Renormalization and resolution of singularities · 2018-11-29 · Introduction Outline 1 Introduction Renormalization in momentum space Renormalization in position space Geometric

Renormalization and resolution of singularities

C. Bergbauer1

joint with R. Brunetti2, D. Kreimer3

1Mainz, SFB 45

2Trento

3IHES and Boston University

October 21, 2009

C. Bergbauer (SFB 45) Renormalization October 21, 2009 1 / 17

Outline

1 IntroductionRenormalization in momentum spaceRenormalization in position spaceGeometric analysis of renormalization

2 Resolution of singularities for subspace arrangementsSimple blow-upsThe wonderful models of De Concini and ProcesiThe Feynman distribution on the smooth model

3 Conclusions and outlook


Introduction

Outline





Introduction Renormalization in momentum space

Feynman graphs and renormalization

We consider the perturbative expansion in terms of Feynmangraphs of a massless scalar Quantum Field Theory such as φ3, φ4

on Euclidean spacetime M = Rd , d = 4,6, . . . .Momentum space Feynman rules associate a (usually divergent)integral to a Feynman graph.

Example:

→∫ dd k|k |2|k−p|2 , p ∈ Rd .

A subtraction of a similarly divergent integral∫ dd k|k |2|k−p|2 −

∫ dk

|k |2|k−p|2

∣∣∣|p|=µ

leads to a finite value. The

subtracted term can be seen as coming from a renormalizedLagrangian.We care only about UV divergences (when momenta get large) inthis talk. For simplicity we restrict to at most logarithmicdivergences.






Example:

→∫ dd k|k |2|k−p|2 , p ∈ Rd .


∫ dk

|k |2|k−p|2

∣∣∣|p|=µ








Example: →∫ dd k|k |2|k−p|2 , p ∈ Rd .


∫ dk

|k |2|k−p|2

∣∣∣|p|=µ










∫ dk

|k |2|k−p|2

∣∣∣|p|=µ










∫ dk

|k |2|k−p|2

∣∣∣|p|=µ




Introduction Renormalization in position space

The Feynman distribution in position space

A Fourier transformation leads to the position space Feynmanrules. The position space propagator is the distributionu0(x) = 1

|x |(d−2)/2 = F( 1|k |2 ) on M = Rd .

In position space, renormalization is an extension problem ofproducts of u0’s at diagonals.Example:

(d = 4) → u0(x1 − x2)u0(x1 − x2) = 1|x1−x2|4

,

notdefined at the diagonal x1 = x2.

Let Γ be a Feynman graph with set of vertices V (Γ) = {1, . . . ,n}and set of edges E(Γ). Let nij be the number of edges between iand j . Then we call

uΓ(x1, . . . , xn) =∏i<j

u0(xi − xj)nij

the Feynman distribution.





|x |(d−2)/2 = F( 1|k |2 ) on M = Rd .

In position space, renormalization is an extension problem ofproducts of u0’s at diagonals.Example:

(d = 4) → u0(x1 − x2)u0(x1 − x2) = 1|x1−x2|4

,

notdefined at the diagonal x1 = x2.


uΓ(x1, . . . , xn) =∏i<j

u0(xi − xj)nij






|x |(d−2)/2 = F( 1|k |2 ) on M = Rd .

In position space, renormalization is an extension problem ofproducts of u0’s at diagonals.Example: (d = 4) → u0(x1 − x2)u0(x1 − x2) = 1

|x1−x2|4, not

defined at the diagonal x1 = x2.


uΓ(x1, . . . , xn) =∏i<j

u0(xi − xj)nij






|x |(d−2)/2 = F( 1|k |2 ) on M = Rd .

