$$\displaystyle\hskip 60.00009pt{}-2{\mathcal{\nabla}}_{n}{\mathcal{\nabla}}_{k}F_{km}+4F_{mk}F_{kn}\Big{]}^{ab}\tilde{A}_{n}^{b}-\frac{1}{2}({\mathcal{\nabla}}_{m}\tilde{A}_{m}^{a}){\mathcal{\nabla}}^{2}({\mathcal{\nabla}}_{n}\tilde{A}_{n}^{a})\ ,$$

$$\mathrm{Tr}(F_{mn}{F}_{nk}{F}_{km})$$

$$\displaystyle\hskip 60.00009pt-\Big{(}\frac{3}{4}[F_{mk},F_{kn}]+\frac{3}{4}F_{r(m}F_{n)r}+\frac{3}{8}F_{rk}F_{rk}\delta_{mn}\Big{)}\Big{]}^{ab}\tilde{A}^{b}_{n}\ ,$$

$${\mathcal{L}}^{(2)}=\frac{1}{2g^{2}}\tilde{A}^{a}_{m}\,(\Delta_{4A})_{mn}^{ab}\,\tilde{A}_{n}^{b}+\frac{1}{2g^{2}}({\mathcal{\nabla}}_{m}\tilde{A}^{a}_{m})\big{(}-{\mathcal{\nabla}}^{2}\big{)}({\mathcal{\nabla}}_{n}\tilde{A}^{a}_{n})\ .$$

$${\Delta }_4$$

$$({\hat{V}}_{rk}{)}_{mn}$$

$$({\hat{N}}_k{)}_{mn}$$

$$(\hat{U}{)}_{mn}$$

$${\Gamma }_1=\frac{1}{2}\log \frac{\det {\Delta }_{4A}}{(\det {\Delta }_{\mathrm{gh}}{)}^2\det H}=\frac{1}{2}\log \det {\Delta }_{4A}-\frac{3}{2}\log \det {\Delta }_{2,0},{\Delta }_{2,0}=-{\nabla }^2,$$

$${\Gamma }_{1\infty }(\Delta )=\frac{1}{2}(\log \det \Delta {)}_{\infty }=-\frac{\log \Lambda }{(4\pi {)}^{d/2}}{B}_d(\Delta ),B_d=\int d^{d}x{b}_d(\Delta ).$$

$$\log \det {\Delta }_{\ell }=-\int d^{d}x{\int }_{\epsilon }^{\infty }\frac{dt}{t}\mathrm{tr}⟨x⟩e^{-t{\Delta }_{\ell }}x,$$

$$\mathrm{tr}⟨x⟩e^{-t{\Delta }_{\ell }}x\equiv \mathrm{tr}K(t;x,x;{\Delta }_{\ell })\simeq \sum _{p\ge 0}\frac{2}{(4\pi {)}^{d/2}\ell }{t}^{(p-d)/\ell }{b}_p({\Delta }_{\ell },d;x).$$

$$\begin{split}&{(\log\det\Delta_{\ell})_{\infty}}=-\frac{2}{(4\pi)^{d/2}}\Big{[}\sum^{d-1}_{p=0}\frac{{B_{p}(\Delta_{\ell})}}{d-p}\Lambda^{d-p}+B_{d}(\Delta_{\ell})\log\frac{\Lambda}{\mu}\Big{]},\\ &B_{p}(\Delta_{\ell})=\int d^{d}x\ b_{p}(\Delta_{\ell})\ .\end{split}$$

$${\Delta }_2=-{\nabla }^2+X,$$

$$b_6({\Delta }_2)$$

$${\Delta }_4={\Delta }_2{\Delta }_2^{\prime },\det {\Delta }_4=\det {\Delta }_2\det {\Delta }_2^{\prime },b_d({\Delta }_2{\Delta }_2^{\prime })=b_d({\Delta }_2)+b_d({\Delta }_2^{\prime }).$$

$$b_6({\Delta }_4)=$$

$$+{\hat{k}}_3{\hat{V}}_{mn}{\hat{V}}_{nk}{\hat{V}}_{km}+{\hat{k}}_4{\hat{V}}_{mn}{\hat{V}}_{mn}\hat{V}+{\hat{k}}_5\hat{V}\hat{V}\hat{V}+{\hat{k}}_6{\hat{V}}_{mn}{\nabla }_{(n}{\nabla }_{k)}{\hat{V}}_{km}$$

