ANEC on stress-tensor states in perturbative 4theory Teresa Bautista1and Lorenzo Casarin23

2025-04-30 0 0 506.45KB 25 页 10玖币

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ANEC on stress-tensor states

in perturbative λ φ4theory

Teresa Bautista1and Lorenzo Casarin2,3

1Department of Mathematics, King’s College London,

The Strand, London WC2R 2LS, UK

2Institut f¨ur Theoretische Physik

Leibniz Universit¨at Hannover

Appelstraße 2, 30167 Hannover, Germany

3Max-Planck-Institut f¨ur Gravitationsphysik (Albert Einstein Institut)

Am M¨uhlenberg 1, D-14476 Potsdam, Germany

October 2022

Abstract

We evaluate the Average Null Energy Condition (ANEC) on momentum eigenstates generated by

the stress tensor in perturbative λ φ4and general spacetime dimension. We ﬁrst compute the

norm of the stress-tensor state at second order in λ; as a by-product of the derivation we obtain

the full expression for the stress tensor 2-point function at this order. We then compute the

ANEC expectation value to ﬁrst order in λ, which also depends on the coupling of the stress-tensor

improvement term ξ. We study the bounds on these couplings that follow from the ANEC and

unitarity at ﬁrst order in perturbation theory. These bounds are stronger than unitarity in some

regions of coupling space.

1teresa.bautista@kcl.ac.uk, 2lorenzo.casarin@{itp.uni-hannover.de, aei.mpg.de}

arXiv:2210.11365v1 [hep-th] 20 Oct 2022

Contents

1 Introduction 2

2 Norm of the state 4

3 ANEC expectation value on the stress-tensor state 8

4 Positivity bounds 11

4.1 Case 2 < d < 4........................................ 12

4.2 Case d=4. ......................................... 14

5 Conclusions and Outlook 16

A Conventions and formulae 18

B Calculation of the eye diagram 18

C Full Euclidean stress-tensor 2-point function to order O(λ2)20

D Wightman functions in momentum space 21

E Massive free scalar 23

1 Introduction

The Average Null Energy Condition (ANEC) states that the integral of the null energy over a

complete null worldline is non-negative,

+∞

−∞

dz−T−−(z)≥0.(1.1)

This is a quantum statement, it holds true at operatorial level. It is satisﬁed in free theory [1],

it has been shown to hold for interacting unitary QFTs with a nontrivial UV ﬁxed point using

ﬁeld-theoretic methods [2], and more generally for any unitary QFT using entropy arguments [3].

The ANEC (1.1) is an inherently Lorentzian concept. In fact, the central ingredient in the proof

of [2] is causality, which more in general is crucial in the analytic conformal bootstrap programme,

recently reviewed in [4, 5]. In this sense, studying the implications of the ANEC lies within the

broad program of determining the consequences of causality and unitarity for QFTs.

The ANEC was shown to encode important information about conformal ﬁeld theories with the

derivation of the ‘conformal collider bounds’ [6], which are bounds on conformal anomalies. To

derive these, the ANEC operator is placed at null inﬁnity, and its expectation values are taken on

a state |ψiwhich generates some energy excitation,

hEi=1

hψ|ψilim

z+→∞ z+

2d−2hψ|

+∞

−∞

dz−T−−(z)|ψi ≥ 0.(1.2)

This then has the interpretation of the energy ﬂux measured per unit angle in the transverse

directions at null inﬁnity, which owing to (1.1) has to be non-negative.

The expectation value (1.2) is computed from 3-point correlators involving the stress tensor,

and their positivity translates into bounds on the quantities which such correlators depend on. In

a CFT, for a momentum eigenstate generated by the stress tensor itself, the expectation value in

d= 4 spacetime dimensions depends on the conformal anomalies a and c, and the ANEC translates

into a lower and an upper bound on their ratio a/c. The bounds thus obtained also happen to be

optimal, given that the ANEC operator commutes with the momentum operator at null inﬁnity.

The ANEC has also been used to place bounds on conformal dimensions of operators [7, 8], in

some cases stronger than the unitary bounds. Furthermore, it has been shown [2,9] that the ANEC

is the ﬁrst of a whole family of positivity conditions, which similarly follow from causality and

unitarity. These take the form of the positivity of light-ray operators [9–11], non-local operators

labeled by a continuous spin J, for which the J= 2 operator is precisely the ANEC operator. Their

positivity therefore generalises the ANEC to continuous spin.

Given how useful the ANEC has proven to be in the context of CFTs, it is natural to explore its

implications for generic QFTs. Since it follows from unitarity and causality, it is tempting to think

that the ANEC could encode interesting constraints on RG ﬂows and be related to monotonicity

theorems. However, the lack of conformal symmetry makes it much more diﬃcult to make general

statements on the correlators. It is therefore useful to start by studying a particular example.

In this paper, we continue the programme initiated in [12] by studying the implications of the

ANEC in the particular example of λφ4in perturbation theory. This is an interacting theory with

a trivial ﬁxed point in d= 4 dimensions and a Wilson-Fisher ﬁxed point in d= 4 −2, and it

is simple enough to allow one to explicitly compute the expectation value of the ANEC operator

at low perturbative orders. Concretely, here we consider a state generated by the stress-tensor,

thereby following the construction of [6] and deriving nontrivial constraints for the parameters of

the theory. For practical purposes we focus on the case 2 < d ≤4, although some of the results

have a more general range of validity.

