SOME RESULTS ON LANDAU POLES AND FEYNMAN DIAGRAM CUT STRUCTURE BY HOPF ALGEBRA WILLIAM DALLAWAY AND KAREN YEATS

2025-05-03 0 0 3.96MB 32 页 10玖币

侵权投诉

SOME RESULTS ON LANDAU POLES AND FEYNMAN

DIAGRAM CUT STRUCTURE BY HOPF ALGEBRA

WILLIAM DALLAWAY AND KAREN YEATS

Abstract. We investigate a system of diﬀerential equations for the beta func-

tion of massless scalar φ4theory and continue the combinatorial investigation

of the cut structure of Feynman diagrams.

1. Introduction

The goal of this work is to analyse the anomalous dimension and the beta func-

tion of a scalar quantum ﬁeld theory using a diﬀerential equation obtained through

combinatorial techniques. We also consider combinatorial questions related to the

coaction of the cut structure of Feynman diagrams.

Broadhurst and Kreimer in [11] derived a non-linear ﬁrst order diﬀerential equa-

tion for the anomalous dimension of a particular approximation to the fermion

propagator in a massless Yukawa theory and a non-linear third order diﬀerential

equation for the scalar φ3case. In [40] one of us developed a more general approach

following the original Broadhurst and Kreimer Yukawa example that quite broadly

builds a non-linear ﬁrst order diﬀerential equation or system of non-linear ﬁrst or-

der diﬀerential equation for anomalous dimensions of quantum ﬁeld theories. The

cost of this level of generality was that much of the complexity and character of

the speciﬁc quantum ﬁeld theory is swept into a catch-all function P, but a beneﬁt

is that the overall shape of the diﬀerential equation is controlled by combinatorial

features of the insertion of Feynman diagrams in the theory. The speciﬁc cases

of the photon propagator of QED in a Baker-Johnson-Wiley gauge and the gluon

propagator in similarly special gauge in massless QCD were studied in [38] and [39]

respectively. These both correspond to single diﬀerential equations in this frame-

work. The massless φ4case, a system of two diﬀerential equations, was set up at

the end of [40] but never subsequently investigated until the present paper.

The diﬀerential equation set up is particularly well suited to understanding when

there are solutions for the anomalous dimension that exist for all values of the

coupling and when there are solutions that exist for all values of the energy scale,

that is, when there are or are not Landau poles. Consequently, these questions are

a focus in the case studied here as well as in the previously studied cases of [38]

and [39].

Finally, let us note that the original two diﬀerential equations of Broadhurst and

Kreimer have recently been very strikingly studied from a resurgence perspective

in [9, 10, 8], and Bellon and collaborators investigated the anomalous dimension in

the Wess-Zumino model [3, 4, 1] in a similar context.

KY is supported by an NSERC Discovery grant and the Canada Research Chairs program.

KY thanks Dirk Kreimer for longstanding collaborations that led to this paper.

arXiv:2210.01164v1 [hep-th] 3 Oct 2022

2 WILLIAM DALLAWAY AND KAREN YEATS

An algebraic and combinatorial approach to the relationship between the cut

structure and subdivergence structure of Feynman diagrams was recently devel-

oped by one of us along with Dirk Kreimer [31]. The cut structure of a Feynman

diagram controls its monodromy around singularities and is related to its infrared

behaviour while the subdivergence structure controls its behaviour under renormal-

ization. However, the set up of [31] required ﬁxing a spanning tree, as motivated by

connections to Cullen and Vogtmann’s Outerspace [16, 5, 29], something which is

unnecessary and at times cumbersome. We rectify this here with a tree-independent

formulation and then look at some of the combinatorial preliminaries necessary for

extending Klann’s calculations [26] on how the cut structure relates to the infrared

divergences.

The basis of all of this are the renormalization Hopf algebras ﬁrst developed

by Connes and Kreimer and ﬁrst we will give a brief account of this. We then

present the diﬀerential equations which will be the main focus of the analysis.

The remainder of the introduction considers extensions of the Hopf algebras and a

coaction studied in [31].

For a mathematical reader uninterested in the physics background, the key things

are the diﬀerential equations (1.1), (1.2) along with a general sense that there is

some motivation to studying them and especially to studying when solutions exist

for all xand all L, and the core and cut coproducts on graphs Deﬁnition 1.7 and

(1.3), and (1.4), along with a general sense that the relationship between them is

meaningful. Everything else in this introduction is motivation.

1.1. Feynman diagrams and renormalization Hopf algebras. Quantum ﬁeld

theory is a framework in which we can understand arbitrary numbers of interacting

particles quantum mechanically. It is the standard way to unify quantum mechanics

and special relativity. In high energy physics, a prototypical experiment consists of

generating known particles, colliding them together at high energy, and investigat-

ing the particles that appear as a result. On the theoretical and mathematical side

of high energy physics, then, we want to better understand the mathematical struc-

tures in quantum ﬁeld theory so as to be able to better calculate the probability

amplitudes of particle interactions as occur in such a collider experiment.

