# Gradient flow and the Wilsonian renormalization group flow

Gradient flow and the Wilsonian renormalization group flow Prog. Theor. Exp. Phys. 2018, 053B02 (9 pages) DOI: 10.1093/ptep/pty050 Gradient flow and the Wilsonian renormalization group flow Hiroki Makino, Okuto Morikawa, and Hiroshi Suzuki Department of Physics, Kyushu University 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan E-mail: hsuzuki@phys.kyushu-u.ac.jp Received March 2, 2018; Revised March 29, 2018; Accepted April 3, 2018; Published May 30, 2018 ................................................................................................................... The gradient ﬂow is the evolution of ﬁelds and physical quantities along a dimensionful param- eter t, the ﬂow time. We give a simple argument that relates this gradient ﬂow and the Wilsonian renormalization group (RG) ﬂow. We then illustrate the Wilsonian RG ﬂow on the basis of the gradient ﬂow in two examples that possess an infrared ﬁxed point, the 4D many-ﬂavor gauge theory and the 3D O(N ) linear sigma model. ................................................................................................................... Subject Index B01, B32, B37 1. Introduction and the basic idea The gradient ﬂow [1–5] is the evolution of ﬁelds and physical quantities along a dimensionful parameter t, the ﬂow time; the ﬂow acts as the “coarse-graining” as t > 0 becomes large. These two features of the gradient ﬂow are common to the Wilsonian renormalization group (RG) ﬂow [6]in a broad sense, provided that the ﬂow time is identiﬁed with the renormalization scale. In fact, it has sometimes been indicated that the gradient ﬂow and the Wilsonian RG ﬂow can be identiﬁed in some ways [7–10]; see also Refs. [11–14] for related studies. In this paper, we give a simple argument that relates the gradient ﬂow and the Wilsonian RG ﬂow; our argument is somewhat similar to that of Ref. [7]. We then illustrate the Wilsonian RG ﬂow on the basis of the gradient ﬂow in two examples that possess an infrared ﬁxed point, the 4D many-ﬂavor gauge theory and the 3D O(N ) linear sigma model. Our idea is very simple. We take the following ﬂow equations for the gauge potential A (x) and for the Dirac ﬁelds ψ(x) and ψ(x): ∂ B (t, x) = D G (t, x), B (t = 0, x) = A (x), (1.1) t μ ν νμ μ μ ∂ χ(t, x) = χ (t, x), χ(t = 0, x) = ψ(x), (1.2) ← − ∂ χ( ¯ t, x) =¯ χ(t, x)  , χ( ¯ t = 0, x) = ψ(x). (1.3) Let us consider the correlation function of operators composed of the ﬂowed ﬁelds: O (t , x ) ···O (t , x ) . (1.4) 1 1 1 N N N Here, the covariant derivative on the gauge ﬁeld is deﬁned by D ≡ ∂ +[B , ·]; the ﬁeld strength is μ μ μ deﬁned by G (t, x) ≡ ∂ B (t, x) − ∂ B (t, x) +[B (t, x), B (t, x)]. The Laplacians on the Dirac ﬁelds are μν μ ν ν μ μ ν ← − ← − ← − deﬁned by  ≡ D D , and  ≡ D D from the covariant derivatives on the Dirac ﬁelds, D = ∂ + B μ μ μ μ μ μ μ ← − ← − a a and D ≡ ∂ − B . We will occasionally use notation such as A (x) = A (x)T by using the generator of μ μ μ μ the gauge group, T . © The Author(s) 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 Funded by SCOAP by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. Let us also suppose that we have a set of (a generally inﬁnite number of) coupling constants {g } with which the correlation function is computed. We consider the mapping in this space of the coupling constants induced by the Wilsonian RG ﬂow, {g }→{g (ξ )}, (1.5) i i where ξ parametrizes the RG ﬂow. This RG ﬂow can be characterized by the scaling relation 2ξ ξ 2ξ ξ O (e t , e x ) ···O (e t , e x ) = Z (ξ ) O (t , x ) ···O (t , x ) , (1.6) 1 1 1 N N N 1 1 1 N N N {g (ξ )} {g } i where Z (ξ ) is the multiplicative renormalization factor and the subscript implies that the correlation function is evaluated with respect to the set of coupling constants. Compare this relation with, for instance, Eqs. (7.10) and (7.15) of Ref. [6]. Note that the ﬂow time has the mass dimension −2 instead of −1. The advantage of this characterization of the Wilsonian RG ﬂow is that this scaling relation itself can be written down even for gauge theory for which the momentum cutoff is incompatible with the gauge invariance (at least naively). In particular, for the one-point function of an operator that does not require the multiplicative renormalization, 2ξ O (e t) = O (t) , (1.7) 1 1 {g (ξ )} {g } where we have omitted the argument x assuming translational invariance in the