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

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JHEP08(2021)142

Published for SISSA by Springer

Received: May 5, 2021 Revised: July 28, 2021 Accepted: August 2, 2021 Published: August 26, 2021

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

Marco Bochicchio,^a Mauro Papinutto^b,a and Francesco Scardino^b,a

aPhysics Department, INFN Roma1, Piazzale A. Moro 2, Roma, I-00185, Italy

bPhysics Department, Sapienza University, Piazzale A. Moro 2, Roma, I-00185, Italy

E-mail: marco.bochicchio@roma1.infn.it,

mauro.papinutto@roma1.infn.it,francesco.scardino@roma1.infn.it

Abstract: We compute, to the lowest perturbative order in SU(N) Yang-Mills theory, n-point correlators in the coordinate and momentum representation of the gauge-invariant twist-2 operators with maximal spin along the p₊ direction, both in Minkowskian and

— by analytic continuation — Euclidean space-time. We also construct the corresponding generating functionals. Remarkably, they have the structure of the logarithm of a functional determinant of the identity plus a term involving the effective propagators that act on the appropriate source fields.

Keywords: Perturbative QCD, Renormalization Group ArXiv ePrint: 2104.13163

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Contents

1 Introduction and physics motivations 1

2 Main results 2

2.1 Balanced and unbalanced twist-2 conformal operators 2 2.2 Minkowskian n-point correlators in the coordinate representation 5

2.2.1 Standard basis 5

2.2.2 Extended basis 7

2.3 Euclidean n-point correlators in the coordinate representation 9

2.3.1 Standard basis 9

2.3.2 Extended basis 11

2.4 Generating functional ofn-point correlators in the coordinate representation 13

2.4.1 Minkowskian standard basis 14

2.4.2 Minkowskian extended basis 15

2.4.3 Euclidean standard basis 16

2.4.4 Euclidean extended basis 16

2.5 Generating functional andn-point correlators in the momentum representation 17

2.5.1 Minkowskian standard basis 17

2.5.2 Minkowskian extended basis 19

2.5.3 Euclidean standard basis 20

2.5.4 Euclidean extended basis 22

3 Plan of the paper 23

4 Twist-2 gluonic operators in Minkowskian space-time 24

4.1 Standard basis 24

4.2 Extended basis 25

5 2-point correlators of twist-2 gluonic operators 26

6 3-point correlators of twist-2 gluonic operators 31

7 n-point correlators of twist-2 gluonic operators 34

8 n-point correlators and twist-2 gluonic operators in Euclidean space-time 40 8.1 Analytic continuation of n-point correlators to Euclidean space-time 40 8.2 Twist-2 gluonic operators in Euclidean space-time 41

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9 Generating functional of n-point correlators in the coordinate

representation 42

9.1 Minkowskian standard basis 43

9.2 Minkowskian extended basis 47

9.3 Euclidean standard basis 48

9.4 Euclidean extended basis 48

10 Generating functional and n-point correlators in the momentum

representation 48

10.1 Minkowskian standard basis 51

10.2 Minkowskian extended basis 53

10.3 Euclidean standard basis 55

10.4 Euclidean extended basis 56

A Notation and Wick rotation 58

B Minkowskian and Euclidean propagators 59

C Identities involving σ^µ and ¯σ^µ 60

D Relation between the spinorial and vectorial bases in Minkowskian

space-time 62

E Complex basis 65

F Jacobi and Gegenbauer polynomials 66

G Matching 2- and 3-point Minkowskian correlators with [1] 68

H Summation trick for 2-point correlators 69

H.1 Standard basis 69

H.2 Extended basis 71

1 Introduction and physics motivations

In the present paper we compute, to the lowest perturbative order in SU(N) Yang-Mills (YM) theory,n-point connected correlators,G⁽ⁿ⁾_conf(x₁, . . . , x_n), in the coordinate representation of the gauge-invariant twist-2 operators with maximal spin along the p₊ direction, both in Minkowskian and — by analytic continuation — Euclidean space-time.

