Euclidean extended basis Similarly, in the extended basis we get:

Γ^E_conf[A^E,A˜^E] =− N²−1

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Explicitly, for the nonvanishing correlators in the balanced sector, we obtain in the mo-mentum representation:

In the unbalanced sector, we get:

hB^Es1(p₁). . .B^Esn(pn)¯B^E_s⁰

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A Notation and Wick rotation

We mostly follow the notation in [7]. We define the Minkowskian metric as:

(g_µν) = diag(1,−1,−1,−1) (A.1)

The light-cone coordinates are:

x^±= x⁰±x³

√2 =x∓ (A.2)

The corresponding Minkowskian (squared) distance is:

|x|² = 2x⁺x⁻−x²_⊥ (A.3)

where:

x²_⊥= (x¹)²+ (x²)² (A.4)

We denote the derivative with respect tox⁺ by:

∂₊= ∂

∂x⁺ =∂_x⁺ = ∂

∂x−

=∂_x− (A.5)

We define the light-like vectors n^µ and ¯n^µ:

nµn^µ= ¯nµn¯^µ= 0 nµn¯^µ= 1 (A.6) that can be parametrized as (n^µ) = ^√1

2(1,0,0,1) and (¯n^µ) = ^√1

2(1,0,0,−1).

The Minkowskian metric can be decomposed into orthogonal and longitudinal parts with respect to the light-like vectors:

gµν =g_µν^⊥ +nµn¯ν+nνn¯µ (A.7) The Euclidean metric is:

(δ_µν) = diag(1,1,1,1) (A.8)

The corresponding Euclidean (squared) distance is:

x² = 2x^zx^z¯+x²_⊥ (A.9)

with:

x^z = x⁴+ix³

√2 = x₄+ix₃

√2 =x_¯z (A.10)

and:

x^z¯= x⁴−ix³

√2 = x4√−ix3

2 =xz (A.11)

We define the Wick rotation by:

x⁰ =x0 → −ix⁴=−ix₄ (A.12)

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and:

p0=p⁰ →ip4 =ip⁴ (A.13)

Eq. (A.12) ensures that exp(iSM) → exp(−S_E), where SM and SE are the Minkowskian and Euclidean actions respectively, with S_E positive definite.

By defining p·x =pµx^µ and hpxi= pµx^µ in Minkowskian and Euclidean space-time respectively, eq. (A.13) ensures that, by the Wick rotation, p·x → hpxi, in such a way that the pairingsp·xandhpxiare actually independent of the Minkowskian and Euclidean metric respectively.

Therefore, by a slight abuse of notation, we also write p·x in Euclidean space-time, instead ofhpxi. Besides, |x|² → −x² and|p|²→ −p².

As a consequence, the Wick rotation of the scalar propagator of massmin Minkowskian space-time:

hφ(x)φ(y)i=

Z d⁴p

(2π)⁴ e^ip·(x−y) i

|p|²−m²+i (A.14)

reads in Euclidean space-time:

hφ^E(x)φ^E(y)i=

Z d⁴p

(2π)⁴e^ip·(x−y) i²

−p²−m² =

Z d⁴p

(2π)⁴e^ip·(x−y) 1

p²+m² (A.15) as it should be. Moreover, the Wick rotation of the light-cone coordinates is:

x⁺=x−→ −ix^z=−ix_z¯ (A.16) and:

x⁻=x₊→ −ix^¯z=−ix_z (A.17) Correspondingly, the Wick rotation of the derivative with respect to x⁺ is:

∂₊ →i∂_z =i ∂

∂x^z (A.18)

