# Inconsistency of Minkowski higher-derivative theories

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## Abstract

We show that Minkowski higher-derivative quantum field theories are generically inconsistent, because they generate nonlocal, non-hermitian ultraviolet divergences, which cannot be removed by means of standard renormalization procedures. By “Minkowski theories” we mean theories that are defined directly in Minkowski spacetime. The problems occur when the propagators have complex poles, so that the correlation functions cannot be obtained as the analytic continuations of their Euclidean versions. The usual power counting rules fail and are replaced by much weaker ones. Self-energies generate complex divergences proportional to inverse powers of D’Alembertians. Three-point functions give more involved nonlocal divergences, which couple to infrared effects. We illustrate the violations of the locality and hermiticity of counterterms in scalar models and higher-derivative gravity.

## Keywords

Dimensional Regularization External Momentum Power Counting Logarithmic Divergence Loop Momentum## 1 Introduction

The ultraviolet structure of quantum field theories is notoriously a fundamental problem in high-energy physics. Nowadays, with the Large Hadron Collider currently running at a center-of-mass energy of 13 TeV, the standard model is experimentally verified at the TeV scale. On the theoretical side, the renormalizability of the standard model, together with the relatively small value found for the Higgs mass, \(m_{H}\simeq 125\) GeV, imply that the model could be valid at energies much higher than the ones investigated so far. The interest in a high-energy modification of the standard model is, therefore, rather limited, on the practical side. As far as quantum gravity is concerned, the situation is different. The negative mass dimension of the coupling constant (compared with the dimensionless gauge couplings of the standard model) makes the Hilbert–Einstein action nonrenormalizable [1, 2, 3, 4, 5]. This fact, together with the difficulties to build up a phenomenology, render the investigation of alternative high-energy structures of quantum gravity more attractive.

*M*is the scale associated with the higher-derivative terms and the dots denote convergent terms. Similar results occur in four dimensions, when vertices carry derivatives. More involved nonlocal structures appear in triangle diagrams. Moreover, gauge symmetries are unable to protect the locality and hermiticity of counterterms. We prove this fact by extending the calculations to a model of higher-derivative quantum gravity [6, 7].

The rules of power counting obeyed by Minkowski higher-derivative theories are much weaker than the standard ones, because a propagator calculated on the pole of another propagator falls off half less rapidly than expected. This property implies that the higher-derivative terms often have an “anti-regulating” effect, in the sense that they enhance divergences rather than suppressing them. The divergences (1.1) can also be related to specific pinch singularities occurring for \(p^{2}\rightarrow 0\), which have no direct analog in standard field theories.

It is well known that, in general (for example, when the free propagators have poles infinitesimally close to the real axis, in the second and fourth quadrants), Minkowski higher-derivative theories are physically unacceptable, because they violate perturbative unitarity. Our results show that when the free propagators also contain poles that are located at finite distances from the real axis, in the first and third quadrants, the theories are in general unacceptable from the mathematical point of view, because they violate both the locality and the hermiticity of counterterms.

The problems we have found do not occur in Euclidean theories, Lee–Wick models [8, 9, 10, 11] or Minkowski theories that are analytically equivalent to their Euclidean versions. In those cases, nonlocal divergences may appear in some intermediate steps of the calculations (examples being the residues of the integrals on the energies, if taken separately), but they cancel out at the end.

The paper is organized as follows. In Sect. 2 we study some key aspects of the higher-derivative Minkowski propagator. In Sect. 3, we calculate the nonlocal divergent part of the bubble diagram in six dimensions and generalize the calculation to the bubble diagram with nontrivial numerators in four dimensions. In Sect. 4, we study the one-loop triangle diagram and provide an interpretation for its nonlocal divergent part. In Sect. 5 we investigate the modified power counting of Minkowski quantum field theories. In Sect. 6, we study a higher-derivative version of quantum gravity in four dimensions and prove that gauge symmetries fail to protect the locality and hermiticity of counterterms. Finally, in Sect. 7 we draw our conclusions.

In Appendix A we show that the dimensional regularization (of the integrals on the space momenta) allows us to apply the residue theorem on the energy integrals, even when they are divergent. In Appendix B we discuss the gauge fixing of Minkowski higher-derivative gravity and show that the Ward–Takahashi–Slavnov–Taylor (WTST) identities [12, 13, 14, 15] are also plagued with nonlocal divergences.

## 2 Higher-derivative propagator

*M*effectively act as \(\pm i\eta \) prescriptions for the propagation of exotic, high-energy excitations on the light cone, with the large lifetimes

Since \(\eta \left( p_{s}\right) \rightarrow 0\) for \(p_{s}\rightarrow +\infty \), there is a pinch singularity of the pole located in the first quadrant with the two poles located in the fourth quadrant, and a similar pinch singularity of the pole located in the third quadrant with the two poles located in the second quadrant. The violations of power counting that we find in the next sections can be traced back to this pinching and ultimately to the presence of both the \(+i\eta \) and the \(-i\eta \) terms at \(p_{s}\gg M\).

