Standard basis Given the bilocal operators:

O(x^A_i , x^B_i ) = Trf₁₁(x^A_i )f_{˙1 ˙1}(x^B_i ) = 1

2f₁₁^a(x^A_i )f^a_{˙1 ˙1}(x^B_i ) (7.1) in the light-cone gauge, the correspondingn-point correlator that is connected in the local limit, x^A_i =x^B_i =x_i, takes the form:

hO(x^A₁, x^B₁). . . O(x^A_n, x^B_n)i= 1 n

1 2ⁿ

n

X

i1=1

. . .

n

X

in=1

i16=i₂6=...6=i_n

hf₁₁^ai1(x^A_i1)f_{˙1 ˙1}^ai2(x^B_i2)i

hf₁₁^ai2(x^A_i2)f^a_{˙1 ˙1}ⁱ³(x^B_i3)i. . .hf₁₁^ain(x^A_in)f_{˙1 ˙1}^ai1(x^B_i1)i (7.2) The factor of _n¹ arises because, if the first index — for example i₁ = 1 — is kept fixed, there are only (n−1)! Wick contractions that contribute to the connected correlator. A nicer — but completely equivalent — formula is written in terms of permutations. If we denote byP_n the set of permutations of 1. . . n, it follows identically:

hO(x^A₁, x^B₁). . . O(x^A_n, x^B_n)i= 1 n

1 2ⁿ

X

σ∈Pn

hf₁₁^aσ(1)(x^A_σ(1))f^a_{˙1 ˙1}^σ(2)(x^B_σ(2))i

hf₁₁^aσ(2)(x^A_σ(2))f^a_{˙1 ˙1}^σ(3)(x^B_σ(3))i. . .hf₁₁^aσ(n)(x^A_σ(n))f_{˙1 ˙1}^aσ(1)(x^B_σ(1))i (7.3) Besides, eq. (B.4) reads:

hf₁₁^a(x_i)f_{˙1 ˙1}^b (x_j)i=−∂_x+ i ∂_x+

j

δ^ab

4π²|x_i−xj|² (7.4) Hence, for the balanced operators with even s, we get in the light-cone gauge:

hOs1(x₁). . .Osn(x_n)i= 1 2ⁿG

5 2

s1−2(∂_xA+ 1 , ∂_xB+

1

). . .G

5 2

sn−2(∂_xA+ n , ∂_xB+

n ) hf₁₁^a1(x^A₁)f^a_{˙1 ˙1}¹(x^B₁). . . f₁₁^an(x^A_n)f_{˙1 ˙1}^an(x^B_n)i

A=B

(7.5) where:

G

5 2

s−2(∂_x⁺

i , ∂_x⁺

j

) = i^s−2Γ(3)Γ(s+ 3) Γ(5)Γ(s+ 1)

s−2

X

k=0

s k

! s k+ 2

!

(−1)^s−k←−

∂^s−k−2

x⁺_i

−

→∂^k_x+ j

= 2i^s−2(s+ 1)(s+ 2) 4!

s−2

X

k=0

s k

! s k+ 2

!

(−1)^s−k←−

∂^s−k−2

x⁺_i

−

→∂^k_x+ j

(7.6)

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It follows from eq. (5.5) that, correspondingly, the n-point correlator contains factors of the form:

−∂^si−ki−2

x⁺_i ∂^kj

x⁺_j∂_x⁺

i ∂_x⁺

j

1 4π²|x_i−xj|²

= 1

4π²(−1)^si−ki(si−ki+kj)! 2^si−ki+kj (x_i−x_j)^s₊^i−ki+kj

(|x_i−x_j|²)^si−ki+kj+1 (7.7) Therefore, we get:

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

N²−1 2ⁿ i^P

n

l=1slΓ(3)Γ(s₁+3)

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

1 n

X

σ∈P_n s_σ(1)−2

X

k_σ(1)=0

. . .

s_σ(n)−2

X

k_σ(n)=0

s_σ(1) k_σ(1)

! s_σ(1) k_σ(1)+2

!

(−1)^{sσ(1)−kσ(1)}. . . s_σ(n) k_σ(n)

! s_σ(n) k_σ(n)+2

!