One longstanding but still important technique for computing these amplitudes

is the loop expansion in Feynman diagram. In this approach amplitudes are com-

puted and studied as given by an inﬁnite series indexed by Feynman diagrams. Very

roughly Feynman diagrams are graphs where the edges represent particles propagat-

ing and the vertices represent particle interactions. Each graph contributes to the

amplitude via its Feynman integral, an integral which can be read oﬀ of the graph.

Particles entering or exiting are represented as unpaired half-edges in the graph.

The loop number of the graph is the dimension of the cycle space of the graph, and

in the loop expansion of an amplitude, the series is organized by increasing number

of loops.

Formally we will deﬁne graphs as follows.

Deﬁnition 1.1. Agraph Gconsists of

•a set H(G) of the half edges of the graph,

•a set partition, V(G), of H(G) into parts of size at least 3, the parts of

which are the vertices of the graph, and

LANDAU POLES AND CUT STRUCTURE BY HOPF ALGEBRA 3

•a set partition, E(G), of H(G) into parts of size at most 2, the parts of

which are the edges of the graph.

This notion of graph is a little diﬀerent from what is usual in graph theory. It

allows multiple edges and self-loops, but does not allow vertices of degree less than

3 – in this formulation the degree of a vertex is its size as a part in the set partition.

Additionally, E(V) may include edges where the size of the part is 2 which we

call internal edges and which correspond to edges in the usual graph theory sense,

but also edges where the size of the part is 1 which we call external edges and

which correspond, as described roughly above, to the particles entering and exiting

the system. This notion of graph is closely related to the usual formulation of a

combinatorial map (see for instance [33]) but without any cyclic ordering of the

half edges at each vertex.

Despite these diﬀerences most graph theory notions can be inherited directly

from their usual versions (see [19] for a standard graph theory introduction), and

so we speak of notions such as connectivity for our graphs without further deﬁnition.

To move from these graphs to our formulation of Feynman diagrams, we will

extract only the algebraic or combinatorial part that we need from the quantum

ﬁeld theory in the following deﬁnition.

Deﬁnition 1.2. Acombinatorial physical theory is a set of half edge types along

with

•a set of pairs of not necessarily distinct half edge types deﬁning the per-

missible edge types,

•a set of multisets of half edge types deﬁning the permissible vertex types,

•an integer called the power counting weight for each edge type and each

vertex type, and

•a nonnegative integer dimension of spacetime.

Typically, the set of half edge types will be ﬁnite. The dimension of spacetime

should be the one at which the theory is renormalizable but not superrenormaliz-

able. The combinatorial meaning of this will be given later.

For example, here are some standard quantum ﬁeld theories in this framework.

•Quantum electrodynamics (QED) has 3 half edge types, a half photon, a

front half fermion, and a back half fermion. There are two edges type,

the pair of two half photons, giving a photon edge which is drawn as a

wiggly line and which has power counting weight 2, and the pair of a front

half fermion and a back half fermion, giving a fermion edge, which by its

construction is oriented and drawn as an edge with an arrow, and has power

counting weight 1. There is one vertex consisting of one of each half edge

type and with weight 0. The dimension of spacetime is 4.

•Scalar φ4theory has just one half edge type and just one edge type consist-

ing of a pair of the half edges drawn as a plain unoriented edge. This edge

type has weight 2. The 4 of φ4indicates that the one vertex is a 4-valent

vertex; that is, the vertex consists of a multiset of 4 copies of the half edge.

It has weight 0. The dimension of spacetime is 4.

For further examples see [40].

Deﬁnition 1.3. AFeynman graph in a given combinatorial physical theory is a

graph in the sense above along with a map from the half edges of the graph to the

4 WILLIAM DALLAWAY AND KAREN YEATS

set of half edge types such that every internal edge is of a permissible edge type

and every vertex is of a permissible vertex type.

The step from this point to actual loop expansions in quantum ﬁeld theory is

a map known as the Feynman rules which associate a Feynman integral to each

Feynman graph. There are many diﬀerent takes on the Feynman rules; for a stan-

dard physics take consider a standard quantum ﬁeld theory text book such as [24]

or [35], while for a more mathematical viewpoint one might see the Feynman rules

as coming from the exponential map on the Lie algebra associated to the renro-

malization Hopf algebra [27], but in any case the details won’t be important for

us.

However, a property of the Feynman integrals that will matter is that in most

interesting cases they are divergent integrals. Because of this it is best to think of

the Feynman rules as mapping not to integrals but to formal integral expressions

[40, 36] so the integrands or diﬀerential forms can be manipulated and compared.

Divergences of a Feynman integral come in a few types. There may be a proper

subgraph that already corresponds to a divergent integral; this is a subdivergence.

Divergences may come when momenta are taken in a limit where they are large,

these are ultraviolet (UV) divergences; or when some aspect of the story becomes

small, for example when two particles become parallel so the angle between them

goes to 0, these are infrared (IR) divergences. The IR divergences are much more

subtle and we will postpone any discussion of them until we have the machinery of

Cutkosky cuts in place. The UV divergences are more straightforward. The power

counting weights in the deﬁnition of combinatorial physical theory are the power to

which the factor given by the corresponding edge or vertex grows as the momenta

get large. Because of this we can determine the overall degree of UV divergence of

a Feynman graph from only these combinatorial considerations.