B Minkowskian and Euclidean propagators

The gluon propagator in the light-cone gauge, n·A=A+= 0, is:

hA^a_µ(x)A^b_ν(y)i=

Z d⁴p

(2π)⁴e^ip·(x−y) −i δ^ab

|p|²+i

g_µν−n_µp_ν+n_νp_µ p·n

(B.1) and in the Feynman gauge:

hA^a_µ(x)A^b_ν(y)i=

Z d⁴p

(2π)⁴ e^ip·(x−y) −i δ^ab

|p|²+ig_µν = δ^ab 4π²

gµν

|x−y|²−i (B.2) Hence, in the light-cone gauge the transverse propagator is:

hA^a(x)A^b(y)i= 0 (B.3)

hA¯^a(x) ¯A^b(y)i= 0 hA^a(x) ¯A^b(y)i=−δ^ab

4π²

1

|x−y|²−i

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We employ eq. (E.2) to work out the 2-point correlators in the light-cone gauge:

hf₁₁^a(x)f₁₁^b (y)i= 0 (B.4)

hf^a_{˙1 ˙1}(x)f^b_{˙1 ˙1}(y)i= 0 hf₁₁^a(x)f^b_{˙1 ˙1}(y)i=−δ^ab

4π²∂_x⁺∂_y⁺ 1

|x−y|²−i and:

h∂₊⁻¹f₁₁^a(x)∂₊⁻¹f₁₁^b (y)i= 0 (B.5) h∂₊⁻¹f^a_{˙1 ˙1}(x)∂₊⁻¹f_{˙1 ˙1}^b (y)i= 0

h∂₊⁻¹f₁₁^a(x)∂₊⁻¹f_{˙1 ˙1}^b (y)i=−δ^ab 4π²

1

|x−y|²−i

The Euclidean propagator in the Feynman gauge follows from the Wick rotation (ap-pendix A):

hA^Ea_µ (x)A^Eb_ν (y)i=

Z d⁴p

(2π)⁴ e^ip·(x−y)δ^ab

p² δ_µν = δ^ab 4π²

δ_µν

(x−y)² (B.6) Moreover, by performing the Wick rotation of eq. (B.4) and (B.5), we obtain in Euclidean space-time:

hf₁₁^Ea(x)f₁₁^Eb(y)i= 0 (B.7)

hf_{˙1 ˙1}^Ea(x)f^Eb_{˙1 ˙1}(y)i= 0 hf₁₁^Ea(x)f^Eb_{˙1 ˙1}(y)i=−δ^ab

4π²∂x^z∂y^z

1 (x−y)² and:

h∂_z⁻¹f₁₁^Ea(x)∂_z⁻¹f₁₁^Eb(y)i= 0 (B.8) h∂_z⁻¹f_{˙1 ˙1}^Ea(x)∂_z⁻¹f^Eb_{˙1 ˙1} (y)i= 0

h∂_z⁻¹f₁₁^Ea(x)∂_z⁻¹f^Eb_{˙1 ˙1} (y)i=−δ^ab 4π²

1 (x−y)² C Identities involving σ^µ and ¯σ^µ

We define the matrix (σ^µ_a˙a):

(σ^µ) = (1, ~σ) (C.1)

by means of the Pauli matrices that satisfy:

hσⁱ, σ^ji= 2i^ijkσ^k (C.2)

{σⁱ, σ^j}= 2δ^ijI We also define:

(¯σ^µ) = (1,−~σ) (C.3)

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and:

σ⁺= 1 +√σ3

2 σ⁻= 1√−σ3

2 (σ^µ_⊥) = (σ¹, σ²) (¯σ_⊥^µ) = (−σ¹,−σ²) (C.4) By means of (σ_a˙^µ_a) we may represent a vector, V_µ, in matrix form:

V=Vµσ^µ=√

2 V₊ V¯ V V−

!

(C.5) with:

V₊= V₀+V₃

√2 V− = V₀−V₃

√2 V = V₁+iV₂

√2

V¯ = V₁−iV₂

√2 (C.6)

in such a way that:

Det(Vaa˙) = 2(V+V−−VV¯) =V^µVµ (C.7) Hence, the Lorentz group is embedded into SL(2,C), and a Lorentz transformation acts as:

V⁰ =LVL¯ (C.8)

withL∈SL(2,C), leaving the determinant invariant.