In fact, our computation matches and extends the previous lowest-order perturbative computation of 2- and 3-point gluonic correlators of twist-2 operators in N = 4 SUSY YM theory [1], by including the unbalanced¹ operators with collinear twist 2 in pure YM

1In our terminology ‘unbalanced’ and ‘balanced’ refers to either the different or the equal number of dotted and undotted indices that the aforementioned operators possess in the spinorial representation respectively. Unbalanced operators are referred to as ‘asymmetric’ in [2] and ‘anisotropic’ in [3].

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theory and, most importantly, by calculating all the n-point correlators in the balanced and unbalanced sectors separately, and the 3-point correlators in the mixed sector as well.

Our physics motivation is threefold.

Firstly, our lowest-order computation has an intrinsic interest in YM theory, and — according to [1] — in theories that extends it, such as its supersymmetric versions and QCD.

Secondly, our computation is preliminary to work out the ultraviolet (UV) asymp- totics [4, 5] — based on the renormalization-group (RG) improvement of perturbation theory — of the above Euclidean n-point correlators.

Thirdly, our computation is an essential ingredient to test the prediction in section 3 of [6] that, by fundamental principles of the large-N ’t Hooft expansion, the generating functional of the nonperturbative leading nonplanar contributions to the aforementioned Euclidean correlators must have the structure of the logarithm of a functional determinant [6] that sums the glueball one-loop diagrams.

Indeed, according to the philosophy of the asymptotically free bootstrap outlined in [6], the RG-improved correlators mentioned above must be asymptotic in the UV [6] to the corresponding nonperturbative correlators involving glueballs. Therefore, to the leading nonplanar order, the generating functional of the former must share with the one of the latter the very same structure of the logarithm of a functional determinant.

As an intermediate step for the program above, we construct the generating functionals of the aforementioned lowest-ordern-point correlators.

Remarkably, they have the structure of the logarithm of a functional determinant of the identity plus a term involving the effective propagators that act on the appropriate source fields.

Hence, according to the argument above, our formulas for the generating functionals are as simple as they can be.

Incidentally, the generating functionals also allow us to compute straightforwardly the n-point correlators in the momentum representation, whose structure is slightly simpler than in the coordinate representation.

2 Main results

2.1 Balanced and unbalanced twist-2 conformal operators

We describe our calculation and the operators that enter it. We compute, to the lowest perturbative order in SU(N) YM theory,n-point connected correlators in Minkowskian space- time of the gauge-invariant twist-2 operators with maximal spin along thep+ direction:

hO₁(x₁). . .O_n(x_n)ilowest order=G⁽ⁿ⁾_conf(x₁, . . . , x_n) (2.1) It has been known for some time that, to the lowest order and the next one,² YM theory is conformal invariant [7], since the beta function only affects the solution of the Callan- Symanzik equation starting from the order of g⁴. More recently, the exact conformal

2In fact, to the order ofg², in the conformal subtraction scheme [7].

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symmetry of QCD at the Wilson-Fisher critical point in d= 4−2dimensions has been exploited [8,9]³ as a computational tool in higher orders of perturbation theory.

Therefore, following [7] we employ operators that have nice transformation properties with respect to the collinear conformal subgroup involving the coordinatex⁺. Alternatively, operators can be also constructed with nice transformation properties [10,11] with respect to the conformal group, whose suitably chosen components restrict to the aforementioned representations of the collinear conformal subgroup.

Primary conformal operators O_j(x), with collinear conformal spin j=s+^τ₂, whereτ is the collinear twist andsthe collinear spin, i.e., the spin projected along thep₊ direction, transform under the action of the generators [7] of the collinear conformal algebra SL(2,R):

[L₀, L∓] =∓L_∓ (2.2)

[L−, L+] =−2L₀ according [7] to:

[L+,O_j(x)] =−∂₊O_j(x) (2.3)

[L−,O_j(x)] = (x^{+ 2}∂++ 2jx⁺)O_j(x) [L₀,O_j(x)] = (x⁺∂₊+j)O_j(x)

where in eq. (2.3) x= (x⁺, x⁻, x¹, x²) is restricted [7] to the linex⁻=x¹=x² = 0. Their conformal descendants, ∂ⁱ₊O_j(x), are obtained by taking derivatives with respect to x⁺, and have the same τ.

