Phase Diagram and Fixed-Point Structure of two dimensional Wess-Zumino Models
We study the phases and fixed-point structure of two-dimensional supersymmetric Wess-Zumino models with one supercharge. Our work is based on the functional renormalization group formulated in terms of a manifestly off-shell supersymmetric flow equation for the effective action. Within the derivative expansion, we solve the flow of the superpotential also including the anomalous dimension of the superfield. The models exhibit a surprisingly rich fixed-point structure with a discrete number of fixed-point superpotentials. Each fixed-point superpotential is characterized by its number of nodes and by the number of RG relevant directions. In limiting cases, we find periodic superpotentials and potentials which confine the fields to a compact target space. The maximally IR-attractive fixed point has one relevant direction, the tuning of which distinguishes between supersymmetric and broken phases. For the Wess-Zumino model defined near the Gaußian fixed point, we determine the phase diagram and compute the corresponding ground-state masses.
Supersymmetry has become a well-received guiding principle in the construction of particle-physics models beyond the standard model. Whereas the resulting phenomenology of these models is often worked out by perturbative analysis, the required breaking of supersymmetry may be of nonperturbative origin; it is therefore typically parameterized into the models by a phenomenological reasoning. For a proper understanding of the underlying dynamical mechanisms of symmetry breaking which is often related to collective condensation phenomena, powerful and flexible nonperturbative methods specifically adapted to supersymmetric theories will eventually be needed.
For many nonperturbative problems in field theory, lattice formulations and simulations have proven successful. As supersymmetry intertwines field transformations with spacetime translations, discretizing spacetime often induces a partial loss of supersymmetry. This problem also goes along with the challenge of properly implementing dynamical fermions on the lattice, currently witnessing significant progress Catterall:2009it (); Giedt:2006pd (); Bergner:2007pu (); Kastner:2008zc (). As completely new territory is entered in these studies, nonperturbative continuum methods which can preserve supersymmetry manifestly can complement the lattice studies, eventually leading to a coherent picture.
A promising candidate for a nonperturbative method is the functional renormalization group (RG) which has been successfully applied to a wide range of nonperturbative problems such as critical phenomena, fermionic systems, gauge theories and quantum gravity, see Aoki:2000wm (); Berges:2000ew (); Litim:1998nf (); Pawlowski:2005xe (); Gies:2006wv (); Sonoda:2007av () for reviews. A number of conceptual studies of supersymmetric theories has already been performed with the functional RG. The delicate point here is, of course, the construction and use of a manifestly supersymmetry-preserving regulator. For instance, a supersymmetric regulator for the four-dimensional Wess-Zumino model has been presented in Vian:1998kv (); Bonini:1998ec (). A functional RG formulation of supersymmetric Yang-Mills theory employing the superfield formalism has been given in Falkenberg:1998bg (); for applications, see also Arnone:2004ey (); Arnone:2004ek (). Recently, general theories of a scalar superfield including the Wess-Zumino model have been investigated with a Polchinski-type RG equation in Rosten:2008ih (), yielding a new approach to supersymmetric nonrenormalization theorems. A Wilsonian effective action for the Wess-Zumino model by perturbatively iterating the functional RG has been constructed in Sonoda:2008dz ().
This work is devoted to the two-dimensional Wess-Zumino model with a general superpotential, exploring the model beyond the realm of perturbative expansions around zero coupling. This is the simplest quantum field theoretic and supersymmetric model where the nonperturbative dynamical aspects of supersymmetry breaking can be studied. The present study details and generalizes our results presented in a recent Letter Gies:2009az (), and builds on our earlier work on supersymmetric quantum mechanics, where we have constructed a manifestly supersymmetric functional RG flow for the anharmonic oscillator Synatschke:2008pv (); see also Horikoshi:1998sw (); Weyrauch:2006aj () for RG studies of supersymmetric quantum mechanics.
