Mellin amplitudes for fermionic conformal correlators
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Received: January 12, 2018 Accepted: February 25, 2018 Published: March 19, 2018
Mellin amplitudes for fermionic conformal correlators
a
Institut f¨ ur Physik, Humboldt-Universit¨ at zu Berlin, IRIS-Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany b Institut f¨ ur Mathematik, Humboldt-Universit¨ at zu Berlin, IRIS-Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany c Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, Am M¨ uhlenberg 1, 14476 Potsdam, Germany d Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad-211019, India e International Centre for Theoretical Sciences, TIFR, Hesaraghatta, Hubli, Bengaluru-560089, India
E-mail: [email protected], [email protected], [email protected] Abstract: We define Mellin amplitudes for the fermion-scalar four point function and the fermion four point function. The Mellin amplitude thus defined has multiple components each associated with a tensor structure. In the case of three spacetime dimensions, we explicitly show that each component factorizes on dynamical poles onto components of the Mellin amplitudes for the corresponding three point functions. The novelty here is that for a given exchanged primary, each component of the Mellin amplitude may in general have more than one series of poles. We present a few examples of Mellin amplitudes for tree-level Witten diagrams and tree-level conformal Feynman integrals with fermionic legs, which illustrate the general properties. Keywords: Conformal Field Theory, AdS-CFT Correspondence ArXiv ePrint: 1711.07929
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2018)106
JHEP03(2018)106
Josua Faller,a Sourav Sarkara,b,c and Mritunjay Vermad,e
Contents 1 Introduction
2
2 Mellin amplitudes for fermionic correlators
3
Review: Mellin representation of scalar correlators
3
2.2
Tensor structures
5
2.3
Definition
10
2.4
Pole structure: fermion-scalar four point function
12
2.4.1
Direct channel
12
2.4.2
Crossed channel
15
2.5
Pole structure: four fermion correlator
3 Witten diagrams
15 18
3.1
Contact Witten diagram
18
3.2
Scalar exchange Witten diagram with two external fermions
19
3.3
Scalar exchange Witten diagram with four external fermions
20
3.4
Spinor exchange Witten diagrams
21
4 Conformal Feynman integrals
22
4.1
Fermion-scalar four point function: contact diagram
23
4.2
Fermion four point function: contact interaction
24
4.3
Fermion-scalar four point function: scalar exchange
25
4.4
Fermion-scalar four point function: fermion exchange
26
4.5
Fermion four point function: scalar propagator
26
5 Discussion
27
A Mellin amplitudes for fermionic correlators
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A.1 Counting tensor structures
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A.2 Tensor structures: fermion four point function
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A.3 Reduced Mellin amplitude and Mellin amplitude for the four fermion correlator 33 A.4 Mixed fermion scalar conformal blocks
34
A.5 u-channel poles in the fermion-scalar four point Mellin a
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