PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.99.025004ARTICLESFormal aspects of field theory, field theory in curved spacea-theorem at large Nfa-THEOREM AT LARGE NfANTIPIN et al.AntipinOleg^{1}DondiNicola Andrea^{2}SanninoFrancesco^{2}ThomsenAnders Eller^{2}Rudjer Boskovic Institute, Division of Theoretical Physics, Bijenička 54, HR-10000 Zagreb, CroatiaCP^{3}-Origins & Danish IAS, University of Southern Denmark, Campusvej 55, Odense M—DK-5230, Denmark7January201915January201999202500413August2018Published by the American Physical Society2019authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP^{3}.

We determine the Jack and Osborn a-function and related metric for gauge-fermion theories to leading order in the large number of fermions and to all orders in the gauge coupling, demonstrating that the strong a-theorem is violated for the minimal choice of the a-function.

Hrvatska Zaklada za Znanost10.13039/5011000044884418H2020 European Research Council10.13039/100010663692194Danmarks Grundforskningsfond10.13039/501100001732DNRF:90

Quantum field theory (QFT) is the language chosen by nature to describe its fundamental laws with the renormalization group (RG) flow connecting physics at different energy scales. Remarkably this flow is thought to be irreversible as encompassed by Cardy’s proposal of the 4-dimensional a-theorem [1], originally inspired by the 2-dimensional proof of the Zamolodchikov c-theorem [2]. The main idea is that one can, in principle, define a monotonically decreasing function from the ultraviolet (UV) to the infrared (IR) along the flow.

Our goal is to push forward the state of the art by computing the a-function to all orders in the couplings, for nonsupersymmetric field theories, exploiting the large number of flavors limit [3].

In order to define the relevant quantities we start by considering a generic theory defined via the bare Lagrangian L0(g0i,ϕ0) for a set of fundamental fields denoted by ϕ0. We restrict ourselves to the case in which the couplings, g0i, are associated to marginal operators. The theory is extended to curved background, changing the metric ημν→γμν(x) so that it is classically invariant under diff×Weyl, and to spacetime-dependent couplings g0i→g0i(x).

Within the extended theory the (renormalized) couplings act as sources for associated composite operators, Oi=δSδgi. If we appropriately renormalize the vacuum energy functional, W=W[γμν,gi], such that it is finite, we can extract renormalized composite operator correlators through functional derivatives of W. To achieve this, usual renormalization is insufficient: the spacetime dependence of the couplings induces new divergences that have to be canceled by new counter terms (CTs) proportional to coupling derivatives. These of course vanish in the limit of constant sources.

We restrict ourselves to only listing the field-independent CTs. The new Lagrangian in d=4-ε reads [4]L˜0(g0i,ϕ0)=L0(g0i,ϕ0)+μ-ελ(g)·R.Of all 4-dimensional CTs built from metric and couplings that can appear in λ·R we are mainly interested in the following ones: λ·R⊃λaE4+12Gij∂μgi∂νgjGμν+12Aij∇2gi∇2gj+12Bijk∂μgi∂μgj∇2gk,where E4 is the Euler density and Gμν is the Einstein tensor. In the minimal subtraction (MS) scheme, the CT coefficients only contain poles in ε; there are no finite parts. These terms are needed to renormalize specific contact divergences in composite operator correlators, so they can be written directly in terms of the renormalized couplings, gi.

In this framework, RG transformations are deeply interconnected with Weyl rescalings: γμν→e-2σγμν,ϕ→eσΔϕϕ.We know that for a general QFT in curved space the Weyl symmetry, when present at the classical level, is anomalous. We take {gi} to couple to the set of all marginal operators defined at gi=0, so there exists a source transformation law such that the field-dependent part of the action is invariant under Weyl transformation [5,6]. The associated operator Δσ≡∫ddxσ(x)[2γμνδδγμν-β^iδδgi] acting on W transforms the field-independent CTs ΔσW=Δσ∫ddxγμ-ελ·R=∫ddxγμ-ε(σβλ·R+∂μσLμ).Here μddμgi=β^i=-ρigiε+βi(g).

ρi is defined such that g0iμ-ρiε is dimensionless, and its index does not count in the summation convention.

The σ-dependent part of Eq. (4) has a tensor expansion analogous to (2): βλ·R⊃aE4+12χijg∂μgi∂νgjGμν+12χija∇2gi∇2gj+12χijkb∂μgi∂μgj∇2gk.Since [μ∂∂μ+∫ddx2γμνδδγμν]W=0 by dimensional analysis, for a global (σ=const) transformation, the integrand of (4) becomes [ε-∫ddyβ^i(y)δδgi(y)]λ·R=βλ·R.From this we derive, e.g., χijg=(ε-β^ℓ∂ℓ)Gij-Gℓj∂iβ^ℓ-Giℓ∂jβ^ℓ,χija=(ε-β^ℓ∂ℓ)Aij-Aℓj∂iβ^ℓ-Aiℓ∂jβ^ℓ,which we will need later.