For a given canonical dimensiond=τ+s, the quasi-partonic [12] operators have mini- mumτ and maximums, with nice mixing under renormalization and conformal properties as above [7,10–15]. Their collinear twistτ does not necessarily coincide [7] with the twist T — defined byd=T+S, whereSis the spin — that refers to the conformal group [10,11]

instead of the collinear subgroup.

In general, local gauge-invariant operators with T = 2 provide the leading contribution to the OPE of two vector currents in massless QCD-like theories⁴ [16]⁵ in Minkowskian space-time near the light-cone.

An infinite family of quasi-partonic operators is constructed as follows. A composite gauge-covariant primary conformal operator, built by two elementary⁶ gauge-covariant primary conformal operators Φ_j₁,Φ_j₂, with collinear conformal spinsj1, j2, has the form [7]:

O^j_l¹^j²(x) = Φ_j₁(x)(i−→

D₊+i←−

D₊)^lP_l^(2j¹^−1,2j²⁻¹⁾

−

→D₊−←− D₊

−

→D₊+←− D₊

!

Φ_j₂(x) (2.4) where P_l^(2j¹^−1,2j²⁻¹⁾ are Jacobi polynomials (appendix F), D₊ is the covariant derivative along thep+direction (appendix A), and the arrows denote the action of the derivative on

3And references therein.

4By massless QCD-like theories we mean asymptotically free gauge theories that are massless to all orders of perturbation theory, such as QCD with massless quarks.

5And references therein.

6In the present paper, we refer to the operators Φj(x) as elementary, since they play the role of elementary constituents, though they may actually be composite operators.

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the right or the left. The corresponding gauge-invariant object is obtained by taking the color trace.

The collinear conformal spin, j, of the operator, O^j_l¹^j²(x), isj=j₁+j₂+l, wherel is the power of the derivative in eq. (2.4). By working out the definition in eq. (2.4), we get:

O_l^j¹^j²(x) =

l

X

k=0

l+ 2j₁−1 k

! l+ 2j₂−1 k+ 2j₂−1

!

(−1)^l−kΦ_j₁(x)←− D^l−k₊ −→

D^k₊Φ_j₂(x)

=

l

X

k=0

O^j_lk¹^j²(x) (2.5)

thus realizing the conformal operator O_l^j¹^j²(x) as a sum ofl+ 1 operators, O_lk^j¹^j²(x), that are not necessarily conformal.

Hence, the composite operators depend on a choice of the elementary conformal operators Φ_j₁,Φ_j₂. We define the standard conformal basis for primary operators with collinear twist 2, where the elementary operators aref₁₁, f_{˙1 ˙1} (section4.1) with conformal spinj= ³₂. In the standard basis the gluonic operators are classified as in [2,10,11]:

Os= Trf11(x)(i−→

D++i←−

D+)^s−2C

5 2

s−2

−

→D+−←− D+

−

→D₊+←− D₊

!

f_{˙1 ˙1}(x) s= 2,4,6, . . . O˜s= Trf11(x)(i−→

D++i←−

D+)^s−2C

5 2

s−2

−

→D+−←− D+

−

→D₊+←− D₊

!

f_{˙1 ˙1}(x) s= 3,5,7, . . .

Ss= 1

√

2Trf₁₁(x)(i−→

D₊+i←−

D₊)^s−2C

5

s−22

−

→D+−←− D+

−

→D₊+←− D₊

!

f₁₁(x) s= 2,4,6, . . . S¯s= 1

√2Trf_{˙1 ˙1}(x)(i−→

D₊+i←−

D₊)^s−2C

5

s−22

−

→D₊−←− D₊

−

→D₊+←− D₊

!

f_{˙1 ˙1}(x) s= 2,4,6, . . . (2.6) by restricting the appropriate conformal multiplet [10, 11] to the components along the p₊ direction, with C_l^α Gegenbauer polynomials (appendix F), which are a special case of Jacobi polynomials.