We introduce the antisymmetric symbols _ab, ^ab [17]:

¹²=−²¹=21=−₁₂= 1 (C.9)

with:

_ac^cb=δ_a^b

^ab =^ac_cd^db (C.10)

that are employed to lower and rise the spinor indices respectively. For example:

ψa=abψ^b Φ^ab =^acbdΦ_cd (C.11) The following identities [17] hold:

σ¯^µaa˙ =^aba˙b˙σ^µ

bb˙ (C.12)

σ_a^µ_a˙σ¯_µ^bb˙ = 2δ_a^bδ^b˙_a˙ σ_a^µ_a˙σ_{µ b}b˙ = 2_aba˙b˙

σ¯^µaa˙ σ¯_µ^bb˙ = 2^aba˙b˙ Besides, we define [17]:

(σ^µν)_a^b = i 4

σ^µ_a˙cσ¯^νcb˙ −σ^ν_a˙cσ¯^µcb˙ (C.13) (¯σ^µν)^a˙_˙

b = i 4

σ¯^µac˙ σ_c^ν_b˙−σ¯^νac˙ σ^µ

cb˙

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with vanishing traces:

Trσ^µν =_ab(σ^µν)^ab = 0 (C.14)

Tr ¯σ^µν =_a˙b˙(¯σ^µν)^a˙b˙ = 0 σµν and ¯σµν satisfy the duality relations:

σ^µν =−i

2^µνρσσρσ (C.15)

σ¯^µν = i

2^µνρσ¯σρσ

where⁰¹²³ = 1. Moreover, the following identities [17] hold:

(σ^µν)_b^a(σµν)_d^c= 2δ_b^cδ_d^a−δ_b^aδ_d^c (C.16) (¯σ^µν)^a˙_b˙(¯σµν)^c˙_d˙= 2δ^c˙_b˙δ^a˙_d˙−δ^a˙_b˙δ^c˙_d˙

(σ^µν)_b^a(¯σµν)^c˙_d˙= 0

D Relation between the spinorial and vectorial bases in Minkowskian space-time

The components ofF_µν, with their s, j, τ assignments, are:

F+µ=F^αβnαg_βµ^⊥ µ, ν = 1,2 s= 1, j= 3

2, τ = 1 (D.1) F−µ=F^αβn¯_αg_βµ^⊥ µ, ν = 1,2 s=−1, j= 1

2, τ = 3 F_µν =F^αβg^⊥_αµg_βν^⊥ µ, ν = 1,2 s= 0, j= 1, τ = 2 F+−=F^αβnα¯n_β s= 0, j= 1, τ = 2

The component with maximal s,F+µ, is well suited (section 2) to build twist-2 operators that are primary [10,11] for the collinear conformal subgroup. In the light-cone gauge:

F_+µ=∂₊A_µ (D.2)

with µ= 1,2. Similarly, twist-2 primary conformal operators can also be built by means of ˜F+µ:

F˜_+µ= ˜F^αβn_αg^⊥_βµ= 1

2^αβρσn_αg_βµ^⊥ F_ρσ =^−β+σg_βµ^⊥F_+σ

=^−β+σg_βµ^⊥g_σν^⊥F₊^ν =−^σβ+−g_βµ^⊥g_σν^⊥F₊^ν =^βσ+−g^⊥_βµg_σν^⊥F₊^ν (D.3) withs= 1, j= ³₂ and τ = 1, where:

F˜^µν = 1

2^µνρσF_ρσ (D.4)

We define:

_µν =^αβρσg_αµ^⊥ g^⊥_βνn¯_ρn_σ =^αβ+−g_αµ^⊥ g_βν^⊥ (D.5)

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withµ, ν = 1,2 and₁₂= 1. Hence:

F˜+µ=µνF₊^ν (D.6)

and:

F˜₊₁ =−F₊₂ F˜₊₂=F₊₁ (D.7)

In the spinorial representation [15]:

F_a˙abb˙ =σ_a^µ_a˙σ^ν_bb˙F_µν (D.8) It turns out that Fµν decomposes [15] into the (1,0)⊕(0,1) representation of the Lorentz group:

F_{aab˙ b˙} = 2(f_aba˙b˙−_abf_a˙b˙) (D.9) where:⁷

f_ab= i

2(σ^µν)_abFµν (D.10)

and:

f_a˙b˙ =−i

2(¯σ^µν)_a˙b˙F_µν (D.11) with:

f¯_ab=f_a˙b˙ (D.12)

Indeed, since ^a˙b˙f_a˙b˙ = 0 and _a˙b˙_a˙b˙=−2, we get from eqs. (D.8) and (D.9):

