Journal Club - 2401.10981

Smoothed asymptotics

We all know that

(∑_(n=0)^N)n=(1/2)N(N+1)

Turn this into a weighted sum

(∑_(n=0)^∞)n Θ[n/N]=(1/2)N(N+1)

Where

Θ->Plot[HeavisideTheta[x],{x,-1,1},ImageSize->Medium]

Replace the regulator by a smooth function:
	η[0]=1
	η[∞]->0
	Schwartz function

Then

(∑_(n=0)^∞)n η[n/N]=(Subscript[C, 1])[η](N^2)-(1/12)+𝒪((1/N))

Where

(Subscript[C, 1])[η]=((∫_0)^∞)dx x η[x]

Example: η[x] =(e^(-x))

η[x_]:=Exp[-x];
(∑_(n=0)^∞)n η[n/N]//Series[#,{N,∞,0}]&//Normal
C1*(N^2)-(1/12)/.C1->Integrate[η[x]x,{x,0,∞}]

Example: η[x]=(e^(-x))Cos[x]

η[x_]:=Exp[-x]Cos[x];
(∑_(n=0)^∞)n η[n/N]//Series[#,{N,∞,0}]&//Normal
C1*(N^2)-(1/12)/.C1->Integrate[η[x]x,{x,0,∞}]

Features:
	1) There is no linear divergence.
	2) Quadratic divergence depends on the regulator (can be made zero -- Enhanced regulators).
	3) There is a universal behaviour (regulator independent): -1/12.

Similar feature to QFT:
	1) Power law divergences are meaningless (regulator dependent).
	2) Log-divergences are universal (regulator independent). 
	3) What are enhanced regulators for QFTs? Are they physical in some sense?

(Number Theory) -> (QFT η-regularization)

What they did: 
	1) Start with one fold irreducible integrals (ILS).
	2) Implement the smoothed regularization to the integrals.
	3) Get that power law divergences are reg. dependent and log-divergences are log-independent.
	4) Gauge invariant regulators -> enhanced regulators

Details

1) Start with one fold irreducible integrals (ILS):

2) Implement the smoothed regularization to the integrals.

3) Get that power law divergences are reg. dependent and log-divergences are log-independent.

where

(Subscript[C, z])[η]=∫(x^z)η[x]dx                       and                        γ[η]=∫(x^z)((dη)/dx)[x]ln[x]dx

4) Gauge invariant regulators:

[25] Starts with the diagrams below and write the vacuum polarization in terms of the integrals.

Requiring gauge invariance puts the following constraint in the integrals

This constrains in terms of the regulators implies:
	i) We need more than one regulator:

ii) The satisfy the following properties:

4) Gauge invariant regulators -> enhanced regulators