Renormalization group in Lifshitz-type theories
Roberto Iengo, Jorge G. Russo and Marco Serone
ISAS-SISSA and INFN, Via Beirut 2-4, I-34151 Trieste, Italy
Institució Catalana de Recerca i Estudis Avancats (ICREA),
Departament ECM and Institut de Ciencies del Cosmos,
Facultat de Física - Universitat de Barcelona,
Diagonal 647, E-08028 Barcelona, Spain.
We study the one-loop renormalization and evolution of the couplings in scalar field theories of the Lifshitz type, i.e. with different scaling in space and time. These theories are unitary and renormalizable, thanks to higher spatial derivative terms that modify the particle propagator at high energies, but at the expense of explicitly breaking Lorentz symmetry. We study if and under what conditions the Lorentz symmetry can be considered as emergent at low energies by studying the RG evolution of the “speed of light” coupling and, for more than one field, of in simple models. We find that in the UV both and generally flow logarithmically with the energy scale. A logarithmic running of persists also at low-energies, if in the UV. As a result, Lorentz symmetry is not recovered at low energies with the accuracy needed to withstand basic experimental constraints, unless all the Lorentz breaking terms, including , are unnaturally fine-tuned to extremely small values in the UV. We expect that the considerations of this paper will apply to any generic theory of Lifshitz type, including a recently proposed quantum theory of gravity by Hoava.
Recently, there has been an increasing interest in non-relativistic quantum field theories where Lorentz invariance is explicitly broken at high energies and hopefully recovered at low energies. In particular, in [2, 3, 4] (see also ), general gauge theories, including non-relativistic extensions of the Standard Model, were proposed and investigated, while in [6, 7] similar constructions were implemented in Yang-Mills theories in 4+1 space-time dimensions and membrane theory. The same type of construction was then extended to four-dimensional quantum gravity in ,111See also [9, 10] for a (partial) list of further works that investigate the proposal of . where it was suggested that the resulting theory may provide a candidate for a renormalizable and unitary quantum theory of gravity which flows in the infrared (IR) to Einstein theory.
The ultraviolet (UV) behavior of all these theories is substantially ameliorated by the presence of higher derivative (in the spatial directions only) quadratic terms that improve the UV behavior of the particle propagator, without introducing ghost-like degrees of freedom that in Lorentz invariant higher derivative theories typically spoil the unitarity of the theory. The proposed theories are of Lifshitz type . In the UV, they exhibit, at the classical level, an anisotropic scaling symmetry under which time and space scale differently: , , where is the critical exponent, equal to one in a Lorentz invariant theory. The renormalizability properties of these theories have been extensively studied in [2, 3] for scalar, fermion and gauge theories. The usual power-counting argument for the renormalizability of a theory does not strictly hold anymore, but it is essentially still valid, provided one substitutes the standard scaling dimensions of the operators by their “weighted scaling dimensions” , i.e. by the dimensions implied by the assignment and . Lifshitz-type theories exhibit at least two qualitative different energy regimes, set by the scale (denoted by ) of the higher derivative operators. We will generally denote as UV regime the energy range , where the theory is manifestly non-Lorentz invariant. This is the proper “Lifshitz” regime, where the effective scaling dimensions of the operators are the weighted ones. We denote as IR the range , where the theory is expected to smoothly reach the “standard” regime, where the operators are classified by their standard scaling dimensions. Weighted relevant operators break the anisotropic scaling symmetry explicitly and, at low energies, the theory is expected to flow to the Lorentz invariant theory with . This is, however, a non-trivial (and obviously crucial) point, since there is no dynamical principle for which Lorentz symmetry should emerge in the IR. As a matter of fact, we find that Lorentz invariance is recovered in the IR only if an unnatural fine tuning of the parameters of the theory ensure that all sources of Lorentz violation in the theory are tiny and below the rather stringent, presently known experimental bounds. These issues, expected on general grounds , are explored by performing a one-loop calculation in concrete, simple, models, which also clarify the IR–UV structure of Lifshitz-type theories.
