# On the c-theorem in more than two dimensions
^{†}^{†}thanks: Preprint SPhT t00/131; proceedings of the TMR Conference
“Non-perturbative Quantum Effects 2000”, Paris, 7-13 Sept. 2000,
D. Bernard and B. Julia Eds., to be published in the J. of High-Energy
Physics.

###### Abstract:

Several pieces of evidence have been recently brought up in favour of the -theorem in four and higher dimensions, but a solid proof is still lacking. We present two basic results which could be useful for this search: i) the values of the putative -number for free field theories in any even dimension, which illustrate some properties of this number; ii) the general form of three-point function of the stress tensor in four dimensions, which shows some physical consequences of the -number and of the other trace-anomaly numbers.

^{†}

^{†}conference: Non-perturbative Quantum Effects 2000

## 1 Introduction

### 1.1 The -theorem in two dimensions

The renormalization-group (RG) flow is defined as the one-parameter motion in the space of (renormalized) coupling constants ,

(1.1) |

with “velocities” given by the beta-functions; the flow corresponds to a change of scale in the field theory which grows towards the infrared.

The Zamolodchikov -theorem[1] holds for unitary, renormalizable quantum field theories in two dimensions; it says that there exists a positive-definite real function of the coupling constants such that:

i) it is monotonically decreasing along the flow,

(1.2) |

ii) it is stationary at the fixed points ,

(1.3) |

iii) at the fixed points, it equals the Virasoro central charge of the corresponding conformal field theory,

(1.4) |

This theorem implies some fundamental properties of the RG flow:

i) The flow necessarily ends into fixed points (or fixed surfaces); there cannot exist limit cycles or strange attractors, which are other possible asymptotic behaviours for the solutions of non-linear differential equations (1.1).

ii)
The fixed points are classified according to the value
of their central charge; we can think the space of theories^{1}^{1}1
Namely, the space of coupling constants.
as a mountain landscape, with
the fixed points located at the tips, the saddles and the
valleys bottoms.

The central charge is a measure of the “number of degrees of freedom” and its decreasing along the RG flow can be viewed as the consequence of “coarse graining”, the integration of high-energy degrees of freedom in the Wilsonian approach to the renormalization group [2]. Note that a theory with an asymptotic limit cycle would have a never-ending infrared flow, with degrees of freedom periodically dying out and coming back; it would be very difficult to make sense of this RG behaviour in a unitary field theory. In conclusion, the -theorem confirms our intuitive understanding of the RG flow.

Let us remark some aspects of the Zamolodchikov proof that will be useful for the forecoming discussion:

i) The inputs of the proof are just “kinematics”, i.e. general properties of Poincare invariance, unitarity and renormalizability – there are no hypotheses on the dynamics of the theory.

ii) The function is finite once the coupling constants are renormalized; moreover, its critical value is uniquely defined, since the trace anomaly is both finite and universal (as any other anomaly). It follows that is defined globally (i.e. non-perturbatively) on the whole space of theories.

### 1.2 The -theorem in higher dimension: motivations and overview of the results

The consequences of the -theorem on the RG flow are so general that it is natural to expect its extension to higher-dimensional field theories. However, more than ten years have passed since the first attempts to a generalization [3][4][5]. First of all, a straightforward extension of the Zamolodchikov argument is not possible [5]. Secondly, in odd dimension , the -function lacks the natural global definition given by the trace anomaly, because the latter is equal to zero (actually, it is very easy to construct functions which are monotonically decreasing along the flow but are discontinuous at fixed points).

It seems that the extension of the -theorem to higher dimension requires a new ingredient, possibly involving the field-theory dynamics. In this respect, we believe that the eventual proof of the theorem could teach us new properties of field theories and of their interaction with gravity (through their stress tensor). Therefore, the interest of the -theorem extends beyond the proof of mandatory properties of the RG flow.

Several works have recently discussed
the -theorem in four dimensions,
by providing new arguments for the proof [6][7]
[8] and by analysing examples of RG flows for
the trace anomaly coefficients , and [9][10].
These are defined by the following expression^{2}^{2}2
The coefficient is included
to normalize the values of and to one for
the free conformal-invariant scalar field.
[11]:

(1.5) |

where is the Euler characteristics, is the square of the Weyl tensor and is the curvature scalar.

-theorem:

has been exactly proven [9]
for the non-trivial RG flows among
supersymmetric gauge theories found by Seiberg, most notably those
in the “conformal window” [12].

-theorem:

cannot be true in general, because
counterexamples are known [5][9]; however, it holds for
the field theories which have a gravity dual theory
according to the AdS/CFT correspondence [13][10],
such as the supersymmetric gauge theories.
In these theories, the ratio is an overall
fixed constant, thus these results also support
the -theorem.
The AdS/CFT correspondence has provided a lot of evidence
for the irreversibility of the RG flow and a proof
of the theorem has been found in this context [10]
(see also Ref.[8]).