## 3 Bubble diagrams

In this section we compute the nonlocal divergent parts of the higher-derivative, one-loop scalar bubble diagrams in six and four dimensions, with trivial and nontrivial numerators.

As usual, the ultraviolet divergent part is a sum of powerlike divergences and logarithmic divergences. The powerlike divergences are less interesting than the logarithmic ones for the purpose of singling out inconsistencies, because they depend on the subtraction scheme and can be removed in a renormalization-group invariant way. The one-loop logarithmic divergences, on the contrary, do not depend on the regularization scheme, and provide meaningful tests of the locality of counterterms. For these reasons, we focus our attention mostly on them. We either use the dimensional regularization technique or a sharp cutoff \(\Lambda _{UV}\) on the space momenta of the loops, according to convenience. We always convert the outcome to the cutoff notation.

### 3.1 Bubble diagram in six dimensions

*D*spacetime dimensions gives the loop integral

Since the higher-derivative theory is relativistically invariant, \(\Sigma (p) \) is expected to be a function of \(p^{2}\) only.^{1} It is then convenient to consider a timelike external momentum, \( p^{2}>0\), and select a Lorentz frame in which \(p^{\mu }\) has only the time component, \(p^{\mu }=\left( p_{0},\mathbf {0}\right) \). Once the loop integral is evaluated, we can retrieve the Lorentz invariant result by means of the replacement \(p_{0}^{2}\rightarrow p^{2}\). The values of the bubble diagram for a spacelike external momentum, \(p^{2}<0\), are obtained by means of the analytic continuation in \(p^{2}\).

The next step is to integrate over the space momentum \(\mathbf {k}\). The angular integration is trivial, because of our choice of the Lorenz frame, and gives the volume \(\Omega _{D-2}\) of the unit sphere in \(D-2\) dimensions.

The divergent part is nonlocal, equal to the sum of a term proportional to \( 1/(p^{2})^{2}\) plus a term proportional to \(1/p^{2}\). Differently from the usual divergences of local theories, which are anti-hermitian, the ones of (3.2) are not, since the coefficients have nontrivial real and imaginary parts. For these reasons, we cannot absorb the divergent part in the usual way, by shifting the bare masses, rescaling the bare fields and adding new local, hermitian terms to the Lagrangian. We cannot even add nonlocal hermitian terms. We conclude that the locality and hermiticity of counterterms are both violated.

*u*poles. For example, consider a typical denominator that is met in the calculation, such as

*u*integrals. However, according to (3.3), the

*u*pole has a positive imaginary part. If we first replace

*u*by \(u-i\epsilon \), with \(\epsilon \) arbitrarily small, then the expansion for large \(k_{s}\) is safe. So doing, no fictitious singularity is generated.

The procedure works as long as the integrand can be arranged so that the powers \(1/(u-i\epsilon )^{n}\) do not mix with the powers \(1/(u+i\epsilon )^{n}\). It can be shown that the bubble diagram has this property. Carrying on the computation to the end, we find (3.2) again.

### 3.2 Bubble diagram in four dimensions

As shown in the previous section, the bubble diagram with unit numerator has nonlocal divergences only in dimensions \(D\geqslant 6\). On the other hand, bubble diagrams with nontrivial numerators may have nonlocal divergences also in four dimensions. In this section we study typical one-loop integrals of this type. Their applications to higher-derivative gravity will be considered in Sect. 6.

Then we remain with the integral on the space momentum \(\mathbf {k}\). The logarithmic nonlocal divergences \(I_{r,n}^{\text {nld}}\) of \( I_{r,n}\) are obtained by expanding the integrand in powers of the absolute value \(k_{s}=|\mathbf {k}|\) and isolating the contributions proportional to \( \mathrm {d}k_{s}/k_{s}\).

*M*. The lowest-order coefficients are

*r*from the numerator. Instead, the residues with \(\left( k-p\right) ^{2}=\) constant have degree of divergence \(\omega _{2}=D-5+r+n\), where the

*n*additional powers come from the numerator, using \(k^{2}\sim 2k\cdot p\). For \(r=1\), \(n=0\), we have \( \omega _{1}=\omega _{2}=D-4\), so ultraviolet logarithmic divergences are expected from both types of residues in \(D=4\). However, formula (3.6) shows that the coefficient \(c_{1,0}\) vanishes. This may be interpreted as an eikonal cancellation between the two types of residues. For \(r=0\), \(n=1\), we have \(\omega _{2}>\omega _{1}=D-4\), so the cancellation cannot occur in \(D=4\). Indeed, \(c_{0,1}\) is different from zero. More generally, cancellations between the ultraviolet divergences of the residues are unlikely to occur for \(n>0\). Indeed, all the coefficients (3.6) with \(n>0\) are nonvanishing.

