Flow-oriented perturbation theory

2025-04-24 0 0 895.05KB 54 页 10玖币

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Nikhef 2022-017 IPARCOS-UCM-23-005

Michael Borinsky,1Zeno Capatti,2Eric Laenen,3,4,5Alexandre Salas-Bernárdez6

1Institute for Theoretical Studies, ETH Zürich, 8092 Zürich, Switzerland

2Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland

3IOP/ITFA, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

4Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands

5ITF, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands

6Universidad Complutense de Madrid, Departamento de Física Teórica and IPARCOS, 28040

Madrid, Spain

Abstract: We introduce a new diagrammatic approach to perturbative quantum ﬁeld

theory, which we call ﬂow-oriented perturbation theory (FOPT). Within it, Feynman graphs

are replaced by strongly connected directed graphs (digraphs). FOPT is a coordinate space

analogue of time-ordered perturbation theory and loop-tree duality, but it has the advan-

tage of having combinatorial and canonical Feynman rules, combined with a simpliﬁed

iε dependence of the resulting integrals. Moreover, we introduce a novel digraph-based

representation for the S-matrix. The associated integrals involve the Fourier transform of

the ﬂow polytope. Due to this polytope’s properties, our S-matrix representation exhibits

manifest infrared singularity factorization on a per-diagram level. Our ﬁndings reveal an

interesting interplay between spurious singularities and Fourier transforms of polytopes.

arXiv:2210.05532v2 [hep-th] 30 Jan 2023

Contents

1 Introduction 2

2 Flow-oriented perturbation theory 4

2.1 Scalar QFT in coordinate space 4

2.2 The triangle diagram in FOPT 5

2.3 Derivation of the general FOPT Feynman rules 11

2.3.1 Cauchy integrations 11

2.3.2 Energy ﬂows and digraphs 12

2.3.3 Canonical cycle basis and admissible paths 14

2.4 FOPT Feynman rules 16

2.5 A simple example: The bubble graph 17

2.6 Routes, cycles and UV singularities 19

3 Finite and long distance singularity structure of FOPT diagrams 21

3.1 Finite distance singularities 22

3.2 Long distance singularities 24

4 The S-matrix and its p-xrepresentation 25

4.1 Derivation of the p-xrepresentation 26

4.2 The ﬂow polytope 28

4.3 Truncated routes 29

4.4 The p-xS-matrix representation 30

4.5 Example: The p-xrepresentation of a triangle diagram 31

4.6 Example: The ﬂow polytope of a pentagon digraph 33

4.7 Polytopes and spurious singularities 34

5 IR singularities in the p-xrepresentation 36

5.1 Collinear singularity on the triangle digraph 36

5.2 General collinear singularities in the p-xS-matrix representation 40

5.3 Soft-collinear singularity of the triangle diagram 41

6 Conclusion 42

A Unitarity, cut integrals and Cutkosky’s Theorem 44

– 1 –

1 Introduction

The perturbative approach to quantum ﬁeld theory provides the basis of our understanding

of the fundamental laws governing high-energy processes. Experimental observables are

computed in an asymptotic formalism as a power series in a coupling whose value should

be small. The coeﬃcients of this perturbative series are represented in terms of Feynman

diagrams. The practical computation of such observables is most often performed within

a covariant approach in momentum space, in which each diagram is manifestly invariant

under the action of the Poincaré group. Nevertheless, since the early days of quantum

electrodynamics, other, non manifestly covariant approaches have been used, such as the

venerable time-ordered perturbation theory (TOPT), which can be derived analogously to

quantum mechanical perturbation theory (see ref. [1,2] for a review).

TOPT involves three-dimensional momentum space loop integrals, with time-ordered

vertices, and is well-suited for a local singularity analysis. Similarly, more recent formula-

tions as loop-tree duality [3–8] and manifestly-causal loop-tree duality [9–14], have led to an

advance in the understanding of the large distance singularity structure of Feynman inte-

grals. In particular the denominator structures that arise therein are usually more directly

identiﬁed with formal graph-theoretic constructs [15–18], which makes the generalization

of the singularity analysis considerably easier.

