A covariant momentum representation for loop corrections in gravity
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Springer
Received: January 20, Revised: March 6, Accepted: May 3, Published: May 26,
2020 2020 2020 2020
Rodrigo Alonso Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, 277-8583 Chiba, Japan Institute for Particle Physics Phenomenology, Department of Physics, Durham University, South Road, Durham DH1 3LE, U.K.
E-mail: [email protected] Abstract: A transformation is introduced in momentum representation to keep a covariant description at every stage of a loop computation in gravity. The procedure treats on equal footing local internal and space-time symmetries althought the complete transformation is known for the former [1] whereas in gravity we solve for the first few orders in an expansion. As an explicit application the one loop UV divergences of Hilbert-Einstein gravity with a cosmological constant and spin 0, 1/2 and 1 matter are computed with functional methods and in a field-covariant formalism. Keywords: Models of Quantum Gravity, Effective Field Theories, Renormalization Group ArXiv ePrint: 1912.09671
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)131
JHEP05(2020)131
A covariant momentum representation for loop corrections in gravity
Contents 1
2 Second order covariant variation of the action 2.1 Hilbert-Einstein and cosmological constant 2.2 Scalars 2.3 Fermions 2.4 Vector boson
3 4 5 6 7
3 Covariant derivative transformation and applications 3.1 Covariant derivative transformation 3.2 Application to the second order variation of the action
9 9 12
4 Evaluation of the operator trace 4.1 Ultraviolet divergences 4.1.1 Single species loops 4.1.2 Mixed contributions in the loop
16 18 19 23
5 Comparison with Schwinger-DeWitt coefficient computation 5.1 Heat kernel in brief 5.2 Covariant momentum representation in brief 5.3 Core computations in heat kernel 5.4 Core computations in covariant momentum representation
26 26 27 28 29
6 Conclusions
30
1
Introduction
The complete quantum theory of gravity stands as one of the most relevant and loftiest goals of theoretical high energy physics. While prospects for the experimental test of our theories of gravity are challenging due to the smallness of the Planck lenght LP , this also means that the low energy theory of gravity can be treated perturbatively in L P to a very, very good approximation. This expansion on a small distance or large mass scale LP = (MP )−1 is the basis of Effective Field Theory (EFT), a scheme in which gravity fits seamlessly [2– 4]. As such quantum corrections in the low energy theory of gravity are well defined and calculable. Computational methods exist since half a century to obtain these corrections; the most developed being the coordinate-representation based heat kernel [5–14]. Within this technique an expansion characterized by Schwinger-DeWitt coefficients is appropriate for the computation of short distance contributions and in particular UV divergences. Momentum representation techniques have also
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