Inspired by Witten’s work on the potential breaking of supersymmetry in the Wess-Zumino model Witten:1982df (), pioneering nonperturbative lattice studies based on Hamiltonian Monte-Carlo methods had early been performed for this model by Ranft and Schiller Ranft1984166 (). More recently Beccaria and coworkers Beccaria:2004ds (); Beccaria:2004pa () re-investigated the phase diagram and the ground-state energy of the model with similar methods. Golterman and Petcher Golterman:1988ta () formulated a lattice action with a partially realized supersymmetry. Another lattice study of the Wess-Zumino model has been performed by Catterall and Karamov Catterall:2003ae (). The results of these studies give a first glimpse into the properties of the phase diagram as will be discussed in Sect. VII.
In the present paper, we use the functional RG equation for the superpotential to study the phase structure of the Wess-Zumino model in two dimensions. In Sect. III, we begin with recalling the off-shell and on-shell effective potentials in a one-loop approximation and comment on certain flaws of the approximations. The main part of this work is concerned with extending and applying the manifestly supersymmetric RG techniques developed in Synatschke:2008pv () in the context of supersymmetric quantum mechanics. The manifestly supersymmetric flow equation for the effective action is constructed in Sect. IV. To first order in a derivative expansion of the effective action, we solve the RG flow equation for the effective superpotential in Sect. V.
In the fixed-point analysis performed in Sect. VI, already a simple polynomial expansion of the superpotential gives access to an infinite number of fixed points with an increasing number of RG relevant directions. Beyond the polynomial expansion, the flow equation for the full superpotential reveals a variety of qualitatively different solutions depending on the initial conditions: we find periodic, sine-Gordon type solutions as well as sigma-model type solutions confining the field values to a finite interval. At next-to-leading order, a nonzero anomalous dimension governs the large-field asymptotics of the fixed-point superpotentials, such that a family of regular fixed-point solutions arises. This family of superpotentials shows oscillating behavior for small fields and a standard asymptotics for large fields, similar to fixed-point potentials for pure bosonic theories in two dimensions Morris:1994jc (); Neves:1998tg (). As a particularity of these supersymmetric models, we identify a new scaling relation between the leading critical exponent of the superpotential flow and the anomalous dimension.
Finally, we study the phase diagram of a particular Wess-Zumino model defined near the Gaußian fixed point in terms of a quadratic superpotential perturbation in Sect. VII. Depending on the initial values of the control parameters of the potential, we observe a quantum phase transition from the supersymmetric to the dynamically broken phase. Following the lattice studies Beccaria:2003ba (); Beccaria:2004ds (); Beccaria:2004pa (), we calculate the critical value of the control parameter for the phase transition as a function of the coupling at the cutoff , but now for all values of the coupling . We also determine the fermionic and bosonic masses in both phases.
Ii Wess-Zumino model
Two-dimensional Wess-Zumino models with one supersymmetry are particular Yukawa models where the self-interaction of the scalar field determines the Yukawa coupling. In an off-shell formulation they contain a scalar field , a Majorana spinor field and an auxiliary field . To maintain supersymmetry in every step of our calculations we combine these fields to one real superfield
The anticommuting parameter in this expansion is a constant Majorana spinor. Supersymmetry transformations are generated by the supercharges
which anticommute on space-time translations, . The transformation rules for the component fields are obtained by comparing coefficients in and read
As usual the term transforms into a total derivative such that its space-time integral is invariant under supersymmetry transformations. The supercharges anticommute with the superderivatives
and up to a sign they obey the same anticommutation rules as the supercharges, . In explicit calculations, one uses Fierz identities which all follow from
where anti-commutes with the . It is useful to keep in mind that for Majorana spinors the fermionic bilinears have the symmetry properties
such that the only Lorentz-invariant bilinear is since . Thus, we choose as Lagrangian density for the free theory the term of
In components it has the form
In this work, we study a class of interacting theories, where the interaction Lagrangian is given by the term of a superpotential ,
In components, has the form
The sum of and defines the off-shell Lagrangian density
which gives rise to an invariant action. As expected for a Euclidean model, this action is unbounded from below and above. After eliminating the auxiliary field via its algebraic equation of motion
we end up with the stable on-shell Lagrangian density
This density is invariant under the nonlinear on-shell supersymmetry transformations
For a polynomial superpotential the supersymmetric Yukawa models defined in Eq. (11) and (13) are perturbatively super-renormalizable. If the leading term in the superpotential contains an even power of , , supersymmetry cannot be broken by quantum corrections. However, if the leading term contains an odd power, supersymmetry may be broken.
