Abstract
We study a just-renormalizable tensorial group field theory of rank six with quartic melonic interactions and Abelian group U(1). We introduce the formalism of the intermediate field, which allows a precise characterization of the leading order Feynman graphs. We define the renormalization of the model, compute its (perturbative) renormalization group flow and write its expansion in terms of effective couplings. We then establish closed equations for the two point and four point functions at leading (melonic) order. Using the effective expansion and its uniform exponential bounds we prove that these equations admit a unique solution at small renormalized coupling.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Gurau and J.P. Ryan, Colored Tensor Models — A review, SIGMA 8 (2012) 020 [arXiv:1109.4812] [INSPIRE].
V. Rivasseau, Quantum Gravity and Renormalization: The Tensor Track, AIP Conf. Proc. 1444 (2011) 18 [arXiv:1112.5104] [INSPIRE].
V. Rivasseau, The Tensor Track: an Update, arXiv:1209.5284 [INSPIRE].
V. Rivasseau, The Tensor Track, III, Fortsch. Phys. 62 (2014) 81 [arXiv:1311.1461] [INSPIRE].
T. Thiemann, Modern canonical quantum General Relativity, Cambridge University Press, Cambridge U.K. (2007).
A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A status report, Class. Quant. Grav. 21 (2004) R53 [gr-qc/0404018] [INSPIRE].
C. Rovelli, Quantum Gravity, Cambridge University Press, (2006).
D.V. Boulatov, A model of three-dimensional lattice gravity, Mod. Phys. Lett. A 7 (1992) 1629 [hep-th/9202074] [INSPIRE].
H. Ooguri, Topological lattice models in four-dimensions, Mod. Phys. Lett. A 7 (1992) 2799 [hep-th/9205090] [INSPIRE].
L. Freidel, Group field theory: An overview, Int. J. Theor. Phys. 44 (2005) 1769 [hep-th/0505016] [INSPIRE].
D. Oriti, The microscopic dynamics of quantum space as a group field theory, arXiv:1110.5606 [INSPIRE].
D. Oriti, The group field theory approach to quantum gravity, gr-qc/0607032 [INSPIRE].
D. Oriti, Quantum gravity as a quantum field theory of simplicial geometry, gr-qc/0512103 [INSPIRE].
D. Oriti, The Group field theory approach to quantum gravity: Some recent results, in The Planck Scale: Proceedings of the XXV Max Born Symposium, J. Kowalski-Glikman et al. eds., AIP: conference proceedings (2009), [arXiv:0912.2441] [INSPIRE].
A. Baratin and D. Oriti, Ten questions on Group Field Theory (and their tentative answers), J. Phys. Conf. Ser. 360 (2012) 012002 [arXiv:1112.3270] [INSPIRE].
T. Krajewski, Group field theories, PoS(QGQGS 2011)005 [arXiv:1210.6257] [INSPIRE].
J. Ambjørn, B. Durhuus and T. Jonsson, Three-dimensional simplicial quantum gravity and generalized matrix models, Mod. Phys. Lett. A 6 (1991) 1133 [INSPIRE].
M. Gross, Tensor models and simplicial quantum gravity in > 2-D, Nucl. Phys. Proc. Suppl. 25A (1992) 144 [INSPIRE].
N. Sasakura, Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A 6 (1991) 2613 [INSPIRE].
V.A. Kazakov, Bilocal Regularization of Models of Random Surfaces, Phys. Lett. B 150 (1985) 282 [INSPIRE].
F. David, A Model of Random Surfaces with Nontrivial Critical Behavior, Nucl. Phys. B 257 (1985) 543 [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2-D Gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
C. Rovelli, The basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum gravity model in the loop representation basis, Phys. Rev. D 48 (1993) 2702 [hep-th/9304164] [INSPIRE].
R. De Pietri, L. Freidel, K. Krasnov and C. Rovelli, Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, Nucl. Phys. B 574 (2000) 785 [hep-th/9907154] [INSPIRE].
M.P. Reisenberger and C. Rovelli, Space-time as a Feynman diagram: The connection formulation, Class. Quant. Grav. 18 (2001) 121 [gr-qc/0002095] [INSPIRE].
A. Perez, The Spin Foam Approach to Quantum Gravity, Living Rev. Rel. 16 (2013) 3 [arXiv:1205.2019] [INSPIRE].
C. Rovelli, Zakopane lectures on loop gravity, PoS(QGQGS 2011)003 [arXiv:1102.3660] [INSPIRE].
A. Baratin and D. Oriti, Group field theory with non-commutative metric variables, Phys. Rev. Lett. 105 (2010) 221302 [arXiv:1002.4723] [INSPIRE].