More specifically, the purpose of this paper is to study the Renormalization Group (RG) evolution at one-loop level of simple scalar field theories of Lifshitz type. In particular, we will calculate the one-loop beta-functions of the weighted marginal operators in the theory and solve the corresponding equations to study the evolution of the associated couplings . Once this is performed, we will focus our attention on a particular weighted relevant operator, , and study the RG evolution of the “speed of light” parameter for (physically, in the low energy theory represents the maximal speed for particles). In order to keep the technical analysis as simple as possible, we will mostly consider scalar field theories in higher dimensions with , namely a theory in spatial dimensions, with quartic derivative couplings as well, and a theory in spatial dimensions. These two theories are among the simplest theories which are i) of Lifshitz type, ii) their -functions are respectively positive and negative in the UV, iii) has a non-trivial running already at one-loop level. These theories are obviously toy laboratories (in particular, the theory is not even stable), yet they manifest, in a simple context, the main features that more complicated and “realistic” models of this sort should exhibit. In both theories, typically shows a logarithmic running in the range , where is the high-energy scale where the theory is anisotropic:
here an particle-dependent constant and a reference scale in the UV range. In eq.(1.1), we schematically denote by the radiative coefficient governing the RG flow, which depends on the coupling constants of the weighted marginal operators. It is also important to investigate what happens in the presence of more than one field, and particularly if and under what conditions their “speed of light” parameters converge to the same value. To address this question, we have also studied the RG evolution of the difference . Under the assumption that , one schematically finds, for ,
After having studied the UV, we move on to analyze the IR regime . We will see that is the characteristic scale below which Lifshitz theories turn into “standard theories”. More precisely, we will explicitly show that the RG evolution of all weighted marginal couplings is essentially frozen below , in complete analogy to the decoupling of a massive particle in a standard quantum field theory. The key point is, of course, that in the IR the relevant propagator term is the usual one, quadratic in the momentum, while the higher derivative terms can be neglected (it is however a delicate point, given that the theory with the usual quadratic propagator is non-renormalizable). Taking into account only the effect of weighted marginal couplings, for we find
which shows that, for sufficiently high , the IR effect of the weighted marginal couplings can be neglected. This is expected, since in the IR the usual classification of operators in terms of canonical rather than weighted dimensions holds. What is marginal in the UV becomes then irrelevant in the IR. We will explicitly show how the -functions smoothly change their behavior going from the UV to the IR by computing them in a momentum subtraction renormalization scheme, where all the decoupling effects are manifest.
However, care has now to be paid for the weighted relevant operators which become standard marginal in the IR, since they can efficiently mediate the UV Lorentz violation to the low-energy theory. Indeed, we will show, by explicitly working out a toy example in 3+1 space-time dimensions, that a logarithmic running like eq.(1.1) (with replaced now by ) still holds in the IR, with depending now on the (standard) relevant couplings and being proportional to any Lorentz symmetry breaking coefficient of the low-energy effective theory, remnant of the Lifshitz-like nature of the UV completion.
In general, then, Lorentz symmetry is not emergent in the IR in theories of Lifshitz type. Recovering Lorentz symmetry would require some dynamical principle keeping all sources of Lorentz violation sufficiently small. The experimental bounds on for ordinary particles are of the order of , which give an idea of the order of magnitude of the fine–tuning that is needed. An indirect bound on for any charged particle is implicitly given by eq.(1.1). In the case of photons, for instance, an experimental constraint on the energy dependence of by the FERMI experiment  gives, taking in eq.(1.1), the following rough constraint on :
Modulo a loop-factor and coefficients of order one, the bound (1.4) can be seen as a bound on for any charged particle. We expect that these fine-tuning problems will affect all generic quantum field theories of Lifshitz type, in particular, the non-relativistic standard model of  and the proposed quantum gravity theory of . In the latter case, after coupling the theory to matter, the problems mentioned above will reappear for the Standard Model particles, where parameters like have tight experimental constraints.
The plan of this paper is as follows. In section 2 we briefly review the main properties of the Lifshitz-like theories, using, for the sake of illustration, a scalar field theory in 3+1 dimensions. In section 3.1 we study the one-loop renormalization of a single scalar field theory with derivative interactions in spatial dimensions; in section 3.2 this analysis is extended to the case of two coupled fields. In section 4.1 we study the one-loop renormalization of a single scalar field in ; the two-field case is dealt with in section 4.2. The analysis in sections 3 and 4 are performed using the minimal subtraction (MS) scheme, suitable for finding the -functions in the UV regime (). In section 5 we study the IR behavior () of Lifshitz-like theories by using the momentum subtraction scheme. After reviewing in section 5.1 the IR behavior of the function in conventional - theory, in section 5.2 we show that the –functions produced by weighted marginal couplings go to zero for . In section 5.3 we discuss the contribution of the weighted relevant operators (marginal in the standard sense) on Lorentz symmetry breaking effects in the IR. We will see that the presence of a non-zero , inherited from the UV, induces a running in at low-energies. The effect is general and we expect that it should apply to any low-energy field theory (i.e. not only to theories of Lifshitz type) perturbed by Lorentz symmetry breaking terms. In section 6 we conclude.
