Regge OPE blocks and light-ray operators
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Springer
Received: May 29, 2020 Accepted: September 16, 2020 Published: November 10, 2020
Regge OPE blocks and light-ray operators
a
Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-Ku, Tokyo 113-0033, Japan b Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
E-mail: [email protected], [email protected], [email protected] Abstract: We consider the structure of the operator product expansion (OPE) in conformal field theory by employing the OPE block formalism. The OPE block acted on the vacuum is promoted to an operator and its implications are examined on a non-vacuum state. We demonstrate that the OPE block is dominated by a light-ray operator in the Regge limit, which reproduces precisely the Regge behavior of conformal blocks when used inside scalar four-point functions. Motivated by this observation, we propose a new form of the OPE block, called the light-ray channel OPE block that has a well-behaved expansion dominated by a light-ray operator in the Regge limit. We also show that the two OPE blocks have the same asymptotic form in the Regge limit and confirm the assertion that the Regge limit of a pair of spacelike-separated operators in a Minkowski patch is equivalent to the OPE limit of a pair of timelike-separated operators associated with the original pair in a different Minkowski patch. Keywords: AdS-CFT Correspondence, Conformal and W Symmetry, Conformal Field Theory ArXiv ePrint: 2005.06343
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)040
JHEP11(2020)040
Nozomu Kobayashi,a,b Tatsuma Nishiokaa and Yoshitaka Okuyamaa
Contents 1 Introduction
1
2 Regge limit in CFT
3
Regge configuration
4
2.2
Adapted coordinates and light-ray translation
5
2.3
Adapted coordinates in embedding space
7
2.4
Lorentzian conformal block in Regge regime
10
2.5
Regge limit and light transform
12
3 Regge conformal block via Lorentzian OPE
14
3.1
OPE block formalism
15
3.2
OPE block in the Regge limit
17
3.3
A holographic view of Regge OPE block and light-ray operator
19
3.4
Regge behavior from OPE block
20
4 Timelike OPE block
23
4.1
Two representations of timelike OPE block
24
4.2
Regge limit of timelike OPE block
26
4.3
More on the equivalence between two timelike OPE blocks
27
5 Light-ray channel OPE block
31
6 Discussion
34
A Notations and normalization
36
A.1 Notations
36
A.2 Normalization
36
B Wightman functions with different operator orderings
41
C Derivations of Regge conformal block
41
C.1 Sub-leading terms in z/¯ z
43
C.2 Regge conformal block
44
D Timelike OPE blocks with continuous spin
–i–
44
JHEP11(2020)040
2.1
1
Introduction
An operator product expansion (OPE) is one of the most fundamental postulates in local quantum field theories [1, 2]. For a given pair of operators Oi (x1 ) and Oj (x2 ) th
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