renormalization by kinematic subtraction and hopf...

Renormalization by kinematic subtractionand Hopf algebras

Erik Panzer1

Institutes of Physics and MathematicsHumboldt-Universitat zu Berlin

Unter den Linden 610099 Berlin, Germany

November 7th, 2013

[email protected]

mailto:[email protected]

Renormalization and algebraic structuresfrom quantum field theory (QFT)

Connes, Kreimer [2, 3]:renormalization in QFT formulated as an algebraicBirkhoff-decomposition of characters on a combinatorial Hopf algebraformulate renormalization group via β-function of local characterscombinatorial Dyson-Schwinger equations (DSE) formulated in termsof Hochschild one-cocycles

Ebrahimi-Fard, Manchon, Menous, Patras et. al.Rota-Baxter algebras [6, 5]BPHZ scheme and exponential renormalization [7, 8]Dynkin operator, logarithmic derivatives [4, 17]

Foissy: Structure of the Hopf algebra HR of rooted trees [10, 11, 9],classification of combinatorial Dyson-Schwinger equations andsystems [12, 13]van Baalen, Kreimer, Uminsky, Yeats: study of non-perturbative(analytic) Dyson-Schwinger equations [18, 19, 20, 16]

E. Panzer (HU Berlin) Kinematic subtraction November 7th, 2013 2 / 32




Foissy: Structure of the Hopf algebra HR of rooted trees [10, 11, 9],classification of combinatorial Dyson-Schwinger equations andsystems [12, 13]

van Baalen, Kreimer, Uminsky, Yeats: study of non-perturbative(analytic) Dyson-Schwinger equations [18, 19, 20, 16]





Foissy: Structure of the Hopf algebra HR of rooted trees [10, 11, 9],classification of combinatorial Dyson-Schwinger equations andsystems [12, 13]van Baalen, Kreimer, Uminsky, Yeats: study of non-perturbative(analytic) Dyson-Schwinger equations [18, 19, 20, 16]


Aims of the talk

1 algebraic features of kinematic subtraction

2 Hochschild-cohomology not only describes DSE, but alsorenormalized characters

3 comparison of different renormalization schemes4 analytic vs. combinatorial descriptions


Aims of the talk

1 algebraic features of kinematic subtraction2 Hochschild-cohomology not only describes DSE, but also

renormalized characters

3 comparison of different renormalization schemes4 analytic vs. combinatorial descriptions


Aims of the talk


renormalized characters3 comparison of different renormalization schemes

4 analytic vs. combinatorial descriptions


Aims of the talk


renormalized characters3 comparison of different renormalization schemes4 analytic vs. combinatorial descriptions


A model of a single scale

Theorem (Universal property)To any linear map L ∈ End(A) on an algebra A exists a unique morphismφ : HR → A of algebras (notation φ ∈ GHR

A ) such that

φ ◦ B+ = L ◦ φ. (1.1)

If A is a Hopf algebra and L ∈ HZ1ε(A) a one-cocycle, φ is Hopf.

Feynman rules φ of QFT map sub graphs to sub integrals, hence

φs (B+(w)) =

∫ ∞0

dζs f

(ζ

s

)φζ(w) for any w ∈ HR . (1.2)

s is a physical parameter (mass or momentum)f is dictated by the graph into which B+ insertsthese integrals typically diverge and are understood formally (as a pairof differential form & domain of integration)




A ) such that

φ ◦ B+ = L ◦ φ. (1.1)



φs (B+(w)) =

∫ ∞0

dζs f

(ζ

s


s is a physical parameter (mass or momentum)

f is dictated by the graph into which B+ insertsthese integrals typically diverge and are understood formally (as a pairof differential form & domain of integration)




A ) such that

φ ◦ B+ = L ◦ φ. (1.1)



φs (B+(w)) =

∫ ∞0

dζs f

(ζ

s


s is a physical parameter (mass or momentum)f is dictated by the graph into which B+ inserts

these integrals typically diverge and are understood formally (as a pairof differential form & domain of integration)




A ) such that

φ ◦ B+ = L ◦ φ. (1.1)



φs (B+(w)) =

∫ ∞0

dζs f

(ζ

s


s is a physical parameter (mass or momentum)f is dictated by the graph into which B+ insertsthese integrals typically diverge and are understood formally (as a pairof differential form & domain of integration)


A model of a single scaleRenormalization by kinematic subtraction

Example (f (ζ) = 11+ζ )

φ (1) = 1, φs ( ) =

∫ ∞0

dζζ + s , φs ( ) =

∫ ∞0

dζζ + s

∫ ∞0

dξξ + ζ

.