x-space. Hence, assuming that the correspondence {O (t)} ⇔{g (ξ )} (1.8) i i arising from Eq. (1.7) is one to one, we can use the one-point functions {O (t)} instead of the coupling constants {g (ξ )}. Of course, this idea is well known for the case of the gauge coupling constant [3]: 2 2 a a g (μ = 1/ 8t) ∝ t G G (t) . (1.9) μν μν In what follows, we illustrate the idea (1.8) in theories in which several coupling constants play an interesting role; we will observe the ﬂow of relevant and irrelevant coupling constants around an RG ﬁxed point through the correspondence (1.8). We hope that our present consideration will be useful for more difﬁcult models for which an infrared non-trivial ﬁxed point can be concluded only non-perturbatively. 2. 4D N -ﬂavor gauge theory and the Banks–Zaks ﬁxed point Our ﬁrst example is the 4D vector-like gauge theory with N -ﬂavor Dirac fermions with the degenerate mass m. As the operators in Eq. (1.8), we take (as the one corresponding to the gauge coupling [3]) 2 2 8(4π) t 1 a a O (t, x) ≡ G (t, x)G (t, x) (2.1) μν μν 3 dim(G) 4 We implicitly assume the presence of the ultraviolet cutoff. Here, we neglect a possible non-trivial mixing of operators under the RG ﬂow, for notational simplicity. 4 a The generators T (a runs from 1 to dim(G)) of the gauge group G are anti-Hermitian and the structure a b abc c acd bcd ab constants are deﬁned by [T , T ]= f T . Quadratic Casimirs are deﬁned by f f = C (G)δ and, for a a b ab a a gauge representation R,tr (T T ) =−T (R)δ and T T =−C (R)1. We also denote tr (1) = dim(R). R 2 R 2/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. and 2 3/2 χ( ¯ t, x)χ (t, x) (4π) t O (t, x) ≡   ≡− χ( ¯ t, x)χ( ˚ t, x). (2.2) ← → 2 dim(R)N 1/2 t χ( ¯ t, x) D / χ(t, x) The ﬂowed gauge ﬁeld and its local products such as O (t, x) do not receive any multiplicative renormalization [4]. On the other hand, although the ﬂowed Dirac ﬁeld is multiplicatively renormal- ized [5], the renormalization of local products is simply determined by the number of Dirac ﬁelds in the product. Thus, O (t, x) in Eq. (2.2) also does not receive multiplicative renormalization because of the division by the expectation value [15]. In the present system, one observes the so-called Banks–Zaks infrared ﬁxed point [16,17]ifone uses the two-loop approximation of the beta function. We introduce the running gauge coupling g ¯ (μ) and the running mass parameter m ¯ (μ) in the MS scheme, respectively, by −b /(2b ) 2 0 b g ¯ (μ) exp − = , (2.3) 2b g ¯ (μ) μ d /(2b ) 0 0 m ¯ (μ) = M 2b g ¯ (μ) , (2.4) where 1 11 4 b = C (G) − T (R)N , (2.5) 0 2 f (4π) 3 3 1 34 20 b = C (G) − 4C (R) + C (G) T (R)N , (2.6) 1 2 2 2 (4π) 3 3 d = 6C (R), (2.7) 0 2 (4π) and and M are RG invariant mass scales. In terms of these running parameters, we have the one-point function √ √ 2 4 g ¯ (1/ 8t) g ¯ (1/ 8t) O (t) =¯ g(1/ 8t) 1 + K (t) + K , (2.8) 1 1 2 2 4 (4π) (4π) where 11 52 4 8 8 K (t) = γ + − 3ln 3 C (G) + − γ − + ln 2 + 16m ¯ (1/ 8t) t T (R)N , (2.9) 1 E 2 E f 3 9 3 9 3 and 2 2 K = 8(4π) −0.0136423(7)C (G) 2 2 [ ] + 0.006 440 134(5)C (R) − 0.008 688 4(2)C (G) T (R)N 2 2 f 2 2 + 0.000 936 117T (R) N . (2.10) Equation (2.8) for the massless case was obtained in Ref. [3] and for general mass cases in Ref. [18]; we have retained only the leading mass correction in Eq. (2.9) (as given in Eq. (2.34) of Ref. [18]). Although this treatment of the mass correction, which is also adopted in Eq. (2.11), is approximate, 3/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. this makes the resulting RG equations (2.12) and (2.13) quite simple and illustrative, so here we content ourselves with this approximate treatment. On the other hand, to the one-loop order, O (t, x) is given by g ¯ (1/ 8t) 1/2 O (t) =¯ m(1/ 8t)t 1 + [3γ + 4 + 2ln 2 − ln(432)] C (R) . (2.11) 2 E 2 (4π) We now take the ﬂow time derivatives of Eqs. (2.8) and (2.11). By using Eqs. (2.3) and (2.4) (or the corresponding RG equations) and eliminating the running parameters in favor of one-point functions, we arrive at d 1 2 3 2 2 t O (t) = b O (t) + b O (t) + 16T (R)N O (t) O (t) , (2.12) 1 0 1 1 1 f 1 2 dt (4π) d 1 t O (t) = [1 + d O (t)] O (t). (2.13) 2 0 1 2 dt 2 From these equations, it is clear that O (t) and O (t) can be used as parameters in the coupling 1 2 constant space. Note that the RG coefﬁcients b , b , and d are universal. In the infrared limit t →∞, 0 1 0 O (t) = 0 corresponds to a relevant coupling O (t)→∞ around the Banks–Zaks ﬁxed point 2 2 at (O (t), O (t)) = (−b /b ,0). 