Os and ˜Os are Hermitian balanced operators with τ = T = 2. They have an equal number of undotted and dotted spinor indices (appendicesC and D):

Os=O1 ˙1...1 ˙1

O˜s= ˜O1 ˙1...1 ˙1 (2.7)

Ss and its Hermitian conjugate, ¯Ss, denoted by the bar superscript, are unbalanced operators withτ = 2. They have a different number of undotted and dotted spinor indices:

Ss=S1111...1 ˙1

S¯s=S˙1 ˙1 ˙1 ˙1...˙11 (2.8)

Besides, we also define the extended conformal basis for primary operators with collinear twist 2, where the elementary operators are D⁻¹₊ f₁₁, D₊⁻¹f_{˙1 ˙1}, with conformal spin j = ¹₂,

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which are nonlocal in general, but local (appendix E) in the light-cone gauge A₊ = 0.

Clearly, gauge invariance ensures that all their correlators are local, as we verify explicitly.

The extended basis is natural in SUSY calculations [14], and includes (nonlocal) operators withτ = 2 ands= 0,1. We have chosen it in YM theory because of the simplicity of the results for the correlators. In the extended basis (section4.2) the gluonic operators are:

As= TrD⁻¹₊ f11(x) (i−→ D++i←−

D+)^sC

1

s2

−

→D+−←− D+

−

→D₊+←− D₊

!

D⁻¹₊ f_{˙1 ˙1}(x) s= 0,2,4, . . . A˜s= TrD⁻¹₊ f₁₁(x) (i−→

D₊+i←− D₊)^sC

1

s2

−

→D+−←− D+

−

→D₊+←− D₊

!

D⁻¹₊ f_{˙1 ˙1}(x) s= 1,3,5, . . .

Bs= 1

√

2TrD₊⁻¹f₁₁(x) (i−→ D₊+i←−

D₊)^sC

1

s2

−

→D₊−←− D₊

−

→D₊+←− D₊

!

D₊⁻¹f₁₁(x) s= 0,2,4, . . . B¯s= 1

√2TrD₊⁻¹f_{˙1 ˙1}(x) (i−→ D₊+i←−

D₊)^sC

1

s2

−

→D₊−←− D₊

−

→D₊+←− D₊

!

D₊⁻¹f_{˙1 ˙1}(x) s= 0,2,4, . . . (2.9)

2.2 Minkowskian n-point correlators in the coordinate representation 2.2.1 Standard basis

We have normalized our operators in such a way that the 2-point correlators in the standard basis are equal for even s:

hOs1(x)Os2(y)i=hSs1(x)¯Ss2(y)i=C_s₁(x, y)δs1s2 (2.10) and for odds:

hO˜s1(x) ˜Os2(y)i=C_s₁(x, y)δ_s₁_s₂ (2.11) with:

C_s(x, y) = 1 (4π²)²

N²−1 4

2^2s+2i^2s−4

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

s−2

X

k1=0 s−2

X

k2=0

s k₁

! s k₁+ 2

! s k₂

! s k₂+ 2

!

(−1)^s−k²^+k¹ (s−k₁+k₂)!(s+k₁−k₂)!

= 1

(4π²)²

N²−1 4

2^2s+2i^2s−4

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

s−2

X

k1=0 s−2

X

k2=0

s k₁

! s k₁+ 2

! s k₂

! s k₂+ 2

!

(−1)^k²^+k¹ 1

2s k1+k2+2

(2.12)

where we omit the i prescription in the propagators in the coordinate representation, in such a way that (appendixA):

1

|x|² (2.13)

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should be read (appendix B):

1

|x|²−i (2.14)

The very same correlators are evaluated by a trick [1] (appendixH):

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 (2.15) Therefore, we have discovered the following — seemingly nontrivial — identity (section 5):

s₁(s₁−1)

(s1+1)(s1+2)δ_s₁_s₂=

s1−2

X

k1=0 s2−2

X

k2=0

s₁ k1

! s₁ k1+2

! s₂ k2

! s₂ k2+2

!

(−1)^k²^+k¹ 1

s1+s2

k1+k2+2

(2.16) We have not found a direct proof of the above identity, but we have verified it numerically.