In position space, renormalization is an extension problem ofproducts of u0’s at diagonals.Example: (d = 4) → u0(x1 − x2)u0(x1 − x2) = 1

|x1−x2|4, not

defined at the diagonal x1 = x2.


uΓ(x1, . . . , xn) =∏i<j

u0(xi − xj)nij



Introduction Geometric analysis of renormalization

Singular and divergent arrangements

uΓ(x1, . . . , xn) =∏

i<j u0(xi − xj)nij defines a priori only a

distribution on the configuration space Mn\ diagonals.Diagonals xi = xj where nij > 0 form the singular arrangement forΓ.Diagonals xi1 = . . . = xim where the full subgraph γ with verticesi1, . . . , im satisfies d rank H1(γ) = 2|E(γ)| form the divergentarrangement. This is where uΓ is not defined as a distribution.

Example: sing. arr. for Γ = K4




uΓ(x1, . . . , xn) =∏







uΓ(x1, . . . , xn) =∏







uΓ(x1, . . . , xn) =∏






Difficulties of renormalization

If there are no subdivergences, for example Γ0 = , (d = 4) :uΓ0(x1, x2) = 1

|x1−x2|4, a simple subtraction suffices:

uΓ0,R = (usΓ0− us

Γ0[ω])

∣∣∣s=1

, where ω is a test function satisfyingω|{x1=x2} = 1.

If the graph is disconnected, for example

Γ = :

theFeynman distribution is a tensor product uΓ = uΓ0 ⊗ uΓ0 ,uΓ,R = uΓ0,R ⊗ uΓ0,R.

If the graph has nested divergences, for example

Γ = ,

several subtractions have to be performed, one at the diagonalsx3 = x4, x2 = x3 = x4, x1 = x2 = x3 = x4 respectively.General solutions for this in momentum space: BPHZ,Zimmermann’s forest formula, Connes-Kreimer Hopf algebras.







Γ0[ω])

∣∣∣s=1


If the graph is disconnected, for example Γ = : theFeynman distribution is a tensor product uΓ = uΓ0 ⊗ uΓ0 ,uΓ,R = uΓ0,R ⊗ uΓ0,R.

If the graph has nested divergences, for example

Γ = ,








Γ0[ω])

∣∣∣s=1



If the graph has nested divergences, for example Γ = ,








Γ0[ω])

∣∣∣s=1



If the graph has nested divergences, for example Γ = ,




Epstein-Glaser renormalization vs. our approach

Starting from a recursive construction of the perturbative S-matrix,Epstein and Glaser show implicitly how to recursively renormalizeFeynman graphs in position space.This problem becomes nontrivial when there are subdivergences.Not all possible extensions are physically consistent, i. e. can beabsorbed by a local Lagrangian.Their work is elegant but difficult to compare with otherapproaches because the extensions have to be made in a certainprescribed order.Our goal: Using resolution of singularities, find ”obvious”extensions suggested by geometry and show they satisfy theEpstein-Glaser conditions and hence lead to a local renormalizedLagrangian. No recursion needed.














Resolution of singularities for subspace arrangements

Outline





Resolution of singularities for subspace arrangements Simple blow-ups

Simple blow-ups

Blowing up a point means replacing it by a projective space (orsphere) of codimension 1.Blowing up a line means replacing it by its projectivized normalbundle, etc.Example: The origin of the arrangement is blown up. Somesubspaces which intersected previously are now made disjoint.

→



Simple blow-ups


→



Simple blow-ups


→


Resolution of singularities for subspace arrangements The wonderful models of De Concini and Procesi

The wonderful models of De Concini and Procesi

By blowing up the arrangement is transformed into a divisor withnormal crossings. There are several ways of doing this. Blowingup strict transforms by increasing order of dimension does notyield the best result here.De Concini and Procesi provide a minimal model. For Γ = Kn itcoincides with the Fulton-MacPherson compactification.The combinatorics of the divisor fit exactly with the physicsapplication and the Connes-Kreimer Hopf algebras.

Example: FM compactification for Γ = K4

















Resolution of singularities for subspace arrangements The Feynman distribution on the smooth model

Laurent expansion of the regularized distribution

Let β : Y → Mn be the minimal model of the divergentarrangement for a graph Γ and E ⊂ Y the exceptional divisor.us

Γ can be pulled back (as distribution density-valued meromorphicfunction of s ∈ C) along β|Y\E . Write ws

Γ = β∗usΓ.