1 2 0 1 3. 3D O(N ) linear sigma model at large N and the Wilson–Fisher ﬁxed point Our second example is the 3D O(N ) linear sigma model that possesses the so-called Wilson–Fisher ﬁxed point [19] in the infrared limit. The gradient ﬂow of an operator in this system in relation to the Wilsonian RG ﬂow was studied in detail in Ref. [10] and the Wilson–Fisher ﬁxed point was observed. Actually, our present study was partially motivated by the study of Ref. [10]. We will consider the RG ﬂow in the 2D coupling constant space in which there is one direction of the relevant operator around the ﬁxed point (in Ref. [10], only 1D space along the irrelevant coupling is considered). We will work out the large-N approximation to the order of our concern. So, we ﬁrst recapitulate the solution of the model in the large-N approximation for later use. 3.1. The solution in the large-N approximation The Euclidean action of the 3D O(N ) linear sigma model is given by 1 1 1 3 2 2 S = d x ∂ φ (x)∂ φ (x) + m φ (x)φ (x) + λ [φ (x)φ (x)] , (3.1) μ i μ i i i 0 i i 2 2 8N where i = 1, ... , N . We introduce the effective action, i.e., the generating functional of the 1PI correlation functions, as (n) 3 3 [φ]= d x ··· d x φ (x ) ··· φ (x ) (x , ... , x ), (3.2) 1 n i 1 i n 1 n 1 n i ···i 1 n n! n=0 (n) where  (x , ... , x ) are the vertex functions. We also introduce the Fourier transformation: 1 n i ···i 1 n (n) (x , ... , x ) 1 n i ···i 1 n 3 3 d p d p 1 n (n) −ip x −···−ip x 3 1 1 n n = ··· e  (p , ... , p )(2π) δ(p + ··· + p ). (3.3) 1 n 1 n i ···i 3 3 1 n (2π) (2π) The computation of this is given in v2 of the arXiv reference in Ref. [15]. We assume b < 0; note that O (t) > 0 by deﬁnition. 1 1 4/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. The large-N approximation in this model is well known and, at the leading order of the approximation, by using the auxiliary ﬁeld method for instance, we have (2) 2 2 (p , p ) = δ (p + M ), (3.4) 1 2 i i i i 1 2 1 1 2 (4) (p , p , p , p ) 1 2 3 4 i i i i 1 2 3 4 ⎡ ⎛ ⎞ ⎤ −1 λ λ 1 1 (p + p ) 0 0 1 2 ⎣ ⎝ ⎠ ⎦ = δ δ 1 +  arctan + (2 ↔ 3) + (2 ↔ 4). i i i i 1 2 3 4 N 8π 2 M (p + p ) 1 2 (3.5) In these expressions, the “physical” mass M is given by the solution to the so-called gap equation, λ 1 2 2 M + M = m + λ , (3.6) 8π 4π with being the momentum cutoff. In the present model, the renormalized parameters in the mass-independent renormalization scheme can be deﬁned as 2 2 2 m = Z m + δm , λ = Z λ. (3.7) m 0 λ 0 0 As Eq. (3.4) shows, there is no need of the wave function renormalization in the leading order of the large-N approximation. We ﬁx the renormalization constants Z , δm , and Z by imposing the m λ following renormalization conditions at the renormalization scale μ: (2) (p , p ) = 0, (3.8) 1 2 i i 1 2 2 2 p =p =0,m =0 1 2 (2) (p , p ) = μ , (3.9) 1 2 i i 1 2 2 2 2 2 p =p =0,m =μ 1 2 (4) (p , p , p , p ) = δ δ + (2 ↔ 3) + (2 ↔ 4). (3.10) 1 2 3 4 i i i i i i i i 1 2 3 4 1 2 3 4 1 2 2 2 2 p ·p =μ δ − μ (1−δ ),m =μ N i j ij ij From the ﬁrst two relations, we have 1 1 λ δm =− λ , Z = 1 + , (3.11) 0 m 4π 8π μ and, from the last renormalization condition, ! " −1 λ λ 3 λ 0 0 = 1 + . (3.12) μ μ 96 μ This gives rise to the beta function λ ∂ λ λ 3 λ β ≡ μ =− + . (3.13) μ ∂μ μ μ 96 μ We note that the slopes of the beta function at two zeros of the beta function (ﬁxed points) are given by λ λ 96 ∗ ∗ β = 0 =−1, β = √ =+1, (3.14) μ μ respectively. 7 ∂ The subscript 0 in (μ ) implies that the derivative is taken while the bare parameters are kept ﬁxed. ∂μ 5/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. On the other hand, from the above relations, we have ! " −1 2 m 1 λ 1 λ 0 0 = 1 + + , (3.15) 2 2 2 2 μ 8π μ μ 4π μ and # $⎡ ⎤ 3 1 3 λ 2 1 + − 2 ∂ m 2 8π 96 μ m ⎣ ⎦ #$ μ =−2 √ . (3.16) 2 2 1 3 λ ∂μ μ μ 0 1 + − 8π 96 μ This RG equation becomes quite simple in terms of the parameter M deﬁned by Eq. (3.6): ∂ M M μ =− . (3.17) ∂μ μ μ 3.2. The ﬂowed system and the RG ﬂow We now examine the picture (1.8) in the present model. We ﬁrst have to introduce the ﬂow equation for the scalar ﬁeld φ (x). The simplest choice is ∂ ϕ (t, x) = ∂ ∂ ϕ (t, x), ϕ (t = 0, x) = φ (x). (3.18) t i μ μ i i i We refer the reader to Ref. [20] for the renormalizability of the ﬂowed scalar theory. With the above choice, the correlation functions of the ﬂowed ﬁeld ϕ (t, x) can be obtained from those of φ (y) i i simply substituting ϕ (t, x) by d p 2 3 ip(x−y) −tp ϕ (t, x) = d y e e φ (y). (3.19) i i (2π) We thus have, for instance, 3 −2p d p e −1/2 0 ϕ (t, x)ϕ (t, x) = Nt + O((1/N ) ) i i 3 2 2 (2π) p + M t 1 M t→0 −1/2 0 → N t − N + O((1/N ) ), (3.20) 3/2 2(2π) 4π −3/2 2 0 ∂ ϕ (t, x)∂ ϕ (t, x) = N t − M ϕ (t, x)ϕ (t, x) + O((1/N ) ), (3.21) μ i μ i i i 3/2 (8π) and 2 2 [ϕ (t, x)ϕ (t, x)] − 1 + ϕ (t, x)ϕ (t, x) i i i i ! " 3 −p d p e −1/2 3 =−N λ t (2π) δ(p + p + p + p ) 0 1 2 3 4 3 2 (2π) p + M t i=1 ⎡ ⎛  ⎞ ⎤ −1 1/2 λ t 1 1 (p + p ) 0 1 2 ⎣ ⎝ ⎠ ⎦ × 1 +  arctan + O((1/N ) ). (3.22) 8π 2 M t (p + p ) 1 2 Note that, in these expressions, momentum variables are dimensionless. 