$$+{\hat{k}}_7{\hat{V}}_{mn}{\nabla }^2{\hat{V}}_{mn}+{\hat{k}}_8{\hat{V}}_{mn}{\nabla }_m{\nabla }_n\hat{V}+{\hat{k}}_9\hat{V}{\nabla }^2\hat{V}+{\hat{k}}_{10}{\hat{V}}_{mn}{\hat{V}}_{nk}{F}_{mk}$$

$$+{\hat{k}}_{11}{F}_{mn}{\nabla }_{(m}{\nabla }_{k)}{\hat{V}}_{kn}+{\hat{k}}_{12}\hat{V}{F}_{mn}{F}_{mn}+{\hat{k}}_{13}{\hat{V}}_{mn}{F}_{mk}{F}_{nk}$$

$$\displaystyle\hskip 20.00003pt{}+\hat{k}_{14}F_{mn}{\mathcal{\nabla}}_{m}\hat{N}_{n}+\hat{k}_{15}\hat{V}_{mn}{\mathcal{\nabla}}_{m}\hat{N}_{n}+\hat{k}_{16}\hat{V}{\mathcal{\nabla}}_{m}\hat{N}_{m}+\hat{k}_{17}\hat{N}_{m}\hat{N}_{m}+\hat{k}_{18}\hat{U}\hat{V}\Big{]}\ .$$

$${\hat{k}}_1$$

$${\hat{k}}_7$$

$${\hat{k}}_{13}$$

$$b_6$$

$$b_6({\Delta }_{4A})$$

$$b_6({\Delta }_{2,0})$$

$$b_6$$

Full text

Imperial-TP-AT-2019-04

**One-loop $\beta$-functions

in 4-derivative gauge theory in 6 dimensions **

Lorenzo Casarin*a,111 lorenzo.casarin@aei.mpg.de and Arkady A. Tseytlinb,c,*222 Also at the Lebedev Institute, Moscow.    tseytlin@imperial.ac.uk

a* Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)

Am Mühlenberg 1, DE-14476 Potsdam, Germany

b Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.

c Institute of Theoretical and Mathematical Physics

Moscow State University, 119991, Russia *

1. 1 Introduction

2. 2 One-loop effective action

3. 3 Heat kernel coefficient $b_{6}(\Delta_{4})$

4. 4 Divergences of 4-derivative 6d gauge theories

5. 4.1 Bosonic theory

6. 4.2 (1,0) supersymmetric theory

7. A Derivation of the expression for $b_{6}(\Delta_{4})$

8. B One-loop divergences in $F^{2}+(\nabla F)^{2}+F^{3}$ theory

1 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 $(\nabla F)^{2}+F^{3}$ 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.111 In 4 dimensions the $F^{2}+(\nabla F)^{2}+F^{3}$ 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].

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 $\beta$-functions in the Euclidean 6d theory with the action222 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 $\operatorname{tr}(t^{a}t^{b})=-T_{R}\delta^{ab},\ [t^{a},t^{b}]=f^{abc}t^{c}$, where $T_{R}={1\over 2}$ in the fundamental representation of $SU(N)$ (we denote the trace in this case as $\operatorname{Tr}{}$) and $T_{R}=C_{2}=N$ in the adjoint representation.

[TABLE]

Here $g$ and $\gamma$ are the two independent dimensionless coupling parameters.333 Two other possible 4-derivative $\nabla F\nabla F$ invariants are related to the above two by the Bianchi identity, e.g., $\ {F}_{mn}{\mathcal{\nabla}}^{2}{F}_{mn}=-2\left({\mathcal{\nabla}}_{m}{F}_{mn}\right)^{2}+4{F}_{mn}{F}_{nk}{F}_{km}+\text{total derivative}$.

In general, the UV logarithmically divergent part of the 6d one-loop effective action $\Gamma_{1}$ in a gauge field background may be written as444 Here $\operatorname{tr}$ 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 $A^{ab}_{m}=f^{acb}A^{c}_{m},\ f_{acd}f_{bcd}=C_{2}\delta_{ab}$.

[TABLE]

where the 1-loop $\beta$-function coefficients $\beta_{2}$, $\beta_{3}$ 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 $\gamma$ (1-loop coefficients do not depend on the overall $g^{2}$ coupling)

[TABLE]

Somewhat surprisingly, the coefficient $\beta_{2A}$ of the $(\nabla F)^{2}$ divergence turns out to be independent of the coupling $\gamma$.