Such logarithmic divergences are independent of the parameter s and thusrenormalizable by a subtraction:

DefinitionFor a renormalization point µ, let Rµ := evµ denote the evaluation ats 7→ µ. The BPHZ- or MOM-renormalized character is

φR := (Rµ ◦ φ)?−1 ? φ = φ?−1µ ? φs . (1.3)

Rµ ◦ φ?−1 are called the counterterms.




φ (1) = 1,

φs ( ) =

∫ ∞0

dζζ + s , φs ( ) =

∫ ∞0

dζζ + s

∫ ∞0

dξξ + ζ

.



φR := (Rµ ◦ φ)?−1 ? φ = φ?−1µ ? φs . (1.3)





φ (1) = 1, φs ( ) =

∫ ∞0

dζζ + s ,

φs ( ) =

∫ ∞0

dζζ + s

∫ ∞0

dξξ + ζ

.



φR := (Rµ ◦ φ)?−1 ? φ = φ?−1µ ? φs . (1.3)





φ (1) = 1, φs ( ) =

∫ ∞0

dζζ + s , φs ( ) =

∫ ∞0

dζζ + s

∫ ∞0

dξξ + ζ

.



φR := (Rµ ◦ φ)?−1 ? φ = φ?−1µ ? φs . (1.3)





φ (1) = 1, φs ( ) =

∫ ∞0

dζζ + s , φs ( ) =

∫ ∞0

dζζ + s

∫ ∞0

dξξ + ζ

.



φR := (Rµ ◦ φ)?−1 ? φ = φ?−1µ ? φs . (1.3)

Rµ ◦ φ?−1 are called the counterterms.E. Panzer (HU Berlin) Kinematic subtraction November 7th, 2013 5 / 32

A model of a single scaleFiniteness

Example (f (ζ) = 11+ζ as before)

φR (1) = 1,

φR ( ) = (id− Rµ)φ ( ) =

∫ ∞0

dζ[ 1ζ + s −

1ζ + µ

]= − ln s

µ.

φR := (Rµ ◦ φ)?−1 ? φ = φ?−1µ ? φs .

Corollary (B+ ∈ HZ1ε is a cocycle: ∆B+ = (id⊗ B+)∆ + B+ ⊗ 1)

The renormalized character φR arises from the universal property of HR :

φR,s (B+(w)) =

∫ ∞0

dζ

f(ζs

)s −

f(ζµ

)µ

φR,ζ(w) for any w ∈ HR .

(1.4)

Proof.Use S ◦ B+ = −S ? B+ and write L =

∫∞0

dζs f(ζs

). . . to deduce

φR ◦ B+ =(

Rµφ?−1 ? φ)◦ B+ = Rµφ?−1 ? φB+ + Rµφ?−1B+

= Rµφ?−1 ? [(id− Rµ) ◦ φ ◦ B+] = (id− Rµ) ◦ L ◦ φR.




φR (1) = 1, φR ( ) = (id− Rµ)φ ( ) =

∫ ∞0

dζ[ 1ζ + s −

1ζ + µ

]= − ln s

µ.

φR := (Rµ ◦ φ)?−1 ? φ = φ?−1µ ? φs .



φR,s (B+(w)) =

∫ ∞0

dζ

f(ζs

)s −

f(ζµ

)µ


(1.4)


∫∞0

dζs f(ζs

). . . to deduce

φR ◦ B+ =(

Rµφ?−1 ? φ)◦ B+ = Rµφ?−1 ? φB+ + Rµφ?−1B+





φR (1) = 1, φR ( ) = (id− Rµ)φ ( ) =

∫ ∞0

dζ[ 1ζ + s −

1ζ + µ

]= − ln s

µ.



φR,s (B+(w)) =

∫ ∞0

dζ

f(ζs

)s −

f(ζµ

)µ


(1.4)


∫∞0

dζs f(ζs

). . . to deduce

φR ◦ B+ =(

Rµφ?−1 ? φ)◦ B+ = Rµφ?−1 ? φB+ + Rµφ?−1B+






φR,s (B+(w)) =

∫ ∞0

dζ

f(ζs

)s −

f(ζµ

)µ


(1.4)


∫∞0

dζs f(ζs

). . . to deduce

φR ◦ B+ =(

Rµφ?−1 ? φ)◦ B+ = Rµφ?−1 ? φB+ + Rµφ?−1B+




This is an algebraic Birkhoff decomposition along the splittingA = A− ⊕A+ induced by the character Rµ:

A− = im Rµ are (infinite) constants independent of s,A+ = im (id− Rµ) are functions that vanish at s = µ.

Lemma (finiteness for logarithmic divergences)If the kernel f (ζ) is continuous on [0,∞) with asymptotic growth

f (ζ)−c−1ζ∼ ζ−1−ε at ζ →∞,

for some ε > 0 and c−1 ∈ K, then φR is finite.

Moreover it is polynomial:

φR,s = ev` ◦ φR, φR : HR → K[x ] where ` := ln sµ. (1.5)




A− = im Rµ are (infinite) constants independent of s,

A+ = im (id− Rµ) are functions that vanish at s = µ.


f (ζ)−c−1ζ∼ ζ−1−ε at ζ →∞,









f (ζ)−c−1ζ∼ ζ−1−ε at ζ →∞,









f (ζ)−c−1ζ∼ ζ−1−ε at ζ →∞,

for some ε > 0 and c−1 ∈ K, then φR is finite. Moreover it is polynomial:



A model of a single scaleAn algebraic recursion

Inserting the polynomial φR,ζ(w) ∈ K[ln ζµ ] into

φR,s (B+(w)) =

∫ ∞0

dζ

f(ζs

)s −

f(ζµ

)µ

φR,ζ(w)

actually supplies the algebraic recursion

φR ◦ B+ = P ◦ F (−∂x ) ◦ φR, (1.6)

where P := id− ev0 annihilates the constant terms and the analytic inputof the kernel f is captured by the operator

F (−∂x ) := −c−1∫

0 +∑

n≥0 cn (−∂x )n ∈ End (K[x ]) and (1.7)

cn−1 :=

∫ ∞0

dζ[f (ζ) + ζf ′(ζ)

] (− ln ζ)n

n!. (1.8)


A model of a single scaleAn algebraic recursion: Examples

ExampleφR (1) = 1

φR ( ) = P ◦[−c−1

∫0 +c0 − . . .

](1) = P

(−c−1x + c0

)= −c−1x

φR ( ) = P ◦[−c−1

∫0 +c0 − c1∂x + . . .

] (−c−1x

)= P

(x2

2 c2−1 − c−1c0x + c1c−1

)=

x2

2 c2−1 − xc−1c0

φR ( ) = P ◦[−c−1

∫0 + . . .

] (x2c2−1

)= −x3

3 c3−1 + x2c2

−1c0 − 2xc2−1c1

RemarkThe Laurent series F (z) ∈ z−1K[[z ]] is the Mellin transform

F (z) =

∫ ∞0

dζ f (ζ) · ζ−z =∑

n≥−1cnzn. (1.9)


A model of a single scaleAn algebraic recursion: Examples

ExampleφR (1) = 1φR ( ) = P ◦

[−c−1

∫0 +c0 − . . .

](1) = P

(−c−1x + c0

)= −c−1x

φR ( ) = P ◦[−c−1

∫0 +c0 − c1∂x + . . .

] (−c−1x

)= P

(x2

2 c2−1 − c−1c0x + c1c−1

)=

x2

2 c2−1 − xc−1c0

φR ( ) = P ◦[−c−1

∫0 + . . .

] (x2c2−1

)= −x3

3 c3−1 + x2c2

−1c0 − 2xc2−1c1

RemarkThe Laurent series F (z) ∈ z−1K[[z ]] is the Mellin transform

F (z) =

∫ ∞0

dζ f (ζ) · ζ−z =∑

n≥−1cnzn. (1.9)


The Hopf algebra of polynomialsFor a field K, the polynomials K[x ] form a commutative connected gradedHopf algebra with the coproduct ∆x = 1⊗ x + x ⊗ 1. Note that:

Prim (K[x ]) = K · xGK[x ]K = {eva : a ∈ K} are evaluations eva := [p 7→ p(a)] with

eva ? evb = eva+b (group law) (1.10)

gK[x ]K := log?

(GK[x ]K

)= K · ∂0 for ∂0 := ∂

∂x

∣∣∣x=0

= (∑

n pnxn 7→ p1)

exp?(a∂0) = evafunctionals α ∈ K[x ]′ induce coboundaries (let P := id− ev0)

δ(α) = P ◦∑n≥0

α

(xn

n!

)∂n

x ∈ HZ1ε ⊂ End (K[x ]) (1.11)

HZ1ε (K[x ]) = K ·

∫0⊕δ (K[x ]′), i.e. the only non-trivial one-cocycle is∫

0 : K[x ]→ K[x ], p =∑

n≥0 pnxn 7→∫ x

0 p(y)dy =∑

n>0pn−1

n xn

(1.12)



Prim (K[x ]) = K · x

GK[x ]K = {eva : a ∈ K} are evaluations eva := [p 7→ p(a)] with


gK[x ]K := log?