2 Renormalizable Lifshitz-like scalar field theories
Unconventional scalar field theories of the Lifshitz type, with higher derivative interactions and higher derivative quadratic terms, have been extensively studied in . Here we briefly review some aspects of the construction that will be useful in what follows and refer the reader to  for more details. As mentioned in the introduction, the key point of the whole construction is to break Lorentz invariance, so that one is allowed to introduce higher derivative terms in the spatial derivatives and quadratic in the fields, without necessarily introducing the dangerous higher time derivative terms that would lead to violations of unitarity. In doing so, the UV behavior of the propagator is improved and theories otherwise non-renormalizable become effectively renormalizable. A useful guiding principle to easily classify and identify the renormalizable theories in this enlarged set-up is achieved by demanding an invariance under “anisotropic”222The word “anisotropic” arises from condensed-matter physics. In all the instances we consider, we assume to be in a so-called “preferred frame”  where spatial rotations, translations and time-reversal are unbroken symmetries. scale transformations:
where parametrizes the spatial directions. The parameter is known as “critical exponent” and, when it equals one, the transformations (2.1) reduce to the usual Lorentz invariant scale transformations. According to eq.(2.1), we can assign to the coordinates and to the fields a “weighted” scaling dimension as follows:
It is straightforward to see that at the quadratic level, modulo total derivative terms, the Lagrangian for a single scalar field, invariant under (2.1), reads
with being a dimensionless coupling and a high-energy scale parametrizing the strength of the higher derivative operator. Due to the improved UV behavior of the propagator resulting from (2.3) when , the usual power-counting argument for the renormalizability of a theory is no longer applicable. The required modification is obtained by substituting the scaling dimensions of the operators by their “weighted scaling dimensions” .333Sometimes in the literature the weighted scaling dimension is introduced as the standard scaling dimension in some non-standard natural units. Although there is nothing wrong in doing so, we prefer to distinguish between from and use the usual natural units. In other words, a theory is renormalizable if all the operators appearing in the Lagrangian have weighted scaling dimensions (not to be confused with the standard scaling dimensions ) which are not greater than . Thus, the second term in (2.3), although manifestly irrelevant in the standard sense, behaves (and should be considered) as a marginal operator in this theory.
It is useful to illustrate this construction with a specific simple example, namely a scalar field in space-time dimensions () and . For simplicity, we also impose a discrete symmetry . The most general renormalizable Lagrangian, invariant under the transformations (2.1), is given by
where is the Laplace operator in the spatial directions. All the operators appearing in (2.4) are weighted marginal. The renormalizability properties of the theory are not changed if the scaling symmetry (2.1) is softly broken by adding weighted relevant operators. They are given by
so that the final Lagrangian is the sum of eqs.(2.4) and (2.5). In conventional scalar field theories in dimensions, the interactions appearing in (2.4) would be non-renormalizable. What renders the theory renormalizable – and the interactions in (2.4) weighted marginal – is the modification of the propagator, which in momentum space now reads
with . The high-energy behavior of the propagator leads to an improvement of the ultraviolet behavior of the theory. As a result, if no coupling of higher dimension is added, the theory is power-counting renormalizable. In the UV the RG evolution of the weighted relevant parameters, such as or in eq.(2.5), will be governed by the RG evolution of a combination of the weighted marginal couplings , and . As we will see, in the IR the Lifshitz-type theory turns into a low-energy effective theory, where the weighted marginal couplings turn back into standard irrelevant ones and do not effectively run anymore. In this regime, if no Lorentz violating parameter appears in the Lagrangian terms with standard dimension , then we effectively recover Lorentz symmetry in the IR, which protects from any possible running. On the other hand, if some Lorentz violating parameter is left (like e.g. in a two-field model), it will still generically induce a running of , governed now by standard marginal couplings.
For the 3+1 dimensional model defined by eqs. (2.4) and (2.5), there is no renormalization of the couplings at one-loop level, due to the fact that the vertices in (2.4) involve at least six fields. For this reason, in what follows we will consider higher dimensional scalar field theories, for which the weighted marginal vertices contain less fields and, as we will see, there is a non-trivial renormalization of the couplings already at one-loop level.
3 UV behavior: a model with
3.1 One scalar particle
We look for a weighted renormalizable scalar field theory with a non-trivial renormalization of the operator at one-loop level. A simple quantum field theory of this sort is obtained in space-time dimensions with anisotropic scaling . The most general renormalizable Lagrangian, up to total derivative terms and including all possible weighted marginal and relevant operators, is
All couplings appearing in , but the mass and , are dimensionless. In order to reduce the number of operators, we have imposed on a discrete symmetry under which . For simplicity, in the following we set .