Deﬁnition 1.4. For a Feynman graph G in a combinatorial physical theory, let

w(a) be the power counting weight of an internal edge or a vertex aof Gand let

Dbe the dimension of spacetime. Then the superﬁcial degree of divergence is

D` −X

e∈E(G)

w(e)−X

v∈V(G)

w(v)

where `is the loop number of the graph.

If the superﬁcial degree of divergence of a graph is nonnegative we say the graph

is (UV) divergent. If it is 0 we say the graph is (UV) logarithmically divergent.

Note that as Dincreases, more graphs are deemed divergent. For the theories of

interest to use there is a special value of Dwhere the superﬁcial degree of divergence

of a connected graph depends only on its external edges. This is the value of Dthat

corresponds to the physical situation of the theory being renormalizable but not

superrenormalizable, and is the value of Dwe typically want in our combinatorial

physical theories.

The problem of how to deal with UV divergences has long been understood

by physicists, and beautifully, one perspective on it can be reformulated using a

combinatorial Hopf algebra.

At this point it is also important to note what notion of subgraph we want.

Deﬁnition 1.5. Asubgraph γof a graph Gis a graph where H(γ)⊆H(G), E(γ)

is a reﬁnement of E(G) restricted to H(γ) and V(γ) is exactly V(G) restricted to

LANDAU POLES AND CUT STRUCTURE BY HOPF ALGEBRA 5

Figure 1. The right hand graph is a subgraph of the left hand

graph in three ways.

H(γ) with the additional property that every part in V(γ) has the same size in

V(G).

For example, consider the graph on the left in Figure 1. It has the graph on

the right in the ﬁgure as a subgraph in three ways. In each case the set of half

edges in the subgraph is the same as in the original graph, and the set of vertices is

thus necessarily also the same, but the set of edges in each subgraph reﬁnes exactly

one of the internal edges of the original graphs by breaking the part of size two

corresponding to the internal edge into two parts of size 1, giving two new external

edges.

Note that this deﬁnition of subgraph guarantees that if a graph is a Feynman

graph then the subgraph will also be a Feynman graph with the inherited mapping

to the half edge types.

We deﬁne the contraction of a connected subgraph γwithin a graph Gdiﬀerently

depending on whether γhas more than two external edges or not. If γhas more

than two external edges, then simply contract all internal edges of γ, so γbecomes

a new vertex made of its external edges. If γhas two external edges, then to avoid

a 2-valent vertex, remove both internal and external edges of γand then join into

an edge the two half-edges originally joined to the two external edges of γ, or leave

as external, the half if there was only one. Subgraphs γwith one external edge will

not come up for us, but they would be treated also by removing all edges. For a

disconnected subgraph γcontract it by contracting each connected component.

As is often done in enumerative combinatorics as well as in quantum ﬁeld theory

we can reduce to considering connected Feynman graphs by the exp-log trans-

formation, and furthermore, using a Legendre transform [25] we can restrict to

considering one particle irreducible (1PI), that is bridgeless in the graph theory

sense. Now we are ready to deﬁne a Hopf algebra structure on the connected 1PI

Feynman diagrams of a combinatorial physical theory.

For the deﬁnition of the renormalization Hopf algebra we need one more prop-

erty for the combinatorial physical theory, we need that the external edges of any

divergent subgraph appear as a vertex or edge type of the theory. If this is not the

case, extend the allowable vertex and edge types as necessary.

Deﬁnition 1.6. Given a combinatorial physical theory as above, let Gbe the set of

connected 1PI Feynman graphs in that theory. Deﬁne the renormalization bialgebra,

H, of the theory as follows. As an algebra H=Q[G] and we identify disconnected

graphs with the monomial of their connected components. The coproduct is deﬁned

on elements of Gby

∆(G) = X

γ⊆G

γdivergent 1PI

γ⊗G/γ

and extended as an algebra homomorphism to H. The counit is deﬁned by (1) =

1 and (G) = 0 for Gwith loop order at least 1 and extended as an algebra

homomorphism. (Note that the subgraphs in the sum do not need to be connected).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

SOMERESULTSONLANDAUPOLESANDFEYNMANDIAGRAMCUTSTRUCTUREBYHOPFALGEBRAWILLIAMDALLAWAYANDKARENYEATSAbstract.Weinvestigateasystemofdierentialequationsforthebetafunc-tionofmasslessscalar4theoryandcontinuethecombinatorialinvestigationofthecutstructureofFeynmandiagrams.1.IntroductionThegoalofthisworkistoan...

展开>> 收起<<

SOME RESULTS ON LANDAU POLES AND FEYNMAN DIAGRAM CUT STRUCTURE BY HOPF ALGEBRA WILLIAM DALLAWAY AND KAREN YEATS.pdf

共32页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

SOME RESULTS ON LANDAU POLES AND FEYNMAN DIAGRAM CUT STRUCTURE BY HOPF ALGEBRA WILLIAM DALLAWAY AND KAREN YEATS

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: