Barrio Luna, J.A. Brochero Cifuentes, C.A. Carrillo Montoya, M. Cepeda, M.
Cerrada, N. Colino, B. De La Cruz, A. Delgado Peris, C. Fernandez Bedoya, J.P.
Fernández Ramos, J. Flix, M.C. Fouz, O. Gonzalez Lopez, S. Goy Lopez, J.M.
Hernandez, M.I. Josa, D. Moran, Á. Navarro Tobar, A. Pérez-Calero Yzquierdo,
J. Puerta Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, M.S. Soares, A.
Triossi, C. Willmott Universidad Autónoma de Madrid, Madrid, Spain
C. Albajar, J.F. de Trocóniz Universidad de Oviedo, Instituto Universitario de
Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Oviedo, Spain
B. Alvarez Gonzalez, J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras,
I. Gonzalez Caballero, J.R. González Fernández, E. Palencia Cortezon, V.
Rodríguez Bouza, S. Sanchez Cruz Instituto de Física de Cantabria (IFCA),
CSIC-Universidad de Cantabria, Santander, Spain
I.J. Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M.
Fernandez, P.J. Fernández Manteca, A. García Alonso, G. Gomez, C. Martinez
Rivero, P. Martinez Ruiz del Arbol, F. Matorras, J. Piedra Gomez, C. Prieels,
T. Rodrigo, A. Ruiz-Jimeno, L. Russo45, L. Scodellaro, N. Trevisani, I. Vila,
J.M. Vizan Garcia University of Colombo, Colombo, Sri Lanka
K. Malagalage University of Ruhuna, Department of Physics, Matara, Sri Lanka
W.G.D. Dharmaratna, N. Wickramage CERN, European Organization for Nuclear
Research, Geneva, Switzerland
D. Abbaneo, B. Akgun, E. Auffray, G. Auzinger, J. Baechler, P. Baillon, A.H.
Ball, D. Barney, J. Bendavid, M. Bianco, A. Bocci, E. Bossini, C. Botta, E.
Brondolin, T. Camporesi, A. Caratelli, G. Cerminara, E. Chapon, G. Cucciati,
D. d’Enterria, A. Dabrowski, N. Daci, V. Daponte, A. David, O. Davignon, A. De
Roeck, N. Deelen, M. Deile, M. Dobson, M. Dünser, N. Dupont, A. Elliott-
Peisert, F. Fallavollita46, D. Fasanella, G. Franzoni, J. Fulcher, W. Funk, S.
Giani, D. Gigi, A. Gilbert, K. Gill, F. Glege, M. Gruchala, M. Guilbaud, D.
Gulhan, J. Hegeman, C. Heidegger, Y. Iiyama, V. Innocente, P. Janot, O.
Karacheban19, J. Kaspar, J. Kieseler, M. Krammer1, C. Lange, P. Lecoq, C.
Lourenço, L. Malgeri, M. Mannelli, A. Massironi, F. Meijers, J.A. Merlin, S.
Mersi, E. Meschi, F. Moortgat, M. Mulders, J. Ngadiuba, S. Nourbakhsh, S.
Orfanelli, L. Orsini, F. Pantaleo16, L. Pape, E. Perez, M. Peruzzi, A.
Petrilli, G. Petrucciani, A. Pfeiffer, M. Pierini, F.M. Pitters, M. Quinto, D.
Rabady, A. Racz, M. Rovere, H. Sakulin, C. Schäfer, C. Schwick, M. Selvaggi,
A. Sharma, P. Silva, W. Snoeys, P. Sphicas47, J. Steggemann, V.R. Tavolaro, D.
Treille, A. Tsirou, A. Vartak, M. Verzetti, W.D. Zeuner Paul Scherrer
Institut, Villigen, Switzerland
L. Caminada48, K. Deiters, W. Erdmann, R. Horisberger, Q. Ingram, H.C.
Kaestli, D. Kotlinski, U. Langenegger, T. Rohe, S.A. Wiederkehr ETH Zurich -
Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland
M. Backhaus, P. Berger, N. Chernyavskaya, G. Dissertori, M. Dittmar, M.
Donegà, C. Dorfer, T.A. Gómez Espinosa, C. Grab, D. Hits, T. Klijnsma, W.
Lustermann, R.A. Manzoni, M. Marionneau, M.T. Meinhard, F. Micheli, P.
Musella, F. Nessi-Tedaldi, F. Pauss, G. Perrin, L. Perrozzi, S. Pigazzini, M.
Reichmann, C. Reissel, T. Reitenspiess, D. Ruini, D.A. Sanz Becerra, M.
Schönenberger, L. Shchutska, M.L. Vesterbacka Olsson, R. Wallny, D.H. Zhu
Universität Zürich, Zurich, Switzerland
T.K. Aarrestad, C. Amsler49, D. Brzhechko, M.F. Canelli, A. De Cosa, R. Del
Burgo, S. Donato, B. Kilminster, S. Leontsinis, V.M. Mikuni, I. Neutelings, G.
Rauco, P. Robmann, D. Salerno, K. Schweiger, C. Seitz, Y. Takahashi, S. Wertz,
A. Zucchetta National Central University, Chung-Li, Taiwan
T.H. Doan, C.M. Kuo, W. Lin, S.X. Liu, S.S. Yu National Taiwan University
(NTU), Taipei, Taiwan
P. Chang, Y. Chao, K.F. Chen, P.H. Chen, W.-S. Hou, Y.y. Li, R.-S. Lu, E.
Paganis, A. Psallidas, A. Steen Chulalongkorn University, Faculty of Science,
Department of Physics, Bangkok, Thailand
B. Asavapibhop, N. Srimanobhas, N. Suwonjandee Çukurova University, Physics
Department, Science and Art Faculty, Adana, Turkey
M.N. Bakirci50, A. Bat, F. Boran, S. Cerci51, S. Damarseckin52, Z.S.
Demiroglu, F. Dolek, C. Dozen, I. Dumanoglu, E. Eskut, G. Gokbulut, Y. Guler,
I. Hos53, C. Isik, E.E. Kangal54, O. Kara, A. Kayis Topaksu, U. Kiminsu, M.
Oglakci, G. Onengut, K. Ozdemir55, S. Ozturk50, A.E. Simsek, U.G. Tok, S.
Turkcapar, I.S. Zorbakir, C. Zorbilmez Middle East Technical University,
Physics Department, Ankara, Turkey
B. Isildak56, G. Karapinar57, M. Yalvac Bogazici University, Istanbul, Turkey
I.O. Atakisi, E. Gülmez, M. Kaya58, O. Kaya59, B. Kaynak, Ö. Özçelik, S.
Ozkorucuklu60, S. Tekten, E.A. Yetkin61 Istanbul Technical University,
Istanbul, Turkey
A. Cakir, K. Cankocak, Y. Komurcu, S. Sen62 Institute for Scintillation
Materials of National Academy of Science of Ukraine, Kharkov, Ukraine
B. Grynyov National Scientific Center, Kharkov Institute of Physics and
Technology, Kharkov, Ukraine
L. Levchuk University of Bristol, Bristol, United Kingdom
F. Ball, E. Bhal, S. Bologna, J.J. Brooke, D. Burns, E. Clement, D. Cussans,
H. Flacher, J. Goldstein, G.P. Heath, H.F. Heath, L. Kreczko, S. Paramesvaran,
B. Penning, T. Sakuma, S. Seif El Nasr-Storey, D. Smith, V.J. Smith, J.
Taylor, A. Titterton Rutherford Appleton Laboratory, Didcot, United Kingdom
K.W. Bell, A. Belyaev63, C. Brew, R.M. Brown, D. Cieri, D.J.A. Cockerill, J.A.
Coughlan, K. Harder, S. Harper, J. Linacre, K. Manolopoulos, D.M. Newbold, E.
Olaiya, D. Petyt, T. Reis, T. Schuh, C.H. Shepherd-Themistocleous, A. Thea,
I.R. Tomalin, T. Williams, W.J. Womersley Imperial College, London, United
Kingdom
R. Bainbridge, P. Bloch, J. Borg, S. Breeze, O. Buchmuller, A. Bundock,
GurpreetSingh CHAHAL64, D. Colling, P. Dauncey, G. Davies, M. Della Negra, R.
Di Maria, P. Everaerts, G. Hall, G. Iles, T. James, M. Komm, C. Laner, L.
Lyons, A.-M. Magnan, S. Malik, A. Martelli, V. Milosevic, J. Nash65, V.
Palladino, M. Pesaresi, D.M. Raymond, A. Richards, A. Rose, E. Scott, C. Seez,
A. Shtipliyski, M. Stoye, T. Strebler, S. Summers, A. Tapper, K. Uchida, T.
Virdee16, N. Wardle, D. Winterbottom, J. Wright, A.G. Zecchinelli, S.C. Zenz
Brunel University, Uxbridge, United Kingdom
J.E. Cole, P.R. Hobson, A. Khan, P. Kyberd, C.K. Mackay, A. Morton, I.D. Reid,
L. Teodorescu, S. Zahid Baylor University, Waco, USA
K. Call, J. Dittmann, K. Hatakeyama, C. Madrid, B. McMaster, N. Pastika, C.
Smith Catholic University of America, Washington, DC, USA
R. Bartek, A. Dominguez, R. Uniyal The University of Alabama, Tuscaloosa, USA
A. Buccilli, S.I. Cooper, C. Henderson, P. Rumerio, C. West Boston University,
Boston, USA
D. Arcaro, T. Bose, Z. Demiragli, D. Gastler, S. Girgis, D. Pinna, C.
Richardson, J. Rohlf, D. Sperka, I. Suarez, L. Sulak, D. Zou Brown University,
Providence, USA
G. Benelli, B. Burkle, X. Coubez, D. Cutts, Y.t. Duh, M. Hadley, J. Hakala, U.
Heintz, J.M. Hogan66, K.H.M. Kwok, E. Laird, G. Landsberg, J. Lee, Z. Mao, M.
Narain, S. Sagir67, R. Syarif, E. Usai, D. Yu University of California, Davis,
Davis, USA
R. Band, C. Brainerd, R. Breedon, M. Calderon De La Barca Sanchez, M. Chertok,
J. Conway, R. Conway, P.T. Cox, R. Erbacher, C. Flores, G. Funk, F. Jensen, W.
Ko, O. Kukral, R. Lander, M. Mulhearn, D. Pellett, J. Pilot, M. Shi, D. Stolp,
D. Taylor, K. Tos, M. Tripathi, Z. Wang, F. Zhang University of California,
Los Angeles, USA
M. Bachtis, C. Bravo, R. Cousins, A. Dasgupta, A. Florent, J. Hauser, M.
Ignatenko, N. Mccoll, W.A. Nash, S. Regnard, D. Saltzberg, C. Schnaible, B.
Stone, V. Valuev University of California, Riverside, Riverside, USA
K. Burt, R. Clare, J.W. Gary, S.M.A. Ghiasi Shirazi, G. Hanson, G.
Karapostoli, E. Kennedy, O.R. Long, M. Olmedo Negrete, M.I. Paneva, W. Si, L.
Wang, H. Wei, S. Wimpenny, B.R. Yates, Y. Zhang University of California, San
Diego, La Jolla, USA
J.G. Branson, P. Chang, S. Cittolin, M. Derdzinski, R. Gerosa, D. Gilbert, B.
Hashemi, D. Klein, V. Krutelyov, J. Letts, M. Masciovecchio, S. May, S. Padhi,
M. Pieri, V. Sharma, M. Tadel, F. Würthwein, A. Yagil, G. Zevi Della Porta
University of California, Santa Barbara - Department of Physics, Santa
Barbara, USA
N. Amin, R. Bhandari, C. Campagnari, M. Citron, V. Dutta, M. Franco Sevilla,
L. Gouskos, J. Incandela, B. Marsh, H. Mei, A. Ovcharova, H. Qu, J. Richman,
U. Sarica, D. Stuart, S. Wang, J. Yoo California Institute of Technology,
Pasadena, USA
D. Anderson, A. Bornheim, J.M. Lawhorn, N. Lu, H.B. Newman, T.Q. Nguyen, J.
Pata, M. Spiropulu, J.R. Vlimant, S. Xie, Z. Zhang, R.Y. Zhu Carnegie Mellon
University, Pittsburgh, USA
M.B. Andrews, T. Ferguson, T. Mudholkar, M. Paulini, M. Sun, I. Vorobiev, M.
Weinberg University of Colorado Boulder, Boulder, USA
J.P. Cumalat, W.T. Ford, A. Johnson, E. MacDonald, T. Mulholland, R. Patel, A.
Perloff, K. Stenson, K.A. Ulmer, S.R. Wagner Cornell University, Ithaca, USA
J. Alexander, J. Chaves, Y. Cheng, J. Chu, A. Datta, A. Frankenthal, K.
Mcdermott, N. Mirman, J.R. Patterson, D. Quach, A. Rinkevicius68, A. Ryd, S.M.
Tan, Z. Tao, J. Thom, P. Wittich, M. Zientek Fermi National Accelerator
Laboratory, Batavia, USA
S. Abdullin, M. Albrow, M. Alyari, G. Apollinari, A. Apresyan, A. Apyan, S.
Banerjee, L.A.T. Bauerdick, A. Beretvas, J. Berryhill, P.C. Bhat, K. Burkett,
J.N. Butler, A. Canepa, G.B. Cerati, H.W.K. Cheung, F. Chlebana, M. Cremonesi,
J. Duarte, V.D. Elvira, J. Freeman, Z. Gecse, E. Gottschalk, L. Gray, D.
Green, S. Grünendahl, O. Gutsche, AllisonReinsvold Hall, J. Hanlon, R.M.
Harris, S. Hasegawa, R. Heller, J. Hirschauer, B. Jayatilaka, S. Jindariani,
M. Johnson, U. Joshi, B. Klima, M.J. Kortelainen, B. Kreis, S. Lammel, J.
Lewis, D. Lincoln, R. Lipton, M. Liu, T. Liu, J. Lykken, K. Maeshima, J.M.
Marraffino, D. Mason, P. McBride, P. Merkel, S. Mrenna, S. Nahn, V. O’Dell, V.
Papadimitriou, K. Pedro, C. Pena, G. Rakness, F. Ravera, L. Ristori, B.
Schneider, E. Sexton-Kennedy, N. Smith, A. Soha, W.J. Spalding, L. Spiegel, S.
Stoynev, J. Strait, N. Strobbe, L. Taylor, S. Tkaczyk, N.V. Tran, L. Uplegger,
E.W. Vaandering, C. Vernieri, M. Verzocchi, R. Vidal, M. Wang, H.A. Weber
University of Florida, Gainesville, USA
D. Acosta, P. Avery, P. Bortignon, D. Bourilkov, A. Brinkerhoff, L. Cadamuro,
A. Carnes, V. Cherepanov, D. Curry, F. Errico, R.D. Field, S.V. Gleyzer, B.M.
Joshi, M. Kim, J. Konigsberg, A. Korytov, K.H. Lo, P. Ma, K. Matchev, N.
Menendez, G. Mitselmakher, D. Rosenzweig, K. Shi, J. Wang, S. Wang, X. Zuo
Florida International University, Miami, USA
Y.R. Joshi Florida State University, Tallahassee, USA
T. Adams, A. Askew, S. Hagopian, V. Hagopian, K.F. Johnson, R. Khurana, T.
Kolberg, G. Martinez, T. Perry, H. Prosper, C. Schiber, R. Yohay, J. Zhang
Florida Institute of Technology, Melbourne, USA
M.M. Baarmand, V. Bhopatkar, M. Hohlmann, D. Noonan, M. Rahmani, M. Saunders,
F. Yumiceva University of Illinois at Chicago (UIC), Chicago, USA
M.R. Adams, L. Apanasevich, D. Berry, R.R. Betts, R. Cavanaugh, X. Chen, S.
Dittmer, O. Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, K. Jung, C.