*m*/

*M*. The lowest-order ones are

*S*(

*p*,

*m*) with the more general ones

We see that the locality and hermiticity of counterterms are violated again. The nonlocal behavior is always of the form \(1/p^{2}\), but it must be recalled that the integrals (3.4) contain *r* powers of \(p^{\mu }\) in the numerator, through the term \((k\cdot p)^{r}\). If we divide by those powers, the true nonlocal behavior of the divergent part is \(\sim 1/(p^{2})^{(2+r)/2}\).

## 4 Triangle diagrams

*Q*is the hard scale and \(\mu \) is the virtuality of external legs.

*u*integral. At the end, the logarithmic divergences are the coefficients of \( \mathrm {d}k_{s}/k_{s}\).

*t*-channel, with an ultraviolet logarithmically divergent coefficient.

It is natural to expect that nonlocal divergences also occur in one-loop box diagrams, pentagon diagrams, etc., and that they overlap the usual logarithmic structures (infrared divergences, small-*x* logarithms, etc.).

We conclude by briefly reporting results concerning the massive case \(m\ne 0 \). In both \(I_{1,2,1}^{\text {nld}}\) and \(I_{2,2,0}^{\text {nld}}\), it is sufficient to replace \(M^{8}\) with \( M^{10}/(M^{2}+im^{2})\) in Eqs. (4.5) and (4.6).

## 5 Power counting for nonlocal divergences

*a*is a vector and

*b*is a constant. The reason why this intuitive argument is not correct is that it overlooks the role played by the \(+i\epsilon \) prescription, which allows the Wick rotation.

On the other hand, the higher-derivative propagator *S*(*k*, *m*) has poles in all quadrants, so the analytic continuation to Euclidean space is not possible. Then the rules of power counting are no longer guaranteed to coincide in Minkowski and Euclidean spaces and actually turn out to be different.

To illustrate this fact, we work out a formula for the degree of divergence of a generic one-loop diagram in Minkowski higher-derivative theories. Assume that the propagators *S*(*k*, *m*) behave like \(1/(k^{2})^{N}\) for large \( |k^{2}|\) and that the vertices contain up to \(N^{\prime }\) derivatives. Consider a one-particle irreducible diagram with *V* vertices, equal to the number of internal lines. We assume that \(V>1\), i.e. exclude the tadpoles, because they are independent of the external momenta and cannot originate nonlocal divergences.

Letting *k* denote the loop momentum, we integrate over the energy \(k_{0}\) by means of the residue theorem. In each residue, \((k-q)^{2}\) is equal to some constant, *q* being a linear combination of external momenta. Making a translation, we can assume that the integrand is evaluated at \(k^{2}=\) constant. Then by Eq. (2.4) the propagator *S*(*k*, *m*) gives a contribution that behaves like \(1/k_{s}\) for large \(k_{s}\), while each one of the other \(V-1\) propagators behaves like \(1/((p-k)^{2})^{N}\sim 1/(p\cdot k)^{N}\), where *p* is also a linear combination of the external momenta. If we use analyticity to assume \(p^{2}>0\), the factors \(1/(p\cdot k)^{N}\) are regular everywhere.

*N*is odd. Then

If \(D=6\), \(N=3\), \(N^{\prime }=0\), \(V=2\), which is the case treated in Sect. 3.1, we have \(\omega _{\text {nl}}^{\prime }=0\), which confirms that there is a logarithmic divergence. The same diagram in four dimensions has no nonlocal divergences (\(\omega _{\text {nl}}^{\prime }=-2\)), unless we equip it with nontrivial numerators. If we take \(N^{\prime }=1\), we raise \(\omega _{\text {nl}}^{\prime }\) to 0, which is confirmed by the nonvanishing coefficient \(c_{0,1}\) of Eq. (3.6). On the other hand, it is not enough to have a vertex with one derivative and a vertex with no derivatives (which can be formally obtained by setting \(N^{\prime }=1/2\)), as the vanishing of the coefficient \(c_{1,0}\) confirms.

In the case of the triangle diagram (\(D=4\), \(N=3\) and \(V=3\)), we may distribute the \(r+2n+t\) derivatives over the three vertices by formally writing \(N^{\prime }=(r+2n+t)/3\). The integrals \(I_{1,2,1}^{\text {nld}}\) and \(I_{2,2,0}^{\text {nld}}\) have \(N^{\prime }=2\) and \(\omega _{\text {nl}}>0\), indeed formulas (4.5) and (4.6) shows that they are divergent. Moreover, \(r+2n+t<4\) implies \(\omega _{\text {nl}}<0\), which agrees with Eq. (4.4). A better agreement can be obtained by improving the power counting as shown in Sect. 4. Indeed, after a residue is evaluated, a \(k^{2}\) factor in the numerator does not provide two powers of \(k_{s}\), but one at most. This is equivalent to setting \(N^{\prime }=(r+n+t)/3\). Then Eq. (4.4) follows in all cases. Moreover, both \(I_{1,2,1}^{\text {nld}}\) and \(I_{2,2,0}^{ \text {nld}}\) have \(N^{\prime }=4/3\), \(\omega _{\text {nl}}=0\), which implies that the ultraviolet divergence is at most logarithmic, as is actually the case.

