Loop Correlators

82 5. Ramond–Neveu–Schwarz Correlators at Loop-Level

with genus g Atick and Sen have been able to calculate all loop correlators of the SO(2) spin system [130, 150, 151]. For the case of 2N spin fields they find

YN

i=1

s⁺(zi)s⁻(wi) ~a

~b

= Θh_~a

~b

i 1 2

PN

i=1∫w^zii~ω Θh_~a

~b

i (~0)

QN

i<jE(zi, zj)E(wi, wj) QN

i,j=1E(zi, wj)

!1/4

. (5.20) This correlator is the key ingredient for deriving correlation functions of R spin fields. The most general form with fermions Ψ^± and spin fields s^±, which is the starting point for calculating general RNS loop-correlators, is given by

Y^N1

i=1

s⁺(y_i)

N2

Y

j=1

s⁻(z_j)

N3

Y

k=1

Ψ⁻(u_k)

N4

Y

l=1

Ψ⁺(v_l) ~a

~b

=

QN1

r<sE(yr, ys) QN2

r<sE(zr, zs) QN1

i=1

QN2

j=1E(z_j, y_i)

!1/4

×

QN3

r<sE(ur, us)QN4

r<sE(vr, vs) QN3

k=1

QN4

l=1E(vl, uk)

! QN2

j=1

QN3

k=1E(u_k, z_j)QN1

i=1

QN4

l=1E(v_l, y_i) QN1

i=1

QN3

k=1E(uk, yi) QN2

j=1

QN4

l=1E(vl, zj)

!1/2

× h Θh_~a

~b

i (~0)i₋1

Θh_~a

~b

i1 2

N1

X

i=1 yi

∫p

~ω− ¹₂

N2

X

j=1 zj

∫p

~ω−

N3

X

k=1 uk

∫p

~ω+

N4

X

l=1 vl

∫p

~ω

. (5.21)

Ramond charge conservation demands that ¹₂(N1−N2)−N3+N4 = 0. Hence, the arbitrary reference point p entering (5.21) through the canonical map drops out. RNS fields in D = 2m dimension are expressed through m copies of an SO(2) spin system that do not interact with each other. Hence, RNS correlation functions at loop-level factorize into m SO(2) correlators of type (5.20) or (5.21).

In the following we use zi as arguments for the SO(2) spin operators and define the shorthand notation Eij ≡ E(zi, zj). Additionally, we abbreviate the generalized Θ func-tions by

Θh_~a

~b

i1 2

hzi

z∫l

~ω+

zj

z∫m

~ω+. . .+^z∫^k

zn

~ωi

≡ Θ^~a_{~bi j ... k}

l m ... n

. (5.22)

Note in particular that the factor 1/2 in the argument of Θ_~b^~a, which is omnipresent for the spin fields, is always implicit.

Considerable simplifications occur for g = 0, i.e. scattering at tree-level. The spin structure dependent Θ functions trivialize, Θ^~a_~b →1, and the prime form reduces to E(z)− E(w) → z−w. In this way, by neglecting the generalized Θ functions and replacing the prime forms Eij by zij one obtains the tree-level correlator from the loop result. The prime forms in the coefficients of the index terms then mimic the tree-level behavior of the correlation function.

5.2.2 Results in Lorentz Covariant Form

With the background on spin systems in mind, we can now calculate RNS correlation func-tionshψ^µ1. . . ψ^µnSα1. . . SαrS^β˙1. . . S^β˙si_~b^~a for specific choices ofµi, αi and ˙βi by formulating

5.2 Loop Correlators 83

the RNS fields inD= 2mdimension in terms ofmspin systems via (5.14) and (5.15). The mspin systems do not interact with each other and hence the resulting correlator factorizes intomseparate correlation function of a single spin system. Then, using the loop correlator formulas (5.20) and (5.21) for the individual SO(2) correlation functions we find a result of the RNS correlation function for the specific choice of the Lorentz indices. However, the final goal is to express the result in covariant form, i.e. in terms of Clebsch–Gordan coefficients that are built from gamma and charge conjugation matrices. Due to the con-servation of Ramond charge in (5.21) then index terms can be viewed as SO(1,2m−1) covariant Ramond charge conserving delta functions, schematically Cαβ ∼ δ(α+β) and (γ^µC)αβ˙ ∼δ(µ+α+ ˙β) where µ, α, β, ˙β are treated as Ramond charge vectors with m components such as µ≡(0,±1,0, . . . ,0) and α≡(±1/2, . . . ,±1/2).