E having normal crossings, wsΓ is given in local coordinates by

wsΓ =

f sΓ

ys1 . . . y

sk

where f sΓ ∈ L1

loc ,1ys

i=δ0(yi)

s − 1+ . . . .

wsΓ has a first order pole along each irreducible component of E .

The residues are Feynman integrals. Where componentsintersect, there will be higher order poles.The intersections of E are explicitly described by nested sets ofdivergent irreducible graphs.






Γ = β∗usΓ.


wsΓ =

f sΓ

ys1 . . . y

sk

where f sΓ ∈ L1

loc ,1ys

i=δ0(yi)

s − 1+ . . . .








Γ = β∗usΓ.


wsΓ =

f sΓ

ys1 . . . y

sk

where f sΓ ∈ L1

loc ,1ys

i=δ0(yi)

s − 1+ . . . .








Γ = β∗usΓ.


wsΓ =

f sΓ

ys1 . . . y

sk

where f sΓ ∈ L1

loc ,1ys

i=δ0(yi)

s − 1+ . . . .








Γ = β∗usΓ.


wsΓ =

f sΓ

ys1 . . . y

sk

where f sΓ ∈ L1

loc ,1ys

i=δ0(yi)

s − 1+ . . . .





The leading coefficient and contractions

Theorem: Let a−n be the leading pole coefficient in the Laurentexpansion of ws

Γ at s = 1.

a−n[1] =∑

N

∏γ∈N

res(γ//N).

Here the N are nested sets of subgraphs, and γ//N the graph γ,with all maximal elements of N contained in γ contracted.Example:

Γ = , a−3 =(

res)3.



The leading coefficient and contractions

Theorem: Let a−n be the leading pole coefficient in the Laurentexpansion of ws

Γ at s = 1.

a−n[1] =∑

N

∏γ∈N

res(γ//N).

Here the N are nested sets of subgraphs, and γ//N the graph γ,with all maximal elements of N contained in γ contracted.Example:

Γ = , a−3 =(

res)3.



Renormalization

One can find extensions of wsΓ onto the divisor E where simply at

each irreducible component the simple pole is removed. There areseveral ways of doing this (renormalization freedom).Theorem: These extensions satisfy the Epstein-Glaser conditionswhich ensure that the extension is physically consistent (providelocal counterterms).



Renormalization

One can find extensions of wsΓ onto the divisor E where simply at

each irreducible component the simple pole is removed. There areseveral ways of doing this (renormalization freedom).Theorem: These extensions satisfy the Epstein-Glaser conditionswhich ensure that the extension is physically consistent (providelocal counterterms).


Conclusions and outlook

Outline







A power counting analysis of Γ and its subgraphs determine thedivergent arrangement of subspaces.The De Concini-Procesi minimal model for this arrangement hasexactly the right combinatorial properties (compare Connes-Kreimer Hopf algebras, Zimmermann’s forest formula) forrenormalization.It encodes the combinatorial subtleties of renormalization in itsgeometry. Renormalization looks therefore simpler and morestraightforward on this model. Renormalization can be done inone go, no recursive recipes needed.Using the Fulton-MacPherson compactification, all graphs in agiven order of perturbation theory can be treated simultaneously,as in Epstein-Glaser.Dimensional regularization in position space is better understood,as is the Hopf algebra approach and recent work on the motivicnature of renormalization.



















References

C. Bergbauer, R. Brunetti and D. Kreimer: Renormalization andresolution of singularities. arXiv:0908.0633C. Bergbauer: Combinatorial and geometric aspects of Feynmangraphs and Feynman integrals, Dissertation (2009), FU BerlinC. Bergbauer: Epstein-Glaser renormalization, the Hopf algebra ofrooted trees and the Fulton-MacPherson compactification ofconfiguration spaces, Diplomarbeit (2004), FU Berlin


Renormalization and resolution of singularities · 2018-11-29 · Introduction Outline 1 Introduction Renormalization in momentum space Renormalization in position space Geometric

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