(GK[x ]K

)= K · ∂0 for ∂0 := ∂

∂x

∣∣∣x=0

= (∑

n pnxn 7→ p1)


δ(α) = P ◦∑n≥0

α

(xn

n!

)∂n

x ∈ HZ1ε ⊂ End (K[x ]) (1.11)

HZ1ε (K[x ]) = K ·


0 : K[x ]→ K[x ], p =∑


0 p(y)dy =∑

n>0pn−1

n xn

(1.12)





gK[x ]K := log?

(GK[x ]K

)= K · ∂0 for ∂0 := ∂

∂x

∣∣∣x=0

= (∑

n pnxn 7→ p1)


δ(α) = P ◦∑n≥0

α

(xn

n!

)∂n

x ∈ HZ1ε ⊂ End (K[x ]) (1.11)

HZ1ε (K[x ]) = K ·


0 : K[x ]→ K[x ], p =∑


0 p(y)dy =∑

n>0pn−1

n xn

(1.12)





gK[x ]K := log?

(GK[x ]K

)= K · ∂0 for ∂0 := ∂

∂x

∣∣∣x=0

= (∑

n pnxn 7→ p1)

exp?(a∂0) = eva

functionals α ∈ K[x ]′ induce coboundaries (let P := id− ev0)

δ(α) = P ◦∑n≥0

α

(xn

n!

)∂n

x ∈ HZ1ε ⊂ End (K[x ]) (1.11)

HZ1ε (K[x ]) = K ·


0 : K[x ]→ K[x ], p =∑


0 p(y)dy =∑

n>0pn−1

n xn

(1.12)





gK[x ]K := log?

(GK[x ]K

)= K · ∂0 for ∂0 := ∂

∂x

∣∣∣x=0

= (∑

n pnxn 7→ p1)


δ(α) = P ◦∑n≥0

α

(xn

n!

)∂n

x ∈ HZ1ε ⊂ End (K[x ]) (1.11)

HZ1ε (K[x ]) = K ·


0 : K[x ]→ K[x ], p =∑


0 p(y)dy =∑

n>0pn−1

n xn

(1.12)


The renormalization group

φR ◦ B+ = P ◦ F (−∂x ) ◦ φR,

Corollary (since P ◦ F (−∂x ) ∈ HZ1ε (K[x ]) is a cocycle)

φR : HR → K[x ] is a morphism of Hopf algebras: ∆ ◦ φR = (φR ⊗ φR) ◦∆.

This means thatφR,a+b = eva+b ◦φR = (eva ?evb)◦φR = (eva ◦φR)?(evb ◦φR) = φR,a ?φR,b.

CorollaryφR = exp? (−xγ) for the anomalous dimension γ := −∂0 ◦ φR ∈ gHR

K ⊂ H ′R .

In other words, log? (φR) = −xγ is linear in x ; φR is completely determinedby its linear coefficients γ.

Example (∆ ( ) = 2 ⊗ + ⊗ and ∆2 ( ) = 2 ⊗ ⊗ )

φR( )=

[−x3

6 γ?3 +

x2

2 γ?2 − γx

]( ) = −x3

3 [γ ( )]3+x2γ ( ) γ ( )−xγ ( )



φR ◦ B+ = P ◦ F (−∂x )︸︷︷︸∈HZ1

ε(K[x ])

◦ φR,





K ⊂ H ′R .


Example (∆ ( ) = 2 ⊗ + ⊗ and ∆2 ( ) = 2 ⊗ ⊗ )

φR( )=

[−x3

6 γ?3 +

x2

2 γ?2 − γx

]( ) = −x3

3 [γ ( )]3+x2γ ( ) γ ( )−xγ ( )







K ⊂ H ′R .


Example (∆ ( ) = 2 ⊗ + ⊗ and ∆2 ( ) = 2 ⊗ ⊗ )

φR( )=

[−x3

6 γ?3 +

x2

2 γ?2 − γx

]( ) = −x3

3 [γ ( )]3+x2γ ( ) γ ( )−xγ ( )


Recursions for γUsing the Mellin transforms, we can calculate γ recursively by

Lemmaγ ◦ B+ = [zF (z)]?γ :=

∑n≥0 cn−1γ

?n

Proof.Exploit φR = exp? (−xγ) and apply −∂x |x=0 to both sides of

φR ◦ B+ = P ◦ F (−∂x ) ◦ φR.