A. Baratin and D. Oriti, Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity, Phys. Rev. D 85 (2012) 044003 [arXiv:1111.5842] [INSPIRE].
D. Oriti, Group field theory as the 2nd quantization of Loop Quantum Gravity, arXiv:1310.7786 [INSPIRE].
D. Oriti, Group Field Theory and Loop Quantum Gravity, arXiv:1408.7112 [INSPIRE].
D. Oriti, J.P. Ryan and J. Thürigen, Group field theories for all loop quantum gravity, New J. Phys. 17 (2015) 023042 [arXiv:1409.3150] [INSPIRE].
R. Gurau, Colored Group Field Theory, Commun. Math. Phys. 304 (2011) 69 [arXiv:0907.2582] [INSPIRE].
R. Gurau, Lost in Translation: Topological Singularities in Group Field Theory, Class. Quant. Grav. 27 (2010) 235023 [arXiv:1006.0714] [INSPIRE].
R. Gurau, The 1/N expansion of colored tensor models, Annales Henri Poincaré 12 (2011) 829 [arXiv:1011.2726] [INSPIRE].
R. Gurau, The complete 1/N expansion of colored tensor models in arbitrary dimension, Annales Henri Poincaré 13 (2012) 399 [arXiv:1102.5759] [INSPIRE].
R. Gurau and V. Rivasseau, The 1/N expansion of colored tensor models in arbitrary dimension, Europhys. Lett. 95 (2011) 50004 [arXiv:1101.4182] [INSPIRE].
V. Bonzom, R. Gurau, A. Riello and V. Rivasseau, Critical behavior of colored tensor models in the large-N limit, Nucl. Phys. B 853 (2011) 174 [arXiv:1105.3122] [INSPIRE].
R. Gurau, Universality for Random Tensors, arXiv:1111.0519 [INSPIRE].
V. Bonzom, R. Gurau and V. Rivasseau, Random tensor models in the large-N limit: Uncoloring the colored tensor models, Phys. Rev. D 85 (2012) 084037 [arXiv:1202.3637] [INSPIRE].
A. Tanasa, Multi-orientable Group Field Theory, J. Phys. A 45 (2012) 165401 [arXiv:1109.0694] [INSPIRE].
S. Dartois, V. Rivasseau and A. Tanasa, The 1/N expansion of multi-orientable random tensor models, Annales Henri Poincaré 15 (2014) 965 [arXiv:1301.1535] [INSPIRE].
M. Raasakka and A. Tanasa, Next-to-leading order in the large N expansion of the multi-orientable random tensor model, Annales Henri Poincaré 16 (2015) 1267 [arXiv:1310.3132] [INSPIRE].
E. Fusy and A. Tanasa, Asymptotic expansion of the multi-orientable random tensor model, arXiv:1408.5725 [INSPIRE].
R. Gurau, The Double Scaling Limit in Arbitrary Dimensions: A Toy Model, Phys. Rev. D 84 (2011) 124051 [arXiv:1110.2460] [INSPIRE].
W. Kaminski, D. Oriti and J.P. Ryan, Towards a double-scaling limit for tensor models: probing sub-dominant orders, New J. Phys. 16 (2014) 063048 [arXiv:1304.6934] [INSPIRE].
S. Dartois, R. Gurau and V. Rivasseau, Double Scaling in Tensor Models with a Quartic Interaction, JHEP 09 (2013) 088 [arXiv:1307.5281] [INSPIRE].
V. Bonzom, R. Gurau, J.P. Ryan and A. Tanasa, The double scaling limit of random tensor models, JHEP 09 (2014) 051 [arXiv:1404.7517] [INSPIRE].
V. Rivasseau, The Tensor Theory Space, Fortsch. Phys. 62 (2014) 835 [arXiv:1407.0284] [INSPIRE].
D. Oriti, Disappearance and emergence of space and time in quantum gravity, Stud. Hist. Philos. Mod. Phys. 46 (2014) 186 [arXiv:1302.2849] [INSPIRE].
B.L. Hu, Can spacetime be a condensate?, Int. J. Theor. Phys. 44 (2005) 1785 [gr-qc/0503067] [INSPIRE].
T.A. Koslowski, Dynamical Quantum Geometry (DQG Programme), arXiv:0709.3465 [INSPIRE].
T. Koslowski and H. Sahlmann, Loop quantum gravity vacuum with nondegenerate geometry, SIGMA 8 (2012) 026 [arXiv:1109.4688] [INSPIRE].