(−1)^{sσ(n)−kσ(n)}

2^{sσ(1)−kσ(1)+kσ(2)}(−1)^{sσ(1)−kσ(1)}(s_σ(1)−k_σ(1)+k_σ(2))! (x_σ(1)−x_σ(2))^s₊^{σ(1)−kσ(1)+kσ(2)} |x_σ(1)−x_σ(2)|^{2sσ(1)−kσ(1)+kσ(2)+1} . . .2^{sσ(n)−kσ(n)+kσ(1)}(−1)^{sσ(n)−kσ(n)}(s_σ(n)−k_σ(n)+k_σ(1))! (x_σ(n)−x_σ(1))^s₊^{σ(n)−kσ(n)+kσ(1)}

|x_σ(n)−x_σ(1)|^{2sσ(n)−kσ(n)+kσ(1)+1} (7.8) where we have set x^A_i = x^B_i = x_i in order to implement the local limit of the bilocal operators. The color factor comes from the contraction of the n Kronecker delta:

N²−1 =δ^aσ(1)aσ(2)δ^aσ(2)aσ(3). . . δ^aσ(n)aσ(1) (7.9) The overall factor of (−1)ⁿoccurs because of the factor of i⁻², which is a partial factor of i^s−2 in eq. (7.6).

After cancelling between themselves the pairs of factors of the kind (−1)^sa−ka in eq. (7.8), and moving out of the sum over the permutations the product of the binomial coefficients, since it is independent of the permutations, we obtain:

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)−kσ(1)+kσ(2)} |x_σ(1)−x_σ(2)|^{2sσ(1)−kσ(1)+kσ(2)+1}

. . . (x_σ(n)−x_σ(1))^s₊^{σ(n)−kσ(n)+kσ(1)} |x_σ(n)−x_σ(1)|^{2sσ(n)−kσ(n)+kσ(1)+1}

(7.10)

Actually, if n is even, eq. (7.10) also holds for the n-point correlator of the operators ˜Os, with the only difference that their collinear spin is odd.

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Otherwise, ifnis odd, the correlator vanishes. To verify it, it suffices to notice that, in the sum over the permutations, for every permutation the inverse permutation also occurs with the opposite sign. For example, for 3-point correlators we get pairs of terms of the kind:

s1−2

X

k1=0 s2−2

X

k2=0 s3−2

X

k3=0

. . .(s₁−k₁+k₂)!(s₂−k₂+k₃)!(s₃−k₃+k₁)!

(x1−x2)^s1−k1+k2(x2−x3)^s2−k2+k3(x3−x1)^s3−k3+k1 +

s1−2

X

k1=0 s2−2

X

k2=0 s3−2

X

k3=0

. . .(s2−k2+k1)!(s3−k3+k2)!(s1−k1+k3)!

(x₂−x₁)^s2−k2+k1(x₃−x₂)^s3−k3+k2(x₁−x₃)^s1−k1+k3 (7.11) Employing the substitutionk_i⁰ =s_i−2−k_i, we obtain for the last term above:

s1−2

X

k1=0 s2−2

X

k2=0 s3−2

X

k3=0

. . .(−1)^s1+s2+s3(s₁−k₁+k₂)!(s₂−k₂+k₃)!(s₃−k₃+k₁)!

(x₁−x₂)^s1−k1+k2(x₂−x₃)^s2−k2+k3(x₃−x₁)^s3−k3+k1 (7.12) that cancels the first term in eq. (7.11).

The same reasoning applies to the n+ 2m+ 1-point correlators of balanced operators:

hOs1(x1). . .Osn(xn) ˜Osn+1(xn+1). . .O˜sn+2m+1(xn+2m+1)i_conn = 0 (7.13) Otherwise, we get:

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

= 1

(4π²)^n+2m

N²−1

2^n+2m 2^{Pn+2ml=1 sl}i^{Pn+2ml=1 sl}Γ(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

!

. . . s_n+2m kn+2m

! s_n+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(1)−x_σ(2))^s₊^{σ(1)−kσ(1)+kσ(2)} |x_σ(1)−x_σ(2)|^{2sσ(1)−kσ(1)+kσ(2)+1}

. . . (x_σ(n+2m)−x_σ(1))^s₊^{σ(n+2m)−kσ(n+2m)+kσ(1)} |x_σ(n+2m)−x_σ(1)|^{2sσ(n+2m)−kσ(n+2m)+kσ(1)+1}

(7.14) For the correlators of the unbalanced operators, we obtain:

hSs1(x₁). . .Ssn(x_n)¯Ss⁰1(y₁). . .S¯s⁰n(y_n)i

= 1 2²ⁿG

5 2

s1−2(∂_x^A+

1 , ∂_x^B+

1

). . .G

5 2

sn−2(∂_x^A+

n , ∂_x^B+

n )G

5 2

s⁰1−2(∂_y^A+

1 , ∂_y^B+

1

). . .G

5 2

s⁰n−2(∂_y^A+

n , ∂_y^B+

n ) 1

2ⁿhf₁₁^a1(x^A₁)f₁₁^a1(x^B₁). . . f₁₁^an(x^A_n)f₁₁^an(x^B_n)f^b_{˙1 ˙1}¹(y^A₁)f_{˙1 ˙1}^b1(y^B₁). . . f_{˙1 ˙1}^bn(y_n^A)f^b_{˙1 ˙1}ⁿ(y_n^B)i

A=B

(7.15)

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The factor of ₂¹_2n arises from the normalization of the color trace in eq. (7.1), while the factor of ₂¹n comes from the normalization of the operators.