6/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. It is convenient to introduce a new ﬁeld variable, t→0 ϕ ˚ (t, x) ≡ ϕ (t, x) → ϕ (t, x) + O(1/N ), (3.23) i i i 3/2 1/2 2(2π) t ϕ (t, x)ϕ (t, x) j j by analogy with Eq. (2.2), which is free from the wave function renormalization. Using this new variable, we deﬁne dimensionless operators: 4(2π) O (t, x) ≡− t [ϕ ˚ (t, x)ϕ ˚ (t, x)] + (N + 2), (3.24) 1 i i 16π 1 3/2 O (t, x) ≡ t ∂ ϕ ˚ (t, x)∂ ϕ ˚ (t, x) − . (3.25) 2 μ i μ i 1/2 N (2π) Then, we have −2 3 −2p d p e 1/2 O (t) = λ t 1 0 3 2 2 (2π) p + M t ! " 3 −p d p e × (2π) δ(p + p + p + p ) 1 2 3 4 3 2 (2π) p + M t i=1 ⎡ ⎛  ⎞ ⎤ −1 1/2 λ t 1 1 (p + p ) 0 1 2 ⎣ ⎝ ⎠ ⎦ × 1 +  arctan (3.26) 8π 2 M t (p + p ) 1 2 t→0 1/2 → K λ t (3.27) −1 λ 1 λ 1 t→∞ 0 0 → K 1 + , (3.28) 3 3/2 M 16π M M t where ! " 3 −p d p e 3 3 K = 32π (2π) δ(p + p + p + p )  0.289 432, (3.29) 1 2 3 4 3 2 (2π) i=1 d p 2 1 3 −p 3 K = 512π e (2π) δ(p + p + p + p ) = , (3.30) 1 2 3 4 3 3/2 (2π) (4π) i=1 and ! " −1 1/2 3 −2p 1 d p e 8 1 O (t) = − M t − (3.31) 2 3 2 2 1/2 8π (2π) p + M t π (2π) t→0 1/2 → Mt (3.32) 1/2 2 3 1 t→∞ → − . (3.33) 1/2 2 π (8π) M t The asymptotic behaviors (3.27) and (3.32) show that the initial condition of the ﬂow is given by the parameters λ and M . In Figs. 1 and 2, we depict the RG ﬂow lines in the space of O (t) and O (t) 0 1 2 t→0 t→∞ 8 1/2 If one sets M → 0 ﬁrst, Eq. (3.26) yields O (t) → K λ t , O (t) → 1.425 96, while from Eq. (3.31), 1 0 1 O (t)≡ 0. 7/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 Fig. 1. The RG ﬂow in the space of O (t) and O (t). The arrows indicate how the point (O (t), O (t)) 1 2 1 2 changes as t increases. The red point is the infrared ﬁxed point. 0.20 0.15 0.10 0.05 0.00 0.0 0.5 1.0 1.5 Fig. 2. Same as Fig. 1, but the region around the horizontal axis is magniﬁed. obtained numerically. We conﬁrmed that the point indicated by the red point [(O (t), O (t)) = 1 2 (1.425 96, 0)] is an infrared ﬁxed point that can be identiﬁed with the Wilson–Fischer ﬁxed point (λ /μ = 96/ 3 in Eq. (3.14)). From the ﬁgures, we see that O (t) basically corresponds to the relevant coupling around the ﬁxed point; O (t) to the irrelevant coupling. Acknowledgements We would like to thank Robert Harlander for helpful remarks. The work of H.S. is supported in part by a JSPS Grant-in-Aid for Scientiﬁc Research Grant Number JP16H03982. 8/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 O O 2 2 PTEP 2018, 053B02 H. Makino et al. Funding Open Access funding: SCOAP . References [1] R. Narayanan and H. Neuberger, J. High Energy Phys. 03, 064 (2006) [arXiv:hep-th/0601210][Search INSPIRE]. [2] M. Lüscher, Commun. Math. Phys. 293, 899 (2010) [arXiv:0907.5491 [hep-lat]] [Search INSPIRE]. [3] M. Lüscher, J. High Energy Phys. 08, 071 (2010); 03, 092 (2014) [erratum] [arXiv:1006.4518 [hep-lat]] [Search INSPIRE]. [4] M. Lüscher and P. Weisz, J. High Energy Phys. 02, 051 (2011) [arXiv:1101.0963 [hep-th]] [Search INSPIRE]. [5] M. Lüscher, J. High Energy Phys. 04, 123 (2013) [arXiv:1302.5246 [hep-lat]] [Search IN SPIRE]. [6] K. G. Wilson and J. Kogut, Phys. Rept. 12, 75 (1974). [7] M. 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# Gradient flow and the Wilsonian renormalization group flow

, Volume Advance Article (5) – May 30, 2018
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### Abstract