Moreover, the only nonvanishing 3-point correlators are:

hOs1(x)Os2(y)Os3(z)i=hOs1(x)Ss2(y)¯Ss3(z)i=C_s₁_s₂_s₃(x, y, z) (2.17) and:

hOs1(x) ˜Os2(y) ˜Os3(z)i=C_s₁_s₂_s₃(x, y, z) (2.18) with:

C_s₁_s₂_s₃(x, y, z) =− 1

(4π²)³(1 + (−1)^s¹^+s²^+s³) 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 (2.19) We also compute the n-point correlators. In the balanced sector, we get:

hOs1(x₁). . .Osn(x_n)i_conn = 1 (4π²)ⁿ

N²−1 2ⁿ 2^P

n l=1sli^P

n l=1sl

Γ(3)Γ(s1+ 3)

Γ(5)Γ(s₁+ 1). . .Γ(3)Γ(sn+ 3) Γ(5)Γ(s_n+ 1)

s1−2

X

k1=0

. . .

sn−2

X

kn=0

s1

k₁

! s1

k₁+ 2

!

. . . sn

k_n

! sn

k_n+ 2

!

(−1)ⁿ n

X

σ∈Pn

(s_σ(1)−k_σ(1)+k_σ(2))!. . .(s_σ(n)−k_σ(n)+k_σ(1))!

(x_σ(1)−x_σ(2))^s₊^σ(1)^−k^σ(1)^+k^σ(2) |x_σ(1)−x_σ(2)|²^s^σ(1)^−k^σ(1)^+k^σ(2)⁺¹

. . . (x_σ(n)−x_σ(1))^s₊^σ(n)^−k^σ(n)^+k^σ(1) |x_σ(n)−x_σ(1)|²^s^σ(n)^−k^σ(n)^+k^σ(1)⁺¹

(2.20)

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The very same formula holds for an even number of operators ˜Os, otherwise the correlators vanish. The nonvanishing correlators in the balanced sector are:

hOs1(x₁). . .Osn(x_n) ˜Osn+1(x_n+1). . .O˜sn+2m(x_n+2m)i_conn

= 1

(4π²)^n+2m

N²−1 2^n+2m 2^P

n+2m l=1 sli^P

n+2m

l=1 slΓ(3)Γ(s₁+ 3)

Γ(5)Γ(s1+ 1). . .Γ(3)Γ(s_n+2m+ 3) Γ(5)Γ(sn+2m+ 1)

s1−2

X

k1=0

. . .

sn+2m−2

X

kn+2m=0

s1

k1

! s1

k1+ 2

!

. . . sn+2m

kn+2m

! sn+2m

kn+2m+ 2

!

(−1)^n+2m n+ 2m

X

σ∈P_n+2m

(s_σ(1)−k_σ(1)+k_σ(2))!. . .(s_σ(n+2m)−k_σ(n+2m)+k_σ(1))!

. . . (x_σ(n+2m)−x_σ(1))^s₊^σ(n+2m)^−k^σ(n+2m)^+k^σ(1) |x_σ(n+2m)−x_σ(1)|²^s^σ(n+2m)^−k^σ(n+2m)^+k^σ(1)⁺¹

(2.21) In the unbalanced sector, we get:

hSs1(x₁). . .Ssn(x_n)¯Ss⁰1(y₁). . .S¯s⁰n(y_n)i= 1 (4π²)²ⁿ

N²−1

2²ⁿ 2^Pⁿ^l=1^s^l^+s⁰^li^Pⁿ^l=1^s^l^+s⁰^l Γ(3)Γ(s₁+ 3)

Γ(5)Γ(s₁+ 1). . .Γ(3)Γ(s_n+ 3) Γ(5)Γ(s_n+ 1)

Γ(3)Γ(s₁⁰ + 3)

Γ(5)Γ(s₁⁰ + 1). . .Γ(3)Γ(s_n⁰ + 3) Γ(5)Γ(s_n⁰ + 1)

s1−2

X

k1=0

. . .

sn−2

X

kn=0

s₁ k₁

! s₁ k₁+ 2

!

. . . sn

k_n

! sn

k_n+ 2

!

s1⁰−2

X

k⁰1=0

. . .

s⁰n−2

X

k⁰n=0

s1⁰

k₁⁰

! s1⁰

k₁⁰ + 2

!

. . . sn⁰

k_n⁰

! sn⁰

k_n⁰ + 2

!

2ⁿ⁻¹ n

X

σ∈P_n

X

ρ∈P_n

(s_σ(1)−k_σ(1)+k_ρ(1)⁰ )!(s_ρ(1)⁰ −k_ρ(1)⁰ +k_σ(2))!