We get the very same correlator by exchanging A and B in all the couples (x^A_i , x^B_i ) and (y_k^A, y_k^B) simultaneously:

hSs1(x1). . .Ssn(xn)¯Ss⁰1(y1). . .S¯s⁰n(yn)i

= 1 2ⁿG

5 2

s1−2(∂_x^B+

1 , ∂_x^A+

1

). . .G

5 2

sn−2(∂_x^B+

n , ∂_x^A+

n )G

5 2

s⁰1−2(∂_y^B+

1 , ∂_y^A+

1

). . .G

5 2

s⁰n−2(∂_y^B+

n , ∂_y^A+

n ) 1

2²ⁿhf₁₁^a1(x^B₁)f₁₁^a1(x^A₁). . . f₁₁^an(x^B_n)f₁₁^an(x^A_n)f^b_{˙1 ˙1}¹(y₁^B)f^b_{˙1 ˙1}¹(y^A₁). . . f^b_{˙1 ˙1}ⁿ(y_n^B)f^b_{˙1 ˙1}ⁿ(y_n^A)i

A=B

(7.16) Indeed, in eq. (7.17) we may conveniently relabel the coordinates,x^A_i →x^B_i ,y_k^A→y^B_k, and vice versa for each iand k, since they coincide in the local limit. Moreover, according to eq. (6.4), G

5

s−22 (∂_xA+, ∂_xB+) in eqs. (7.15) and (7.16) is symmetric for the exchange of its arguments, because the collinear spin is even.

We evaluate the Wick contractions:

hf₁₁^a1(x^A₁). . . f₁₁^an(x^A_n)f_{˙1 ˙1}^b1(y^A₁). . . f^b_{˙1 ˙1}ⁿ(y_n^A)f₁₁^a1(x^B₁). . . f₁₁^an(x^B_n)f^b_{˙1 ˙1}¹(y₁^B). . . f^b_{˙1 ˙1}ⁿ(y_n^B)i

A=B

(7.17) We exploit the symmetry above: we only perform the Wick contractions involving the pairing ofx^A_i with y^A_k and of x^B_i withy_k^B for anyi, k, since all the remaining contractions provide the very same result due to the symmetry, and can be taken into account by a symmetry factor that we compute momentarily.

Besides, since we only are interested in the connected correlator, once x^A_i has been contracted with some y_k^A, x^B_i cannot be contracted with y_k^B, because the corresponding contribution to the correlator is not connected.

Hence, we construct the correlator as follows: we contract all the x^A_i with they_k^A and all the x^B_i0 with they_k^B0 withi6=i⁰ ifk=k⁰and k6=k⁰ifi=i⁰, in such a way that we build a single connected loop.

This is realized by summing over two sets of independent permutations arranged in such a way that no disconnected piece may be created: firstly, we contract x^A_i

1 with y^A_k

1, secondly, we contract y^B_k1 with x^B_i2 fori1 6=i2, then, we contract x^A_i2 with y_k^A₂ for k2 6=k1, afterwards, we contracty^B_k

2 withx^B_i3 fori₃ 6=i₂6=i₁ and so on, until we arrive atx^B_i1, which we contract with the last remaining y_k^A_n with kn 6=kn−1 6=. . . 6=k1, in order to close the loop. We end up with a chain that looks like:

X

i16=i26=i3...6=in

X

k16=k26=k3...6=kn

hf₁₁^ai1(x^A_i1)f^b_{˙1 ˙1}^k1(y_k^A₁)ihf_{˙1 ˙1}^bk1(y_k^B₁)f₁₁^ai2(x^B_i2)i (7.18) hf₁₁^ai2(x^A_i2)f_{˙1 ˙1}^bk2(y_k^A₂)ihf_{˙1 ˙1}^bk2(y_k^B₂)f₁₁^ai3(x^B_i3)i. . .hf₁₁^ain(x^A_in)f^b_{˙1 ˙1}^kn(y_k^A_n)ihf_{˙1 ˙1}^bkn(y^B_kn)f₁₁^ai1(x^B_i1)i Yet, now we are creating a redundancy, since we also are summing on the possiblenchoices

JHEP08(2021)142

of the starting point of the loop. Therefore, we divide the sum by a factor of n:

1 n

X

i16=i26=i3...6=in

X

k16=k26=k3...6=kn

hf₁₁^ai1(x^A_i1)f^b_{˙1 ˙1}^k1(y^A_k1)ihf^b_{˙1 ˙1}^k1(y^B_k1)f₁₁^ai2(x^B_i2)i (7.19) hf₁₁^ai2(x^A_i2)f_{˙1 ˙1}^bk2(y_k^A₂)ihf_{˙1 ˙1}^bk2(y_k^B₂)f₁₁^ai3(x^B_i3)i. . .hf₁₁^ain(x^A_in)f^b_{˙1 ˙1}^kn(y_k^A_n)ihf_{˙1 ˙1}^bkn(y^B_kn)f₁₁^ai1(x^B_i1)i A nicer — but completely equivalent — formula is written in terms of permutations. It follows identically:

1 n

X

σ∈Pn

X

ρ∈Pn

hf₁₁^aσ1(x^A_σ1)f_{˙1 ˙1}^bρ1(y^A_ρ1)ihf^b_{˙1 ˙1}^ρ1(y_ρ^B₁)f₁₁^aσ2(x^B_σ2)ihf₁₁^aσ2(x^A_σ2)f_{˙1 ˙1}^bρ2(y_ρ^A₂)ihf^b_{˙1 ˙1}^ρ2(y^B_ρ2)f₁₁^aσ3(x^B_σ3)i . . .hf₁₁^aσn(x^A_σn)f^b_{˙1 ˙1}^ρn(y_ρ^A_n)ihf_{˙1 ˙1}^bρn(y_ρ^B_n)f₁₁^aσ1(x^B_σ1)i (7.20) All the remaining contractions are obtained from this formula by exchanging the coor-dinates in each couple, (x^A_i , x^B_i ) and (y^A_k, y^B_k), for each i and k. There are 2²ⁿ of such exchanges.

However, the actual degeneration factor is 2²ⁿ⁻¹. Indeed, the extra factor of ¹₂ comes from the fact that the simultaneous exchange of the coordinates in each couple, (x^A_i , x^B_i ) and (y^A_k, y^B_k), yields a contraction that has already been counted due to the symmetry of eqs. (7.15) and (7.16) with respect of the simultaneous exchange of A with B in all the coordinate pairs.

Hence, by combining the degeneration factor of 2²ⁿ⁻¹ with the factor of ₂¹_n from the normalization of the operators, the overall factor of 2ⁿ⁻¹ survives. It follows:

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

N²−1

2²ⁿ 2^Pnl=1sl+s0li^Pnl=1sl+s0l 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))^{sσ(1)−kσ(1)+k}

0ρ(1)

+

|x_σ(1)−y_ρ(1)|^{2sσ(1)−kσ(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)|^2s

0ρ(1)−k⁰_ρ(1)+kσ(2)+1

. . . (x_σ(n)−y_ρ(n))^{sσ(n)−kσ(n)+k}

0 ρ(n)

+

|x_σ(n)−y_ρ(n)|^{2sσ(n)−kσ(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)|^2s

ρ(n)0 −k⁰_ρ(n)+k_σ(1)+1 (7.21)

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7.2 Extended basis

Similarly, in the extended basis, we get:

hAs1(x₁). . .Asn(x_n)i= 1 in the light-cone gauge, where:

G It follows from eqs. (B.5) and (5.5) that, correspondingly, the n-point correlator contains factors of the form:

−∂^si−ki where now the overall factor of (−1)ⁿ occurs because of the extra minus sign in eq. (7.24) with respect to eq. (7.7).

The very same formula holds for an even number of operators ˜As, otherwise the cor-relators vanish. We obtain as well:

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

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Analogously, for the unbalanced operators in the extended basis, 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∈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))^{sσ(1)−kσ(1)+k}

0ρ(1)

+

|x_σ(1)−y_ρ(1)|^{2sσ(1)−kσ(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)|^2s

0ρ(1)−k⁰_ρ(1)+k_σ(2)+1

. . . (x_σ(n)−y_ρ(n))^{sσ(n)−kσ(n)+k}

0ρ(n)

+

|x_σ(n)−y_ρ(n)|^{2sσ(n)−kσ(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)|^2s

ρ(n)0 −k⁰_ρ(n)+kσ(1)+1 (7.27)

8 n-point correlators and twist-2 gluonic operators in Euclidean space-time

8.1 Analytic continuation of n-point correlators to Euclidean space-time

Im Dokument n-point correlators of twist-2 operators in SU(N ) Yang-Mills theory to the lowest perturbative order (Seite 36-42)