Prog. Theor. Exp. Phys. 2018, 053B02 (9 pages) DOI: 10.1093/ptep/pty050 Gradient flow and the Wilsonian renormalization group flow Hiroki Makino, Okuto Morikawa, and Hiroshi Suzuki Department of Physics, Kyushu University 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan E-mail: hsuzuki@phys.kyushu-u.ac.jp Received March 2, 2018; Revised March 29, 2018; Accepted April 3, 2018; Published May 30, 2018 ................................................................................................................... The gradient ﬂow is the evolution of ﬁelds and physical quantities along a dimensionful param- eter t, the ﬂow time. We give a simple argument that relates this gradient ﬂow and the Wilsonian renormalization group (RG) ﬂow. We then illustrate the Wilsonian RG ﬂow on the basis of the gradient ﬂow in two examples that possess an infrared ﬁxed point, the 4D many-ﬂavor gauge theory and the 3D O(N ) linear sigma model. ................................................................................................................... Subject Index B01, B32, B37 1. Introduction and the basic idea The gradient ﬂow [1–5] is the evolution of ﬁelds and physical quantities along a dimensionful parameter t, the ﬂow time; the ﬂow acts as the “coarse-graining” as t > 0 becomes large. These two features of the gradient ﬂow are common to the Wilsonian renormalization group (RG) ﬂow [6]in a broad sense, provided that the ﬂow time is identiﬁed with the renormalization scale. In fact, it has sometimes been indicated that the gradient ﬂow and the Wilsonian RG ﬂow can be identiﬁed in some ways [7–10]; see also Refs. [11–14] for related studies. In this paper, we give a simple argument that relates the gradient ﬂow and the Wilsonian RG ﬂow; our argument is somewhat similar to that of Ref. [7]. We then illustrate the Wilsonian RG ﬂow on the basis of the gradient ﬂow in two examples that possess an infrared ﬁxed point, the 4D many-ﬂavor gauge theory and the 3D O(N ) linear sigma model. Our idea is very simple. We take the following ﬂow equations for the gauge potential A (x) and for the Dirac ﬁelds ψ(x) and ψ(x): ∂ B (t, x) = D G (t, x), B (t = 0, x) = A (x), (1.1) t μ ν νμ μ μ ∂ χ(t, x) = χ (t, x), χ(t = 0, x) = ψ(x), (1.2) ← − ∂ χ( ¯ t, x) =¯ χ(t, x)  , χ( ¯ t = 0, x) = ψ(x). (1.3) Let us consider the correlation function of operators composed of the ﬂowed ﬁelds: O (t , x ) ···O (t , x ) . (1.4) 1 1 1 N N N Here, the covariant derivative on the gauge ﬁeld is deﬁned by D ≡ ∂ +[B , ·]; the ﬁeld strength is μ μ μ deﬁned by G (t, x) ≡ ∂ B (t, x) − ∂ B (t, x) +[B (t, x), B (t, x)]. The Laplacians on the Dirac ﬁelds are μν μ ν ν μ μ ν ← − ← − ← − deﬁned by  ≡ D D , and  ≡ D D from the covariant derivatives on the Dirac ﬁelds, D = ∂ + B μ μ μ μ μ μ μ ← − ← − a a and D ≡ ∂ − B . We will occasionally use notation such as A (x) = A (x)T by using the generator of μ μ μ μ the gauge group, T . © The Author(s) 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 Funded by SCOAP by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. Let us also suppose that we have a set of (a generally inﬁnite number of) coupling constants {g } with which the correlation function is computed. We consider the mapping in this space of the coupling constants induced by the Wilsonian RG ﬂow, {g }→{g (ξ )}, (1.5) i i where ξ parametrizes the RG ﬂow. This RG ﬂow can be characterized by the scaling relation 2ξ ξ 2ξ ξ O (e t , e x ) ···O (e t , e x ) = Z (ξ ) O (t , x ) ···O (t , x ) , (1.6) 1 1 1 N N N 1 1 1 N N N {g (ξ )} {g } i where Z (ξ ) is the multiplicative renormalization factor and the subscript implies that the correlation function is evaluated with respect to the set of coupling constants. Compare this relation with, for instance, Eqs. (7.10) and (7.15) of Ref. [6]. Note that the ﬂow time has the mass dimension −2 instead of −1. The advantage of this characterization of the Wilsonian RG ﬂow is that this scaling relation itself can be written down even for gauge theory for which the momentum cutoff is incompatible with the gauge invariance (at least naively). In particular, for the one-point function of an operator that does not require the multiplicative renormalization, 2ξ O (e t) = O (t) , (1.7) 1 1 {g (ξ )} {g } where we have omitted the argument x assuming translational invariance in the x-space. Hence, assuming that the correspondence {O (t)} ⇔{g (ξ )} (1.8) i i arising from Eq. (1.7) is one to one, we can use the one-point functions {O (t)} instead of the coupling constants {g (ξ )}. Of course, this idea is well known for the case of the gauge coupling constant [3]: 2 2 a a g (μ = 1/ 8t) ∝ t G G (t) . (1.9) μν μν In what follows, we illustrate the idea (1.8) in theories in which several coupling constants play an interesting role; we will observe the ﬂow of relevant and irrelevant coupling constants around an RG ﬁxed point through the correspondence (1.8). We hope that our present consideration will be useful for more difﬁcult models for which an infrared non-trivial ﬁxed point can be concluded only non-perturbatively. 2. 4D N -ﬂavor gauge theory and the Banks–Zaks ﬁxed point Our ﬁrst example is the 4D vector-like gauge theory with N -ﬂavor Dirac fermions with the degenerate mass m. As the operators in Eq. (1.8), we take (as the one corresponding to the gauge coupling [3]) 2 2 8(4π) t 1 a a O (t, x) ≡ G (t, x)G (t, x) (2.1) μν μν 3 dim(G) 4 We implicitly assume the presence of the ultraviolet cutoff. Here, we neglect a possible non-trivial mixing of operators under the RG ﬂow, for notational simplicity. 4 a The generators T (a runs from 1 to dim(G)) of the gauge group G are anti-Hermitian and the structure a b abc c acd bcd ab constants are deﬁned by [T , T ]= f T . Quadratic Casimirs are deﬁned by f f = C (G)δ and, for a a b ab a a gauge representation R,tr (T T ) =−T (R)δ and T T =−C (R)1. We also denote tr (1) = dim(R). R 2 R 2/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. and 2 3/2 χ( ¯ t, x)χ (t, x) (4π) t O (t, x) ≡   ≡− χ( ¯ t, x)χ( ˚ t, x). (2.2) ← → 2 dim(R)N 1/2 t χ( ¯ t, x) D / χ(t, x) The ﬂowed gauge ﬁeld and its local products such as O (t, x) do not receive any multiplicative renormalization [4]. On the other hand, although the ﬂowed Dirac ﬁeld is multiplicatively renormal- ized [5], the renormalization of local products is simply determined by the number of Dirac ﬁelds in the product. Thus, O (t, x) in Eq. (2.2) also does not receive multiplicative renormalization because of the division by the expectation value [15]. In the present system, one observes the so-called Banks–Zaks infrared ﬁxed point [16,17]ifone uses the two-loop approximation of the beta function. We introduce the running gauge coupling g ¯ (μ) and the running mass parameter m ¯ (μ) in the MS scheme, respectively, by −b /(2b ) 2 0 b g ¯ (μ) exp − = , (2.3) 2b g ¯ (μ) μ d /(2b ) 0 0 m ¯ (μ) = M 2b g ¯ (μ) , (2.4) where 1 11 4 b = C (G) − T (R)N , (2.5) 0 2 f (4π) 3 3 1 34 20 b = C (G) − 4C (R) + C (G) T (R)N , (2.6) 1 2 2 2 (4π) 3 3 d = 6C (R), (2.7) 0 2 (4π) and and M are RG invariant mass scales. In terms of these running parameters, we have the one-point function √ √ 2 4 g ¯ (1/ 8t) g ¯ (1/ 8t) O (t) =¯ g(1/ 8t) 1 + K (t) + K , (2.8) 1 1 2 2 4 (4π) (4π) where 11 52 4 8 8 K (t) = γ + − 3ln 3 C (G) + − γ − + ln 2 + 16m ¯ (1/ 8t) t T (R)N , (2.9) 1 E 2 E f 3 9 3 9 3 and 2 2 K = 8(4π) −0.0136423(7)C (G) 2 2 [ ] + 0.006 440 134(5)C (R) − 0.008 688 4(2)C (G) T (R)N 2 2 f 2 2 + 0.000 936 117T (R) N . (2.10) Equation (2.8) for the massless case was obtained in Ref. [3] and for general mass cases in Ref. [18]; we have retained only the leading mass correction in Eq. (2.9) (as given in Eq. (2.34) of Ref. [18]). Although this treatment of the mass correction, which is also adopted in Eq. (2.11), is approximate, 3/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. this makes the resulting RG equations (2.12) and (2.13) quite simple and illustrative, so here we content ourselves with this approximate treatment. On the other hand, to the one-loop order, O (t, x) is given by g ¯ (1/ 8t) 1/2 O (t) =¯ m(1/ 8t)t 1 + [3γ + 4 + 2ln 2 − ln(432)] C (R) . (2.11) 2 E 2 (4π) We now take the ﬂow time derivatives of Eqs. (2.8) and (2.11). By using Eqs. (2.3) and (2.4) (or the corresponding RG equations) and eliminating the running parameters in favor of one-point functions, we arrive at d 1 2 3 2 2 t O (t) = b O (t) + b O (t) + 16T (R)N O (t) O (t) , (2.12) 1 0 1 1 1 f 1 2 dt (4π) d 1 t O (t) = [1 + d O (t)] O (t). (2.13) 2 0 1 2 dt 2 From these equations, it is clear that O (t) and O (t) can be used as parameters in the coupling 1 2 constant space. Note that the RG coefﬁcients b , b , and d are universal. In the infrared limit t →∞, 0 1 0 O (t) = 0 corresponds to a relevant coupling O (t)→∞ around the Banks–Zaks ﬁxed point 2 2 at (O (t), O (t)) = (−b /b ,0). 1 2 0 1 3. 