B. Dittrich and M. Geiller, A new vacuum for Loop Quantum Gravity, arXiv:1401.6441 [INSPIRE].
B. Dittrich, F.C. Eckert and M. Martin-Benito, Coarse graining methods for spin net and spin foam models, New J. Phys. 14 (2012) 035008 [arXiv:1109.4927] [INSPIRE].
B. Bahr, B. Dittrich, F. Hellmann and W. Kaminski, Holonomy Spin Foam Models: Definition and Coarse Graining, Phys. Rev. D 87 (2013) 044048 [arXiv:1208.3388] [INSPIRE].
B. Dittrich, M. Martín-Benito and E. Schnetter, Coarse graining of spin net models: dynamics of intertwiners, New J. Phys. 15 (2013) 103004 [arXiv:1306.2987] [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, Nonperturbative Quantum Gravity, Phys. Rept. 519 (2012) 127 [arXiv:1203.3591] [INSPIRE].
D. Benedetti and J. Henson, Spacetime condensation in (2+1)-dimensional CDT from a Hořava-Lifshitz minisuperspace model, arXiv:1410.0845 [INSPIRE].
J. Mielczarek, Big Bang as a critical point, arXiv:1404.0228 [INSPIRE].
J. Magueijo, L. Smolin and C.R. Contaldi, Holography and the scale-invariance of density fluctuations, Class. Quant. Grav. 24 (2007) 3691 [astro-ph/0611695] [INSPIRE].
S. Gielen, D. Oriti and L. Sindoni, Cosmology from Group Field Theory Formalism for Quantum Gravity, Phys. Rev. Lett. 111 (2013) 031301 [arXiv:1303.3576] [INSPIRE].
S. Gielen, D. Oriti and L. Sindoni, Homogeneous cosmologies as group field theory condensates, JHEP 06 (2014) 013 [arXiv:1311.1238] [INSPIRE].
L. Sindoni, Effective equations for GFT condensates from fidelity, arXiv:1408.3095 [INSPIRE].
S. Gielen and D. Oriti, Quantum cosmology from quantum gravity condensates: cosmological variables and lattice-refined dynamics, New J. Phys. 16 (2014) 123004 [arXiv:1407.8167] [INSPIRE].
S. Gielen, Perturbing a quantum gravity condensate, Phys. Rev. D 91 (2015) 043526 [arXiv:1411.1077] [INSPIRE].
J. Ben Geloun and V. Bonzom, Radiative corrections in the Boulatov-Ooguri tensor model: The 2-point function, Int. J. Theor. Phys. 50 (2011) 2819 [arXiv:1101.4294] [INSPIRE].
J. Ben Geloun and V. Rivasseau, A Renormalizable 4-Dimensional Tensor Field Theory, Commun. Math. Phys. 318 (2013) 69 [arXiv:1111.4997] [INSPIRE].
J. Ben Geloun and V. Rivasseau, Addendum to ‘A Renormalizable 4-Dimensional Tensor Field Theory’, Commun. Math. Phys. 322 (2013) 957 [arXiv:1209.4606] [INSPIRE].
J. Ben Geloun and E.R. Livine, Some classes of renormalizable tensor models, J. Math. Phys. 54 (2013) 082303 [arXiv:1207.0416] [INSPIRE].
J. Ben Geloun, Renormalizable Models in Rank d ≥ 2 Tensorial Group Field Theory, Commun. Math. Phys. 332 (2014) 117 [arXiv:1306.1201] [INSPIRE].
T. Krajewski, Schwinger-Dyson Equations in Group Field Theories of Quantum Gravity, arXiv:1211.1244 [INSPIRE].
M. Raasakka and A. Tanasa, Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory, arXiv:1306.1022 [INSPIRE].
T. Krajewski and R. Toriumi, Polchinski’s equation for group field theory, Fortsch. Phys. 62 (2014) 855 [INSPIRE].
S. Carrozza, D. Oriti and V. Rivasseau, Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions, Commun. Math. Phys. 327 (2014) 603 [arXiv:1207.6734] [INSPIRE].
D.O. Samary and F. Vignes-Tourneret, Just Renormalizable TGFT’s on U(1)d with Gauge Invariance, Commun. Math. Phys. 329 (2014) 545 [arXiv:1211.2618] [INSPIRE].
S. Carrozza, D. Oriti and V. Rivasseau, Renormalization of a SU(2) Tensorial Group Field Theory in Three Dimensions, Commun. Math. Phys. 330 (2014) 581 [arXiv:1303.6772] [INSPIRE].