-theorem:

the decreasing of along the flow can be easily
proven, but this does not imply the theorem,
because the function cannot be globally defined
in the space of theories [14].
Actually, is not well defined at fixed points,
because it corresponds to a scheme-dependent term in the trace
anomaly (1.5): is the Weyl
variation of the local term
in the effective action [15].
This problem can be cured by assuming a proportionality
between and , as proposed in Ref.[7];
however, this amounts to a
strong dynamical hypothesis on the effective action.

In conclusion, the most promising formulation of the theorem in four dimensions involves the coefficient of the Euler term in the trace anomaly (as first suggested in Ref. [3]).

## 2 The anomaly as a measure of degrees of freedom

In this Section, we suppose as a working hypothesis that the -theorem is true in any even dimension (the Euler term is always present in the trace anomaly [16]); we want to understand how the -number is actually measuring the number of degrees of freedom in field theory. To this extent, it is interesting to compute the value of in several free theories and study its dependence on spin and dimension [17].

We consider the free conformal invariant theories of the scalar
, Dirac fermion and antisymmetric tensor
fields^{3}^{3}3This is the -form field,
with for conformal invariance at the
classical level [17]. and compute their trace
anomalies on the -dimensional sphere .
We use the well-known zeta-function regularization of the
Euclidean partition function, given by the determinant of
the Laplacian (the Hodge-de Rahm operator [17])
acting on the respective fields.
Under a scale transformation of the metric,
,
the variation of the partition function is:

(2.1) |

where

(2.2) |

is the zeta function associated to the Laplacian, whose eigenvalues are denoted by .

) |
---|

The trace anomaly in any even dimension contains the Euler term we are interested in, plus a number of terms which are Weyl-covariant polynomials of the Weyl tensor and its derivatives [16]. These additional terms vanish on the geometry of the sphere, which is related to Euclidean space by a Weyl transformation; therefore, the trace anomaly (2.1) is completely given by the Euler term.

Thus, we can write the following equation for the trace anomaly on any conformally-flat space :

(2.3) |

where is the Euler characteristic in dimensions (), is the normalization constant for and is computed on [17].

Equation (2.3) determines the values of once the normalization is chosen. We first consider the normalization (call in this case); this is a rather natural choice, because becomes the proportionality constant between two universal pure numbers which are and scale independent: a topological number on the r.h.s. of (2.3) and the regularized number of modes of the Laplacian on the l.h.s. (which can also be thought of as the number of “effective zero modes” [17]).

The anomaly number divided by the number of field components , , is found to decrease with the dimension and to vanish in the limit [17]:

(2.4) |

with

(2.5) |

The behaviour of is consistent with the known fact
that these free theories
become semiclassical in the limit of large dimensionality^{4}^{4}4
This can be seen by putting the theories on a space-time lattice.:
the anomaly is a quantum effect and should go
to zero (once properly normalized).

We now discuss the use of as a measure of degrees of freedom in the spirit of the -theorem; we should use another normalization in (2.3), such that the scalar field is counted the same value in any dimension, say:

(2.6) |

(This determines the value of in (1.5)). The values of in this normalization are reported in Table 1 together with the ratios per field component,

(2.7) |

We find that the ratios do not approach for large , but actually grow like and for the fermion and antisymmetric tensor fields, respectively. The measure of degrees of freedom given by is very different from the classical value , even when the theories become semiclassical; the higher-spin fields are weighted much more than the lower-spin ones, as is already apparent in . This is the main result of the work [17]. The same qualitative enhancement is found [17] for the other coefficient in the trace anomaly (1.5) and for the gravitational chiral anomaly [18].

This result for the -counting is rather counter-intuitive but does not directly imply an obstruction for the -theorem in higher dimensions: it does not lead to contradictions in the RG flows checked so far. It is a peculiar behaviour that one should keep in mind for further investigations of the -theorem.

## 3 The three stress-tensor correlator in four dimensions.

The previous discussions have shown the important role of the trace anomaly in the various attempts to extend the -theorem above two dimensions. In this respect, a better understanding of the physical consequences of the trace anomaly is very useful. Since the and coefficients of the trace anomaly (1.5) are scheme-independent quantities, it is possible to relate them to finite, scheme-independent amplitudes of the stress-tensor correlators and thus to physical quantities in flat space [19].

### 3.1 Two-dimensional preliminaries

Let us first review the relation of the trace anomaly to two-dimensional correlators and its key role for the dispersive proof of the -theorem [5]. In two dimensions, the two-point correlator of the stress tensor can be written in momentum space as follows:

(3.1) |

where the form of the tensor structure is required by conservation, i.e. by Diffeomorphism invariance. The dimensional analysis shows that the scalar amplitude has dimension and therefore is finite in perturbation theory and scheme independent. At fixed points, it becomes:

(3.2) |

This Equation gives the desired relation of the anomaly coefficient with the scheme-independent correlator, which plays an important role in the conformal field theory [20] ( in coordinate space).