3D O(N ) linear sigma model at large N and the Wilson–Fisher ﬁxed point Our second example is the 3D O(N ) linear sigma model that possesses the so-called Wilson–Fisher ﬁxed point [19] in the infrared limit. The gradient ﬂow of an operator in this system in relation to the Wilsonian RG ﬂow was studied in detail in Ref. [10] and the Wilson–Fisher ﬁxed point was observed. Actually, our present study was partially motivated by the study of Ref. [10]. We will consider the RG ﬂow in the 2D coupling constant space in which there is one direction of the relevant operator around the ﬁxed point (in Ref. [10], only 1D space along the irrelevant coupling is considered). We will work out the large-N approximation to the order of our concern. So, we ﬁrst recapitulate the solution of the model in the large-N approximation for later use. 3.1. The solution in the large-N approximation The Euclidean action of the 3D O(N ) linear sigma model is given by 1 1 1 3 2 2 S = d x ∂ φ (x)∂ φ (x) + m φ (x)φ (x) + λ [φ (x)φ (x)] , (3.1) μ i μ i i i 0 i i 2 2 8N where i = 1, ... , N . We introduce the effective action, i.e., the generating functional of the 1PI correlation functions, as (n) 3 3 [φ]= d x ··· d x φ (x ) ··· φ (x ) (x , ... , x ), (3.2) 1 n i 1 i n 1 n 1 n i ···i 1 n n! n=0 (n) where  (x , ... , x ) are the vertex functions. We also introduce the Fourier transformation: 1 n i ···i 1 n (n) (x , ... , x ) 1 n i ···i 1 n 3 3 d p d p 1 n (n) −ip x −···−ip x 3 1 1 n n = ··· e  (p , ... , p )(2π) δ(p + ··· + p ). (3.3) 1 n 1 n i ···i 3 3 1 n (2π) (2π) The computation of this is given in v2 of the arXiv reference in Ref. [15]. We assume b < 0; note that O (t) > 0 by deﬁnition. 1 1 4/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. The large-N approximation in this model is well known and, at the leading order of the approximation, by using the auxiliary ﬁeld method for instance, we have (2) 2 2 (p , p ) = δ (p + M ), (3.4) 1 2 i i i i 1 2 1 1 2 (4) (p , p , p , p ) 1 2 3 4 i i i i 1 2 3 4 ⎡ ⎛ ⎞ ⎤ −1 λ λ 1 1 (p + p ) 0 0 1 2 ⎣ ⎝ ⎠ ⎦ = δ δ 1 +  arctan + (2 ↔ 3) + (2 ↔ 4). i i i i 1 2 3 4 N 8π 2 M (p + p ) 1 2 (3.5) In these expressions, the “physical” mass M is given by the solution to the so-called gap equation, λ 1 2 2 M + M = m + λ , (3.6) 8π 4π with being the momentum cutoff. In the present model, the renormalized parameters in the mass-independent renormalization scheme can be deﬁned as 2 2 2 m = Z m + δm , λ = Z λ. (3.7) m 0 λ 0 0 As Eq. (3.4) shows, there is no need of the wave function renormalization in the leading order of the large-N approximation. We ﬁx the renormalization constants Z , δm , and Z by imposing the m λ following renormalization conditions at the renormalization scale μ: (2) (p , p ) = 0, (3.8) 1 2 i i 1 2 2 2 p =p =0,m =0 1 2 (2) (p , p ) = μ , (3.9) 1 2 i i 1 2 2 2 2 2 p =p =0,m =μ 1 2 (4) (p , p , p , p ) = δ δ + (2 ↔ 3) + (2 ↔ 4). (3.10) 1 2 3 4 i i i i i i i i 1 2 3 4 1 2 3 4 1 2 2 2 2 p ·p =μ δ − μ (1−δ ),m =μ N i j ij ij From the ﬁrst two relations, we have 1 1 λ δm =− λ , Z = 1 + , (3.11) 0 m 4π 8π μ and, from the last renormalization condition, ! " −1 λ λ 3 λ 0 0 = 1 + . (3.12) μ μ 96 μ This gives rise to the beta function λ ∂ λ λ 3 λ β ≡ μ =− + . (3.13) μ ∂μ μ μ 96 μ We note that the slopes of the beta function at two zeros of the beta function (ﬁxed points) are given by λ λ 96 ∗ ∗ β = 0 =−1, β = √ =+1, (3.14) μ μ respectively. 7 ∂ The subscript 0 in (μ ) implies that the derivative is taken while the bare parameters are kept ﬁxed. ∂μ 5/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. On the other hand, from the above relations, we have ! " −1 2 m 1 λ 1 λ 0 0 = 1 + + , (3.15) 2 2 2 2 μ 8π μ μ 4π μ and # $⎡ ⎤ 3 1 3 λ 2 1 + − 2 ∂ m 2 8π 96 μ m ⎣ ⎦ #$ μ =−2 √ . (3.16) 2 2 1 3 λ ∂μ μ μ 0 1 + − 8π 96 μ This RG equation becomes quite simple in terms of the parameter M deﬁned by Eq. (3.6): ∂ M M μ =− . (3.17) ∂μ μ μ 3.2. The ﬂowed system and the RG ﬂow We now examine the picture (1.8) in the present model. We ﬁrst have to introduce the ﬂow equation for the scalar ﬁeld φ (x). The simplest choice is ∂ ϕ (t, x) = ∂ ∂ ϕ (t, x), ϕ (t = 0, x) = φ (x). (3.18) t i μ μ i i i We refer the reader to Ref. [20] for the renormalizability of the ﬂowed scalar theory. With the above choice, the correlation functions of the ﬂowed ﬁeld ϕ (t, x) can be obtained from those of φ (y) i i simply substituting ϕ (t, x) by d p 2 3 ip(x−y) −tp ϕ (t, x) = d y e e φ (y). (3.19) i i (2π) We thus have, for instance, 3 −2p d p e −1/2 0 ϕ (t, x)ϕ (t, x) = Nt + O((1/N ) ) i i 3 2 2 (2π) p + M t 1 M t→0 −1/2 0 → N t − N + O((1/N ) ), (3.20) 3/2 2(2π) 4π −3/2 2 0 ∂ ϕ (t, x)∂ ϕ (t, x) = N t − M ϕ (t, x)ϕ (t, x) + O((1/N ) ), (3.21) μ i μ i i i 3/2 (8π) and 2 2 [ϕ (t, x)ϕ (t, x)] − 1 + ϕ (t, x)ϕ (t, x) i i i i ! " 3 −p d p e −1/2 3 =−N λ t (2π) δ(p + p + p + p ) 0 1 2 3 4 3 2 (2π) p + M t i=1 ⎡ ⎛  ⎞ ⎤ −1 1/2 λ t 1 1 (p + p ) 0 1 2 ⎣ ⎝ ⎠ ⎦ × 1 +  arctan + O((1/N ) ). (3.22) 8π 2 M t (p + p ) 1 2 Note that, in these expressions, momentum variables are dimensionless. 