Exampleγ ( ) = γ ◦ B+ (1) = c−1γ

?0 (1) = c−1

γ ( ) = γ ◦ B+ ( ) = c−1γ?0 ( ) + c0γ ( ) = c−1c0

γ ( ) = γ ◦ B+ ( ) = c0γ ( ) + c1γ ⊗ γ (1⊗ + 2 ⊗ + ⊗ 1)

= 2c1 [γ ( )]2 = 2c2−1c1


Analytic regularizationRegulate divergences by a parameter z ∈ C, resulting in Feynman ruleszφ : HR → A taking values in Laurent series A = K[z−1, z ]]:

zφs ◦ B+ :=

∫ ∞0

dζs f

(ζ

s

)ζ−z

zφζ . (1.13)

LemmaFor any forest w ∈ HR , the regularized Feynman character is

zφs(w) = s−z|w | ∏v∈V (w)

F (z |wv |). (1.14)

Example

zφs( )=s−zF (z) zφs( )=s−2zF (z)F (2z) zφs( )=s−3z [F (z)]2 F (3z)



zφs ◦ B+ :=

∫ ∞0

dζs f

(ζ

s

)ζ−z

zφζ . (1.13)



F (z |wv |). (1.14)

Example

zφs( )=s−zF (z)

zφs( )=s−2zF (z)F (2z) zφs( )=s−3z [F (z)]2 F (3z)



zφs ◦ B+ :=

∫ ∞0

dζs f

(ζ

s

)ζ−z

zφζ . (1.13)



F (z |wv |). (1.14)

Example

zφs( )=s−zF (z) zφs( )=s−2zF (z)F (2z)

zφs( )=s−3z [F (z)]2 F (3z)



zφs ◦ B+ :=

∫ ∞0

dζs f

(ζ

s

)ζ−z

zφζ . (1.13)



F (z |wv |). (1.14)

Example

zφs( )=s−zF (z) zφs( )=s−2zF (z)F (2z) zφs( )=s−3z [F (z)]2 F (3z)


Analytic regularization

Renormalizing as before, the finiteness implies the existence of

φR = limz→0 zφR , equivalently im (zφR) ⊂ K[[z ]]. (1.15)

Example (cancellation of poles)Use antipodes S( ) = − + , S ( ) = − and Rµ ◦ zφ

?−1 = Rµ ◦ zφ ◦ S in

zφR,s ( ) = s−2zF (z)F (2z)− µ−zF (z)s−zF (z)

− µ−2z [F (z)F (2z)− F (z)F (z)]

=(s−z − µ−z)F (z)

[(s−z + µ−z)F (2z)− µ−zF (z)

]=

[−c−1 ln s

µ+O (z)

]·[

c0 −c−1

2 ln sµ

+O (z)

]






?−1 = Rµ ◦ zφ ◦ S in


− µ−2z [F (z)F (2z)− F (z)F (z)]

=(s−z − µ−z)F (z)

[(s−z + µ−z)F (2z)− µ−zF (z)

]

=

[−c−1 ln s

µ+O (z)

]·[

c0 −c−1

2 ln sµ

+O (z)

]






?−1 = Rµ ◦ zφ ◦ S in


− µ−2z [F (z)F (2z)− F (z)F (z)]

=(s−z − µ−z)F (z)

[(s−z + µ−z)F (2z)− µ−zF (z)

]=

[−c−1 ln s

µ+O (z)

]·[

c0 −c−1

2 ln sµ

+O (z)

]


Analytic regularizationAlgebraic characterization of finiteness

The scale dependence zφs = zφµ ◦ θ−z` is dictated by the grading

θt :=∑n≥0

(Yt)n

n!∈ Aut (HR) ,w 7→ et|w | ·w where Yw = |w | ·w . (1.16)

LemmaFor any character zφ ∈ G

HRA of Laurent series A = K[z−1, z ]] let

zφs := zφ ◦ θ−z`, the following conditions are equivalent:

1 Finiteness: im (zφR) ⊂ K[[z ]] (so limz→0 zφR exists)2 im [zφ ◦ (S ? Y )] ⊂ 1

zK[[z ]]




θt :=∑n≥0

(Yt)n




zφs := zφ ◦ θ−z`, the following conditions are equivalent:1 Finiteness: im (zφR) ⊂ K[[z ]] (so limz→0 zφR exists)

2 im [zφ ◦ (S ? Y )] ⊂ 1zK[[z ]]




θt :=∑n≥0

(Yt)n




zφs := zφ ◦ θ−z`, the following conditions are equivalent:1 Finiteness: im (zφR) ⊂ K[[z ]] (so limz→0 zφR exists)2 im [zφ ◦ (S ? Y )] ⊂ 1

zK[[z ]]




θt :=∑n≥0

(Yt)n





zK[[z ]]