As a starting point we make an ansatz for the correlation function with a minimal set of Clebsch–Gordan coefficients. The cardinality of this set is determined by group theory as described in Chapter 3.3.1. Each of the index terms is accompanied by a z-dependent coefficient consisting of prime forms and generalized Θ functions. The results obtained for special choices of µ_i, α_i and ˙β_i have to be matched with this ansatz. It is most economic to first look at configurations (µi, αi,β˙i) where only one tensor is non-zero. Then the loop-level result (5.21) directly yields the coefficient for the respective index term. In some cases, however, it is not possible to make all Clebsch–Gordan coefficients vanish except for one.

More than one index term then contribute for every choice of µi, αi, ˙βi. In this case it can be helpful to switch to different Lorentz tensors that are (anti-)symmetric in some vector or spinor indices. Otherwise Fay’s trisecant identity [131] has to be used to determine the unknown coefficients. Sign issues can be resolved by calculating certain limits zi → zj

at tree-level using the RNS OPEs (3.10) and (3.11) and comparing the expression to the result of the arising lower-point correlator. Alternatively, the signs can be read off from the respective tree-level correlation function. As the prime formsEij reduce tozij the signs in the loop correlator must be the same as in the tree-level version.

Let us illustrate this procedure with an easy example, the calculation of the correlation function hψ^µψ^νψ^λSαSβi^~a_~b in D = 6 dimensions. Table 3.3 shows that four independent Clebsch–Gordan coefficients exist for this correlator. A convenient ansatz is

ψ^µ(z1)ψ^ν(z2)ψ^λ(z3)Sα(z4)Sβ(z5)~a

~b = F1(z) (γ^µνλC)αβ

+F2(z)η^µν(γ^λC)αβ +F3(z)η^µλ(γ^νC)αβ +F4(z)η^νλ(γ^µC)αβ. (5.23) The task is now to determine F₁, F₂, F₃, F₄ by making clever choices for µ, ν, λ, α, β. The coefficient F1 can easily be obtained by setting µ = 0, ν = 2, λ = 4. As the metric η is diagonal all index terms apart from γ^µνλ vanish for this configuration. Then, by means of (5.14), the NS fermions in terms of the SO(2) spin system fields become

ψ^µ=0(z1) = 1

√2 Ψ⁺₁(z1) + Ψ⁻₁(z1) , ψ^ν=2(z2) = 1

√2 Ψ⁺₂(z2) + Ψ⁻₂(z2) ,

84 5. Ramond–Neveu–Schwarz Correlators at Loop-Level

ψ^λ=4(z₃) = 1

√2 Ψ⁺₃(z₃) + Ψ⁻₃(z₃)

, (5.24)

and we choose for the spin fields

Sα=1(z4) =s⁺₁(z4)s⁺₂(z4)s⁺₃(z4), Sβ=1(z5) =s⁺₁(z5)s⁺₂(z5)s⁺₃(z5). (5.25) Hence, we have to calculate

1 2√

2

(Ψ⁺₁ + Ψ⁻₁)(z₁)s⁺₁(z₄)s⁺₁(z₅)~a

~b

(Ψ⁺₂ + Ψ⁻₂)(z₂)s⁻₂(z₄)s⁻₂(z₅)~a

~b

×

(Ψ⁺₃ + Ψ⁻₃)(z3)s⁻₃(z4)s⁻₃(z5)~a

~b. (5.26) Due to Ramond charge conservations in (5.21) Ψ⁺₁(z₁), Ψ⁻₂(z₂) and Ψ⁻₃(z₃) drop out and we obtain the coefficient F1 up to a sign:

F₁ =± Θ_~b^~a[^{1 1}_{4 5}] Θ_~b^~a[^{2 2}_{4 5}] Θ_~b^~a[^{3 3}_{4 5}] E₄₅^3/4 2√

2

Θ_~b^~a(~0)3

(E14E15E24E25E34E35)^1/2 . (5.27) The coefficient F2 can be determined in a similar way by setting µ = ν = 0, λ = 2 and α= 1, β = 4. No other tensors than η^µν(γ^λC)αβ contribute as the metric is diagonal and γ^µνλ totally antisymmetric. The NS fermions for this index choice are expressed through

ψ^µ=0(z1) = 1

√2 Ψ⁺₁(z1) + Ψ⁻₁(z1) , ψ^ν=0(z₂) = 1

√2 Ψ⁺₁(z₂) + Ψ⁻₁(z₂) , ψ^λ=2(z3) = 1

√2 Ψ⁺₂(z3) + Ψ⁻₂(z3)

, (5.28)

while the spin fields are given by

Sα=1(z4) =s⁺₁(z4)s⁺₂(z4)s⁺₃(z4), Sβ=4(z5) =s⁻₁(z5)s⁺₂(z5)s⁻₃(z5). (5.29) This time the correlator

1 2√

2

(Ψ⁺₁ + Ψ⁻₁)(z1) (Ψ⁺₁ + Ψ⁻₁)(z2)s⁺₁(z4)s⁻₁(z5)~a

~b

×

(Ψ⁺₂ + Ψ⁻₂)(z3)s⁺₂(z4)s⁺₂(z5)~a

~b

s⁺₃(z4)s⁻₃(z5)~a

~b (5.30)

has to be evaluated. Ψ⁻₂ in the second correlator drops out because of Ramond charge conservation, while in the first spin system the two in-equivalent fermion configurations Ψ⁺₁(z1) Ψ⁻₁(z2) and Ψ⁻₁(z1) Ψ⁺₁(z2) contribute. Consequently the total result forF2 consists of two terms:

F2 =±Θ_~b^~a[^{3 3}_{4 5}] Θ^~a_~b[⁴₅] E14E25Θ_~b^~a[^{1 1 4}_{2 2 5}] +E15E24Θ_~b^~a[^{1 1 5}_{2 2 4}] 2√

2

Θ_~b^~a(~0)3

E₁₂(E₁₄E₁₅E₂₄E₂₅E₃₄E₃₅)^1/2E₄₅^1/4 . (5.31)

5.2 Loop Correlators 85

The remaining coefficients F3 and F4 follow fromF2 by permutation of the vector indices and the labels (1,2,3) in the prime forms and Θ functions. The signs of the individual coefficients are fixed by requiring that in the limit z1 →z2 the expression

η^µνz₁₂⁻¹

ψ^λ(z3)Sα(z4)Sβ(z5)~a

~b (5.32)

must emerge. Accordingly, the limit z₄ →z₅ has to give rise to (γ^ρC)αβz₄₅^−1/4

ψ^µ(z1)ψ^ν(z2)ψ^λ(z3)ψρ(z5)~a

~b. (5.33)

Next we consider a second and more complicated example, the six-point function hψ^µ(z₁)ψ^ν(z₂)S_α(z₃)S_β(z₄)S_γ(z₅)S_δ(z₆)i^~a_~b. From Chapter 3.3 it is known that six inde-pendent index terms exist for this correlator in six dimensions. We choose to work with the expression (γ^µC)αβ(γ^νC)γδ and its five relatives coming from permutations of the spinor indices. For this correlation function it is not possible to choose the indices for the spin system correlator in such a way that all but one tensor vanish. By either using the concrete representation of gamma matrices in Appendix A.4 or understanding the index terms as Ramond charge conserving delta functions one finds that, e.g., for

ψ^µ=0(z1) = 1

√2 Ψ⁺₁(z1) + Ψ⁻₁(z1) , ψ^ν=2(z2) = 1

√2 Ψ⁺₂(z2) + Ψ⁻₂(z2)