6/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. It is convenient to introduce a new ﬁeld variable, t→0 ϕ ˚ (t, x) ≡ ϕ (t, x) → ϕ (t, x) + O(1/N ), (3.23) i i i 3/2 1/2 2(2π) t ϕ (t, x)ϕ (t, x) j j by analogy with Eq. (2.2), which is free from the wave function renormalization. Using this new variable, we deﬁne dimensionless operators: 4(2π) O (t, x) ≡− t [ϕ ˚ (t, x)ϕ ˚ (t, x)] + (N + 2), (3.24) 1 i i 16π 1 3/2 O (t, x) ≡ t ∂ ϕ ˚ (t, x)∂ ϕ ˚ (t, x) − . (3.25) 2 μ i μ i 1/2 N (2π) Then, we have −2 3 −2p d p e 1/2 O (t) = λ t 1 0 3 2 2 (2π) p + M t ! " 3 −p d p e × (2π) δ(p + p + p + p ) 1 2 3 4 3 2 (2π) p + M t i=1 ⎡ ⎛  ⎞ ⎤ −1 1/2 λ t 1 1 (p + p ) 0 1 2 ⎣ ⎝ ⎠ ⎦ × 1 +  arctan (3.26) 8π 2 M t (p + p ) 1 2 t→0 1/2 → K λ t (3.27) −1 λ 1 λ 1 t→∞ 0 0 → K 1 + , (3.28) 3 3/2 M 16π M M t where ! " 3 −p d p e 3 3 K = 32π (2π) δ(p + p + p + p )  0.289 432, (3.29) 1 2 3 4 3 2 (2π) i=1 d p 2 1 3 −p 3 K = 512π e (2π) δ(p + p + p + p ) = , (3.30) 1 2 3 4 3 3/2 (2π) (4π) i=1 and ! " −1 1/2 3 −2p 1 d p e 8 1 O (t) = − M t − (3.31) 2 3 2 2 1/2 8π (2π) p + M t π (2π) t→0 1/2 → Mt (3.32) 1/2 2 3 1 t→∞ → − . (3.33) 1/2 2 π (8π) M t The asymptotic behaviors (3.27) and (3.32) show that the initial condition of the ﬂow is given by the parameters λ and M . In Figs. 1 and 2, we depict the RG ﬂow lines in the space of O (t) and O (t) 0 1 2 t→0 t→∞ 8 1/2 If one sets M → 0 ﬁrst, Eq. (3.26) yields O (t) → K λ t , O (t) → 1.425 96, while from Eq. (3.31), 1 0 1 O (t)≡ 0. 7/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053B02 H. Makino et al. 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 Fig. 1. The RG ﬂow in the space of O (t) and O (t). The arrows indicate how the point (O (t), O (t)) 1 2 1 2 changes as t increases. The red point is the infrared ﬁxed point. 0.20 0.15 0.10 0.05 0.00 0.0 0.5 1.0 1.5 Fig. 2. Same as Fig. 1, but the region around the horizontal axis is magniﬁed. obtained numerically. We conﬁrmed that the point indicated by the red point [(O (t), O (t)) = 1 2 (1.425 96, 0)] is an infrared ﬁxed point that can be identiﬁed with the Wilson–Fischer ﬁxed point (λ /μ = 96/ 3 in Eq. (3.14)). From the ﬁgures, we see that O (t) basically corresponds to the relevant coupling around the ﬁxed point; O (t) to the irrelevant coupling. Acknowledgements We would like to thank Robert Harlander for helpful remarks. The work of H.S. is supported in part by a JSPS Grant-in-Aid for Scientiﬁc Research Grant Number JP16H03982. 8/9 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053B02/5021515 by Ed 'DeepDyve' Gillespie user on 21 June 2018 O O 2 2 PTEP 2018, 053B02 H. Makino et al. Funding Open Access funding: SCOAP . References [1] R. Narayanan and H. Neuberger, J. High Energy Phys. 03, 064 (2006) [arXiv:hep-th/0601210][Search INSPIRE]. [2] M. Lüscher, Commun. Math. Phys. 293, 899 (2010) [arXiv:0907.5491 [hep-lat]] [Search INSPIRE]. [3] M. Lüscher, J. High Energy Phys. 08, 071 (2010); 03, 092 (2014) [erratum] [arXiv:1006.4518 [hep-lat]] [Search INSPIRE]. [4] M. Lüscher and P. Weisz, J. High Energy Phys. 02, 051 (2011) [arXiv:1101.0963 [hep-th]] [Search INSPIRE]. [5] M. Lüscher, J. High Energy Phys. 04, 123 (2013) [arXiv:1302.5246 [hep-lat]] [Search IN SPIRE]. [6] K. G. Wilson and J. Kogut, Phys. Rept. 12, 75 (1974). [7] M. Lüscher, PoS LATTICE 2013, 016 (2014) [arXiv:1308.5598 [hep-lat]] [Search INSPIRE]. [8] A. Kagimura, A. Tomiya, and R. Yamamura, arXiv:1508.04986 [hep-lat] [Search INSPIRE]. [9] R. Yamamura, Prog. Theor. Exp. Phys. 2016, 073B10 (2016) [arXiv:1510.08208 [hep-lat]] [Search INSPIRE]. [10] S. Aoki, J. Balog, T. Onogi, and P. Weisz, Prog. Theor. Exp. Phys. 2016, 083B04 (2016) [arXiv:1605.02413 [hep-th]] [Search INSPIRE]. [11] S. Aoki, K. Kikuchi, and T. Onogi, Prog. Theor. Exp. Phys. 2015, 101B01 (2015) [arXiv:1505.00131 [hep-th]] [Search INSPIRE]. [12] S. Aoki, J. Balog, T. Onogi, and P. Weisz, Prog. Theor. Exp. Phys. 2017, 043B01 (2017) [arXiv:1701.00046 [hep-th]] [Search INSPIRE]. [13] S. Aoki and S. Yokoyama, Prog. Theor. Exp. Phys. 2018, 031B01 (2018) [arXiv:1707.03982 [hep-th]] [Search INSPIRE]. [14] S. Aoki and S. Yokoyama, arXiv:1709.07281 [hep-th] [Search INSPIRE]. [15] H. Makino and H. Suzuki, Prog. Theor. Exp. 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Progress of Theoretical and Experimental PhysicsOxford University Press

Published: May 30, 2018

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