For 1.⇒ 2. differentiate zφR = zφ ◦ (S ? θ−z`) (µ = 1); ⇐ inductively:




θt :=∑n≥0

(Yt)n





zK[[z ]]

For 1.⇒ 2. differentiate zφR = zφ ◦ (S ? θ−z`) (µ = 1); ⇐ inductively:

zφ ◦(

S ? Y n+1)

= zφ ◦ (S ? Y n) ◦ Y + [zφ ◦ (S ? Y )] ? [zφ ◦ (S ? Y n)]




θt :=∑n≥0

(Yt)n





zK[[z ]]

The anomalous dimension can be derived from the regularized character by

γ = −∂`|`=0φR = limz→0

[z · zφ ◦ (S ? Y )] = Res zφ ◦ (S ? Y ). (1.17)


Minimal subtraction

DefinitionMinimal subtraction splits A = A− ⊕A+ into poles A− = z−1K[z−1] andholomorphic A+ = K[[z ]] along the projection

RMS : A� A−,∑

anzn 7→∑n<0

anzn. (1.18)

Since RMS is not a character (only Rota-Baxter), the Birkhoffdecomposition of a (regularized) character zφ ∈ G

HRA entails

Bogoliubov map (R-operation): φ = φ+(

zφ− ⊗ zφ)◦ ∆

counterterms: zφ− = −RMS ◦ φrenormalized character: zφ+ = (id− RMS) ◦ φ = zφ− ? zφ ∈ G

HRA+

Finiteness is trivial, let φ+ := limz→0 zφ+ denote the physical limit.

zφ+ ( ) = (id− RMS) zφs ( ) = (id− RMS) s−zF (z) = s−zF (z)−c−1

z

φ+ ( ) = c0 − c−1 ln s


Minimal subtraction


RMS : A� A−,∑

anzn 7→∑n<0

anzn. (1.18)


HRA entails


zφ− ⊗ zφ)◦ ∆

counterterms: zφ− = −RMS ◦ φ

renormalized character: zφ+ = (id− RMS) ◦ φ = zφ− ? zφ ∈ GHRA+



z

φ+ ( ) = c0 − c−1 ln s


Minimal subtraction


RMS : A� A−,∑

anzn 7→∑n<0

anzn. (1.18)


HRA entails


zφ− ⊗ zφ)◦ ∆


HRA+



z

φ+ ( ) = c0 − c−1 ln s


Minimal subtraction


RMS : A� A−,∑

anzn 7→∑n<0

anzn. (1.18)


HRA entails


zφ− ⊗ zφ)◦ ∆


HRA+



zφ+ ( ) = c0 − c−1 ln s


Minimal subtraction


RMS : A� A−,∑

anzn 7→∑n<0

anzn. (1.18)


HRA entails


zφ− ⊗ zφ)◦ ∆


HRA+



z︸︷︷︸zφ−( )

φ+ ( ) = c0 − c−1 ln s


Minimal subtractionDimensional regularization and locality

To obtain dimensionless regularized characters, choose a µ and replace sby s

µ = e`. Then φ+ ( ) = c0 − c−1`.

Example

zφ+ ( ) = (id− RMS)[

zφs/µ ( ) + zφ− ( ) zφs/µ ( )]

= (id− RMS)

[e−2z`F (z)F (2z)−

c−1z e−z`F (z)

]= e−2z`F (z)F (2z)−

c−1z e−z`F (z) +

c2−1

2z2 −c−1c0

2z

φ+ ( ) =c2−12 `2 − 2c−1c0`+ c2

0 +32c−1c1.

Observation: The counterterms zφ− ( ) = − c−1z and zφ− ( ) =

c2−1

2z2 −c−1c0

2zare independent of `.




µ = e`. Then φ+ ( ) = c0 − c−1`.

Example

zφ+ ( ) = (id− RMS)[

zφs/µ ( ) + zφ− ( ) zφs/µ ( )]

= (id− RMS)

[e−2z`F (z)F (2z)−

c−1z e−z`F (z)

]

= e−2z`F (z)F (2z)−c−1

z e−z`F (z) +c2−1

2z2 −c−1c0

2z

φ+ ( ) =c2−12 `2 − 2c−1c0`+ c2

0 +32c−1c1.


c2−1

2z2 −c−1c0





µ = e`. Then φ+ ( ) = c0 − c−1`.

Example

zφ+ ( ) = (id− RMS)[

zφs/µ ( ) + zφ− ( ) zφs/µ ( )]

= (id− RMS)

[e−2z`F (z)F (2z)−

c−1z e−z`F (z)

]= e−2z`F (z)F (2z)−

c−1z e−z`F (z) +

c2−1

2z2 −c−1c0

2z

φ+ ( ) =c2−12 `2 − 2c−1c0`+ c2

0 +32c−1c1.


c2−1

2z2 −c−1c0





µ = e`. Then φ+ ( ) = c0 − c−1`.