Iii One-loop perturbation theory
Let us first discuss the effective potential in one-loop approximation, which can be set up in both the on-shell or the off-shell formulation. It is well-known that the one-loop on-shell potential becomes artificially complex for nonconvex classical potentials Coleman:1973jx (). This problem is avoided in the off-shell one-loop formulation: here, keeping first the auxiliary field in the one-loop calculation and subsequently eliminating it by its quantum equation of motion corresponds to a resummation of higher-order terms in the on-shell formulation. We expect that this potential is a better approximation to the exact effective potential as compared to the one-loop on-shell potential. Indeed, we find a real and stable effective potential based on the off-shell calculation. Similar observations can be found in Murphy:1983ag (). The corresponding problem in one-dimensional supersymmetric systems has been carefully analyzed by Bergner (see Bergner:2009 () and references therein).
iii.1 On-shell effective potential
To calculate the one-loop potential in the on-shell formulation, we need the fluctuation operators and for the fermion and the remaining scalar, respectively. For a homogeneous background field playing the role of a mean field, these operators read
where is the classical potential for the scalar field. In a finite box of size , the momentum takes the values with integer-valued . The fermionic integration yields the Pfaffian of ,
We shall assume that the Pfaffian has a fixed sign, such that we may replace the Pfaffian by the square root of the determinant. In a perturbative approach and in the broken phase (with vanishing Witten index), this assumption is justified.
For an operator with eigenvalues the derivative of the zeta function at the origin is
with the last sum, which involves the MacDonald function, approaching zero exponentially fast with increasing box sizes. Thus, in the thermodynamic limit the one-loop effective potential in the zeta-function scheme is
with . The energy scale is fixed by a renormalization condition. If we use a momentum cutoff regularization instead of the -function regularization then we obtain the same result with replaced by the cutoff .
The effective potential becomes complex for nonconvex classical potentials . To be specific, let us choose
For negative , the one-loop effective potential is real. For positive , it becomes complex for small fields . For fields slightly bigger than , the potential is real and negative. This signals the failure of the approximation since the effective potential must be non-negative in a supersymmetric theory. Depending on the sign of the renormalized , we find both a supersymmetric phase characterized by a non-vanishing expectation value and and a phase with and broken supersymmetry, see Fig. 1. Here, the renormalization conditions are chosen such that the minimum agrees with the renormalized value in the supersymmetric phase. In the broken phase, we use a simple renormalization condition by fixing all parameters at the cutoff, . In both cases, the renormalization condition could alternatively be formulated in terms of Coleman-Weinberg renormalization conditions by fixing the curvature of the potential at the minimum to the physical mass.
iii.2 Off-shell effective potential
In the off-shell formulation, the fluctuations of both scalars (including the auxiliary field) and the fermion are taken into account. The off-shell Lagrangian (11) gives rise to the fluctuation operators in the background of both a and an mean field,
where the mean fields (as the argument of and ) and are assumed to be homogeneous. It follows that the -function regularized one-loop off-shell potential reads
with . With a momentum cutoff regularization, we obtain the same result with denoting the cutoff. To eliminate the auxiliary field , we must solve the transcendental gap equation
and insert the solution for back into . For an arbitrary , the gap equation always has a real solution leading to a real effective potential Bartels:1983wm (). Concerning supersymmetry breaking, we find the same qualitative result as with the on-shell calculation: the sign of the renormalized determines the phase of the system.
Note that the on-shell and off-shell potentials (18) and (21) have similar forms. In Fig. 1 the two potentials are compared with each other and also with the classical potential . Of course, the same renormalization conditions are used for the on- and off-shell potentials. For positive , the scale parameter has been adjusted such that the potentials take their minima at the same value .