Mills, T. Roy, M.B. Tonjes, N. Varelas, H. Wang, X. Wang, Z. Wu The University
of Iowa, Iowa City, USA
M. Alhusseini, B. Bilki69, W. Clarida, K. Dilsiz70, S. Durgut, R.P.
Gandrajula, M. Haytmyradov, V. Khristenko, O.K. Köseyan, J.-P. Merlo, A.
Mestvirishvili71, A. Moeller, J. Nachtman, H. Ogul72, Y. Onel, F. Ozok73, A.
Penzo, C. Snyder, E. Tiras, J. Wetzel Johns Hopkins University, Baltimore, USA
B. Blumenfeld, A. Cocoros, N. Eminizer, D. Fehling, L. Feng, A.V. Gritsan,
W.T. Hung, P. Maksimovic, J. Roskes, M. Swartz, M. Xiao The University of
Kansas, Lawrence, USA
C. Baldenegro Barrera, P. Baringer, A. Bean, S. Boren, J. Bowen, A. Bylinkin,
T. Isidori, S. Khalil, J. King, G. Krintiras, A. Kropivnitskaya, C. Lindsey,
D. Majumder, W. Mcbrayer, N. Minafra, M. Murray, C. Rogan, C. Royon, S.
Sanders, E. Schmitz, J.D. Tapia Takaki, Q. Wang, J. Williams, G. Wilson Kansas
State University, Manhattan, USA
S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y. Maravin, D.R. Mendis, T. Mitchell,
A. Modak, A. Mohammadi Lawrence Livermore National Laboratory, Livermore, USA
F. Rebassoo, D. Wright University of Maryland, College Park, USA
A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng, N.J. Hadley, S. Jabeen,
G.Y. Jeng, R.G. Kellogg, J. Kunkle, A.C. Mignerey, S. Nabili, F. Ricci-Tam, M.
Seidel, Y.H. Shin, A. Skuja, S.C. Tonwar, K. Wong Massachusetts Institute of
Technology, Cambridge, USA
D. Abercrombie, B. Allen, A. Baty, R. Bi, S. Brandt, W. Busza, I.A. Cali, M.
D’Alfonso, G. Gomez Ceballos, M. Goncharov, P. Harris, D. Hsu, M. Hu, M.
Klute, D. Kovalskyi, Y.-J. Lee, P.D. Luckey, B. Maier, A.C. Marini, C. Mcginn,
C. Mironov, S. Narayanan, X. Niu, C. Paus, D. Rankin, C. Roland, G. Roland, Z.
Shi, G.S.F. Stephans, K. Sumorok, K. Tatar, D. Velicanu, J. Wang, T.W. Wang,
B. Wyslouch University of Minnesota, Minneapolis, USA
A.C. Benvenuti${}^{\textrm{\textdagger}}$, R.M. Chatterjee, A. Evans, S. Guts,
P. Hansen, J. Hiltbrand, Sh. Jain, S. Kalafut, Y. Kubota, Z. Lesko, J. Mans,
R. Rusack, M.A. Wadud University of Mississippi, Oxford, USA
J.G. Acosta, S. Oliveros University of Nebraska-Lincoln, Lincoln, USA
K. Bloom, D.R. Claes, C. Fangmeier, L. Finco, F. Golf, R. Gonzalez Suarez, R.
Kamalieddin, I. Kravchenko, J.E. Siado, G.R. Snow, B. Stieger State University
of New York at Buffalo, Buffalo, USA
C. Harrington, I. Iashvili, A. Kharchilava, C. Mclean, D. Nguyen, A. Parker,
J. Pekkanen, S. Rappoccio, B. Roozbahani Northeastern University, Boston, USA
G. Alverson, E. Barberis, C. Freer, Y. Haddad, A. Hortiangtham, G. Madigan,
D.M. Morse, T. Orimoto, L. Skinnari, A. Tishelman-Charny, T. Wamorkar, B.