Our first aim is to renormalize the theory at one-loop level and study the RG flows of the weighted marginal couplings and in the UV. The coupling , although weighted marginal, is one-loop finite, since there is no way to extract four powers of external spatial momentum from the tadpole graph given by the quartic couplings appearing in . Similarly, the wave function renormalization of is trivial at one-loop level, 2-loops). Once the RG flows for and are solved, we will study the evolution of the weighted relevant coupling .444Strictly speaking, the RG evolution of , as determined in a physical scheme, starts at two-loop order, even in presence of the quartic derivative interaction, since the momentum-dependence of the one-loop graph (which is a tadpole) is trivial. However, we can still define a running coupling by adding a fictitious momentum in the loop, seen as the momentum carried by the composite operator . This is a standard trick. See e.g.  for the completely analogous case of the one-loop RG evolution of the mass parameter in the usual theory. We regularize the theory using a variant of dimensional regularization applied only to the spatial directions () and thus renormalize using a minimal subtraction scheme where only the poles in are subtracted, with no finite term.
The superficial degree of divergence of a graph is easily computed by looking at the weighted scaling of a graph. The one-loop corrections to the coupling come from a one-loop graph with 3 insertions of the coupling and from another graph with one insertion of and . See figure 1. Due to some unusual properties of these theories, we report, in detail, the computation of the divergence term of the graph in fig. 1. A divergence can only arise when all the momenta of the vertex are taken in the internal lines, so that we can set to zero all external momenta :
where is a geometrical factor taking into account all possible channels and
is the propagator for . Here and in the following . After Wick rotating (), we can rewrite as
In writing the last equality of eq. (3.4), we have expanded the term in the exponential, since insertions of these terms lower the divergence of the graph. It is straightforward to check that the divergence arises only from the leading, –independent term. The integral (3.5) is easily done by going to radial coordinates and changing variables . Performing the integrals we get
where is the area of the sphere. Using the same techniques, we can compute the graph as well. By denoting the Lagrangian counterterm canceling the divergences coming from the graphs , we then get
where we find convenient to express the result in terms of the usual loop factor for Lorentz invariant theories in space-time dimensions, . Similar manipulations allow to compute , the coefficient of the counterterm :
where a dot stands for a derivative with respect to and is a given UV reference scale. Note that the effective couplings of the theory are
The solutions of the RG equations (3.9) are
with , . Since a Landau pole appears at the scale
the range in which eqs.(3.11) are reliable is .
Having found the RG evolution of the weighted marginal couplings and , we can now go on and study the evolution of the weighted relevant coupling . Its function reads
Equation (3.14) shows that in the UV regime has a logarithmic RG running, governed by the coupling .
We expect that in any generic, weakly-coupled quantum field theory of Lifshitz type, including also theories with gauge fields and matter in , the running of will be qualitatively similar to (3.14), i.e. with a logarithmic dependence on the energy in the UV regime.
3.2 Two scalar particles
We will now show that theories with anisotropic scalings generically lead to different “speed of light” parameters (defined as coefficients of ) associated with different particles. More precisely, we will show that the RG evolution of the difference , even in the most optimistic case when is an attractive fixed point, is generally too slow to give with the needed accuracy. A severe fine-tuning in the UV for seems to be inevitable.
In what follows we consider an extension of the single field model defined by the Lagrangian density (3.1) to two fields and , imposing the symmetry , . The Lagrangian is given by
where are two copies of the Lagrangian appearing in (3.1) for the fields and , and is an additional potential:
As can be seen, the Lagrangian contains a number of new interactions, which considerably complicate the analysis. In particular, the one-loop renormalization of the couplings , , and involves diagrams with all possible combinations of three insertions of the quartic couplings , and as well as diagrams with one insertion of any of the order six terms and one insertion of any of the quartic couplings. Fortunately, at one-loop level, as in the single field model considered before, the renormalization of , , , and does not involve the , , , couplings and therefore we do not need to compute the associated Feynman diagrams. In analogy to eq. (3.11), there will always be a choice of boundary conditions for the couplings at such that the model is stable all the way down to the far UV.
