Gradient flows for $\beta$ functions via multi-scale renormalization group equations

Renormalization schemes and cutoff schemes allow for the introduction of various distinct renormalization scales for distinct couplings. We consider the coupled renormalization group flow of several marginal couplings which depend on just as many renormalization scales. The usual $\beta$ functions describing the flow with respect to a common global scale are assumed to be given. Within this framework one can always construct a metric and a potential in the space of couplings such that the $\beta$ functions can be expressed as gradients of the potential. Moreover the potential itself can be derived explicitly from a prepotential which, in turn, determines the metric. Some examples of renormalization group flows are considered, and the metric and the potential are compared to expressions obtained elsewhere.


Introduction
Originally multi-scale renormalization group (RG) flows were introduced to deal with physical problems involving distinct energy scales [1]. On the other hand it is plausible to consider multiscale RG flows motivated by purely formal arguments: In dimensional regularization marginal couplings (i.e. dimensionless in d = 4) acquire a dimension d−4 which requires the introduction of a scale µ, and in perturbation theory the corresponding renormalized couplings depend on t ≡ log(µ 2 /µ 2 0 ) where µ 0 serves to define initial conditions for the running couplings. In the presence of several marginal couplings g a , a = 1 . . . n g , it is standard to introduce a single scale µ common to all couplings, since this allows to construct RG equations for Green functions with respect to an overall change of scale. However, a priori it is allowed and possible to introduce as many parameters µ i or τ i ≡ log(µ 2 i /µ 2 0i ), i = 1...n g . An overall change of scale can still be defined provided all τ i are related to an overall scale t.
In the presence of an ultraviolet (UV) cutoff Λ the renormalization group can also be used to describe the running of bare couplings with Λ keeping the renormalized couplings fixed. A UV cutoff Λ must not necessarily be universal: Consider, for example, a momemtum space cutoff of propagators which decrease rapidly for p 2 > Λ 2 . A priori it is possible to chose different cutoffs for different fields. Although the number of fields (counting multiplets as single fields) does not necessarily coincide with the number of marginal couplings one obtains again the possibility to introduce n g parameters τ i now defined as τ i ≡ log(Λ 2 i /µ 2 0 ). Distinct momentum space cutoffs can also be introduced in the form of distinct form factors attached to the vertices corresponding to marginal couplings, as it happens automatically in the case of compositeness. Actually the socalled gradient flow in field space (not to be confused with the here considered gradient flow for couplings/β functions), originally introduced for gauge fields on a lattice [2], serves also as a UV cutoff for correlation functions of composite operators and could be generalized to distinct cutoffs for distinct couplings. Finally Pauli-Villars regularization allows for several distinct cutoffs as well.
Subsequently we will use the idea of n g scales τ i independently from whether these refer to renormalization points µ i or to UV cutoffs Λ i .
Computing the radiative corrections to vertices associated to n g marginal couplings the various couplings and scales will mix at least in higher loop order. Consequently, in general each coupling g a will depend on each scale τ i leading to a system of β functions Assuming as many couplings g a as scales τ i and linearly independent β i a (g) this set of partial derivatives can formally be inverted to give ∂τ i ∂ga (g). On the other hand it remains possible to define a universal overall scale (or a cutoff) t with respect to which the properties of a physical system change unless it is scale invariant. Varying t the couplings g a satisfy standard (although scheme dependent) RG equations ∂ga ∂t = β a (g). We will assume that the scales τ i are proportional to t such that where the constants C i may differ from 1 for different scales τ i . But since these drop out (cancel) in the interesting quantities below we will consider C i = 1.
It is the aim of the present paper to show that the concept of different scales τ i leads naturally to the definition of a gradient flow In addition we find that the potential Φ(g) is related to a prepotential P via In principle such a prepotential can always be constructed if one solves the system of coupled RG equations for g a (t), inserts the solutions into the potential Φ(g(t)), integrates with respect to t and re-expresses t in terms of g a (t). In practice these steps are hardly feasable, whereas within the present approach the prepotential is related to the metric η ab (see the next section) which allows for its construction. The possibility to express β functions in terms of a metric η ab (g) and a potential Φ(g) was observed first by Wallace and Zia [3,4] for a multi-component ϕ 4 theory. The consideration of Weyl consistency conditions for local couplings in a gravitational background in dimensional regularization led Osborn and Jack to explicit expressions for a metric η ab (g) and a potential Φ(g) [5][6][7][8][9]; the symmetry of the metric matrix is possibly spoiled, however, in higher order in perturbation theory.
A candidate η ab Z for a metric is the correlation function of two composite operators to the couplings g a , g b respectively. Such a metric was introduced by Zamolodchikov [10] in order to show the irreversibility of the RG flux in d = 2 dimensional field theory where the positivity of η ab Z can be shown.
Couplings g a can be considered as sources for composite operators O a , at least if promoted to local quantities g a (x). Then a functional G(g a ) can be defined such that derivatives of G(g a ) with respect to g a generate correlation functions of operators O a [24]. This allows to relate the Zamolodchikov metric η ab We are not very precise here since, within the present framework of multiple scales, we find a somewhat different expression for the metric η ab in (1.3).
The starting point of our approach is purely algebraic and could find applications for RG flows beyond quantum field theory. We will compare, however, our results for gradient flows in some simple field theory models to those obtained elsewhere.

