Two-point and three-point Green's functions of 4-quark operators that support the traceless representations of the flavor SU(Nf) group

The file contains perturbative results (up to next-to-leading order) for the two-point and three-point Green's functions of 4-quark operators that support the traceless representations of the flavor SU(Nf) group. This includes all DF=2 operators, and a subset of DF=1 and DF=0 operators. We also provide the next-to-leading order conversion matrices between a selected version of GIRS and MSbar scheme. For more details, see our paper: https://arxiv.org/abs/2511.04305 In particular, the file contains: 1. The MSbar-renormalized two-point Green’s functions (involving products of two four-quark operators) and three-point Green’s functions (involving products of one 4-quark and two bilinear operators) for the parity-conserving (Q_i^{S=+-1}) and parity-violating (calQ_i^{S=+-1}) 4-quark operators (i=1,...,5) [For notation see our paper: https://arxiv.org/abs/2511.04305] (a) GF2ptMSbar[Qi, Qj, plus/minus] = <Q_i^{S=+-1}(x) [Q_j^{S=+-1}(y)]^dagger>, GF2ptMSbar[calQi, calQj, plus/minus] = <calQ_i^{S=+-1}(x) [calQ_j^{S=+-1}(y)]^dagger>, where i,j = 1,2,3,4,5. E.g., GF2ptMSbar[Q1,Q4,plus] GF2ptMSbar[calQ2,calQ3,minus] (b) GF3ptMSbar[OGamma, Qi, OGamma, plus/minus] = <O_Gamma(x) Q_i^{S=+-1}(y) O_Gamma(w)>, where i = 1,2,3,4,5 and possible values of OGamma are: S, P, V, A, T (scalar, pseudoscalar, vector, axial vector, tensor) GF3ptMSbar[OGamma, calQi, OGamma', plus/minus] = <O_Gamma(x) calQ_i^{S=+-1}(y) O_Gamma'(w)>, where i = 1,2,3,4,5. OGamma, OGamma' differ by gamma_5; thus, possible values of (OGamma,OGamma') are: (S,P), (V,A), (T,T') [(scalar, pseudoscalar), (vector, axial vector), (tensor, tensor 'prime')]. E.g., GF3ptMSbar[V,Q1,V,minus] GF3ptMSbar[S,calQ2,P,minus] The expressions depend on the 4-vector scales z=(x-y), z'=(y-w). Depending on the bilinear operators used, the three-point functions depend on the Lorentz indices nu[1] for V, A, and nu[1],nu[2] for T. Also, the expressions depend on flavor structures (see Eqs. (32-33) of arXiv:2406.08065 for the convention of all flavor indices). 2. The integrated (over timeslices) MSbar-renormalized two-point and three-point Green’s functions: (a) GF2ptMSbarInt[Qi, Qj, plus/minus] = int d^3 vec{z} GF2ptMSbar[Qi, Qj, plus/minus](vec{z},t), GF2ptMSbarInt[calQi, calQj, plus/minus] = int d^3 vec{z} GF2ptMSbar[calQi, calQj, plus/minus](vec{z},t). E.g., GF2ptMSbarInt[Q1,Q4,plus] GF2ptMSbarInt[calQ2,calQ3,minus] (b) GF3ptMSbarInt[OGamma, Qi, OGamma, plus/minus] = int d^3 vec{z} d^3 vec{z'} GF3ptMSbar[OGamma, Qi, OGamma, plus/minus]((vec{z},t),(vec{z'},t')), GF3ptMSbarInt[OGamma, calQi, OGamma', plus/minus] = int d^3 vec{z} d^3 vec{z'} GF3ptMSbar[OGamma,calQi,OGamma',plus/minus]((vec{z},t),(vec{z'},t')). E.g., GF3ptMSbarInt[V,Q1,V,minus] GF3ptMSbarInt[S,calQ2,P,minus] Results for the Green's functions involving V, A, T are given for spatial components. The expressions depend on the time scale t (In the three-point functions, we set t'=t). 4. 5x5 conversion matrices between GIRS and MSbar for the 4 traceless representations of the flavor SU(Nf) group: (Pcons, plus), (Pcons, minus), (Pviol, plus), (Pviol, minus): CGIRSMSbar[Pcons, minus], etc. Notation: 'Log[x]' is the natural logarithm of x (logarithm to base e), 'EulerGamma' is the Euler's gamma: gamma_E = 0.5772156649..., 'gsqoverpisq16' stands for gMSbar^2/(16 Pi^2), where gMSbar is the MSbar-renormalized coupling constant, 'Nc' is the number of colors, 'Nf' is the number of quark flavors, 'Cf' is the Casimir operator in the fundamental representation: Cf = (Nc^2 - 1)/(2 Nc), 'f[i], fp[i], fpp[i]' are flavor indices, 'delf[..., ...]' is Kronecker's delta for flavor indices, 'zsq' is the square of the distance between x and y: z^2 = (x-y)^2, 'zpsq' is the square of the distance between y and w: zp^2 = (y-w)^2, 'zpluszpsq' is the square of the distance between x and w: (z+zp)^2 = (x-w)^2, 'mubarsqtsq' is the product of 'mubarsq' and 'tsq', 'mubarsq' is the square of the MSbar renormalization scale mubar, 'tsq' is the time separation between the temporal components of (x and y) and (y and w): tsq = (x_4 - y_4)^2 = (y_4 - w_4)^2, 'z[nu[i]], zp[nu[i]], zpluszp[nu[i]]' are the nu[i] components of the 4-vectors (x-y), (y-w), (x-w), respectively.