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We construct an explicit example of unitarity violation in fermionic quantum field theories in noninteger dimensions. We study the two-point correlation function of four-fermion operators. We compute the one-loop anomalous dimensions of these operators in the Gross-Neveu-Yukawa model. We find that at one-loop order, the four-fermion operators split into three classes with one class having negative norms. This implies that the theory violates unitarity, following the definition in Ref.

Conformal field theories (CFTs) have always been an area of active research due to their rich mathematical structure and physical applications. In unitary theories conformal symmetry imposes severe constraints on the spectrum of operator dimensions. It is believed that these dimensions can be determined with the help of the conformal bootstrap technique

The standard technique for the calculation of the operator dimensions, the so-called

The aim of this article is to demonstrate the existence of the negative norm states in the

The article is organized as follows: In Sec.

In order to continue our discussion, we then compute in Sec.

The GNY model describes an interacting fermion-boson system with the Lagrangian given by the following expression

The model has an infrared stable fixed point in

Let us consider an infinite system of four-fermion local operators in

Before taking a closer look at correlators of the operators

Note that in even dimensions,

In noninteger dimensions, however, the situation is different. There exists an infinite number of

The renormalized operators

At the critical point

Let us write the correlator

At leading order only the two Feynman diagrams shown in Fig.

A nontrivial relation in integer dimensions

Feynman diagrams of leading order.

We point out that

So far our calculations are rather general and can be applied to any fermionic theory in noninteger dimensions. In order to continue our study of norm states in a conformal theory, according to Eq.

Feynman diagrams for calculating anomalous dimensions of evanescent operators.

Additional Feynman diagrams contributing to the anomalous dimensions of physical operators.

Then it is straightforward to compute these one-loop diagrams and obtain the anomalous dimension matrix

It is clear from the explicit expression of the anomalous dimension matrix that the physical and evanescent operators decouple at one-loop order. We can therefore study them separately and find the conformal basis in each case.

Let us write the physical operators in conformal basis as

The conformal basis for evanescent operators, denoted as

Anomalous dimensions (AD) of the different physical and evanescent operators.

In order to find the negative norm states of the theory, we have to consider correlation functions between operators of the same anomalous dimension. According to Eq.

With the orthogonality condition checked at the one-loop order, let us now focus on the evanescent operators in the conformal basis. We write the correlator as

The matrices

It is possible to show that all the leading principal minors of

In the large

On the contrary, for small values of

For the sake of clarity one should mention that the mixing with physical operators is here neglected. But it can be shown that the effect of the physical operators is at order

This is because

We then conclude that the negative norm states are an integral part of the GNY model in

We have demonstrated the existence of negative norm states in the Gross-Neveu-Yukaw model in

It is now clear that unitarity violation occurs in both the scalar and fermionic case. In addition, a recent study also reveals that unitarity is violated in noninteger dimensional nonrelativistic conformal field theory

We can’t see any way to consistently remove these negative norm states from the fermionic field theory in noninteger dimensions. They have no effect, however, on theories in integer dimensions where all the negative norm states vanish.

One should mention, however, that although the loss of unitarity prohibits imposing extra constraints while applying the bootstrap technique, the “non-unitary bootstrap” technique, which has no reliance on unitarity, still works

It would be a natural extension of our current study to compute the two-loop anomalous dimension matrix and investigate how the operators in the conformal basis at the two-loop order further classify the negative and positive norm states. It would be interesting to investigate the appearance of negative norm states in other fermion/scalar conformal field theories as well.

The authors are grateful for insightful discussions with Vladimir Braun and Alexander Manashov. Y. J. acknowledges the Deutsche Forschungsgemeinschaft for support under Grant No. BR 2021/7-1.

We provide some key steps in our calculations. The Feynman diagrams in Fig.

We then calculate

The cyclic property of the trace together with the anticommutation relation between gamma matrices allows us to first conclude that

By setting the default ordering of

Here we have reduced a trace with

It is clear that

We proceed to calculate

The recurrence relation can be expressed as

Then the recursive behavior in Eq.

The Feynman diagrams in the first row of Fig.