. . .(s_σ(n)−k_σ(n)+k_ρ(n)⁰ )!(s_ρ(n)⁰ −k_ρ(n)⁰ +k_σ(1))!

(x_σ(1)−y_ρ(1))^s^σ(1)^−k^σ(1)^+k

0ρ(1)

+

|x_σ(1)−y_ρ(1)|²^s^σ(1)^−k^σ(1)^+k

0ρ(1)+1

(y_ρ(1)−x_σ(2))^s

0ρ(1)−k⁰_ρ(1)+k_σ(2) +

|y_ρ(1)−x_σ(2)|²^s

0ρ(1)−k⁰_ρ(1)+kσ(2)+1

. . . (x_σ(n)−y_ρ(n))^s^σ(n)^−k^σ(n)^+k

0 ρ(n)

+

|x_σ(n)−y_ρ(n)|²^s^σ(n)^−k^σ(n)^+k

0ρ(n)+1

(y_ρ(n)−x_σ(1))^s

0

ρ(n)−k_ρ(n)⁰ +k_σ(1) +

|y_ρ(n)−x_σ(1)|²^s

ρ(n)0 −k⁰_ρ(n)+k_σ(1)+1 (2.22) 2.2.2 Extended basis

We normalize our operators in such a way that the 2-point correlators in the extended basis are equal for even s:

hAs1(x)As2(y)i=hB¯s1(x)¯Bs2(y)i=A_s₁(x, y)δs1s2 (2.23) and for odds:

hA˜s1(x) ˜As2(y)i=A_s₁(x, y)δs1s2 (2.24)

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with:

A_s(x, y) = 1 (4π²)²

N²−1

4 2^2si^2s (x−y)^2s₊ (|x−y|²)^2s+2

s

X

k1=0 s

X

k2=0

s k₁

! s k₁

! s k₂

!

(−1)^s−k²^+k¹(s−k1+k2)!(s+k1−k2)!

= 1

(4π²)²

N²−1

4 2^2si^2s(2s)! (x−y)^2s₊ (|x−y|²)^2s+2

s

X

k1=0 s

X

k2=0

s k1

! s k1

! s k2

!

(−1)^k²^+k¹ 1

2s k1+k2

(2.25)

The very same correlators are evaluated by a trick [1] (appendixH):

A_s(x, y) = 1 (4π²)²

N²−1

4 2^2s(−1)^s(2s)! (x−y)^2s₊

(|x−y|²)^2s+2 (2.26) Therefore, we have discovered the following — seemingly nontrivial — identity (section 5):

δs1s2 =

s1

X

k1=0 s2

X

k2=0

s1

k1

! s1

k1

! s2

k2

! s2

k2

!

(−1)^k²^+k¹ 1

s1+s2

k1+k2

(2.27)

We have not found a direct proof of the above identity, but we have verified it numerically.

Moreover, the only nonvanishing 3-point correlators are:

hAs1(x)As2(y)As3(z)i=hAs1(x)Bs2(y)¯Bs3(z)i=A_s₁_s₂_s₃(x, y, z) (2.28) and:

hAs1(x) ˜As2(y) ˜As3(z)i=A_s₁_s₂_s₃(x, y, z) (2.29) with:

A_s₁_s₂_s₃(x, y, z) =− 1

(4π²)³(1+(−1)^s¹^+s²^+s³)N²−1

8 i^s¹^+s²^+s³2^s¹^+s²^+s³

s1

X

k1=0 s2

X

k2=0 s3

X

k3=0

s1

k₁

! s1

k₁

! s2

k₂

! s2

k₂

! s3

k₃

! s3

k₃

!

(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

(2.30) We also compute the n-point correlators. In the balanced sector, we get:

hAs1(x₁). . .Asn(x_n)i_conn= 1 (4π²)ⁿ

N²−1 2ⁿ 2^P

n l=1sli^P

n l=1sl

s1

X

k1=0

. . .

sn

X

kn=0

s1

k₁

!2

. . . sn

k_n

!2

(−1)ⁿ n

X

σ∈Pn

(s_σ(1)−k_σ(1)+k_σ(2))!. . .(s_σ(n)−k_σ(n)+k_σ(1))!