The constraints that we obtain from the ANEC depend on the spacetime dimension, the coupling

λ, the improvement-term coupling ξ, and the energy of the state; they are trivially satisﬁed in the

free case. The constraints are similar to the unitarity constraint that follows from demanding

positivity of the norm of the state. Setting the renormalization scale equal to the energy of the

state we obtain bounds for the couplings at such energy. The ANEC turns out to be in most cases

more stringent than unitarity. The evaluation of the norm of the state and of the ANEC correlator

is solid; however, the analysis of the bounds is more speculative given that they follow from the

edge of validity of perturbation theory, and require higher-order corrections to be conﬁrmed.

The outline of the paper is as follows. In section 2 we compute the norm of the state generated

by the stress tensor up to order O(λ2), which follows from the Wightman 2-point stress-tensor

correlator. In section 3 we compute the ANEC expectation value on the same state up to order

O(λ), by ﬁrst computing the Wightman 3-point correlator of the stress tensor and then turning it

into an expectation value of the ANEC operator at null inﬁnity. Finally in section 4 we present and

discuss the resulting constraints, together with the unitarity constraint following from positivity of

the norm of the state. Several of the intermediate expressions for the correlators are listed in the

appendices, including the expression for the full stress-tensor 2-point function to O(λ2) and details

of the derivation of the Wightman function in momentum space from the Euclidean correlators. In

appendix E, we compare with the case of the free massive scalar.

2 Norm of the state

Our starting point is the Euclidean action in d= 4 −2εdimensions

SE=ZddxEh1

2(∂φ)2+1

4!λφ4i,(2.1)

where subscripts E (L) indicates Euclidean (Lorentzian) signature. In the following we will drop

them whenever it is clear from the context.

The Euclidean stress-energy tensor derived from (2.1) reads

Tµν =∂µφ ∂νφ−1

2(∂φ)2δµν −ξ∂µ∂ν−δµν ∂2φ2−λ

4! φ4δµν .(2.2)

It includes the improvement term with a real parameter ξ. As we shall conﬁrm with our calculation,

its addition is necessary to construct a renormalizable energy-momentum tensor at the quantum

hTµν Tαβi(0) :hTµν Tαβ i(1) :

hTµν Tαβi(2) :a b c

d e f

Figure 1: Diagrams for the 2-point function. The thick dot represents a stress tensor insertion. The

left one has indices µν; the right one αβ. The arrow represents the ﬂow of the external momentum

level [13]. Tracelessness of the stress tensor (when λ= 0 or d= 4) is achieved when ξ=ξd:= d−2

4(d−1) .

We want to evaluate the norm of the stress-tensor momentum eigenstate

|ε·Ti=εµν Zddx e−i q x0Tµν (x)|0i, q > 0 (2.3)

on which we will evaluate the ANEC operator. This state has vanishing spatial momentum pi=~

and energy p0=q > 0. We introduced also a complex symmetric polarization tensor ε. Conservation

of the stress tensor allows us to consider purely space-like polarization, ε0µ= 0. We write the norm

N=hε·T|ε·Ti

=ε∗µν εαβ Zddx eiqx0hTµν (x)Tαβ(0)i=ε∗µν εαβhTµν (q,~

0) Tαβ(−q,~

0)i,

(2.4)

where the correlator involved is the Wightman 2-point function. In the second step we have dropped

a factor of the spacetime volume, since it cancels with an analogous contribution from the ANEC

3-point function in the expression for the normalized energy ﬂux (1.2).

We start by constructing the Euclidean correlator,

hTµν (p)Tαβ(−p)iE=hTµν Tαβ iE(0) +λhTµν Tαβ iE(1) +λ2hTµν TαβiE(2) +O(λ3),(2.5)

where the dependence on pin the rhs is understood. The Feynman diagrams to order O(λ2) are

shown in ﬁgure 1. Except for the eye diagram (2)f, all integrals in the other diagrams’ contributions

can be treated with two-propagator integral technology, summarized in appendix A. The eye diagram

is considerably more complicated, nonetheless it can be computed exactly; details are in appendix B.

For the purpose of this paper, we could disregard the terms with tensorial dependence on the

external momentum, since they vanish when contracted with the polarization tensor, εµν pν= 0

when pµ= (q,~

0). However, we provide the result for the full Euclidean 2-point function with

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摘要：

ANEConstress-tensorstatesinperturbative4theoryTeresaBautista1andLorenzoCasarin2;31DepartmentofMathematics,King'sCollegeLondon,TheStrand,LondonWC2R2LS,UK2InstitutfurTheoretischePhysikLeibnizUniversitatHannoverAppelstrae2,30167Hannover,Germany3Max-Planck-InstitutfurGravitationsphysik(AlbertEinst...

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ANEC on stress-tensor states in perturbative 4theory Teresa Bautista1and Lorenzo Casarin23

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