The total values of $\beta_{2}$, $\beta_{3}$ in a 6d renormalizable model containing the gauge theory (1.1) minimally coupled to the ordinary-derivative “matter” fields – $N_{0}$ real scalars, $N_{\frac{1}{2}}$ Weyl fermions, $N_{1}$ YM vectors and $N_{T}$ self-dual tensors (interacting with $A_{m}$ as in [12]) are then [1, 12]555 Here all the fields are taken for simplicity in the adjoint representation; in the case of other representations one is to rescale the numbers $N_{s}$ by $T_{R}/C_{2}$. We corrected misprints in [1] mentioned in [12]. Note that the vector $N_{1}$ terms here are formal: they indicate the 6d YM contribution in the absence of higher-derivative terms in (1.1). In the combined $F^{2}+(\nabla F)^{2}+F^{3}$ theory discussed below in Appendix B the values of $\beta_{2}$ and $\beta_{3}$ are the same as in the theory (1.1) without the YM $F^{2}$ term.

[TABLE]

Note that for the ordinary spin 0, 1/2, 1 fields their contributions to $\beta_{3}$ are proportional to the number of dynamical degrees of freedom. The same is true also for the 4-derivative gauge theory (1.1) with $\gamma=0$: $\beta_{3A}=9$ is the number of d.o.f. of a 4-derivative gauge vector in 6d.666 While the 2-derivative YM vector in $d$ dimensions has ${1\over 2}2(d-1)-1=d-2$ dynamical d.o.f., for the 4-derivative gauge vector in (1.1) one finds ${1\over 2}4(d-1)-1=2d-3$, i.e. 5 in $d=4$ and 9 in $d=6$. As a consequence one should get $\beta_{3}=0$ in a supersymmetric theory; this is consistent with the non-existence of a super-invariant containing $\operatorname{tr}(F_{mn}F_{nk}F_{km})$. Indeed, for the standard 2-derivative 6d (1,0) SYM theory ($N_{1}=1,\ N_{\frac{1}{2}}=1$) and for the scalar (hyper) multiplet ($N_{0}=4,\ N_{\frac{1}{2}}=1$) one finds

[TABLE]

Since $\nabla_{m}F_{mn}=0$ 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]

[TABLE]

In the $(1,0)$ supersymmetric 4-derivative gauge theory with the action given by the super-extension [5] of $\operatorname{tr}\left({\mathcal{\nabla}}_{m}{F}_{mn}\right)^{2}$ (containing also interacting $\not{\nabla}^{3}$ Weyl fermion and three $\nabla^{2}$ scalars) we will find below that

[TABLE]

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 $\beta$-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 $\Delta_{4}=\nabla^{4}+...$ in a gauge field background. While $b_{6}$ is available for the 2-derivative $\Delta_{2}$ operators [17], its expression for $\Delta_{4}$ was not known so far. The main new technical result of this paper is the computation of $b_{6}(\Delta_{4})$. We shall use the same strategy as employed previously in [2] to obtain $b_{4}(\Delta_{4})$ from the known expression for $b_{4}(\Delta_{2})$ by considering special factorized cases of the operator $\Delta_{4}$.

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 the vanishing conformal anomaly.777This is the 6d counterpart of the known fact of cancellation of the conformal anomaly in the 4d system of $\cal N$=4 conformal supergravity coupled to 4 vector $\cal N$=4 multiplets [18, 19]. 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 one-loop divergences in the bosonic gauge theory (1.1) and its (1,0) supersymmetric extension (with $\gamma=0$). Details of the derivation of $b_{6}(\Delta_{4})$ are described in Appendix A. In Appendix B we discuss divergences of the combined 2- and 4-derivative ${1\over g^{2}}[\kappa^{2}F^{2}+(\nabla F)^{2}+\gamma F^{3}]$ gauge theory and its (1,0) supersymmetric version: adding $F^{2}$ does not change the $\beta$-functions (1.3) for $g$ and $\gamma$ but leads to the $\gamma$-dependent $\beta$-function for $\kappa$.

2 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]). Expanding the invariants in (1.1) near a classical background $A_{m}^{a}\to{A}_{m}^{a}+\tilde{A}_{m}^{a}$ we get

[TABLE]

where $F_{mn}$ and $\nabla_{m}$ 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

[TABLE]

The second term here can be cancelled by adding a gauge-fixing ($\nabla_{m}\tilde{A}_{m}=f(x)$) term averaged with the operator $-\nabla^{2}$. The 4-derivative operator $\Delta_{4A}$ acting on $\tilde{A}^{a}_{m}$ can be written in the following “symmetric” form

[TABLE]

where $\hat{V}_{rk}$, $\hat{N}_{k}$, $\hat{U}$ are local covariant matrices in the internal $(a,m),(b,n)$ indices reading

[TABLE]

The operator that appears in the effective action after path-integral is performed (i.e. $\Delta_{4A}$ in (2.3)) should be self-adjoint and this is so for (2.4) with (2.5).888 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 coefficients $\hat{V}_{mn}^{\dagger}=\hat{V}_{mn}$, $\ \hat{N}_{m}^{\dagger}=-\hat{N}_{m}$, $\ \hat{U}^{\dagger}=\hat{U}$ where $\dagger$ is transposition if the field is real, and hermitian conjugation if the field is complex.

The 1-loop effective action is then given by

[TABLE]

where $\Delta_{\rm gh}=-\nabla^{2}$ is the ghost operator and $H=-\nabla^{2}$ 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 Seeley-DeWitt coefficient $B_{d}$999 Here we ignore boundary terms. Note also that in the dimensional regularization one is to replace $\log\Lambda\to-\frac{1}{{\rm d}-d}$ where $d$ is integer and ${\rm d}<d$ is its analytic continuation.

[TABLE]

The values of $b_{p}$ for 2-derivative Laplacian $\Delta_{2}$ (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 $\Delta_{4}$ only $b_{2}$ and $b_{4}$ were found so far [27, 2, 28, 29]. Thus to compute the divergent part of (2.6) we need first to determine the coefficient $b_{6}$ for $\Delta_{4}$ in (2.4). This will be the subject of the next section and Appendix A.

3 Heat kernel coefficient $b_{6}(\Delta_{4})$

In general, given an elliptic differential operator $\Delta_{\ell}$ of an even order $\ell$ in $d$ dimensions one has

[TABLE]

where tr is the trace over internal indices of the operator. The heat kernel has an asymptotic expansion for $t\to 0$ so that (see, e.g., [28, 25, 29])

[TABLE]

The Seeley-DeWitt coefficients $b_{p}$ are local invariant expressions of dimension $p$ constructed out of the background metric and gauge field, exhibiting an explicit dependence on the spacetime dimension $d$ when $\ell\neq 2$ (below we shall consider them up to total derivative terms). In the following, we shall not indicate explicitly some of the arguments of $b_{p}$. Using the proper-time cutoff $\varepsilon=\Lambda^{-\ell}$ we obtain for the divergent part of (3.1)101010 Note that the form of (3.3) is universal for any order $\ell$ of the differential operator – that is the reason for the above normalization of the Seeley-DeWitt coefficients.

[TABLE]

The renormalization scale $\mu$ in $\log$ will be sometimes left implicit below. For example, for the 2-derivative operator defined on a vector bundle with the covariant derivative $\nabla_{m}$ and the curvature $F_{mn}=[\nabla_{m},\nabla_{n}]$ one has111111 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 $(A_{m},\nabla_{m},F_{mn})$ with an implicit understanding that the connection in the differential operators $\Delta_{\ell}$ may take more general values that in a particular representation of a gauge group.

[TABLE]

[TABLE]

To find $b_{6}(\Delta_{4})$ for the operator in (2.4) we will use the same idea as in [2] and consider several special cases of factorized operators $\Delta_{4}$ for which

[TABLE]

This factorization Ansatz is only true for the coefficient $b_{p}=b_{d}$, i.e.  with the index $p$ equal to the spacetime dimension $d$. This is related to the fact that only the logarithmically divergent term of the expansion (3.3) is universal between different regularizations, while the power-like divergences are regularization-dependent.

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 ($\hat{V}\equiv\hat{V}_{mm}$)

[TABLE]

As mentioned above, in contrast to what happens in the case of $\Delta_{2}$ 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 finds

[TABLE]

4 Divergences of 4-derivative 6d gauge theories

Let us now apply the above general expression (3.7), (3.8) for $b_{6}(\Delta_{4})$ to the gauge theories of interest.

4.1 Bosonic theory

Starting with the explicit form of the coefficient functions (2.4), (2.5) in the operator $\Delta_{4A}$ 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)121212 In applying (3.7) to the gauge field case, the trace there is acting on the full internal index structure of the operator $\Delta_{4A}$, i.e. involving both spacetime and gauge indices (cf. footnote 11).

[TABLE]

Thus finally

[TABLE]

Comparing to (1.2) we end up with the values of the one-loop $\beta$-function coefficients $\beta_{2A}$, $\beta_{3A}$ quoted in (1.3). It is remarkable that the divergence proportional to $(\nabla F)^{2}$ turned out to be independent of the parameter $\gamma$: various terms in $b_{6}$ in (3.7) generically do give $\gamma$-dependent $(\nabla F)^{2}$ contributions and they cancel out only when combined together weighted with the $\hat{k}_{i}$ coefficients in (3.8).

The corresponding RG equations for the renormalized couplings $g(\mu)$ and $\gamma(\mu)$ in (1.1) may be written as ($t={1\over(4\pi)^{3}}\log\mu^{2}$, $C_{2}({\rm SU}(N))=N$)

[TABLE]

The flow of $g$ is independent of the parameter $\gamma$ and the sign of $\beta_{2A}$ corresponds to asymptotic freedom. The fixed points of the flow of $\gamma$ are the solutions of $\beta_{\gamma}=0$, i.e. $\gamma_{1}\simeq-0.897,\ \gamma_{2}\simeq 0.059,\ \gamma_{3}\simeq 0.838$. Since $\beta_{\gamma}>0$ for $\gamma<\gamma_{1}$ or $\gamma_{2}<\gamma<\gamma_{3}$, we have that $\gamma_{1}$ and $\gamma_{3}$ are attractive fixed points of the flow. 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 $h^{2}=\gamma^{-1}g^{2}$ 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 $\beta$-functions for the combined YM plus 4-derivative gauge theory with {\cal L}={1\over g^{2}}\big{[}\kappa^{2}F^{2}+(\nabla F)^{2}+{\gamma}F^{3}\big{]}.

4.2 (1,0) supersymmetric theory

Let us now consider the 6d supersymmetric version of the theory (1.1) constructed in [5]. In this case $\gamma=0$ since, in general, there is no supersymmetric extension of the $F^{3}$ term.131313 This can be easily understood using, e.g., the standard $N=1$ 4d superspace formulation: the YM field strength $F_{mn}$ is part of the spinor superfield strength $W_{\alpha}$ and thus constructing an invariant cubic in $W_{\alpha}$ is not possible. The field content includes the 4-derivative gauge field $A_{m}$, the 3-derivative 6d Weyl spinor $\Psi$, and the three 2-derivative real scalars $\Phi_{I}$ ($I=1,2,3$).141414 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. In total, one has $9+3$ bosonic and $3\times 4$ 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 action151515 Our notation differ significantly from that of [5] (where, e.g., the scalar kinetic term is defined using $\epsilon^{ij}$ to raise the indices and thus implicitly is negative definite). Here, the Dirac matrices $\Gamma_{m}$ are $8\times 8$ hermitian complex matrices satisfying $\Gamma_{(m}\Gamma_{n)}={1\over 2}\{\Gamma_{m},\Gamma_{n}\}=\delta_{mn}$ and $\Gamma_{mn}\equiv\Gamma_{[m}\Gamma_{n]}$.

[TABLE]

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 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.161616 In [5] the opposite overall sign was chosen so that their coupling is related to ours by $g^{2}\to-g^{2}$. This translates into the opposite sign of the $\beta$-function for $g$ in (4.17). Note that here there is thus no “preferred” choice of the sign of the action (redefining the scalars $\Phi_{I}\to i\Phi_{I}$ leads to imaginary $\Phi^{3}$ interaction, i.e. to non-hermiticity of the action). For a review of related issues in higher-derivative theories see [30].

The 4-derivative operator for the fluctuations of the gauge field is given by (2.4), (2.5) with $\gamma=0$, i.e. it is \Delta^{(0)}_{4A}\equiv\Delta_{4A}\big{|}_{\gamma=0}, while the 3-derivative fermion and the 2-derivative scalar operators in gauge field background may be written as171717 In the first form of $\Delta_{3\Psi}$ the derivative in the second term acts all the way to the right while in the term term it acts only on $F_{mn}$.