Note that the off-shell potential contains resummed contributions from higher order in . The reason is that the solution of (22) contains terms of higher order in and inserting the solution back into the effective potential generates terms to all orders of in the effective potential. By expanding the off-shell action in , the first-order result agrees again with the complex on-shell effective potential. The effective resummation contained in the off-shell action is such that the effective potential becomes real and non-negative everywhere, in particular, at those points where the classical potential is not convex. For , both one-loop potentials predict a phase with broken symmetry and unbroken supersymmetry. For , we find a phase with unbroken symmetry and broken supersymmetry.
Iv Supersymmetric RG flow
In this section, we will construct a manifestly supersymmetric flow equation in the off-shell formulation. Our approach is based on the functional RG formulated in terms of a flow equation for the effective average action , i.e., the Wetterich equation, Wetterich:1992yh ()
Here, is a scale-dependent effective action; it interpolates between the microscopic or classical action for , with being the microscopic UV scale, and the full quantum effective action , being the standard generating functional for 1PI correlation functions. The interpolating scale denotes an infrared IR regulator scale below which all fluctuations with momenta smaller than are suppressed. For , all fluctuations are taken into account and we arrive at the full solution of the quantum theory in terms of the effective action . The Wetterich equation defines an RG trajectory in the space of action functionals with the classical action serving as initial condition.
In Eq. (23), we encounter the second functional derivative of ,
where the indices summarize field components, internal and Lorentz indices, as well as spacetime or momentum coordinates. In the present case, we have where is not a superfield, but merely a collection of fields. The momentum-dependent regulator function in Eq. (23) establishes the IR suppression of modes below . In the general case, three properties of the regulator are essential: (i) which implements the IR regularization, (ii) which guarantees that the regulator vanishes for , (iii) which serves to fix the theory at the classical action in the UV. Different functional forms of correspond to different RG trajectories manifesting the RG scheme dependence, but the end point remains invariant.
The regularization preserves supersymmetry if the regulator contribution to the action is supersymmetric, see below. As the regulator needs to be quadratic in the fields in order to maintain the one-loop structure of the flow, a general supersymmetric quadratic form can be constructed as a term of a superfield operator . Here, is a function of the two invariant and commuting operators and . Since powers of boil down to
where is the standard Laplacian, any invariant and quadratic regulator action is the superspace integral of
Expressed in component fields, we find
In momentum space, is replaced by and the operators take the explicit form
where . Comparison with Eq. (20) reveals that plays the role of a momentum-dependent supersymmetric mass term, whereas can be viewed as a deformation of the momentum dependence of the kinetic term.
This choice of the regulator guarantees a supersymmetric RG trajectory; i.e., for a supersymmetric initial condition , the solution to the flow equation will remain manifestly supersymmetric for all including the endpoint . This does not only hold for the exact solution, but is also valid for truncated effective actions, provided the truncation is built from supersymmetric field operators.
V Local potential approximation
Various systematic and consistent approximation schemes for the construction of can be devised with the flow equation. In this work, we use the derivative expansion which is based on the underlying assumption that the fully interacting theory remains sufficiently local if formulated in the given set of field variables. In order to preserve supersymmetry, we expand the effective action in powers of super-covariant derivatives in the off-shell formulation. This expansion allows for a systematic and unique classification of all possible operators. A truncation of the effective action to a finite derivative order leads to a closed set of equations for the expansion parameters.
In this section, we concentrate on the leading-order derivative expansion: the so-called local potential approximation. Here, the truncated effective Lagrangian is given by Eq. (11) with a scale-dependent superpotential , such that the truncated effective action reads
The derivation of the flow equation for the superpotential parallels the corresponding one for supersymmetric quantum mechanics given in a previous work Synatschke:2008pv (). Within the approximation of constant mean fields, the second functional derivative of the effective action plus regulator is
where the operators on the diagonal read
The inverse of the operator defined in Eq. (30) can be written as follows,
where we have abbreviated
The flow equation for is obtained by projecting both sides of this equation onto the term linear in the auxiliary field. This yields
where we have introduced . Integrating with respect to and dropping an irrelevant constant leads to
This flow equation for the superpotential has exactly the same structure as the corresponding flow equation in supersymmetric quantum mechanics Synatschke:2008pv (). This is not surprising since supersymmetric quantum mechanics can be obtained from this model through dimensional reduction.