Since ΔσW has to be finite by construction, χg and χa must be finite too. This implies that (7) and (8) can be interpreted as RG equations for the CTs Gij and Aij. The same reasoning is independently valid for the term proportional to ∂μσ in (4). The constraints originating from the requirement of finiteness

These relations can be also recovered by imposing the Weyl variation to be Abelian: [Δσ1,Δσ2]W=0.

were found in [4], and lead to nontrivial relations between CTs. First off 8∂ia˜=(χijg+(∂iwj-∂jwi))βj,8a˜≡8a+wiβi,where wi is a 1-form parametrizing a renormalization scheme redundancy, and a˜ coincides with a at fixed points (FPs). It follows that a˜ satisfies the gradient flow equation 8μda˜dμ=8βi∂ia˜=χijgβiβj,which suggests viewing χijg as a metric in the space of couplings.

Under the assumption that the metric χijg is positive definite a˜ decreases monotonically along the RG flow. The positivity of the metric χijg has been established at sufficiently small couplings to the highest known order in perturbation theory, and it has been conjectured to hold nonperturbatively. Up to an arbitrariness in the definition of χg (to be discussed later), this constitutes the strong version of the a-theorem conjecture, see e.g., [7]. In contrast, the weak version of the a-theorem has been proven[8] and states that the quantity Δa˜=(a˜UV-a˜IR)=Δa=(aUV-aIR)>0 for any RG flow between physical FPs, where a˜UV(IR) is evaluated at the corresponding FPs.

Another consistency relation, derived with the same procedure as (9), turns out to be particularly useful: χijg=-2χija+χ¯ijkaβk-βℓ∂ℓVij-∂iβℓVℓj-∂jβℓViℓ,where Vij≡ρkgkA¯ijk(1),A¯ijk≡∂kAij-12Bikj-12Bjki,χ¯ijka≡(ε-β^ℓ∂ℓ)A¯ijk-A¯ℓjk∂iβ^ℓ-A¯iℓk∂jβ^ℓ-A¯ijℓ∂kβ^ℓ,and Aijk(1) is the residue of the 1/ε pole. Thus, χg can be computed from CTs needed to renormalize contact divergences of marginal operators.

In this article, we consider Yang-Mills theories with Nf vectorlike fermions in the 1/Nf expansion. We compute the metric χijg and the a-function a˜ to leading order (LO) in 1/Nf but to all orders in the gauge coupling.

PRELIMINARIES TO DETERMINE THE METRICS

Using the finiteness of n-point functions in the renormalized theory we show how to determine the CTs Aij and Bijk in regular flat space and constant gi. We start with the 2-point function δδgi(x)δδgj(y)W=i⟨[Oi(x)][Oj(y)]⟩+⟨δ[Oj(y)]δgi(x)⟩,which must be finite. Working with the flat metric γμν=ημν and following [4], we define [Oi]c≡[Oi]|∂μg=0=∂ihaOa0.The last equality embodies that the standard constant-coupling operators [Oi]c can always be expanded in terms of some functions ha(gi) and coupling independent operators Oa0. Only the CTs in L˜0 have dependence on ∂μgi, so when the limit of spacetime-independent couplings is taken, we have δ[Oj(y)]δgi(x)=Kijk[Ok]cδd(x-y)+μ-εAij∂4δd(x-y),where Kijk≡∂i∂jha∂gk∂ha. Also in this limit, the Fourier-transformed 2-point function is defined by Γij(p)=i∫ddxe-ip·x⟨[Oi(x)]c[Oj(0)]c⟩.The renormalized and finite 2-point function from (13) takes the form ΓijR(p)=Γij(p)+μ-εAij(p2)2+Kijk⟨[Ok]c⟩.This relation can be used to extract the Aij CT from the momentum-dependent part of Γij(p) in flat space and constant couplings.

Continuing on to the 3-point function, it is given by δδgi(x)δδgj(y)δδgk(z)W=-⟨[Oi(x)][Oj(y)][Ok(z)]⟩+i∑cyc⟨δ[Oj(y)]δgi(x)[Ok(z)]⟩+i⟨δ2[Ok(z)]δgi(x)δgj(y)⟩,the sum being over cyclic permutations of i, j, k and x, y, z. The Fourier-transformed 3-point function is defined by Γijk(p,q)=-∫ddxddye-ip·xe-iq·y⟨[Oi(x)]c[Oj(y)]c[Ok(0)]c⟩,to allow for computations in momentum space. The CTs of the renormalized 3-point function (18) can be determined from the CTs (2) and the relation (15). Taking for simplicity ⟨[Oi]c⟩=0, the finite 3-point function with constant couplings and flat metric is ΓijkR(p,q)=Γijk(p,q)+KijℓΓℓk(p+q)+KjkℓΓℓi(p)+KkiℓΓℓj(q)+μ-ε(A¯ijkp2q2+A¯jkiq2(p+q)2+A¯kij(p+q)2p2)+12μ-ε(Bijk(p+q)4+Bjkip4+Bkijq4).From here A¯ijk can be extracted by, say, considering the term proportional to p2q2, which does not receive contributions from any other CTs.

