OPE of Green functions of chiral currents
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Springer
Received: July 1, Revised: September 10, Accepted: September 17, Published: October 22,
2020 2020 2020 2020
Tom´ aˇs Kadav´ y, Karol Kampf and Jiˇr´ı Novotn´ y Institute of Particle and Nuclear Physics, Charles University, V Holeˇsoviˇck´ ach 2, 180 00 Prague 8, Czech Republic
E-mail: [email protected], [email protected], [email protected] Abstract: In this paper, we investigate the high-energy behavior of two-point and threepoint Green functions of the QCD chiral currents and densities using the framework of the operator product expansion in the chiral limit. In detail, we study the contributions of the quark, gluon, quark-gluon and four-quark condensates to all the relevant non-vanishing three-point correlators. Keywords: Chiral Lagrangians, Effective Field Theories, Nonperturbative Effects, Perturbative QCD ArXiv ePrint: 2006.13006
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)142
JHEP10(2020)142
OPE of Green functions of chiral currents
Contents 1 Introduction 1.1 Notation
1 3 5 6 6 7 9 9 10 13
3 Operator product expansion and QCD condensates 3.1 Operator product expansion 3.2 QCD condensates 3.3 Fock-Schwinger gauge 3.4 Propagation of nonlocal condensates 3.5 Comparison with literature 3.6 Translation invariance
15 15 15 17 18 19 21
4 Perturbative contribution 4.1 General remarks 4.2 Results
22 22 23
5 Quark condensate 5.1 General remarks 5.2 Results
29 29 30
6 Gluon condensate 6.1 General remarks 6.2 Results
32 32 35
7 Quark-gluon condensate 7.1 General remarks 7.2 Results
37 37 40
8 Four-quark condensate 8.1 General remarks 8.2 Results
41 41 45
9 Summary
46
–i–
JHEP10(2020)142
2 Green functions of chiral currents 2.1 Classification 2.2 Chiral Ward identities 2.2.1 Green functions of Set 1 2.2.2 Green functions of Set 2 2.3 Tensor decomposition 2.3.1 Green functions of Set 1 2.3.2 Green functions of Set 2
47
B Note on Fock-Schwinger gauge
48
C Derivation of propagation formulas C.1 Preliminaries C.1.1 Equations of motion C.1.2 Minimal Lorentz structure of nonlocal QCD condensates C.1.3 Expansion of quark and gluon fields C.2 Derivation of propagation formula (3.9a) C.2.1 Propagation of quark condensate C.2.2 Propagation of quark-gluon condensate C.2.3 Propagation of four-quark condensate C.3 Derivation of propagation formula (3.9b) C.4 Derivation of propagation formula (3.9c) C.4.1 Propagation of quark-gluon condensate C.4.2 Propagation of four-quark condensate
49 50 50 51 51 52 53 53 54 56 57 57 57
D OPE for two-point Green functions D.1 Perturbative contribution D.2 Quark condensate D.3 Gluon condensate D.4 Quark-gluon condensate D.5 Four-quark condensate
59 60 60 61 63 66
E On decompositions of the hV V Ai and hAAAi Green functions E.1 hV V Ai Green function E.2 hAAAi Green function
69 70 73
F On decompositions of the hAAV i and hV V V i Green functions F.1 hAAV i Green function F.2 hV V V i Green function
75 76 80
1
Introduction
There is countless experimental evidence showing that Quantum ch
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