S. Carrozza, Tensorial methods and renormalization in Group Field Theories, arXiv:1310.3736 [INSPIRE].
J. Ben Geloun, Two and four-loop β-functions of rank 4 renormalizable tensor field theories, Class. Quant. Grav. 29 (2012) 235011 [arXiv:1205.5513] [INSPIRE].
J. Ben Geloun, Asymptotic Freedom of Rank 4 Tensor Group Field Theory, arXiv:1210.5490 [INSPIRE].
J. Ben Geloun and D.O. Samary, 3D Tensor Field Theory: Renormalization and One-loop β-functions, Annales Henri Poincaré 14 (2013) 1599 [arXiv:1201.0176] [INSPIRE].
D. Ousmane Samary, β-functions of U(1)d gauge invariant just renormalizable tensor models, Phys. Rev. D 88 (2013) 105003 [arXiv:1303.7256] [INSPIRE].
S. Carrozza, Discrete Renormalization Group for SU(2) Tensorial Group Field Theory, arXiv:1407.4615 [INSPIRE].
A. Baratin, S. Carrozza, D. Oriti, J. Ryan and M. Smerlak, Melonic phase transition in group field theory, Lett. Math. Phys. 104 (2014) 1003 [arXiv:1307.5026] [INSPIRE].
R. Gurau and J.P. Ryan, Melons are branched polymers, Annales Henri Poincaré 15 (2014) 2085 [arXiv:1302.4386] [INSPIRE].
D. Benedetti, J. Ben Geloun and D. Oriti, Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model, JHEP 03 (2015) 084 [arXiv:1411.3180] [INSPIRE].
V. Rivasseau, From perturbative to constructive renormalization, Princeton series in physics, Princeton University Press, Princeton U.S.A. (1991).
T. Delepouve and V. Rivasseau, Constructive Tensor Field Theory: The T 43 Model, arXiv:1412.5091 [INSPIRE].
R. Gurau, The 1/N Expansion of Tensor Models Beyond Perturbation Theory, Commun. Math. Phys. 330 (2014) 973 [arXiv:1304.2666] [INSPIRE].
T. Delepouve, R. Gurau and V. Rivasseau, Universality and Borel Summability of Arbitrary Quartic Tensor Models, arXiv:1403.0170 [INSPIRE].
V.A. Nguyen, S. Dartois and B. Eynard, An analysis of the intermediate field theory of T 4 tensor model, JHEP 01 (2015) 013 [arXiv:1409.5751] [INSPIRE].
H. Grosse and R. Wulkenhaar, Renormalization of ϕ 4 theory on noncommutative R 4 in the matrix base, Commun. Math. Phys. 256 (2005) 305 [hep-th/0401128] [INSPIRE].
H. Grosse and R. Wulkenhaar, Progress in solving a noncommutative quantum field theory in four dimensions, arXiv:0909.1389 [INSPIRE].
H. Grosse and R. Wulkenhaar, Self-Dual Noncommutative ϕ 4 -Theory in Four Dimensions is a Non-Perturbatively Solvable and Non-Trivial Quantum Field Theory, Commun. Math. Phys. 329 (2014) 1069 [arXiv:1205.0465] [INSPIRE].
H. Grosse and R. Wulkenhaar, Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity, arXiv:1406.7755 [INSPIRE].
D.O. Samary, Closed equations of the two-point functions for tensorial group field theory, Class. Quant. Grav. 31 (2014) 185005 [arXiv:1401.2096] [INSPIRE].
D.O. Samary, C.I. Pérez-Sánchez, F. Vignes-Tourneret and R. Wulkenhaar, Correlation functions of just renormalizable tensorial group field theory: The melonic approximation, arXiv:1411.7213 [INSPIRE].
G. Gallavotti and F. Nicolò, renormalization theory in four-dimensional scalar fields. I, Commun. Math. Phys. 100 (1985) 545 [INSPIRE].
V. Rivasseau, Constructive Matrix Theory, JHEP 09 (2007) 008 [arXiv:0706.1224] [INSPIRE].
G. ’t Hooft, Rigorous Construction of Planar Diagram Field Theories in Four-dimensional Euclidean Space, Commun. Math. Phys. 88 (1983) 1 [INSPIRE].
V. Rivasseau, Construction and Borel Summability of Planar Four-dimensional Euclidean Field Theory, Commun. Math. Phys. 95 (1984) 445 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1501.02086v2
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Lahoche, V., Oriti, D. & Rivasseau, V. Renormalization of an Abelian tensor group field theory: solution at leading order. J. High Energ. Phys. 2015, 95 (2015). https://doi.org/10.1007/JHEP04(2015)095
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2015)095