Parametric Feynman integration is still the present-day standard for the numerical

evaluation of multi-loop integrals [19–22], especially in Euclidean kinematic regimes. In this

context practically all relevant Feynman diagrams can be integrated readily [23]. However,

numerical methods based on four-dimensional [24–31] and especially three-dimensional rep-

resentations [14,15,32–36] oﬀer a promising alternative due to their manifest IR singularity

cancellation features, and their native adaption to Minkowski space kinematics. Moreover,

these three-dimensional representations allow for more eﬃcient and robust treatment of

integrals near thresholds.

In contrast to the momentum space formulation, coordinate space methods have re-

ceived considerably less attention. Indeed, while ﬁeld theories are almost always phrased

in terms of coordinate space Lagrangians, scattering theory is naturally formulated in mo-

mentum space. Nevertheless, many fundamental physical principles ﬁnd a beautiful and

revealing formulation in coordinate space. Unitarity, for example, can be concisely en-

coded through the largest time equation [37,38]. Coordinate space methods also have a

prominent role in axiomatic formulations of Quantum Field Theory [39]. In the context of

renormalization group invariants at higher loop coordinate space methods are more pow-

erful than their momentum space counterpart [40–44]. Recently, cutting rules for eikonal

diagrams [45,46] have also been generalized using coordinate space Green’s functions,

showing that these rules express and emphasize causality. Other interesting developments

include light-cone ordered perturbation theory and work on the coordinate space analysis

of infrared divergences of Feynman diagrams [47–50].

In this paper we develop a new representation of coordinate space Green’s functions

based on the concept of energy ﬂow: ﬂow-oriented perturbation theory (FOPT). Much like

TOPT, FOPT is canonical, in the sense that there is a well-deﬁned Feynman integral for

– 2 –

each energy-ﬂow-oriented graph, without ambiguity. We derive the FOPT representation

in sect. 2from four dimensional covariant coordinate space rules, starting with the deriva-

tion of the FOPT representation for the scalar triangle diagram, followed by the general

treatment, which holds for any massless scalar diagram, independent of the number of

edges incident to each internal vertex. We then summarise the FOPT representation in

a concise set of Feynman rules and show that it has the UV behaviour which is expected

from covariant analyses.

However, the FOPT representation comes with two inherent caveats: the external data

is given in coordinate space, but, ultimately, a momentum space encoding of such data is

needed for the computation of scattering cross-sections. Additionally, as we will explain in

sect. 3, ﬁnite distance singularities play a quite intricate role in the FOPT representation

and the presence of IR singularities is not manifest. Motivated by this, we adjust perspec-

tive in sect. 4: we (partially) transform the FOPT representation back to momentum space.

We call the resulting hybrid representation, which eﬀectively describes S-matrix elements,

the p-xS-matrix representation. It is hybrid in the sense that external kinematics are given

in momentum space while internal integrations are performed in three-dimensional coor-

dinate space. These internal coordinate space integrals are covariant three-dimensional

Feynman integrals, modulated by the Fourier transform of a certain polytope associated

to the underlying ﬂow-oriented Feynman diagram.

This polytope turns out to be an instance of the well-studied ﬂow polytope. This type of

polytope has close connections to the representation theory of arithmetic groups, diagonal

harmonics, to Schubert polynomials and other mathematical structures [51–56]. Here, we

add a new application of the ﬂow polytope to this list: the study of Feynman integrals. This

paper is hence an addition to the growing literature on the use of polytopes in the study of

analyticity properties and evaluation techniques in quantum ﬁeld theory [23,57–65]. The

inherent ﬁniteness of such polytopes’ Fourier transforms gives a concise explanation for the

cancellation of spurious singularities in our new p-xS-matrix representation. In general,

the cancellation of such spurious singularities is a not a well-understood phenomenon as we

will discuss in sect. 4.7. Additionally, the Fourier transform of the ﬂow polytope expresses

the oscillating behaviour of the S-matrix integrand which, especially at large distances,

becomes crucial for determining the singular structure of the S-matrix itself.

Equipped with the necessary knowledge of the ﬂow polytope’s properties, we discuss

IR singularities in the p-xrepresentation of the S-matrix in sect. 5and identify a coordinate

space analogue of collinear and soft singularities. Subsequently, we detail the factorization

properties of these singular regions for diagrams contributing to the S-matrix. Our ob-

servation is that, in the p-xS-matrix representation, IR factorization, which is expected

from physical amplitudes [66], is in fact already present at the diagram level. The general

discussion is again supported by the pedagogical treatment of the triangle diagram.

Finally, we observe that the original FOPT representation has some interesting fea-

tures in the context of unitarity-cut integrals and Cutkosky’s theorem and the largest time

equation [37,67]. For instance, virtual and real contributions to cross sections can readily

be put under the same integration measure. We leave an in-depth analysis of these ob-

servations for a future work, hence we relegate these aspects to appx. A, where we sketch

– 3 –

some of these ideas for the interested reader.

2 Flow-oriented perturbation theory

We start by introducing ﬂow-oriented perturbation theory (FOPT) for a massless scalar

quantum ﬁeld theory, and derive the FOPT Feynman rules. To motivate them, we discuss

the one-loop triangle diagram in some detail. A number of useful concepts for FOPT

graphs, such as their completion, cycles and routes are introduced and explained.

2.1 Scalar QFT in coordinate space

The massless coordinate-space Feynman propagator for a scalar ﬁeld in D= 4 dimensional

space time (with the mostly-minus metric) reads

∆F(z) = Zd4p

(2π)4e−ip·zi

p2+iε =1

(2π)2

−z2+iε .(2.1)

As usual, the Feynman rules provide a recipe to translate a graph Gwith sets of edges

E, internal vertices Vint and external vertices Vext into an integral. Recall that external

vertices are deﬁned by the requirement that there is only one adjacent vertex to each of

them. The usual coordinate-space Feynman rules (see for instance [68, Ch. 10.1] or [69,

Ch. 6.1]) read

1. Associate a coordinate vector to each internal or external vertex. We label the lo-

cation of external vertices with xa,a∈Vext, and that of internal vertices with yv,

v∈Vint.

2. For each internal edge e={v, v0}multiply by a Feynman propagator ∆F(ze) =

∆F(−ze), where zeis the diﬀerence of the coordinates of the vertices to which the

edge eis incident. For example, if eis an internal edge (none of the two vertices

deﬁning it is external), then ze=yv−yv0. If vis instead an external vertex, then

ze=xv−yv0.

3. For each interaction vertex multiply by a factor −ig.

4. For each internal vertex v∈Vint integrate over all values of the components of yv, i.e.

over all possible locations of the internal vertex in 4-dimensional Minkowski space.

The resulting expression is a function of the external coordinates {xa}a∈Vext . To be explicit,

the application of the coordinate-space Feynman rules to a generic graph Gcontributing

to a Green’s function of a massless scalar theory gives

AG(x1, . . . , x|Vext|) = (−ig)|Vint|

(2π)2|E|

Y

v∈Vint Zd4yv

Y

e∈E

−z2

e+iε .(2.2)

One integrates over the four dimensional Minkowski space location of each internal vertex.

Accounting for symmetry factors results in an expression for the scalar n-point correlation

– 4 –

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Nikhef2022-017IPARCOS-UCM-23-005Flow-orientedperturbationtheoryMichaelBorinsky,1ZenoCapatti,2EricLaenen,3;4;5AlexandreSalas-Bernárdez61InstituteforTheoreticalStudies,ETHZürich,8092Zürich,Switzerland2InstituteforTheoreticalPhysics,ETHZürich,8093Zürich,Switzerland3IOP/ITFA,UniversityofAmsterdam,Scienc...

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