^a˙b˙σ_a^µ_a˙σ^ν_bb˙Fµν = 2f_ab^a˙b˙_a˙b˙ (D.13) that coincides with eq. (D.10) by the antisymmetry of F_µν and the definition of σ_ab^µν in eq. (C.13). Similarly, we obtain eq. (D.11). It follows that:

f11= i

2(σ^µν)₁₁Fµν (D.14)

where:

(σ^µν)₁₁= i

4^c˙d˙(σ^µ_{1 ˙c}σ^ν_{1 ˙d}−σ^ν_{1 ˙c}σ_{1 ˙}^ν_d)

= i

4(σ_{1 ˙1}^µσ_{1 ˙2}^ν −σ^µ_{1 ˙2}σ^ν_{1 ˙1}−σ_{1 ˙1}^ν σ_{1 ˙2}^µ +σ_{1 ˙2}^ν σ^µ_{1 ˙1})

= i

2(σ_{1 ˙1}^µσ_{1 ˙2}^ν −σ^µ_{1 ˙2}σ^ν_{1 ˙1}) (D.15) From the definition of the matrices (σ_a˙^µ_a) (appendixC):

(σ_a˙⁺_a) = 1

√2 2 0 0 0

!

= 2

√2(δa1δ_a˙1˙ ) (σ_a˙⁻_a) = 1

√2 0 0 0 2

!

= 2

√2(δ_a2δ_a˙2˙ ) (σ_a˙¹_a) = 0 1

1 0

!

= (δ_a1δ_a˙2˙ +δ_a2δ_a˙1˙ ) (σ_a˙²_a) = 0 −i

i 0

!

=i(−δ_a1δ_a˙2˙ +δ_a2δ_a˙˙1) (D.16)

7We write eqs. (D.10) and (D.11) in the notation of appendixC, as opposed to the one in [15].

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it follows thatσ^µ

1 ˙1is nonvanishing only forµ= +, andσ^µ

1 ˙2is nonvanishing only forµ= 1,2.

Hence, by employing the antisymmetry of Fµν, we obtain:

f11= i 2

2i

2σ_{1 ˙1}⁺σ_{1 ˙2}^ν

F+ν = i 22i

2

√2 2

σ_{1 ˙2}¹ F+1+σ²_{1 ˙2}F+2

(D.17) Therefore:

f₁₁=− 1

√

2(F₊₁−iF₊₂) f_{˙1 ˙1} =− 1

√2(F₊₁+iF₊₂) (D.18)

We can now build the dictionary from the spinorial to the vectorial basis of the twist-2 operators:

f11f_{˙1 ˙1} = 1

2(F+1−iF+2) (F+1+iF+2) (D.19)

= 1 2

F₊₁F₊₁+F₊₂F₊₂−i(F₊₁F₊₂−F₊₂F₊₁)

=−1 2

g_⊥^µνF_+µF_+ν +i^µνF_+µF_+ν

and:

f11f11= 1

2(F+1−iF+2) (F+1−iF+2) (D.20)

= 1 2

F₊₁+iF˜₊₁ F₊₁+iF˜₊₁

In principle, the unbalanced operators with τ = 2 in the vectorial basis should be con-structed by means of the tensor:

1 2

F+µ+iF˜+µ F+ν+iF˜+ν

(D.21)

and its Hermitian conjugate, with µ, ν = 1,2. However, a simple computation shows that all the components of the operators above are actually proportional tof11f11 and its Hermitian conjugate respectively. Indeed:

1 2

F+2+iF˜+2 F+2+iF˜+2

= 1

2(F+2+iF+1) (F+2+iF+1) (D.22)

=−1

2(F+1−iF+2) (F+1−iF+2) =−f₁₁f11

and:

1 2

F₊₁+iF˜₊₁ F₊₂+iF˜₊₂= i

2(F₊₁−iF₊₂) (F₊₁−iF₊₂) =if₁₁f₁₁ (D.23)

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It follows that in the standard basis:

Os=−1 and in the extended basis:

As=−1

The complex basis is defined by means of:

A= A1√+iA2

2

A¯= A1√−iA2

2 (E.1)

