Determination of auxiliary scale in dimensional regularization

My questions are in italics. In the article [1] a dimensional regularization is presented on an electrostatic example of an infinite wire with constant linear charge density λ. It is shown that the direct computation of the scalar potential gives infinity: ϕ(x)=λ4πϵ0∫∞−∞dl|x−l|=λ4πϵ0∫∞−∞dl(x2+y2+(z−l)2)12= =λ4πϵ0∫∞−∞du√x2+y2+u2=∞ But with dimensional regularization in the modified minimal subtraction scheme we get … Read more

Showing I=∫d3k∫dk01k4I=\int d^3k\int dk^0\frac{1}{k^4} to be logarithmically divergent

Consider a momentum integral of the form I=∫d3k∫dk01k4 where k2=(k0)2−(→k)2 and the integral over k0 runs from −∞ to +∞. This integral is common in QFT and I want to show that this integral is logarithmically divergent. The usual trick is to define k0=ik4 so that the integral becomes I=∫d3k∫dk41((k4)2+(→k)2)2. Now I can go to … Read more

Question about infinite sum in quantum field

I read from some books of number theory that ∞∑n=11ns=−112,when s=−1. Now is there such a result ∞∑n=11ns=π,when s=1,or ∞∑n=11ns=cπ,when s=1,where c is a rational number ? I get a similar result in mathematics by analogue, I suspect the result may have some interpretation in physics. Answer The true fact is the following. Consider ζ(s):=+∞∑n=11nswith s∈C and Res>1. That function, with the said … Read more

Dirac delta function defined in Zee’s Quantum Field Theory book

This is from Appendix 1 of the first chapter of Zee’s Quantum Field Theory in a Nutshell: I am not sure whether it is correct to call this the Dirac delta function. Sure, the integral over all space is 1, and it is sharply peaked at x=0. But its width doesn’t approach 0 when K→∞. … Read more

Cutoff regularization: Why not cutoff exactly at the momentum reached in an experiment?

So far I have only actually calculated dimensional regularization and I just know about the idea of cutoff regularization. From what I understand, as the name suggests, you just ignore momentum as from some high value and integrate the virtual momenta until this arbitrary chosen value. Depending on which value you chose, and thereby to … Read more

Length path integral

Let’s consider a 2-dimensional Euclidean plane. The length between two points a and b can be defined in the following way: (ab):=inf where the infimum is taken over all paths \gamma joining a and b, \delta_{ab} is the Euclidean metric on the plane and the parametrization is chosen so that \gamma(\tau=0)=a \quad \& \quad \gamma(\tau=1)=b. … Read more

Inconsistency in regularization with parallel and perpendicular momenta

In deriving the axial anomaly Peskin and Schroeder use dimensional regularization, continuing loop momenta to 4−ϵ dimenstions. The loop momenta can now be split into pieces “parallel” to d=0,1,2,3 and those “perpendicular” to d=0,1,2,3, ℓ=ℓ∥+ℓ⊥ Furthermore, they define γ5 as, γ5≡iγ0γ1γ2γ3 with this definition γ5 commutes with γμ in the extra dimensions and so, \require{cancel} … Read more

Is Wick rotation of loop integrals legitimate?

In Feynman diagram calculations, we seem to invariably Euclideanise loop integrals in order to exploit the resulting spherical symmetry. This Wick rotation is simply a deformation of the contour; providing we avoid all poles and providing our integrand falls off at infinity sufficiently fast, this is legitimate. See, for instance, Figure 6.1 of Peskin & … Read more

How can Weinberg assume that PbP_b acts as derivative?

In QM of finitely many degrees of freedom it is well known that due to the Stone-Von Neumann theorem, the CCR [Qi,Pj]=iδij leads to a unique representation up to unitary equivalence, on which Pj acts as the derivative Pj↦−i∂j. Now, in Weinberg’s QFT book volume 1, chapter 9, he considers a general quantum mechanical system … Read more

Dirac matrices in dimensional regularization, get correct order epsilon

Let us work in dimension D=4−2ϵ. In 4-dimension, we can write Tr[AB], where A and B are string of gamma matrices, as ∑mTr[A Γm]Tr[B Γm], where Γm={1,γ5,γμ,γ5γμ,σμν} are complete set of gamma matrices spanning the dirac space in 4-dim. As it is well-known, generalizing this to non-integer D dimension causes difficulties since γ5 (defined as γ5=iγ0γ1γ2γ3 in … Read more