Gradient flow from multiple scales
As stated in the Introduction we consider n g marginal couplings g a depending on n g scales τ i . We assume that the matrix of partial derivatives ∂ga ∂τ i (g) can be inverted such that ∂τ i ∂ga (g) exists, and that eq (1.2) holds.
We consider a prepotential P (τ (g)) (omitting indices of g a and τ i if these appear as arguments of functions); its total derivative with respect to an overall scale t will be identified with the potential Φ(τ (g)): assumed to be known. Next we consider the derivative of (2.1) with respect to g a : Due to (1.2) the second term on the right hand side of (2.3) vanishes. The first term on the right hand side of (2.3) can be rewritten as The metric (2.6) is manifestly symmetric and covariant under redefinitions g → g (g). Note that η ab differs from ∂ 2 P ∂ga∂g b ; the difference are terms of the form ∂P ∂τ i ∂ 2 τ i ∂ga∂g b . From (2.6) positivity of the metric depends now on the positivity of ∂ 2 P ∂τ j ∂τ i and properties of ∂τ i ∂ga on which we cannot make general statements.
Independently from the positivity of η ab the above arguments allow to formulate a potential flow for a general system of β functions. We obtain no constraints on terms in the β functions in the form of Weyl consistency conditions as in dimensional regularization [5][6][7][8][9]20]. The explicit construction of the above gradient flow from a given set β functions with respect to an overall scale t requires, however, to consider some subtleties.
Given a set of n g β functions β a the first task is to find n g independent solutions of (1.2) for for nonzero constants C i which may all be taken as 1 since a constant rescaling of τ i cancels in η ab .
If the system is not degenerate there exist n g independent solutions for τ i (g) which involve arbitrary functions of n g − 1 expressions ϕ k (g); ϕ k (g) are independent solutions of the set of corresponding homogeneous (C i = 0) equations (2.7). In cases where the lowest order terms of β a are of the form β a = b a g n a + . . . (with n an integer = 1, no sum over a) it is natural to take τ i (g) = −δ a i 1 ba(n−1) g 1−n a + . . . such that τ i (g) = t to lowest order, and to construct the higher order terms subsequently. (If the β functions are known to a given order in perturbation theory it can be useful to supplement them with formally higher order terms in g to find analytic expressions for ∂τ i ∂ga satisfying (2.7). Explicit expressions for τ i (g) which require to integrate ∂τ i (g) ∂ga are actually never required.) In other cases of β a one has some freedom in the construction of ∂τ i ∂ga , but such redefinitions in the space of τ i drop out in the final quantities which depend on g a only.
With ∂τ i ∂ga (g) and its inverse ∂ga ∂τ i (g) at hand one can proceed with the construction of a metric η ab . η ab has to satisfy integrability conditions which can be derived as follows. Consider the following derivatives of the prepotential P (τ (g)): which imply the integrability conditions In order to solve (2.9) it can be helpful to expand the derivatives such that (2.9) becomes In the last step we have used Given ∂τ i ∂ga (g) and its inverse ∂ga ∂τ i (g) it is straightforward to compute L ad b from the last term in (2.12).
Note that there are more integrability conditions (2.11) than those which follow from (2.5) alone and read However not all (symmetric) solutions η ab of (2.14) guarantee that η ab is covariant under redefinitions g → g (g). On the other hand this is guaranteed by solutions η ab of (2.11); it suffices to contract the last two terms in (2.8) with ∂τ i ∂g d . Once a metric satisfying (2.11) has been obtained a potential Φ(g) can be found by integration of (2.5), and a prepotential can be found by integration of (2.8).
Again the solutions of the system of partial differential differential equations (2.11) are not unique. In the considered cases we found no obstruction for diagonal metrics η ab ∼ δ ab f a (g), but such ansätze do not always lead to the simplest expressions for the diagonal elements f a (g) of η ab . These ambiguities are not related to redefinitions in the space of couplings since redefinitions would also affect the β functions; these have been taken as fixed inputs, however. In the next Section we consider some examples.

