Renormalization Ch.1

Renormalization Ch.1

Quark self-energy s.1.1I would like to calculate a one loop correction to a quark self-energy which is de ined as 1PR

part of the fermion propagator:

𝑆 (𝑝,𝑚) = 𝑖/𝑝 − 𝑚 + 𝑖

/𝑝 − 𝑚 [𝑖Σ (𝑝,𝑚 )] 𝑖/𝑝 − 𝑚

+ 𝑖/𝑝 − 𝑚 [𝑖Σ (𝑝,𝑚 )] 𝑖

/𝑝 − 𝑚 [𝑖Σ (𝑝,𝑚 )] 𝑖/𝑝 − 𝑚 +…

= 𝑖/𝑝 − 𝑚 1 − [𝑖Σ (𝑝,𝑚 )] 𝑖

/𝑝 − 𝑚 = 𝑖/𝑝 − 𝑚 + Σ (𝑝,𝑚 ) . (1)

Thus the correction has the form:

Σ( ) (𝑝) = −𝑖 [𝑑𝑙] (𝑖𝑔𝛾 𝑡 ) 𝑖/𝑝 − /𝑙 − 𝑚 𝑖𝑔𝛾 𝑡 𝐷 (𝑙) , (2)

where 𝐷 (𝑙) is a gluon propagator. Here I would like to consider a particular form of the planargauge:

𝐷 (𝑙) = −𝑖𝛿𝑙 + 𝑖0 𝑔 −

𝑙 𝑣 + 𝑙 𝑣𝑙 ⋅ 𝑣 , (3)

so that𝑣 = 1 and I put the externalmomentum𝑝 = (𝑝 ⋅ 𝑣) 𝑣 . The convenienceof this de initionis that no other 𝛾-structures are generated except /𝑝:

Σ = 𝑖𝑔 𝐶 [𝑑𝑙] 𝛾 /𝑝 − /𝑙 + 𝑚(𝑝 − 𝑙) − 𝑚 + 𝑖0

𝛾 1𝑙 + 𝑖0 𝑔 −

𝑙 𝑝 + 𝑙 𝑝𝑙 ⋅ 𝑝 . (4)

For the sake of completeness I would like to consider a “covariant” and “non covariant” parts sep-arately. The former one reads as follows:

Σ = 𝑖𝑔 𝐶 [𝑑𝑙] /𝑝 − /𝑙 (2 − 𝒟) +𝑚 𝒟(𝑝 − 𝑙) − 𝑚 + 𝑖0 (𝑙 + 𝑖0)

, (5)

the latter isΣ = −𝑖𝑔 𝐶 [𝑑𝑙]

/𝑙 /𝑝 − /𝑙 + 𝑚 /𝑝 + /𝑝 /𝑝 − /𝑙 + 𝑚 /𝑙(𝑝 − 𝑙) − 𝑚 + 𝑖0 [𝑙 + 𝑖0] (𝑙 ⋅ 𝑝)

. (6)

Averaging over all directions transverse to 𝑝

/𝑙 → /𝑝𝑝 (𝑝 ⋅ 𝑙) = /𝑝

2𝑝 𝑙 + 𝑝 −𝑚 − 𝑝 − 2𝑝 ⋅ 𝑙 + 𝑙 − 𝑚 → /𝑝2𝑝 𝑙 + 𝑝 −𝑚 , (7)

we obtain the following expressions:

Σ = 𝑖𝑔 𝐶 𝑗( ) + 𝑚𝑝 𝑗( ) − 𝑗( ) 1 − 𝒟

2 /𝑝 + 𝒟𝑗( )𝑚 , (8)

Σ = −2𝑖𝑔 𝐶 2𝑗( ) + 𝑗( ) /𝑝 + 𝑗( )𝑚 , (9)

C: \Users \Grisha \Documents \Active work \Munchen \Renormalization \Planar gauge 2.tex p.1

Appendix Ch.2

where the integrals 𝑗( ) are de ined in the Appendix.