Here we are interested in superpotentials for which the map has winding number zero as these potentials allow for dynamical supersymmetry breaking. For a polynomial this is the case if tends asymptotically to an even power, . Then, the highest power of is odd. This implies that the mass-like regulator does not screen but merely shift possible zeroes of . Thus we may set without spoiling the IR properties of the flow.
In the present local-potential approximation, the simple cutoff function turns out to be technically very convenient, since the momentum integration in Eq. (36) can be performed analytically,
In order to calculate the bosonic potential , we only need the derivative of the superpotential. The corresponding flow is
This equation exhibits a particularity for any finite value of , as the sign of the flow depends on whether is smaller or larger than . For large , we generally expect both and to be large and positive for symmetric systems. In this case, the flow for large tends to deplete the height of the potential. Of course, for , we expect the denominator of Eq. (38) to win out over the numerator, such that the flow vanishes at large field amplitudes. For small , or, more generally, in the vicinity of local or global minima of , there can be an inner domain where . For convex potentials with , the flow is negative here, resulting in the tendency to flatten out this inner part of the potential . As the curvature is related to the masses of the excitations, the flow shows a clear tendency to small masses if an inner domain with exists. As it will turn out later, this is a characteristic property of the supersymmetry-broken phase.
Let us note in passing that the regulator used in the present section can lead to artificial divergences at higher orders in the derivative expansion, e.g., when a wave function renormalization is included. Then a stronger regulator in the IR is needed; see, App. A for calculations at next-to-leading order in the derivative expansion.
Vi Fixed-point structure
In this section, we investigate the fixed-point structure of the RG flow. We first concentrate on the local-potential approximation and later include next-to-leading-order terms in the derivative expansion. In fact, it turns out that there is a qualitative difference of the fixed-point superpotentials between the different orders. Similar observations are known from two-dimensional bosonic theories Morris:1994jc (); Neves:1998tg () and are a particularity of two-dimensional systems. Still, the local-potential flow is interesting in its own right. Its IR flow is also quantitatively relevant for studies of the phase diagram, see Sect. VII.
Since RG fixed-point studies require a scaling form of the flow equation, we switch to dimensionless quantities and defined by and . In two dimensions, a scalar field is dimensionless such that no dimensionful rescaling is required. The flow equation (37) for the dimensionless quantities reads
with . The fixed points are characterized by . In the following, we solve the fixed-point equation by various methods.
vi.1 Polynomial expansion
For small values of the field, a polynomial approximation for is justified. If is an even function at the cutoff scale then it remains even at all scales. Its expansion reads . The dimensionless couplings relate to the bare couplings through and . is dimensionless, therefore we have .
Expanding both sides of the flow equation in terms of , a comparison of coefficients leads to the following system of coupled ordinary differential equations:
Note that only enters the right-hand side of the flow equation (39); in particular, the lowest coupling constant does not enter the equations for the higher order couplings.
At a fixed point, the coupling constants, marked by an asterisk, become scale invariant such that the left-hand sides in Eq. (40) vanish. The corresponding system of equations
can be solved iteratively due to the triangular form of the system of flow equations (40). At a fixed point, is a polynomial of order in . Because of the symmetry of the system of equations, these polynomials are odd. In particular, such that the projection of any fixed-point solution on the subspace defined by the couplings and fall onto the curve depicted in Fig. 2. Inserting into the first equation in (40) yields . Later, we shall see that defines an IR unstable direction near a fixed point.
Let us truncate the polynomial expansion of the flow equation for and keep only terms up to order in (39). This is equivalent to keeping the lowest fixed point equations which yield for . In the th equation, the higher-order coupling occurs which we set to zero at the fixed point, .111This prescription is not unique. Alternatively, we could set , for instance, equal to its perturbative one-loop value. In any case, the choice used here is self-consistent in the sense that the equations of are closed and do not depend on further input. This leads to the polynomial equation
where the solutions are to be inserted. With Mathematica, we have checked up to order that all roots of the odd polynomial of order are real.