In the light-cone gauge, it follows from eq. (D.18) that:

f11=−∂₊A¯

f_{˙1 ˙1} =−∂₊A (E.2)

Hence, the operators in the standard basis are:

Os= Tr∂+A(x)(i¯ −→

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in the light-cone gauge, and analogously in the extended basis:

As= Tr ¯A(x)(i−→

∂₊+i←−

∂₊)^sC

1

s2

−

→∂₊−←−

∂₊

−

→∂++←−

∂+

! A(x)

A˜s= Tr ¯A(x)(i−→

∂++i←−

∂+)^sC

1

s2

−

→∂+−←−

∂+

−

→∂₊+←−

∂₊

! A(x)

Bs= 1

√2Tr ¯A(x)(i−→

∂₊+i←−

∂₊)^sC

1

s2

−

→∂₊−←−

∂₊

−

→∂++←−

∂+

! A(x)¯ B¯s= 1

√2TrA(x)(i−→

∂++i←−

∂+)^sC

1

s2

−

→∂+−←−

∂+

−

→∂₊+←−

∂₊

!

A(x) (E.4)

F Jacobi and Gegenbauer polynomials

We work out the formulas for the Jacobi and Gegenbauer polynomials that are employed in the present paper.

For x real, the Jacobi polynomials,P_l^(α,β)(x), admit the representation [18]:

P_l^(α,β)(x) =

l

X

k=0

l+α k

! l+β k+β

!x−1 2

kx+ 1 2

l−k

(F.1) withα, β real andl a natural number. Moreover, they satisfy the symmetry property:

P_l^(α,β)(−x) = (−1)^lP_l^(β,α)(x) (F.2) The Gegenbauer polynomials, C_l^α0(x), are a special case of the Jacobi polynomials:

C_l^α0(x) = Γ(l+ 2α⁰)Γ(α⁰+¹₂) Γ(2α⁰)Γ(l+α⁰+¹₂)P^(α

0−¹

2,α⁰−¹

2)

l (x) (F.3)

Therefore, they satisfy the symmetry property:

C_l^α0(−x) = (−1)^lC_l^α0(x) (F.4) From now on, we set:

x= b−a

a+b (F.5)

in such a way that:

x−1 2

kx+ 1 2

l−k

= (−1)^l−k a^l−kb^k

(a+b)^l (F.6)

Hence, eq. (F.1) becomes:

P_l^(α,β)(x) =

l

X

k=0

l+α k

! l+β k+β

!

(−1)^l−k a^l−kb^k

(a+b)^l (F.7)

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From now on, we restrict α, β to the natural numbers and, correspondingly, α⁰ to the positive half-integers andJ to the natural numbers.

By employing the identity:

l+α it follows from eq. (F.7) that:

P_l^(α,β)(x) = (l+β)!(l+α)! Corrispondingly, eq. (F.3) reads:

C_l^α0(x) = Γ(l+ 2α⁰)Γ(α⁰+ ¹₂) Specializing the above equation to α⁰ = ⁵₂, we obtain:

C to be compared with eq. (F.9).

JHEP08(2021)142

G Matching 2- and 3-point Minkowskian correlators with [1]

We verify that our results for the 2- and 3-point Minkowskian correlators of the balanced operators with even collinear spin in the standard basis coincide with the ones in [1] up to the different normalization of the operators.

Starting from eq. (2.15):

C_s(x, y) = 1 (4π²)²

N²−1 4

2^2s+2

(4!)²(−1)^s(s−1)s(s+ 1)(s+ 2)(2s)! (x−y)^2s₊

(|x−y|²)^2s+2 (G.1) we get:

hOs1(x)Os2(y)i=δ_s1s2 1

(8π²)²(N²−1)2^2s1(−1)^s1 1 2⁴

Γ(s₁+ 3) 3²Γ(s₁−1)

Γ(2s₁+ 2) 2s₁+ 1

(x−y)^2s₊¹ (|x−y|²)^2s1+2

(G.2) in terms of gamma functions. Besides, we rewrite the above correlator in terms ofj=s−1,⁸ to match the notation of eq. (2.11) in [1]:

hOs1(x)Os2(y)i=δj1j2

1

(8π²)²(−1)^j1+1(N²−1)2^2j1−3Γ(j1+ 4) 3²Γ(j₁)

Γ(2j1+ 4) j₁+³₂

(x−y)^2j₊¹⁺² (|x−y|²)^2j1+4

(G.3) This is the very same result in [1] up to the overall factor of σj1σj2, which is missing as

— contrary to eq. (2.2) in [1] — we have defined the operatorsOs in eq. (4.7) without the factor of σ_j in front.

