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Scaling Solutions in the Derivative Expansion PHYSICAL REVIEW D 98, 016013 (2018) Scaling solutions in the derivative expansion † N. Defenu1,* and A. Codello2,3, 1Institut für Theoretische Physik, Universität Heidelberg, D-69120 Heidelberg, Germany 2CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark 3INFN-Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy (Received 24 February 2018; published 23 July 2018) Scalar field theories with Z2 symmetry are the traditional playground of critical phenomena. In this work, these models are studied using functional renormalization group (FRG) equations at order ∂2 of the derivative expansion and, differently from previous approaches, the spike plot technique is employed to find the relative scaling solutions in two and three dimensions. The anomalous dimension of the first few universality classes in d ¼ 2 is given, and the phase structure predicted by conformal field theory is recovered (without the imposition of conformal invariance), while in d ¼ 3 a refined view of the standard Wilson-Fisher fixed point is found. Our study enlightens the strength of shooting techniques in studying FRG equations, suggesting them as candidates to investigate strongly nonperturbative theories even in more complex cases. DOI: 10.1103/PhysRevD.98.016013 I. INTRODUCTION a new kind of nonperturbative expansion and to overcome the traditional limitations of diagrammatic techniques. Since the discovery of the phenomenon of universality In this paper, we use the flow equations for the EAA at and its explanation in terms of the renormalization group order Oð∂2Þ of the derivative expansion to investigate (RG) [1], one of the main goals of statistical and quantum scaling solutions (i.e., functional RG fixed points) of field theory has been the classification of all universality single-component scalar theories with Z2 symmetry in classes, i.e., the determination of fixed points of the RG two and three dimensions. Our approach generalizes the flow. Even though the phase diagram of scalar field theories spike plot technique already employed to solve the local has been subject to intense investigations over the decades, potential approximation (LPA) in [7–10] and refines the we still lack a complete picture of theory space even in the O ∂2 study [11] where a power like regulator and simplest cases, such as that of single-component scalar ð Þ numerical relaxation methods were employed. For related theories in two or three dimensions. This is not a surprise studies, but from a Polchinski equation perspective, see since the problem is inherently nonperturbative, but what is [12] and reference therein; for a proper time RG study also missing is an easy way to render and visualize the see [13]. complex landscape of theory space, since this is generally Our technique provides an extension of the shooting an infinite functional space. method previously employed in the literature to describe In recent years, the functional renormalization group single isolate fixed points in d>2 [14] to the investigation (FRG) [2,3] has shown its versatility and strength as a of the whole phase diagram of the d 2 case, where nonperturbative RG technique in many applications [4–6]. ¼ infinitely many fixed points exist. Our analysis paves the In this approach, the traditional RG is extended to work in way for the systematic application of shooting techniques the functional space of the effective average action (EAA), to high-order truncation in derivative expansions for more i.e., the scale-dependent generator of the one-particle complex field theories. In particular, we foresee possible irreducible vertexes of the theory, allowing us to pursue application of the present technique to OðNÞ symmetric field theories, where strongly nonperturbative fixed point exist [15,16], which cannot be investigated by means of *[email protected] other techniques. Indeed, standard solution procedures for † [email protected] flow equations rely on relaxation methods (which need fine-tuned initial conditions in presence of multiple fixed Published by the American Physical Society under the terms of points), polynomial expansions (which assume analytic the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to form for the effective potential) or full scale dependent the author(s) and the published article’s title, journal citation, solution (which need approximate knowledge of the final and DOI. Funded by SCOAP3. form for the effective potential). In contrast our numerical 2470-0010=2018=98(1)=016013(13) 016013-1 Published