[TABLE]

Here $i\not{{\mathcal{\nabla}}}^{3}$ is the cube of the Dirac operator $\Delta_{{1\Psi}}=-i\not{\nabla}=-i\Gamma^{m}\nabla_{m}$ whose square is

[TABLE]

As a result, the one-loop effective action of the supersymmetric theory (4.7) is the following generalization of the bosonic case (2.6)

[TABLE]

Here the contributions of the ghost and gauge-averaging operators in (2.6) got canceled against the contribution of the three scalars $\Phi_{I}$. We also used that $\det\Delta_{\Psi}$ is defined for the Dirac 6d spinors so that the factor ${1\over 2}$ accounts for the fact that the fermion $\Psi$ is a Weyl spinor. As a result, the coefficient of the log divergent part of the effective action (2.7) is given by (cf. (4.1))

[TABLE]

Setting $\gamma=0$ in (4.2) gives

[TABLE]

To compute the fermionic contribution, let us first construct a 4-derivative operator by taking the product of $\Delta_{3\Psi}$ in (4.8) with the standard Dirac operator $\Delta_{1\Psi}=-i\not{\nabla}$

[TABLE]

$\Delta_{4\Psi}$ is then a 4-order operator of the form (2.4) with the coefficients181818 Notice that this operator is not self-adjoint, i.e. the symmetry requirements in footnote 8 are not satisfied.

[TABLE]

Applying the general expression for $b_{6}(\Delta_{4})$ 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 $\Delta_{1\Psi}$ one obtains (4.9), for which $b_{6}$ can then found from (3.5), we end up with

[TABLE]

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

[TABLE]

This is the same result as quoted in (1.2), (1.7). The resulting renormalized coupling in (4.7) is thus (cf. (2.7), (4.7))

[TABLE]

corresponding to an asymptotically free behaviour. This agrees with the (recently revised) result of [5] (cf. footnote 16). Note that the computation of the $\beta$-function in [5] was done in the scalar field $\Phi_{I}$ 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 ordinary-derivative (1,0) 6d SYM theory

[TABLE]

where $\Psi$ is a Weyl spinor, $\Phi_{I}$ are 3 auxiliary fields (cf. (4.7)) and $\kappa$ is a mass scale. The analog of the one-loop effective action in a gauge field background (4.10) here is

[TABLE]

Using (3.5) we get

[TABLE]

As a result, the one-loop logarithmic divergence is given by (2.7) with

[TABLE]

Once again, the $F^{3}$ divergence cancels, and (4.21) implies the value of $\beta_{2}=-20$ in (1.2), (1.5). Since here ${\mathcal{\nabla}}_{m}F_{mn}=0$ is an equation of motion, the divergence (4.21) vanishes on-shell, i.e. the (1,0) 6d SYM theory is finite on-shell191919 The coefficient in (4.21) here is, in fact, gauge-dependent, see also [31]. 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 ${\mathcal{\nabla}}_{m}F_{mn}=0$ (which is a special on-shell background also in this theory). Then $\Delta_{3\Psi}$ in (4.8) becomes simply $(\Delta_{1\Psi})^{3}=i\not{\nabla}^{3}$ and also the vector field operator in (2.4), (2.5) (with $\gamma=0$) becomes a square of the standard YM operator in (4.19), i.e. $\Delta_{4A}=(\Delta_{2A})^{2}$. As a result, the effective action (4.10) reduces to

[TABLE]

