There are three self energy diagrams for gluon at one loop as shown in Figure 1

QFT II Problem Set 12.

FS 2019 Prof. M. Grazzini https://www.physik.uzh.ch/en/teaching/PHY552/FS2019.html Due by: 27/5/2019

Exercise 1. Renormalization in QCD at one loop: The gluon Propagators

In this exercise series we will learn how to compute the UV divergent diagrams at one loop in QCD and how to renormalize these. For this we will absorb these divergencies into the physical quantities and write the bare Lagrangian parameters in terms of renomalized ones such as masses and coupling constants. The divergent diagrams include the corrections to the propagators and the corrections to the vertices. In this exercise we will look at the gluon propagator only.

There are three self energy diagrams for gluon at one loop as shown in Figure 1.

Figure 1: One loop gluon self energy

(a) Let Π^µν express the one loop gluon self energy. Write down an expression for each diagram contributing to Π^µν, i.e. Π^µν₁ , Π^µν₂ and Π^µν₃ by applying Feynman rules.

(b) Compute the gluonic contribution, i.e. Π^µν₃ in d-dimensions.

• First work out the numerator and write Π^µν₃ as:

Π^µν₃ = g² 2 N_cδ_cd

Z d^dk (2π)^d

A^µν

k²(p+k)² (1) wherek and pare the loop and external momentas respectively.

Keeping the tensor structure apply Feynman parametrization and make the necessary shifting of the loop momentum. Note that the terms linear in the loop momentum are odd and therefore vanishes:

Z d^dk (2π)^d

k^µ

(k²−∆)² = 0 (2)

This means you can make the replacement:

k^µk^ν = k²

d g^µν+ terms linear in k which vanish (3) You should find:

Π^µν₃ = g² 2 N_cδ_cd

Z dx

Z d^dk (2π)^d

g^µν

2 +^4d−6_d

k²+ 2∆ + 5p²

+M^µν

(k²−∆²)² (4)

where

M^µν = p^µp^ν

p² ∆(4d−6) +p²(d−6)

, ∆ =−p²x(1−x) (5) 1

• Now make the Wick rotation to perform the loop integration in the Euclidean space.

Note that:

Z d^dk (2π)^d

k²

(k²+ ∆)² = (2−)∆

−1 ∆⁻(4π)Γ() (6) Z d^dk

(2π)^d 1

(k²+ ∆)² = ∆⁻(4π)Γ() (7)

• Finally you need to perform the integration over the Feynman parameter x. Your result should be

Π^µν₃ = ig²

2 N_Cδ_cdp²

g^µν(19−12) +p^µp^ν

p² (−22 + 14)

(8) Γ() (4π)(−p²)⁻ B(2−,2−)

1− (9)

whereB(2−,2−) is the Beta function:

B(2−,2−) = Z 1

0

dx x¹⁻(1−x)¹⁻ (10) (c) Now compute the ghost contribution Π^µν₂ following the same steps before. (Do not forget

the minus sign for the fermion and the ghost loop!) You should find:

Π^µν₂ = 1 2

g²NCδcd

16π² (4π)(−p²)⁻Γ()

g^µνp²+ 2(1−)p^µp^ν B(2−,2−)

1− (11)

(d) Show that the sum of Π^µν₂ and Π^µν₃ can be brought to the form:

Π^µν(p) = p²gµν−pµpν

Π(p²) (12) wherep is the momentum of the external gluon. This implies that:

Π_µνp^µ= 0. (13)

which is a consequence of the Ward identity in QCD and tells that the gluons are trans- verse. It is important to notice that individually the second and third diagrams cannot be brought to this form. This means to express a physical gluon propagator we need the ghost contribution!