, (5.34)

and

Sα=1(z3) =s⁺₁(z3)s⁺₂(z3)s⁺₃(z3), Sβ=3(z4) =s⁺₁(z4)s⁻₂(z4)s⁻₃(z4),

Sγ=3(z5) =s⁺₁(z5)s⁻₂(z5)s⁻₃(z5), Sδ=2(z6) =s⁻₁(z6)s⁻₂(z6)s⁺₃(z6) (5.35) the spin system correlator contributes to both z coefficients of (γ^µC)αβ(γ^νC)γδ and (γ^µC)αγ(γ^νC)βδ. The result of

1 2√

2

(Ψ⁺₁ + Ψ⁻₁)(z1)s⁺₁(z3)s⁺₁(z4)s⁺₁(z5)s⁻₁(z6)~a

~b

×

(Ψ⁺₂ + Ψ⁻₂)(z2)s⁺₂(z3)s⁻₂(z4)s⁻₂(z5)s⁻₂(z6)~a

~b

×

s⁺₃(z3)s⁻₃(z4)s⁻₃(z5)s⁺₃(z6)~a

~b (5.36)

must thus be split into two parts using Fay’s trisecant identity. Evaluating (5.36) we find

±Θ^~a_~b[^{1 1 6}_{3 4 5}] Θ_~b^~a[^{2 2 3}_{4 5 6}] Θ_~b^~a[^{3 6}_{4 5}] E12E16E23E45, (5.37) where we have taken out the pre-factor

1 2√

2

Θ^~a_~b(~0)3

(E13E14E15E16E23E24E25E26)^−1/2

E₁₂(E₃₄E₃₅E₃₆E₄₅E₄₆E₅₆)^1/4 . (5.38)

86 5. Ramond–Neveu–Schwarz Correlators at Loop-Level

Now, using the version (C.8) of Fay’s trisecant identity with

∆ =~ 1 2

Z z2

z1

~ω+1 2

Z z3

z6

~ω (5.39)

this becomes

∓Θ^~a_~b[^{1 1 5 6}_{2 2 3 4}] Θ^~a_~b[^{3 5}_{4 6}] Θ_~b^~a[^{3 6}_{4 5}] E15E16E23E24

±Θ^~a_~b[^{1 1 4 6}_{2 2 3 5}] Θ^~a_~b[^{3 4}_{5 6}] Θ_~b^~a[^{3 6}_{4 5}] E14E16E23E25. (5.40) At this point we cannot decide which of these terms belongs to the tested index terms but one has to do further calculations of spin system correlators. Making a different choice for the indices, e.g. µ = 0, ν = 2, α = δ = 4, β = 3, γ = 2, probes again (γ^µC)αβ(γ^νC)γδ

but this times in combination with (γ^µC)βδ(γ^νC)αγ. Evaluating this configuration and splitting the result as before yields

∓Θ^~a_~b[^{1 1 5 6}_{2 2 3 4}] Θ^~a_~b[^{3 5}_{4 6}] Θ_~b^~a[^{3 6}_{4 5}] E15E16E23E24

±Θ^~a_~b[^{1 1 3 5}_{2 2 4 6}] Θ^~a_~b[^{3 4}_{5 6}] Θ_~b^~a[^{3 6}_{4 5}] E13E15E24E26. (5.41) From the comparison of (5.40) and (5.41) we can now conclude that the first term must be the coefficient of (γ^µC)αβ(γ^νC)γδ. The second expression in (5.40) is consequently the coefficient of (γ^µC)αγ(γ^νC)βδ, while the Glebsch–Gordan coefficient (γ^µC)βδ(γ^νC)αγ has to come with the second term in (5.41).

Progressing in this way it is possible to evaluate all loop correlators for which one fails to separate the individual index terms in the spin system calculations by appropriate choices for the Lorentz indices. In addition, one obtains the relative signs between the different index terms. The over-all sign can again be determined by looking at certain limits or by comparing with the tree-level result.

Im Dokument Correlators of Ramond-Neveu-Schwarz Fields in String Theory (Seite 99-104)