Due to the underlying symmetry, the remaining fixed points of the truncated system come in pairs . Hence, we find independent nontrivial solutions to the fixed-point equations in addition to the Gaußian fixed point where all couplings vanish. As discussed in the next subsection, only one of these solutions belongs to an infrared stable fixed point with all but one eigenvalues in the stability matrix being positive (i.e., all but one critical exponents being negative). It turns out that this infrared-stable fixed point corresponds to the largest root of Eq. (42), as indicated in Fig. 2. With increasing order of the polynomial truncation the root belonging to the IR-stable fixed point converges to . Roots belonging to any other fixed point are bounded by
In Fig. 3 (left panel), we have plotted the values for the two lowest coefficients and at the infrared-stable fixed point in the polynomial expansion for different orders of truncations. We observe a rapid convergence with increasing order of the polynomial approximation. This suggests that the polynomial approximation to the superpotential in the local potential approximation has acceptable convergence properties.
In Fig. 3 (right panel), we have plotted the inverse ratio of successive couplings at the infrared-stable fixed point for increasing truncation order. This facilitates an estimate of the radius of convergence for a series expansion of in powers of ,
An extrapolation leads to the approximate value such that for the polynomial converges to the IR-stable fixed point solution . In Table 1, the coefficients in the polynomial approximations to at the IR-stable fixed point are listed. Note that the fixed-point values are generically regulator dependent, and thus are not directly related to physical quantities.
|Coefficients at IR-fixed point|
vi.1.1 Stability analysis and critical exponents
Whereas the values of the fixed-point couplings are regulator dependent, the critical exponents are universal and give rise to a classification of the fixed points. The critical exponents are defined as the negative eigenvalues of the stability matrix at the fixed point,
where we have set , , and labels the different critical exponents and eigendirections . Critical exponents with positive real part correspond to RG relevant directions, whereas exponents with negative real part mark irrelevant directions. Inserting the flow of into that of , cf. Eq. (40), we obtain
We observe that the 00-component of the stability matrix at any fixed-point yields, . This together with the fact that the remainder of the first column vanishes, , implies that is always an eigendirection of with corresponding critical exponent . Note that this result is manifestly regulator independent and thus universal. We conclude that any fixed point of the superpotential in the local-potential approximation has at least one RG relevant direction. In analogy with potential flows near the Wilson-Fisher fixed point of Ising-like systems, we introduce the following notion for the leading critical exponent corresponding to this relevant direction:
Even though plays the same role for the superpotential flow as the critical exponent does for the potential flow in Ising-like systems, it should be stressed that does not correspond to the scaling exponent of the correlation length (as does in Ising-like systems). We will later see that quantifies certain properties of the phase diagram.
Depending on whether we study the UV or IR flow of the system, the fixed points have a different meaning. Towards the UV, any of the fixed points which we have found can be used to define a UV completion of the model in the sense of Weinberg’s asymptotic safety scenario Weinberg:1976xy (). All relevant directions emanating from the fixed point span the critical hypersurface. The dimensionality of this critical surface, i.e., the number of critical exponents with , corresponds to the number of physical parameters which have to be fixed in order to unambiguously define the flow towards the IR.222For marginal directions with , the flow in the fixed-point regime has to be studied beyond linear order. Depending on the sign of the first non-vanishing order, these directions are again either marginally relevant or irrelevant. Once these initial conditions to the flow are provided, any other quantity or correlation function can be predicted within the theory. Since , we conclude that any UV completion has at least one physical parameter.