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 $\beta$-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 $\beta$-functions in the 6d gauge theory (1.1) coupled to standard fermions was performed.202020We are grateful to I. Klebanov for drawing our attention to this paper. After correcting a mistake in the original version of this paper we found that our result (1.3), (1.4) for the $\beta$-functions of the theory (1.1) coupled to fermions is in full agreement with the one-loop $\beta$-functions in [33].212121The translation between the notation in [33] and ours is as follows. Instead of $(\nabla_{m}F_{mn})^{2}$ in (1.1) the action in [33] contained $(\nabla_{k}F_{mn})^{2}$ with the two invariants related as in footnote 3. As a result, the couplings $g_{1}$ and $g_{2}$ in [33] are related to ours as $g_{1}=g,\ \ g_{2}=3g(1+\gamma)$ (using also that $g_{2}\to-g_{2}$ due to apparent sign difference in notation for $F_{mn}$). For the gauge theory (1.1) coupled to Weyl fermions in generic representation our result (1.3), (1.4) for the $\beta$-functions reads (cf. (4.5), (4.6)): $\beta_{g}\equiv{dg\over dt}=-{1\over 120}C_{2}\beta_{2}$,   $\beta_{\gamma}\equiv{d\gamma\over dt}=-{1\over 120}C_{2}(2\gamma\beta_{2}+{2\over 3}\beta_{3})g^{2}$,   $\beta_{2}=249+N_{1\over 2},\ \beta_{3}=9-900\gamma+{405\over 2}\gamma^{3}-4N_{1\over 2},\ \ \ N_{1\over 2}={T_{R}\over C_{2}}N_{f}$. Then the $\beta$-functions for the above $g_{1}$ and $g_{2}$, i.e. $\beta_{g_{1}}={dg_{1}\over dt}=\beta_{g},\ \ \beta_{g_{2}}={dg_{2}\over dt}=3\beta_{g}(1+\gamma)+3g\beta_{\gamma}$ match the expressions in [33].

Appendix A Derivation of the expression for $b_{6}(\Delta_{4})$

The operator that we shall consider is

[TABLE]

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

[TABLE]

The general expression for its coefficient $b_{6}$ including only independent invariants may be written as ($V\equiv V_{mm}$)

[TABLE]

where the trace is over internal indices and $k_{i}$ are real coefficients.222222 The relations between the $k_{i}$ and $\hat{k}_{i}$ in (3.8) are, using (A.2), $\hat{k}_{6}=k_{6}+\frac{1}{2}k_{15}-\frac{1}{4}k_{17},\ \hat{k}_{8}=k_{8}+\frac{1}{2}k_{16},\ \hat{k}_{10}=k_{10}-\frac{1}{2}k_{15}+\frac{1}{4}k_{17},\ \hat{k}_{11}=k_{11}+\frac{1}{2}k_{14},\ \hat{k}_{15}=k_{15}-k_{17},\ \hat{k}_{16}=k_{16}+k_{18}$ with $\hat{k}_{i}=k_{i}$ otherwise. Their values in $d=6$ found below are

[TABLE]

To determine $k_{i}$ we shall exploit the factorization property (3.6), i.e.

[TABLE]

where $b_{6}(\Delta_{2})$ is given by (3.5). As was already remarked in Section 3, such factorization applies here because we are considering $b_{6}$ in 6 spacetime dimensions.

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 $\Delta_{2}=-\nabla^{2}+X$ and $\Delta_{2}^{\prime}=-\nabla^{2}+X^{\prime}$ their product is given by (A.1) with

[TABLE]

Using (3.5) and comparing with (A.3) gives

[TABLE]

Next, let us assume that

[TABLE]

Here $\nabla_{m}K_{n}=\partial_{m}K_{n}+[A_{m},K_{n}]$ ($K_{m}$ is in the adjoint representation of the gauge group). The coefficient functions in the corresponding operator $\Delta_{4}=\Delta_{+}\Delta_{-}$ in (A.1) read

[TABLE]

Using (3.5) and the relations

[TABLE]

[TABLE]

one can compute $b_{6}(\Delta_{\pm})$ and then compare to $b_{6}(\Delta_{4})$ in (A.5).

It is enough to consider the following special cases:

Abelian gauge group, $\nabla_{n}K_{m}=\partial_{n}K_{m}$, $[F_{mn},K_{k}]=0$. In (A.5) we consider the terms with $\partial^{r}K_{m}$, $r=0,1,4$ that can always be uniquely cast into the form

[TABLE]

Then comparing also the coefficients of $F_{nm}K^{2}\partial_{n}K_{m}$ and $F_{nm}K_{m}\partial_{n}K^{2}$ (the latter does not actually appear) one obtains

[TABLE] 2. 2.

$K_{n}$ constrained by $\nabla_{m}K_{n}=0,$ implying $2{\mathcal{\nabla}}_{[k}{\mathcal{\nabla}}_{m]}K_{n}=[F_{km},K_{n}]=0.$ This leads to a number of nontrivial relations, e.g.,  $\operatorname{tr}([K_{m},K_{n}]F_{nk}F_{km})=0$. All the remaining invariants can be uniquely written as a combination of

[TABLE]

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)

[TABLE] 3. 3.