Our 3-point correlators of balanced operators with even collinear spin read in eq. (2.19):

hOs1(x)Os2(y)Os3(z)i=− 1 (4π²)³2

2 4!

3N²−1

8 i^s1+s2+s32^s1+s2+s3 (s1+ 1)(s1+ 2)(s2+ 1)(s2+ 2)(s3+ 1)(s3+ 2)

s1−2

X

k1=0 s2−2

X

k2=0 s3−2

X

k3=0

s1

k₁

! s1

k₁+ 2

! s2

k₂

! s2

k₂+ 2

! s3

k₃

! s3

k₃+ 2

!

(s₁−k₁+k₂)!(s₂−k₂+k₃)!(s₃−k₃+k₁)!

(x−y)^s₊^1−k1+k2 (|x−y|²)^s1+1−k1+k2

(y−z)^s₊^2−k2+k3 (|y−z|²)^s2+1−k2+k3

(z−x)^s₊^3−k3+k1 (|z−x|²)^s3+1−k3+k1

(G.4) Employing:

s k

! s k+ 2

!

= s(s−1) (s+ 2)(s+ 1)

s−2 k

! s+ 2 k+ 2

!

(G.5)

8j=s−1 in this section should not be confused with the conformal spin in the rest of the present paper.

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we obtain:

hOs1(x)Os2(y)Os3(z)i=− 1 (4π²)³2

2 4!

3 N²−1

8 i^s1+s2+s32^s1+s2+s3 s₁(s₁−1)s₂(s₂−1)s₃(s₃−1)

s1−2

X

k1=0 s2−2

X

k2=0 s3−2

X

k3=0

s₁−2 k1

! s₁+ 2 k1+ 2

! s₂−2 k2

! s₂+ 2 k2+ 2

! s₃−2 k3

! s₃+ 2 k3+ 2

!

(s1−k1+k2)!(s2−k2+k3)!(s3−k3+k1)!

(x−y)^s₊^1−k1+k2 (|x−y|²)^s1+1−k1+k2

(y−z)^s₊^2−k2+k3 (|y−z|²)^s2+1−k2+k3

(z−x)^s₊^3−k3+k1

(|z−x|²)^s3+1−k3+k1 (G.6) that in terms of j₁, j₂, j₃ reads:

hOs1(x)Os2(y)Os3(z)i=− 1 (8π²)³

1

2²3³(N²−1)i^j1+j2+j3+32^j1+j2+j3 j₁(j₁+ 1)j₂(j₂+ 1)j₃(j₃+ 1)

j1−1

X

k1=0 j2−1

X

k2=0 j3−1

X

k3=0

j₁−1 k1

! j₁+ 3 k1+ 2

! j₂−1 k2

! j₂+ 3 k2+ 2

! j₃−1 k3

! j₃+ 3 k3+ 2

!

(j1+ 1−k1+k2)!(j2+ 1−k2+k3)!(j3+ 1−k3+k1)!

(x−y)^j₊^1+1−k1+k2 (|x−y|²)^j1+2−k1+k2

(y−z)^j₊^2+1−k2+k3 (|y−z|²)^j2+2−k2+k3

(z−x)^j₊^3+1−k3+k1

(|z−x|²)^j3+2−k3+k1 (G.7) This is the very same result of eq. (2.22) in [1] up to the overall factor ofσj1σj2σj3, which is missing because of the aforementioned different normalization of the operators.

H Summation trick for 2-point correlators

We compute the 2-point correlators by means of the technique in [1].

H.1 Standard basis

Im Dokument n-point correlators of twist-2 operators in SU(N ) Yang-Mills theory to the lowest perturbative order (Seite 58-71)