Wang, A. Wisecarver, D. Wood Northwestern University, Evanston, USA
S. Bhattacharya, J. Bueghly, T. Gunter, K.A. Hahn, N. Odell, M.H. Schmitt, K.
Sung, M. Trovato, M. Velasco University of Notre Dame, Notre Dame, USA
R. Bucci, N. Dev, R. Goldouzian, M. Hildreth, K. Hurtado Anampa, C. Jessop,
D.J. Karmgard, K. Lannon, W. Li, N. Loukas, N. Marinelli, I. Mcalister, F.
Meng, C. Mueller, Y. Musienko35, M. Planer, R. Ruchti, P. Siddireddy, G.
Smith, S. Taroni, M. Wayne, A. Wightman, M. Wolf, A. Woodard The Ohio State
University, Columbus, USA
J. Alimena, B. Bylsma, L.S. Durkin, S. Flowers, B. Francis, C. Hill, W. Ji, A.
Lefeld, T.Y. Ling, B.L. Winer Princeton University, Princeton, USA
S. Cooperstein, G. Dezoort, P. Elmer, J. Hardenbrook, N. Haubrich, S.
Higginbotham, A. Kalogeropoulos, S. Kwan, D. Lange, M.T. Lucchini, J. Luo, D.
Marlow, K. Mei, I. Ojalvo, J. Olsen, C. Palmer, P. Piroué, J. Salfeld-Nebgen,
D. Stickland, C. Tully, Z. Wang University of Puerto Rico, Mayaguez, USA
S. Malik, S. Norberg Purdue University, West Lafayette, USA
A. Barker, V.E. Barnes, S. Das, L. Gutay, M. Jones, A.W. Jung, A. Khatiwada,
B. Mahakud, D.H. Miller, G. Negro, N. Neumeister, C.C. Peng, S. Piperov, H.
Qiu, J.F. Schulte, J. Sun, F. Wang, R. Xiao, W. Xie Purdue University
Northwest, Hammond, USA
T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA
K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Kilpatrick, Arun Kumar, W. Li, B.P.
Padley, R. Redjimi, J. Roberts, J. Rorie, W. Shi, A.G. Stahl Leiton, Z. Tu, A.
Zhang University of Rochester, Rochester, USA
A. Bodek, P. de Barbaro, R. Demina, J.L. Dulemba, C. Fallon, T. Ferbel, M.
Galanti, A. Garcia-Bellido, J. Han, O. Hindrichs, A. Khukhunaishvili, E.
Ranken, P. Tan, R. Taus Rutgers, The State University of New Jersey,
Piscataway, USA
B. Chiarito, J.P. Chou, A. Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart,
M. Heindl, E. Hughes, S. Kaplan, S. Kyriacou, I. Laflotte, A. Lath, R.
Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, D. Sheffield,
S. Somalwar, R. Stone, S. Thomas, P. Thomassen University of Tennessee,
Knoxville, USA
H. Acharya, A.G. Delannoy, J. Heideman, G. Riley, S. Spanier Texas A&M
University, College Station, USA
O. Bouhali74, A. Celik, M. Dalchenko, M. De Mattia, A. Delgado, S. Dildick, R.
Eusebi, J. Gilmore, T. Huang, T. Kamon75, S. Luo, D. Marley, R. Mueller, D.
Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock,
USA
N. Akchurin, J. Damgov, F. De Guio, S. Kunori, K. Lamichhane, S.W. Lee, T.
Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang, A.
Whitbeck Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij University of
Virginia, Charlottesville, USA
M.W. Arenton, P. Barria, B. Cox, G. Cummings, R. Hirosky, M. Joyce, A.
Ledovskoy, C. Neu, B. Tannenwald, Y. Wang, E. Wolfe, F. Xia Wayne State
University, Detroit, USA
R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa, S. Zaleski University
of Wisconsin - Madison, Madison, WI, USA
J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De Bruyn, L. Dodd, F.
Fiori, C. Galloni, B. Gomber76, M. Herndon, A. Hervé, U. Hussain, P. Klabbers,
A. Lanaro, A. Loeliger, K. Long, R. Loveless, J. Madhusudanan Sreekala, T.
Ruggles, A. Savin, V. Sharma, W.H. Smith, D. Teague, S. Trembath-reichert, N.