Example

zφ+ ( ) = (id− RMS)[

zφs/µ ( ) + zφ− ( ) zφs/µ ( )]

= (id− RMS)

[e−2z`F (z)F (2z)−

c−1z e−z`F (z)

]= e−2z`F (z)F (2z)−

c−1z e−z`F (z) +

c2−1

2z2 −c−1c0

2z︸︷︷︸zφ−( )φ+ ( ) =

c2−12 `2 − 2c−1c0`+ c2

0 +32c−1c1.


c2−1

2z2 −c−1c0



Minimal subtractionlocal characters and the β-function

DefinitionA Feynman rule zφ ∈ G

HRA is called local :⇔ its MS counterterm

zφ−,s = (zφ ◦ θ−z`)− is independent of ` ∈ K.

Theorem

zφ ∈ GHRA is local ⇔ the inverse counterterms zφ

?−1− : HR → K[ 1

z ] arepoles of only first order on im(S ? Y ), equivalently

β := limz→0

[z · zφ?−1

− ◦ (S ? Y )]

= −Res(

zφ− ◦ Y)∈ gHR

K (1.19)

exists. The physical limit of MS-renormalized local characters is

φ+ = exp? (−`β) ?(ε ◦ φ+

). (1.20)

Here ε ◦ φ+ = ev`=0 ◦ φ+ ∈ GHRK denote the constant terms.






Theorem


?−1− : HR → K[ 1


β := limz→0

[z · zφ?−1

− ◦ (S ? Y )]

= −Res(


K (1.19)

exists.

The physical limit of MS-renormalized local characters is

φ+ = exp? (−`β) ?(ε ◦ φ+

). (1.20)







Theorem


?−1− : HR → K[ 1


β := limz→0

[z · zφ?−1

− ◦ (S ? Y )]

= −Res(


K (1.19)

exists. The physical limit of MS-renormalized local characters is

φ+ = exp? (−`β) ?(ε ◦ φ+

). (1.20)



Minimal subtractionThe scattering formula

LemmaThe vector space im (S ? Y ) generates HR as a free commutative algebra.

CorollaryCounterterms zφ− of local characters are completely determined by theirfirst order poles zφ

?−1− ◦ (S ? Y ) = β

z . Explicitly,

zφ?−1− = ε+

β ◦ Y−1

z +

[(β ◦ Y−1) ? β] ◦ Y−1

z2 +O(

z−3). (1.21)

From β = −Res(

zφ− ◦ Y)

and zφ− ( ) = − c−1z , zφ− ( ) =

c2−1

2z2 −c−1c0

2z weknow β ( ) = c−1, β ( ) = c−1c0. Now we can check

zφ?−1− ( ) =

β(

12

)z +

[β ( )]2

2z2 =c−1c0

2z +c2−1

2z2 = zφ− (− + ) .


Comparing the MOM and MS schemeslocality and finiteness

We renormalized zφ ∈ GHRA in the MOM- and MS-schemes to construct

two renormalized characters φR, φ+ : HR → K[x ]:

MOM MSdefined by kinematics regulator dependent

projection character Rµ ∈ GAK Rota-Baxter RMS ∈ End(A)finiteness conditional zφ+ = (id− RMS) · · ·

locality counterterm zφ?−1µ conditional

RGE φR = exp? (−xγ) φ+ = exp? (−xβ) ?(ε ◦ φ+

)generator γ = Res zφ ◦ (S ? Y ) β = Res zφ

?−1− ◦ (S ? Y )

Lemma1 φR is finite ⇔ zφ is local

2 φ+ = (ε ◦ φ+) ? φR, equivalently ∆φ+ =(φ+ ⊗ φR

)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ




two renormalized characters φR, φ+ : HR → K[x ]:MOM MS

defined by kinematics regulator dependent

projection character Rµ ∈ GAK Rota-Baxter RMS ∈ End(A)finiteness conditional zφ+ = (id− RMS) · · ·




?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ





defined by kinematics regulator dependentprojection character Rµ ∈ GAK Rota-Baxter RMS ∈ End(A)

finiteness conditional zφ+ = (id− RMS) · · ·locality counterterm zφ

?−1µ conditional



?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ






finiteness conditional zφ+ = (id− RMS) · · ·




?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ






finiteness conditional built-in




?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ






finiteness conditional built-inlocality counterterm zφ

?−1µ conditional



?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ






finiteness conditional built-inlocality built-in conditional



?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ








)

generator γ = Res zφ ◦ (S ? Y ) β = Res zφ?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ









?−1− ◦ (S ? Y )



)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ









?−1− ◦ (S ? Y )

Lemma1 φR is finite ⇔ zφ is local2 φ+ = (ε ◦ φ+) ? φR, equivalently ∆φ+ =

(φ+ ⊗ φR

)◦∆

3 β ?(ε ◦ φ+

)=(ε ◦ φ+

)? γ


Dyson-Schwingerequations (DSEs)

Definition (simplified)A perturbation series X (α) is the solution of a DSE

X (α) = 1 + αB+

(X 1+σ(α)

)=:∑n≥0

xnαn ∈ HR [[α]] (1.22)

with coupling constant α. The correlation function is

G(α) := φR (X (α)) ∈ K[x ][[α]]. (1.23)

Theorem (Foissy)∆X (α) =

∑n≥0 X 1+nσ ⊗ xnα

n generates a sub Hopf algebra.

Corollary (in the MOM scheme)

Ga+b(α) =(φR,a ⊗ φR,b

)∆X (α) = Ga(α) · Gb (α · Gσ

a (α))


Dyson-Schwinger equations (DSEs)RGE for correlation functions

Changing the scale s or renormalization point µ is equivalent to anadjustment of the coupling α.

Infinitesimally, with γ := γ (X (α)):

G` (α) · γ [αGσ` (α)]) = −∂`G`(α) = γ(α) · (1 + σα∂α) G`(α). (1.24)

Example (Mellin transforms)Consider renormalized characters of the form φR ◦ B+ = P ◦ F (−∂x ) ◦ φR:

−∂`G`(α) = −∂`[1 + αφR ◦ B+

(X 1+σ(α)

)]= [zF (z)]−∂`

(G1+nσ` (α)

)reduces with −∂`Gnσ

` = γ (1 + nσ + σα∂α) at ` = 0 to

γ(α) =∑n≥0

cn−1 [γ (1 + nσ + σα∂α)]n .



Changing the scale s or renormalization point µ is equivalent to anadjustment of the coupling α. Infinitesimally, with γ := γ (X (α)):



−∂`G`(α) = −∂`[1 + αφR ◦ B+

(X 1+σ(α)

)]= [zF (z)]−∂`

(G1+nσ` (α)


` = γ (1 + nσ + σα∂α) at ` = 0 to

γ(α) =∑n≥0

cn−1 [γ (1 + nσ + σα∂α)]n .






−∂`G`(α) = −∂`[1 + αφR ◦ B+

(X 1+σ(α)

)]

= [zF (z)]−∂`

(G1+nσ` (α)


` = γ (1 + nσ + σα∂α) at ` = 0 to

γ(α) =∑n≥0

cn−1 [γ (1 + nσ + σα∂α)]n .






−∂`G`(α) = −∂`[1 + αφR ◦ B+

(X 1+σ(α)

)]= [zF (z)]−∂`

(G1+nσ` (α)

)

reduces with −∂`Gnσ` = γ (1 + nσ + σα∂α) at ` = 0 to

γ(α) =∑n≥0

cn−1 [γ (1 + nσ + σα∂α)]n .






−∂`G`(α) = −∂`[1 + αφR ◦ B+

(X 1+σ(α)

)]= [zF (z)]−∂`

(G1+nσ` (α)


` = γ (1 + nσ + σα∂α) at ` = 0 to

γ(α) =∑n≥0

cn−1 [γ (1 + nσ + σα∂α)]n .


Dyson-Schwinger equations (DSEs)RGE for correlation functions: physical examples

Example (fermion propagator of Yukawa theory, [1, 20])Summation of all iterated self-insertions of the one-loop-correctionamounts to σ = −2 and

F (z) =1

z(1− z)=∑

n≥−1zn, thus γ(α)− γ(α) (1− 2α∂α) γ(α) = α

which is solved in terms of the complementary error function.

Example (photon propagator of quantum electrodynamics, [18, 14])The setup is analogous, but σ = −1 yields different solutions in terms ofthe Lambert W -function.


Summary

MOM-renormalized Feynman rules have rich algebraic structure

MS and β = Res zφ(S ? Y ) ←→ MOM and φR = exp? (−xγ)

Mellin transforms reduce all analysis to combinatorics of seriesa way to non-perturbative formulations


Summary

MOM-renormalized Feynman rules have rich algebraic structureMS and β = Res zφ(S ? Y ) ←→ MOM and φR = exp? (−xγ)



Summary


Mellin transforms reduce all analysis to combinatorics of series

a way to non-perturbative formulations


Summary




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