This sets up our computation of the Aij and A¯ijk CTs which, through (8) and (11), will allow us to obtain the metric χijg.

LARGE <inline-formula><mml:math display="inline"><mml:msub><mml:mi>N</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:math></inline-formula> METRIC AND <inline-formula><mml:math display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula>-FUNCTION

Consider a theory with large number of fermions, Nf, charged under a simple (non-)Abelian gauge group; L0=-14g02F0,μνaF0a,μν+∑n=1NfiΨ¯n,0γμ(∂μ-iA0μ)Ψn,0.Since the theory has a single (gauge) coupling we can suppress all coupling indices i,j,k,… on the CTs. We now move to determine the leading 1/Nf contribution to the 2- and 3-point correlation functions to all orders in perturbation theory, keeping the coupling K≡g2NfTR/(4π2) fixed in order to prepare for the large Nf limit.

The leading 1PI correction to the gauge field 2-point function is given by the amputated fermion loop iΠμν(p)=ip2Δμν(p)μ-εΠ0(p2),Π0(p2)=-NfTR2π2Γ2(2-ε2)Γ(ε2)Γ(4-ε)(-4πμ2p2)ε/2,with the transverse projector Δμν(p)=ημν-pμpν/p2. One may then extract the LO contribution to the gauge field renormalization or equivalently to the coupling renormalization, setting K0=ZA-1K,ZA=1-2K3ε+O(1/Nf).The beta function associated to the coupling is given by to LO in 1/Nfβ(K)=23K2+O(1/Nf).

Following this renormalization convention, the operator associated with K is found to be [OK]c=NfS2(R)16π2K2μ-εF02+O(1/Nf).Only the F2-term contributes at LO to the 2- and 3-point functions.

At LO in the 1/Nf expansion, 2- and 3-point correlation functions are computed by dressing the gluon propagators with fermion bubble chains as shown in Fig. 1. For the 2-point function, we have ΓKK(p)=id(G)2K2ZA2∫ddk(2π)d-iΔμν(k)k2[1-Π0(k)]Vνρ(k,k+p)×-iΔρσ(k+p)(k+p)2[1-Π0(k+p)]Vσμ(k+p,k),where Vμν(p,q)=p·qημν-qμpν is the momentum dependent Feynman rule stemming from a F2 vertex and d(G)≡Nc2-1 is the number of gauge bosons in the loop. Here we summed over every number of bubble insertions, which for the purpose of finding the divergent part is equivalent to using dressed gauge propagators. Extracting the (p2)2 dependent part of ΓKK(p) and setting p→0, the integral may be evaluated using elementary methods. The A CT is then determined from the finiteness of the renormalized 2-point function (17) to be A=-3d(G)64π2K2divHa(ε)ZA2K0,whereHa(x)=(1-x3)(240-240x+90x2-15x3+x4)Γ(4-x)60(4-x)(6-x)Γ(1+x2)Γ3(2-x2).Here “div” is taken to mean the divergent part of the expression as ε→0.

110.1103/PhysRevD.99.025004.f1

LO 2- and 3-point functions. The crosses represent insertions of the [Og]c composite operator.

Similarly, we evaluate the divergent part of the p2q2 term in the 3-point function ΓKKK(p,q)(20) which allows us to determine the A¯ CT in (12). We find A¯=d(G)64π2K3divH¯a(ε)ZA3K0,whereH¯a(x)=(80-60x+13x2-x3)xΓ(4-x)120(4-x)Γ(1+x2)Γ3(2-x2).