Examples
First we consider a system of 3 two-loop β functions for gauge couplings where fermion loops generate mixings at the two-loop level as in the Standard Model. We maintain the notation g 1 , g 2 , g 3 of the previous sections where g a are related to the usual gauge couplings α by g a = αa 4π . The β functions are written as In the Standard Model we have [25] b 10 = 41 It is fairly easy to find τ i (g) which satisfy (2.7) to the considered order with C i = 1 and τ i = t to lowest order: The quantities ∂τ i ∂ga (g) and ∂ga ∂τ i (g) can now be obtained straightforwardly. The integrability conditions (2.11) admit solutions corresponding to an expansion of the metric η ab around the unit matrix: (

3.4)
With this metric one finds a potential Φ(g) of the form (3.5) By construction Φ(g) can be derived from a prepotential P (g) as in (2.1), Φ(g) = ∂P (g) ∂ga β a , with P (g) = 1 6 (g 2 1 + g 2 2 + g 2 3 ) − 1 36 It is remarkable that the prepotential P (g) does not depend on the mixing terms in the β functions.
The metric (3.4) and the potential (3.5) differ from the ones for the same system of β functions in [22] where the potential consists in quartic terms in g a only (to two-loop order). They differ also from the metric η JO obtained by Jack and Osborn from Weyl consistency conditions [6]. In the space of gauge couplings their metric η JO is also diagonal, but of the form η aa JO ∼ Na g 2 a with constants N a to two-loop order. As a consequence consistency conditions among the two-loop terms of the β functions (in dimensional regularisation and minimal subtraction) can be derived, see also [26]. We found, however, that an expansion of η ab around η aa JO cannot satisfy the integrability conditions (2.11). (We recall that the metric η ab JO is not guaranteed to be symmetric to higher loop order.) Here, on the other hand, we obtain the potential from a simple prepotential.
The other example is more involved already to one-loop order. It concerns a scalar with quartic self interaction and a Yukawa coupling to a Fermion, like the Higgs-top sector of the Standard Model with a quartic Higgs coupling λ|H| 4 and a top quark Yuhawa coupling h t . Our notation is (3.7) The general one-loop β functions are where in the Standard Model The general solution of eq. (2.7) (again with C i = 1) for τ i (g) is of the form where F i (X) is an arbitrary function of (The argument of the root w is positive for b 3 < 0, b 1 > 0 as in the Standard Model.) We have studied various ansätze for F i (X) without observing substantial differences in the final results (since related by redefinitions of τ i ); subsequently we consider the simplest possibility Among the solutions of the integrability conditions (2.11) for the metric η ab we discuss the one which allow for expansions of the potential Φ(g) and the prepotential P (g) in powers of couplings (without logarithms or dilogarithms). This metric is off-diagonal and, using β 2 from (3.8), can be written as (β 2 − a 1 g 1 g 2 )(2b 1 g 2 + (b 2 − a 1 )g 1 ) , The corresponding potential Φ(g) is 14) It can be derived as in (2.1) from the prepotential (3.15) Note that the matching of the various coefficients in Φ(g) = β 1 ∂P (g) ∂g 2 is highly nontrivial, and that the expression for P (g) is actually somewhat simpler than the one for Φ(g). But both expressions for the metric and the potential differ considerably from the ones in [6] and [22].

Conclusions
Using the formalism of multi-scale RG equations we have shown how a potential flow for a set of n g couplings and corresponding β functions can be constructed. Since the metric is not necessarily positive the flow is not necessarily irreversible. This cannot be expected, however, since the formalism holds equally for systems with limit cycles.
A particular feature of the present construction is that the potential Φ(g) derives always from a prepotential P (g) as in (1.4), related to the metric as in (2.6). Contracting (1.3) with β a and using (1.4) one obtains β a η ab β b = d 2 P (g(t)) dt 2 (4.1) which may be helpful for the study of global features of the RG flow. A holographic formulation of the RG flow via Hamilton-Jacobi equations for generic quantum field theories leads always to a gradient flow for β functions [27]. Conversely a gradient flow for β functions is a pre-requisit for a holographic formulation of the RG flow. The present approach may thus find applications in this direction, but also in contexts beyond quantum field theory.
In order to extend the range of possible applications of the present formalism it will be useful to generalise it towards non-marginal couplings such as mass terms. Then, within mass dependent subtraction schemes, the β functions may depend explicitely on the scale(s) which cases require further studies.
Finally the present approach requires as many scales τ i as couplings g a . If this assumption is relaxed the reversibility of the matrices of partial derivatives and/or the construction of a metric η ab imply constraints on the β functions which merit further investigations.