by the American Physical Society N. DEFENU and A. CODELLO PHYS. REV. D 98, 016013 (2018) procedure is unbiased and does not assume any further defined as t ¼ logðk=k0Þ, where k0 is an arbitrary refer- knowledge of the theory under consideration apart for its ence scale. global symmetry, neither it introduces further approxima- After inserting the ansatz (1) into the flow equation (2) tion rather than the EEA ansatz necessary to derive the flow and performing the appropriate projection one can derive equations. the beta functionals βV and βZ for the running effective potential and field-dependent wave-function renormaliza- II. DERIVATIVE EXPANSION tion function The derivative expansion is an approximation scheme ∂ V βd V00;Z where the EAA is expanded in powers of the field spatial t k ¼ Vð k kÞ d 00 000 0 00 derivatives [2,3,17,18]. This scheme is usually employed ∂tZk ¼ βZðVk;Vk ;Zk;Zk;ZkÞ: ð3Þ for matter field theories on flat space, where it becomes a series expansion in powers of the momentum. The deriva- The explicit form of the beta functionals can be obtained in tive expansion has been very successful in drawing phase arbitrary dimension and for arbitrary cutoff functions. The diagrams and computing accurate universal quantities, beta functional for the effective potential follows directly especially critical exponents. by evaluating (2) at a constant field configuration and reads If one considers a one-component scalar field, in d Z -dimensional flat space, with a 2 symmetry, the deriva- d 1 1 2 βV ¼ d Qd½Gk∂tRk: ð4Þ tive expansion for the EAA to order O ∂ reads 2 ð Þ ð4πÞ2 2 Z d 1 2 4 Γk½ϕ¼ d x VkðϕÞþ ZkðϕÞð∂ϕÞ þ Oð∂ Þ: ð1Þ In (4) the regularized propagator at the constant field 2 configuration ϕ is defined as The effective potential VkðϕÞ and the wave-function 1 Z ϕ G x; ϕ ; renormalization function kð Þ are arbitrary functions of kð Þ¼ 00 ð5Þ ZkðϕÞx þ V ðϕÞþRkðxÞ the field ϕ. The derivative expansion has been carried to k higher order in d 3 [17], where a beta function study was ¼ while the Q functionals are defined as performed. The flow equations for these functions are Z derived from the exact RG equation satisfied by the EAA ∞ 1 n−1 [2,3], Qn½f ≡ dxx fðxÞ: ð6Þ ΓðnÞ 0 1 ∂ Γ ϕ Γð2Þ ϕ R −1∂ R : t k½ ¼2 Trð k ½ þ kÞ t k ð2Þ Deriving the flow equation for the wave-function renorm- alization function is more involved. Taking the second For translational invariant systems the trace in latter functional derivative of (2) with respect to the fields it is equation represents a momentum integral and the cutoff possible to write down the flow equation for the two-point function Rk is a momentum-dependent mass term intro- function of the EAA in momentum space. Introducing on duced into the effective action in order to freeze the low the rhs of this equation the vertices of the EAA (1) and energy excitations responsible for the appearance of infra- extracting the Oðp2Þ terms one obtains, after some algebra, red (IR) divergences. The effective action explicitly the following beta functional for the wave-function renorm- depends on a scale k and the renormalization “time” is alization function 000 2 0 2 d ðVk Þ 2 2 ðZkÞ 2d þ 1 3 ðd þ 2Þðd þ 4Þ β Qd G G0 ∂ R Qd G G00∂ R Qd G ∂ R Z ¼ d f ½ k k t kþ þ1½ k k t kg þ d þ1½ k t kþ ð4πÞ2 2 2 ð4πÞ2 2 2 4 000 0 2 2 Vk Zk 3 2 Qd G G0 ∂ R Qd G G00∂ R 2Qd G ∂ R d 2 Qd G G0 ∂ R × ð þ2½ k k t kþ þ3½ k k t kÞ þ d f ½ k t kþð þ Þð þ1½ k k t k 2 2 ð4πÞ2 2 2 00 2 Zk 1 2 Qd G G00∂ R − Qd G ∂ R : þ þ2½ k k t kÞg þ d ½ k t k ð7Þ 2 ð4πÞ2 2 2 Equations (4) and (7) represent the flow equations for been chosen, the integrals in (4) and (7) can be performed. 2 VkðϕÞ and ZkðϕÞ for general cutoff function at Oð∂ Þ of the In this way one obtains a system of partial differential derivative expansion. Once an appropriate cutoff shape has equations for VkðϕÞ and ZkðϕÞ in the variables k and ϕ. 016013-2 SCALING SOLUTIONS IN THE DERIVATIVE EXPANSION PHYS. REV. D 98, 016013 (2018) Finally the flow equations are obtained introducing the Conditions (11) and (14) are a direct consequence of the Z2 dimensionless variables. The dimensionless quantities are symmetry. The condition (14) is obtained absorbing a defined as factor Zkð0Þ into the field redefinition. Thus the only unspecified condition remaining is the v00ð0Þ value. In −1 d 2 ð −1Þ d σ η ϕ ¼ Zk k 2 φ;VkðϕÞ¼k vðφÞ; Eq. (12) and are real values to be determined by requiring the global existence of the scaling solutions, Z ϕ Z ζ φ ; kð Þ¼ k ð Þ condition that, as we will see, reduces to a finite set the allowed functional fixed points.
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