The 1/ε pole of A and A¯ can be extracted using ZAK0=K. Since both Ha and H¯a are regular they are expanded as power series and then resummed in the coupling K at the simple pole in a manner similar to [9]. Inserting these results back into Eqs. (8), (11), and (12) yields χa=-d(G)32π2K2∂K[KHa(23K)]=-d(G)32π2K2(1-53K+49108K2+⋯),χg=d(G)16π2K2∂K[KHa(23K)-19K2H¯a(23K)]=d(G)16π2K2(1-53K+25108K2+⋯).Our result for χa agrees with [10] to all orders and to O(K2) with [11]. Both metrics also agree with [4] to O(K). Notice that the LO result only distinguishes the Abelian and non-Abelian theory through an overall normalization because gauge self-interactions are subleading in 1/Nf. The theories also share the LO in 1/Nf beta function, Eq. (24), with corresponding Landau pole. Using (9), we can now derive the LO a-function (the contribution from the one-form wi vanishes) a˜-a˜free=∫dKK212χg=d(G)192π2[KHa(23K)-19K2H¯a(23K)],where K is the large Nf coupling defined previously and a˜free is the free field theory result: a˜free=190(8π)2(11Nf+62d(G))≈11Nf90(8π)2.

Finally, in Fig. 2 we plot the metric χg and the a-function a˜-a˜free. We conclude that the metric is not positive definite for all values of K and thus, the a-function is not monotonic, violating the strong version of the a-theorem. For completeness, the function χa is plotted in Fig. 3.

210.1103/PhysRevD.99.025004.f2

The LO in 1/Nf metric and a-function.

310.1103/PhysRevD.99.025004.f3

The LO in 1/Nf function χa.

One natural interpretation to restore the strong version of the a-theorem is that the flow towards the IR should start at K no higher than ≈0.8 in the UV. In fact, to this order in 1/Nf, the underlying theory is UV incomplete and should be considered as an effective field theory that could be trusted up to a maximum value of the energy corresponding to K≈0.8. In other words monotonicity of the a-function gives us a sense of how far in the UV the theory can be pushed as an effective field theory. As the underlying theory is UV incomplete, we cannot impose the weak version of the a-theorem. We notice however that if we were to impose it, we can extend the validity of the theory to K≈2.6 where the a-function becomes slightly negative. In the interval 3<K<6 we find that a is positive albeit not monotonic. In between K>6 and K=15/2 we find that a is negative with the LO metric and the a-function having a pole at K=15/2.

Different versions of the a-function can be obtained by redefining a with an arbitrary function f(K) parametrizing the RG scheme change [12]: a˜′=a˜+f(K)β2 and simultaneously modifying the metric to χg′=χg+8[β∂Kf(K)+2f(K)∂Kβ] so that (10) is invariant. To LO in 1/Nf, we have a˜′=a˜+4K4f(K)9χg′=χg+16K[K∂f(K)3∂K+43f(K)].Other proposals have been used in e.g., [8,13,14]. It is not known if other versions of the a-function are monotonic outside perturbation theory. Nevertheless, we have shown that the Jack and Osborn version [4] is not monotonic.

SUBLEADING CORRECTIONS AND OUTLOOK

At LO in 1/Nf, no UV fixed point can emerge and the theory is therefore at best viewed as an effective field theory. However, the situation becomes intriguing upon considering the 1/Nf corrections. In particular, it has been argued in favor of the existence of an interacting UVFP for gauge-fermion theories due to the interplay between the leading and the subleading terms in 1/Nf. Although the result is not as well established as the discovery of asymptotic safety in four dimensions in the Veneziano-Witten limit [15], it has nevertheless led to a number of phenomenological [16–18] and theoretical investigations [19,20] culminating in the conformal window 2.0 [21]. According to the studies above, the UVFP for the fundamental representation occurs at KQCD*=4NfTRα*=3-exp[-pNfNc+k],KQED*=4Nfα*=152-0.0117e-15π2Nf/7,where p=16TR and k=15.86+2.63/Nc2. The UVFP is expected to appear above some critical number of flavors Nfcrit above which the large Nf expansion is reliable. We notice that, for both Abelian and non-Abelian theories, the UVFP occurs for K>0.8 and thus the strong version of the a-theorem is necessarily violated. Intriguingly, the non-Abelian fixed point occurs very close to the apparent loss of validity of the weak a-theorem. Of course, very near to the UVFP one must include the missing 1/Nf corrections to the a-function. For the Abelian case the alleged UVFP occurs at a pole of the a-function.

If an UVFP exists for the non-Abelian case, only the weak version of the a-theorem survives, the reason being that the 1/Nf corrections cannot change the nonmonotonic character of the a-function away from the UVFP.

We elucidated the dynamics of large Nf gauge-fermion theories by determining important properties such as the metric and a-function, for the first time, to all orders in perturbation theory. Our results can be tested via first principle lattice simulations, and can be further extended to multiple-couplings theories at large Nf[22–24].

ACKNOWLEDGMENTS

We thank Colin Poole and Vladimir Prochazka for discussion and helpful comments. The work of O. A. is partially supported by the Croatian Science Foundation Project No. 4418 as well as the H2020 CSA Twinning Project No. 692194, RBI- T-WINNING while N. A. D., F. S., and A. E. T. are partially supported by the Danish National Research Foundation Grant No. DNRF:90.

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