Taking for simplicity, after a straightforward but lengthy computation, we get
where all couplings have been rescaled by a factor while keeping the same notation for the couplings for simplicity (i.e. we omit hats). The functions for the couplings are easily computed to be
The RG equations (3.17) do not admit a simple analytic solution in general. A class of exact solutions is however obtained by substituting the ansatz
in eqs.(3.17) and solving the (now algebraic) equations for and :
In terms of , and , the running for and is given by
The system (3.20) is under-constrained (3 equations for 4 variables), so we fix one of the couplings, say , and look for solutions for and for the other couplings . Taking would simply rescale the solutions , so that the RG evolution of and is unaffected. A sufficient condition to get a (semi)positive definite interaction requires and . We find seven solutions, one of which is unstable and is disregarded. The remaining six solutions are
As can be seen from , all six solutions correspond to couplings (and ) which grow in the UV in all cases. The deviation , instead, increases in the UV for 1), 3), 4) and 6), is constant in the case 2) and decreases in the UV in 5). Note that solution 1) reproduces the RG evolution (3.11) for a single field.
We can also perturb the solutions found above and study the RG evolution of the fluctuations at the linear level. We focus on the case 1), since it is the only one giving rise to simple analytical results. We look at linear perturbations around the solutions putting
and consider as a fluctuation. In this way, we get
From eqs.(3.24) we see that the fixed-point is stable, with all fluctuations decreasing towards the IR. A similar study can be done for the other solutions, which are not all IR stable. From eq.(3.24) we see that it is not enough to start with at to ensure that near . In addition, one has to ensure that also other perturbations around some fixed point are fine-tuned to zero.
In section 5 we will argue that in the IR the RG evolution of , as given by the contributions of only weighted marginal couplings, essentially stops. Given the experimental bounds on for ordinary particles mentioned before, recovering Lorentz invariance in the IR with enough accuracy requires fine-tuning of , and in the UV to extremely small values. We illustrate this point in figures 2(a) and 2(b), where the running of over 40 orders of magnitude for an arbitrary given choice of boundary conditions is shown.
3.3 Case of particle with no self-interactions
The two-scalar model can be adapted to the case where one of the particles, say , has no self-interaction terms, i.e. . This case represents a situation of physical interest: in electrodynamics the photon has no self-interaction term and one may wonder if the speed of light will still significantly depend on the energy. Here we will show that the energy dependence of either speed-of-light parameters or does not rely on the presence or absence of self-interaction terms.
Consider the beta functions (3.17). An exact solution can be found by setting
We are left with two equations
Next, we write , so that the second equation takes the form
One obvious solution is which requires or . The solution with is not physical, since . Setting leads to a solution similar to the class of solutions considered in (3.19),
Now consider solutions where is not constant. Writing
eq. (3.27) becomes
We can now integrate this equation and obtain
We have chosen as the ultra-high energy scale at which , so that is defined in the interval . The integral (3.32) defines and hence and . It can be expressed in terms of hypergeometric functions, but its explicit expression is not needed in order to see the qualitative behavior of the couplings in the regime . In this limit, and hence one approaches the constant solution (3.28) discussed above. For large , one has the behavior,
Hence both and decrease towards the IR.
4 UV behavior: a UV free model with ,
In this section we look for a weighted renormalizable scalar field theory which is UV free. In order to simplify our analysis, we consider a model which is weighted-counting renormalizable in spatial dimensions. Although the model is manifestly an unphysical theory, having an unbounded potential, this instability does not affect the RG equations, which can then be formally studied and hopefully seen as a toy laboratory for more complicated stable UV free theories, such as Yang-Mills (YM) theories. As in section 3, we first study the one-loop RG evolution of a single field, which will enable us to compute the RG evolution of . Then we shall consider a two-field model, where will also be considered.
4.1 One scalar
The most general renormalizable Lagrangian reads
As in section 2, we use a dimensional regularization in the spatial directions only. We will add counterterms to subtract the poles in only, with no finite term. We study the theory in the energy range , where the analysis is reliable.
The wave-function renormalization in the theory is non-trivial and hence the relevant one-loop graphs to compute are those associated with the two and three point functions. These graphs are computed exactly along the same lines followed in section 3, so we will just report the results for the functions and the anomalous dimension of . We find and
where , . The effective coupling of the theory is
The RG equations are easily solved in terms of . One has
with , and thus
The coupling increases in the UV, while and the effective coupling are UV free.
Let us now study the evolution of . Its associated is
where . Hence
The RG evolution of is again governed by the weighted marginal couplings of the theory. Interestingly enough, logarithmically increases in the UV, as in the model of section 3.
4.2 Two scalars
We can add a further scalar to the model and, for simplicity, we impose a discrete symmetry under which . The Lagrangian is given by
The computation of the functions for , , , , and is straightforward, although a bit laborious, mainly because we keep in general. We find