General unconstrained $K_{n}$, comparing the terms with one $K_{m}$ or two of them contracted together. A basis of such tensors contains

[TABLE]

In this case we obtain (the two $KKFF$ terms give the same equation)

[TABLE]

The final system of equations is given by (A.7), (A.14), (A.16) and (A.18). This system is over-determined, with the unique solution for $k_{i}$ given by (A.4). That some of the equations are actually redundant gives a non-trivial consistency check of the calculation. We also checked some of the coefficients $k_{i}$ by explicit diagrammatic calculations of the corresponding UV divergences.

Appendix B One-loop divergences in $F^{2}+(\nabla F)^{2}+F^{3}$ 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)

[TABLE]

Here $\Delta_{2A}$ 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. (3.3))

[TABLE]

The expression for $b_{4}$ is known for both for $\Delta_{2}$ (3.4) and $\Delta_{4}$ (A.1) operators [27, 35]232323 Here tr and $F_{mn}$ are the general trace and the curvature on the bundle, cf. footnote 11. The factor $\sqrt{\pi}$ comes from an overall $\Gamma[\frac{d}{4}]$ in the expression of the heat kernel coefficient $b_{4}(\Delta_{4},d)$.

[TABLE]

The coefficient $b_{4}$ controls the logarithmic divergences in the corresponding 4d theory where their computation was done in [2] (see also [4]). For the operators in (B.1) we get in $d=6$ (here $\operatorname{tr}$ is in the adjoint representation and $F_{mn}$ is the gauge field strength)242424 $b_{4}(\Delta^{\prime}_{4A})$ has two sources of dependence on the space-time dimension $d$: the operator itself and the coefficients in $b_{4}$ in (B.5).The gauge fixing contributions are independent of $d$. As a result, in 4d theory the coefficient $\beta_{1A}$ in (B.12) below is given by $\beta_{1A}=-2(43+108\gamma+27\gamma^{2})$ (cf. [2, 4]).

[TABLE]

Similarly, using (3.5) and (3.7), (3.8) we find

[TABLE]

As a result, the total values of the coefficients of the quadratic and logarithmic divergences in (B.2) in $d=6$ are (omitting field-independent terms)

[TABLE]

where $\beta_{2A}$ and $\beta_{3A}$ in (B.11) are the same as in (1.3). Ignoring non-universal quadratic divergence (absent in dimensional regularization), the logarithmic renormalization of $\kappa$ is controlled by $\beta_{\kappa,A}$ with the RG equation (cf. (4.5), (4.6))252525 Recall that the coefficient of the YM term is chosen as $\kappa^{2}\over g^{2}$, cf. (4.18).

[TABLE]

Near both the attractive fixed points $\gamma_{1}\simeq-0.897$ and $\gamma_{3}\simeq 0.838$ of $\beta_{\gamma}$ in (4.6), the r.h.s  of (B.13) is negative and thus $\kappa^{2}\to 0$ in the UV.

Let us now consider the log divergence in the (1,0) supersymmetric extension of this bosonic model, i.e. the $(1,0)$ SYM combined with the (1,0) theory (4.7). Here the operators in the 1-loop effective action (4.10) get $\kappa$-dependent terms as in (B.1) (with $\gamma=0$)

[TABLE]

where $\Delta_{1\Psi}=-i\not{\nabla}$ and $\Delta_{2\Phi}=-\nabla^{2}$. Explicitly, we get (cf. (4.10), (B.1))

[TABLE]

For the gauge field and scalar determinants the expressions for $b_{4}$ and $b_{6}$ are given by (B.6)–(B.9) with $\gamma=0$ while for the fermion contribution we get as in (4.13),

[TABLE]

As a result, the analog of (B.12) is

[TABLE]

where $\beta_{2\,(1,0)}$ is the same as in (4.16), (1.7). Since the combination $\beta_{\kappa\,(1,0)}+\frac{1}{60}\beta_{2A}$ is negative, as a result of (B.13) we do not have asymptotic freedom in the supersymmetric case.

As a final comment, notice also that on $\nabla_{m}F_{mn}=0$ background (B.15) becomes the following generalization of (4.2)

[TABLE]

i.e. the sum of contributions of massless (1,0) SYM, its massive analog, and a massive analog of scalar multiplet.