Off criticality, the amplitude satisfies the dispersion relation:

(3.3) |

In this Equation, is a positive-definite dimensionless spectral measure [5], whose critical limit is:

(3.4) |

Using this measure, one can obtain another proof of the Zamolodchikov theorem, as follows [5]: off-criticality, the measure contains a delta term plus a smooth positive function peaked at , where is the typical mass scale of the theory:

(3.5) |

In the infrared limit , the peak will move to infinity and the smooth function will go to zero in a weak sense (i.e. as a distribution); therefore, the coefficient of the remaining delta-function is identified with the central charge of the infrared theory: . On the other hand, in the ultraviolet limit , the smooth function should go to a delta function which adds up to the first term of (3.5), such that the total integral gives the central charge of the ultra-violet theory: .

These properties of the spectral measure imply the following sum rule, which is an equivalent form of the -theorem:

(3.6) | |||||

In this Equation, we also wrote the expression of the sum rule in coordinate space [21].

The previous analysis can be extended to four dimensions [5][19] starting from:

(3.7) |

where is the polynome in (3.1) and is an analogous expression projecting on spin-two intermediate states. The two amplitudes now have zero dimension, thus they are superficially divergent and scheme dependent; their critical limits are:

(3.8) |

where is the coefficient of the scheme-dependent term in the trace anomaly (1.5) and is the renormalization scale. These expressions explicitly show the scheme dependences: ; it follows that the two-point function, although positive definite, cannot be used for proving the -theorem in four dimensions [5][22].

### 3.2 The three-point function

The three-point function has the following structure [19]:

(3.9) |

where the tensors have dimensions greater or equal to four. This results has been obtained by solving the Ward identities for Diffeomorphism invariance starting from a general expansion involving basic polynomes; the involved tensor algebra can be overcomed by using algebraic programs.

In Equation (3.9), amplitudes are proper of the three-point function, while are linked to the two-point function. Two among the amplitudes match the Euler and Weyl anomalies at criticality:

(3.10) |

These limits are obtained by solving the Ward identity for the Weyl symmetry of the critical theory, which is anomalous according to (1.5). The amplitudes in (3.10) have dimension because the corresponding tensors are six-dimensional, and thus are scheme independent. Equations (3.10) give the expected relation between the anomaly coefficients and the scheme-independent correlations in four dimensions; the corresponding two-dimensional relation is given by Eq.(3.2).

Other scheme-independent amplitudes are non-vanishing at criticality (see also the analysis of Ref.[23]); two further amplitudes of zero dimension account for the scheme dependence of the three-point function, including that pertaining to [19].

Each amplitude in the expansion (3.9) can be singled out by projecting the three-point function with the help of the dual tensor basis defined by:

(3.11) |

where the non-degenerate scalar product is obtained by contracting the six indices.

### 3.3 Results and Conclusions

Let us now discuss some consequences of the general expression (3.9) of the three-point function:

i) It disentangles the kinematic properties of field theory, such as Poincaré, Weyl and Bose symmetry, from the dynamics encoded in the scalar amplitudes.

ii) The imaginary part of any scheme-independent amplitude describes a physical quantity such as a scattering or a decay process.

iii) The results (3.9, 3.10) amount to a re-derivation of the trace anomaly within the dispersive renormalization, in close analogy to the well-know analysis of the chiral triangle of Ref.[24]; incidentally, the relations (3.9, 3.10) can be practically useful for deriving the trace anomaly by Feynman diagram calculations.

iv) We can write sum rules for the RG flows of the and coefficients in close analogy with the two-dimensional case described by Eqs.(3.3-3.6). For the amplitude, we write (similar expressions can be written for at ):

(3.12) |

where the measure reduces at criticality to (cf. (3.10)):

(3.13) |

The properties of this measure are very similar to that of its two-dimensional counterpart (3.3): it is a finite dimensionless function of the renormalized coupling constants (i.e. of the mass scale off-criticality ), which satisfies an homogeneous RG equation; note, however, the dependence on two variables rather then one.

Following the same steps as in Section 3.1, we arrive to the sum rule:

(3.14) | |||||

In this Equation, we have set the second momentum to zero, in order to let the measure to depend on the ratio only.

Note that the sum rule (3.14) is not enough to prove , because the measure is not manifestly positive definite. A positivity condition for the three-point function has been proposed in Ref.[23], following from the (quantum) weak-energy condition, but its consequences on and remain to be explored. Manifestly positive amplitudes occur in the four-point function, which could also be analysed using the same tools.

In conclusion, we hope that the general expansion (3.9) and its dispersive analysis will be useful for further investigations.

###### Acknowledgments.

The authors thank the organizers of this Conference for the opportunity to present this work; we also thank all our partners of this European network for the many occasions of informal discussion and collaboration. This work is supported in part by the European Community Network grant FMRX-CT96-0012.## References

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