(e) Now compute the fermionic contribution Π^µν₁ . Take the fermions running in the loop massless so that you can write:

Π^µν₁ =− g² 4π²

δab

2 nfΓ() (4π)

g^µνp²−p^µp^ν

(−p²)⁻B(2−,2−) (14) (f) We define the renomalized wave function for the gluon field as:

A=Z₃^1/2A_R (15)

which leads to renormalised propagator at leading order:

D_ab^µν = δ_ab(−ig^µν)

Z3p² (16)

2

Then up to second order in perturbation theory the gluon propagator is:

S_ab^µν(p) = δ_ab(−ig^µν)

Z₃p² + δ_ac(−ig^µσ)

Z₃p² Π^cd_στ(p)δ_db(−ig^{τ ν})

Z₃p² (17)

Using the transversality:

Π^cd_στ(p) =δ^cd

−g_στ +p_σp_τ p²

Π(p²) (18)

Now using this and expressing S_ab^µν(p) as a geometric sum we have:

S_ab^µν(p) = δ_ab(−ig^µν)

Z3p²+ Π(p²) (19)

Now combine your results for Π(p²) from part iv) to determineZ3 so that the divergences cancel:

Z₃= 1− α_s 4π 1

2 3n_f− 5

3N_C

+ finite (20)

Exercise 2. 1-loop renormalised QCD coupling constant

To renormalise QCD one re-scales the fields and parameters in such a way that the Lagrangian can be written: L_R+L_c.t, where L_R has the form of the original ‘bare’ Lagrangian L₀, but with each term replaced by it’s corresponding renormalised one, andL_c.tcontains the respective counter-terms. In particular, under the re-scaling:

A^a_µ=Z

1 2

3A^a_µR ψ=Z

1 2

2ψ_R g=Z_gg_R

the bare quark-quark-gluon vertex is transformed:

gψγ¯ ^µT^aψA^a_µ−→ g_Rµ^εψ¯_Rt^aA^a_µRγ^µψ_R Z_gZ₂Z

1 2

3

| {z }

≡Z_1F

,

where as usual theµ^ε was introduced to render the coupling dimensionless in dimensional reg- ularisation.

(a) Using the definition Zi = 1 +δZi for i= 2,3,1F, expand the renormalisation factors to O(g²_R) and show that

g=

1−δZ2−1

2δZ3+δZ1F

gR µ^ε.

(b) Then, with

Z3 = 1−gR

4π 2

2 3nf −5

3Nc

1

ε+ finite

+O g⁴_R

, (21)

Z2 = 1−gR

4π 2

CF

1

ε+ finite

+O g_R⁴

, (22)

Z_1F = 1−g_R 4π

2

[N_c+C_F] 1

ε+ finite

+O g_R⁴

, (23)

3

whereCF = (N_c²−1)/2Nc, show that this has the explicit form g=

1− g²_R 16π²

11

6 N_C −1

3n_f 1

ε+ finite

g_R µ^ε, up to higher orders ing_R.

Exercise 3. The 1-loop Beta function in QCD

The Beta function β(g_R) in renormalised QCD determines how the coupling parameter g_R(µ) depends on the renormalisation scale µvia the following differential equation:

∂gR

∂lnµ =µ∂gR

∂µ =β(g_R) (24)

where the derivatives are taken withg held constant.

(a) With this definition use the result of Exercise 2 part (b) to show that:

β =− 11

3 N_C−2 3n_f

g_R³

16π² +O(g_R⁵) (b) By defining a scale ΛQCD via the condition:

µ→Λlim_QCD 1 g²_R(µ) = 0

solve equation 24 explicitly and show thatα_S(µ) :=g²_R/4π satisfies:

α_S(µ) = 4π

11

3 N_C−²₃n_f ln

µ² Λ²_QCD

(c) Classify all the different possible behaviours ofαS(µ) as a function of NC and n_f. What is the form ofα_S(µ) in the case of QCD? Why is this so significant?

Exercise 4. Renormalization invariance of the β-function In terms of the α=g²/4π the β-function is defined by

∂α

∂lnµ² =β(α) and has the following expansion:

β(α) =−α

β0

α 4π

+β1

α 4π

2

+O α³

.

Show that the first two coefficients of the expansion are renormalization scheme invariant, i.e.

β₀^A=β₀^B, β₁^A=β₁^B, for any two schemesA and B.

Hint. You may start from the fact thatα^B =α^A 1 +O α^A . 4