. . . (x_σ(n)−x_σ(1))^s₊^σ(n)^−k^σ(n)^+k^σ(1) |x_σ(n)−x_σ(1)|²^s^σ(n)^−k^σ(n)^+k^σ(1)⁺¹

(2.31)

(11)

JHEP08(2021)142

The very same formula holds for an even number of operators ˜As, otherwise the correlators vanish. The nonvanishing correlators in the balanced sector are:

hAs1(x₁). . .Asn(x_n) ˜Asn+1(x_n+1). . .A˜sn+2m(x_n+2m)i_conn

= 1

(4π²)^n+2m

N²−1 2^n+2m 2^P

n+2m l=1 sli^P

n+2m l=1 sl

s1

X

k1=0

. . .

sn+2m

X

kn+2m=0

s1

k1

!2

. . . sn+2m

kn+2m

!2

(−1)^n+2m n+ 2m

X

σ∈P_n+2m

(s_σ(1)−k_σ(1)+k_σ(2))!. . .(s_σ(n+2m)−k_σ(n+2m)+k_σ(1))!

. . . (x_σ(n+2m)−x_σ(1))^s₊^σ(n+2m)^−k^σ(n+2m)^+k^σ(1) |x_σ(n+2m)−x_σ(1)|²^s^σ(n+2m)^−k^σ(n+2m)^+k^σ(1)⁺¹

(2.32) In the unbalanced sector, we get:

hBs1(x1). . .Bsn(xn)¯Bs1⁰(y1). . .B¯s⁰n(yn)i= 1 (4π²)²ⁿ

N²−1 2²ⁿ 2^P

n

l=1sl+s⁰li^P

n l=1sl+s⁰l

s1

X

k1=0

. . .

sn

X

kn=0 s1⁰−2

X

k⁰1=0

. . .

s⁰n

X

k⁰n=0

s₁ k₁

!2

. . . sn

k_n

!2

s₁⁰ k₁⁰

!2

. . . sn⁰

k_n⁰

!2

2ⁿ⁻¹ n

X

σ∈Pn

X

ρ∈Pn

(s_σ(1)−k_σ(1)+k_ρ(1)⁰ )!(s_ρ(1)⁰ −k_ρ(1)⁰ +k_σ(2))!

. . .(s_σ(n)−k_σ(n)+k_ρ(n)⁰ )!(s_ρ(n)⁰ −k_ρ(n)⁰ +k_σ(1))!

(x_σ(1)−y_ρ(1))^s^σ(1)^−k^σ(1)^+k

0ρ(1)

+

|x_σ(1)−y_ρ(1)|²^s^σ(1)^−k^σ(1)^+k

0ρ(1)+1

(y_ρ(1)−x_σ(2))^s

0ρ(1)−k⁰_ρ(1)+kσ(2)

+

|y_ρ(1)−x_σ(2)|²^s

0ρ(1)−k⁰_ρ(1)+k_σ(2)+1

. . . (x_σ(n)−y_ρ(n))^s^σ(n)^−k^σ(n)^+k

0ρ(n)

+

|x_σ(n)−y_ρ(n)|²^s^σ(n)^−k^σ(n)^+k

0ρ(n)+1

(y_ρ(n)−x_σ(1))^s

0ρ(n)−k_ρ(n)⁰ +k_σ(1) +

|y_ρ(n)−x_σ(1)|²^s

ρ(n)0 −k⁰_ρ(n)+kσ(1)+1 (2.33) 2.3 Euclidean n-point correlators in the coordinate representation

2.3.1 Standard basis

After the Wick rotation (appendixA and section8), we obtain in the standard basis:

C_s^E(x, y) = 1 (4π²)²

N²−1 4

2^2s+2

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

s−2

X

k1=0 s−2

X

k2=0

s k₁

! s k₁+ 2

! s k₂+ 2

! s k₂

!

(−1)^s−k²^+k¹

(s−k1+k2)!(s+k1−k2)! (2.34)

which is equivalent to:

C_s^E(x, y) = 1 (4π²)²

N²−1 4

2^2s+2

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

((x−y)²)^2s+2 (2.35)