Let us inquire which theories have no nonlocal divergences at one loop, i.e. when \(\omega _{\text {nl}}<0\) for every \(V>1\). Formula (5.1) shows that this happens when \(D-2<N-2N^{\prime }\) and \(N\geqslant N^{\prime }\). All the scalar and fermion theories with nonderivative interactions satisfy these conditions for *N* sufficiently large, in arbitrary dimensions. For example, in four-dimensional scalar models with nonderivative interactions it is sufficient to take \(N=3\). As far as the fermions are concerned, assume that their propagators \(S_{F}(k,m)\) behave as \(k^{\mu }/(k^{2})^{N}\) for large \( |k^{2}|\). Then we can attach their numerator \(k^{\mu }\) to a nearby vertex, so the arguments given above apply with \(N^{\prime }\rightarrow N^{\prime }+1 \). The higher-derivative theories of gauge fields have \(N^{\prime }=2N-1\), while those of gravity have \(N^{\prime }=2N\), so neither of the two satisfies the conditions for having no nonlocal divergences at one loop. Both are expected to violate the locality of counterterms, if their propagators have poles in the first or third quadrants. In the next section we study the case of gravity explicitly.

## 6 Higher-derivative gravity

In this section we use the results of the previous ones to work out the nonlocal divergences of the graviton two-point function in a relatively simple model of four-dimensional higher-derivative gravity with complex poles. We simplify the calculations as much as possible by choosing a specific Lagrangian and a convenient gauge fixing. The loop integrals are linear combinations of the scalar integrals (3.4).

*R*, \(R^{2}\) and \(R_{\mu \nu }R^{\mu \nu }\). However, it is not suitable for our investigation, because its propagators do not have poles in the first or third quadrants. The simplest model with the features we need is the one with Lagrangian

*a*) encodes the nonlocal divergences of the graviton self-energy. Diagram (

*b*) encodes the nonlocal renormalization of the \(h_{\mu \nu }\) transformation, which is necessary to derive the corrections to the Ward identities satisfied by (

*a*), as explained in Appendix B.

### 6.1 Results

*b*) as explained in Appendix B. It is easy to show that (6.7) does satisfy (B.5), which provides a good check of the result.

Coherently with what we found in the previous sections, the divergences are nonlocal and truly complex. It is impossible to subtract them away by means of reparametrizations and (local as well as nonlocal) field redefinitions that preserve hermiticity.

In conclusion, Minkowski higher-derivative theories of gravity violate the locality and hermiticity of counterterms, when the propagators have poles in the first or third quadrants. Gauge symmetries are unable to protect those properties.

If gravity is coupled to matter, we expect to find similar behaviors in the matter sector. In particular, if the kinetic terms of the matter fields have the same numbers of higher derivatives as the gravitational sector has, the power counting is the same.

## 7 Conclusions

We have shown that Minkowski higher-derivative quantum field theories whose propagators have complex poles are generically inconsistent, because they generate nonlocal, non-hermitian ultraviolet divergences. Bubble diagrams, for example, contain logarithmic divergences multiplied by inverse powers of D’Alembertians. Triangle diagrams present more involved nonlocal divergences, where ultraviolet effects mix with standard infrared effects.

Contrary to intuitive expectations, the introduction of higher-derivative terms in the Lagrangian does not have a regulating effect, because the constraints coming from power counting are much weaker. Indeed, the contribution of one propagator calculated on the pole of another propagator does not decay fast enough. This unusual behavior can also be explained by the appearance of pinch singularities, unrelated to the usual absorptive parts of amplitudes, which occur because the extra excitations introduced by the higher derivatives come with effective prescriptions of both signs.

We have extended the calculations to higher-derivative quantum gravity and proved, in particular, that gauge symmetries are unable to protect the locality and hermiticity of counterterms. The problems we have outlined add up to the well-known problems that higher-derivative Minkowski theories have with perturbative unitarity.

## Footnotes

- 1.
To be rigorous, one should use an ultraviolet regularization that preserves Lorentz invariance, such as the dimensional regularization, instead of a cutoff on the space momenta. However, we are only interested in the logarithmic divergences, which, as already noted, are independent of this choice.

## Notes

### Acknowledgements

One of us (D.A.) is grateful to M. Piva for useful discussions.

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