Within the local-potential approximation at order of the polynomial expansion, the critical exponents are given in Table 2 for all 17 fixed-point potentials. By using a different regulator, we check in App. B that the regulator dependence of the relevant positive critical exponents is rather small (up to 10% or much less), which confirms the reliability of the present truncation. Classifying the fixed-point potentials by the slope of the potential as a function of at , the number of relevant directions increases as the slope decreases. The different fixed-point potentials in the local-potential approximation – if they persist to higher truncation orders – thus correspond to different UV completions of the present system with increasing physical parameters. These different UV completions thus define different nonperturbatively renormalized Wess-Zumino models in two-dimensions.
As for the flow towards the IR, the fixed points can generically be related with critical points in the phase diagram of the system. Since the relevant directions are IR repulsive, fine-tuning the relevant direction to the fixed point corresponds to tuning the system onto its critical point. In this sense, the relevant direction corresponding to with is similar to the temperature parameter in Ising-like systems (or a mass parameter in O()-type relativistic models). For instance, in the domain of attraction of the maximally IR-stable fixed point with only as relevant direction, the tuning of distinguishes between the supersymmetric and symmetry-broken phases of the model. More generally, if a system is in the domain of attraction of a fixed-point with relevant directions, the phases of broken and unbroken supersymmetry are separated by an -dimensional hypersurface in the space of couplings.
There is one important difference to Ising-like systems: the coupling associated with the one common relevant direction does not feed back into the flow of the higher-order couplings. Therefore, the remaining couplings are attracted towards the maximally IR-stable fixed point for any regular trajectory irrespective of the flow of .333Irregular trajectories can run to infinity at a finite value of . These divergences are either physically meaningless or signal the breakdown of the truncation. We conclude that the maximally IR-stable fixed point governs the flow towards the IR of in the domain, where the polynomial expansion is valid.
Let us finally mention that the critical exponent receives corrections at higher orders in the derivative expansion, cf. Eq. (57). Still the relevance of the maximally IR-stable fixed point for the IR flow of the potential persists.
In Table 3, we collected the eigenvalues (negative critical exponents) for the maximally IR-stable fix point for different truncations.
|eigenvalues of the stability matrix|
vi.2 Solving of the nonlinear differential equation
For large values of the scalar field, the polynomial truncation is not valid anymore. Hence, we consider here the full nonlinear ordinary differential equation that describes the fixed point potential in the local-potential approximation. In the polynomial approximation, the potential and contain the IR-unstable coupling which does not flow into the fixed point. Upon differentiation of the fixed-point equation for ,
with respect to , we arrive at the following fixed-point equation for the -independent function ,
Since does not appear in Eq. (48), we expect to find a fully IR-stable solution to this nonlinear differential equation (in addition to further solutions with IR unstable directions).
As before, we consider odd solutions which are fixed by the initial conditions and a finite value for . In fact, we find a continuum of oscillatory solutions that are defined for all values of . In addition, we identify solutions that hit the singular line of the differential equation (48) but can be continued without cusps. Solutions of the second class exist only in a finite range.
vi.2.1 Oscillating solutions
The initial point is an extremum of the solution. The differential equation (48) implies that this extremum is indeed a minimum such that approaches the axis away from . Actually it must intersect the axis at some point. Otherwise, it would possess a maximum at some with or it would monotonically approach the axis without crossing it. But a maximum with contradicts the differential equation (48) which implies that . Also, if approached the axis monotonically from below then near the axis the differential equation would imply . Therefore, the slope of would increase with increasing such that finally would intersect the axis.
We conclude that must intersect the axis and we may assume that this happens at , such that the solution starts off at the minimum at and hits the axis at the origin. This solution extends to an odd solution owing to the symmetry of the differential equation and hence has a maximum at with . Because of the translational invariance and the symmetry of the fixed point equation the solution must be symmetric relative to , i.e. . This proves that every solution with is periodic and takes its values between and . The lines repel solutions oscillating in the strip as long as the slope is less than . Solutions with hit the singularity at .
In a similar fashion, one argues that there exists a second class of solutions of the differential equation having just one minimum with or having just one maximum with . Solution in this class are not periodic and never hit the singular line . Since they cannot be continuous and antisymmetric they are discarded.
vi.2.2 Comparison with the polynomial expansion
In the preceding subsection, we have seen in the polynomial expansions of the truncated system that the slope