Renormalization factors s.1.2There are two ways for further processing of Eqs.(8): either remove regularization after inte-

gration and then expand the result near mass-shell, or irstly expand the integrands near mass-shell and then remove regularization after integration. These two possibilities are not equal sincethe 𝑝 → 𝑚 limit does not commutewith the𝒟 → 4 limit. The former case is known as “off-shell”renormalization while the latter is referred to as “on-shell”. The difference is clear for 𝑗( ) integral(15) which is proportional to:

𝐼( ) (𝑎, 𝑧) = ‖𝑎‖ 𝑑𝑦𝑦 (𝑦 + �̄�𝑧) , (10)

where 𝑧 = 𝑚 − 𝑝 /𝑚 . The result for the regularized integrand in given by Eq. (16). Howeverif one expands the integrand with respect to 𝑧 before integration, then only a part of Eq. (16)appears:

𝐼( ) (𝑎, 𝑧) → ‖𝑎‖1 − 2𝑎 𝐹 (1, 𝑎; 2𝑎; 𝑧). (11)

It is clear that the last expression corresponds to the contribution of the “hard” region in themethod of expansion by regions, thus it acquires additional “infrared” singularities in compari-son with the full result (16).

Ifwe consider onlyΣ , theoff-shell renormalization results in the standard𝑀𝑆 renormalizationfactors:

𝑍 = 1 − �̃�𝜖 , 𝑍 = 1 − 3

𝜖 �̃�, (12)where �̃� = 𝐶 𝛼 / (4𝜋) and𝑚 = 𝑍 𝑚, hence. The covariant part of the “on-shell” renormaliza-tion leads to the factors well known form QED:

𝑍 = 𝑍 = 1 + �̃� −3𝜖 − 4 . (13)

It is remarkable the self-energy in the planar gauge Σ = Σ + Σ yields the result for 𝑍 dif-ferent from that of the covariant gauge (13):

𝑍 = 1 + �̃� 3𝜖 − 8 . (14)

Appendix Ch.2

𝑗( ) =[d𝑙]

𝑙 (𝑝 − 𝑙) − 𝑚= 𝑑𝑦

[d𝑙][𝑙 − 𝑦 𝑝 − 𝑦 (𝑚 − 𝑝 )]

= 𝑖 𝑚𝜇

𝒟/‖2 − 𝒟/2‖ 𝑑𝑦

𝑦 𝒟/ (𝑦 + �̄�𝑧) 𝒟/ = 𝑖 𝑚𝜇

𝒟/𝐼( ) 2 − 𝒟

2 , 𝑧 , (15)

where 𝑧 = 𝑚 − 𝑝 /𝑚 and

𝐼( ) (𝑎, 𝑧) = ‖𝑎‖ 𝑑𝑦𝑦 (𝑦 + �̄�𝑧)

= (1 − 𝑧) 𝑧 ‖1 − 𝑎‖2𝑎 − 1‖ + ‖𝑎‖1 − 2𝑎 𝐹 (1, 𝑎; 2𝑎; 𝑧). (16)

C: \Users \Grisha \Documents \Active work \Munchen \Renormalization \Planar gauge 2.tex p.2

Appendix Ch.2

This result is easy to check by comparison with a tadpole:

𝐼( ) 2 − 𝒟2 , 1 = −‖1 − 𝒟/2‖. (17)

The second integral is as follows:

𝑗( ) =[d𝑙]

𝑚 (𝑙 − 𝑚 + 𝑖0) = −𝑖 𝑚𝜇

𝒟/‖1 − 𝒟/2‖. (18)

The third master integral has the form:

𝑗( ) =[d𝑙]

(−2𝑙 ⋅ 𝑝) (𝑝 − 𝑙) − 𝑚 + 𝑖0= 𝑑𝑦

[d𝑙]

𝑙 + 𝑝 1 − (1 + 𝑦) − 𝑚

= 𝑖‖2 − 𝒟2 ‖

d𝑦(𝑚 − 𝑝 + 𝑦 𝑝 ) 𝒟/ = 𝑖 𝑚

𝜇

𝒟/𝐼 2 − 𝒟

2 , 𝑧 , (19)

where the following integral de ined:

𝐼( ) (𝑎, 𝑏) = ‖𝑎‖ d𝑦(𝑧 + 𝑦 �̄�) = ‖𝑎‖

2𝑎 − 1 𝐹 (1, 𝑎; 𝑎 + 1/2; 𝑧) , (20)

where again 𝑧 = 𝑚 − 𝑝 /𝑚 .

C: \Users \Grisha \Documents \Active work \Munchen \Renormalization \Planar gauge 2.tex p.3