Woods †: Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at UFMS, Nova Andradina, Brazil
6: Also at Universidade Federal de Pelotas, Pelotas, Brazil
7: Also at Université Libre de Bruxelles, Bruxelles, Belgium
8: Also at University of Chinese Academy of Sciences, Beijing, China
9: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
10: Also at Joint Institute for Nuclear Research, Dubna, Russia
11: Now at British University in Egypt, Cairo, Egypt
12: Now at Cairo University, Cairo, Egypt
13: Also at Purdue University, West Lafayette, USA
14: Also at Université de Haute Alsace, Mulhouse, France
15: Also at Erzincan Binali Yildirim University, Erzincan, Turkey
16: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
17: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
18: Also at University of Hamburg, Hamburg, Germany
19: Also at Brandenburg University of Technology, Cottbus, Germany
20: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary,
Debrecen, Hungary
21: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
22: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary, Budapest, Hungary
23: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India
24: Also at Institute of Physics, Bhubaneswar, India
25: Also at Shoolini University, Solan, India
26: Also at University of Visva-Bharati, Santiniketan, India
27: Also at Isfahan University of Technology, Isfahan, Iran
28: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
29: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
30: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
31: Also at Riga Technical University, Riga, Latvia, Riga, Latvia
32: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
33: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
34: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
35: Also at Institute for Nuclear Research, Moscow, Russia
36: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
37: Also at Institute of Nuclear Physics of the Uzbekistan Academy of
Sciences, Tashkent, Uzbekistan
38: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
39: Also at University of Florida, Gainesville, USA
40: Also at Imperial College, London, United Kingdom
41: Also at P.N. Lebedev Physical Institute, Moscow, Russia
42: Also at California Institute of Technology, Pasadena, USA
43: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
44: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
45: Also at Università degli Studi di Siena, Siena, Italy
46: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy,
Pavia, Italy
47: Also at National and Kapodistrian University of Athens, Athens, Greece
48: Also at Universität Zürich, Zurich, Switzerland
49: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria,
Vienna, Austria
50: Also at Gaziosmanpasa University, Tokat, Turkey
51: Also at Adiyaman University, Adiyaman, Turkey
52: Also at Şırnak University, Sirnak, Turkey
53: Also at Istanbul Aydin University, Application and Research Center for
Advanced Studies (App. & Res. Cent. for Advanced Studies), Istanbul, Turkey
54: Also at Mersin University, Mersin, Turkey
55: Also at Piri Reis University, Istanbul, Turkey
56: Also at Ozyegin University, Istanbul, Turkey
57: Also at Izmir Institute of Technology, Izmir, Turkey
58: Also at Marmara University, Istanbul, Turkey
59: Also at Kafkas University, Kars, Turkey
60: Also at Istanbul University, Istanbul, Turkey
61: Also at Istanbul Bilgi University, Istanbul, Turkey
62: Also at Hacettepe University, Ankara, Turkey
63: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
64: Also at IPPP Durham University, Durham, United Kingdom
65: Also at Monash University, Faculty of Science, Clayton, Australia
66: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA
67: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
68: Also at Vilnius University, Vilnius, Lithuania
69: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey
70: Also at Bingol University, Bingol, Turkey
71: Also at Georgian Technical University, Tbilisi, Georgia
72: Also at Sinop University, Sinop, Turkey
73: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
74: Also at Texas A&M University at Qatar, Doha, Qatar
75: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea
76: Also at University of Hyderabad, Hyderabad, India


[2]#2^  #1

# Multi-scale hierarchy from multidimensional gravity

Kirill A. Bronnikova,b,c,1, Arkady A. Popovd,2, Sergey G. Rubinc,d,3

(a Center fo Gravitation and Fundamental Metrology, VNIIMS,Ozyornaya ulitsa
46, Moscow 119361, Russia
b Institute of Gravitation and Cosmology, RUDN University, ulitsa Miklukho-
Maklaya 6, Moscow 117198, Russia
c National Research Nuclear University MEPhI (Moscow Engineering Physics
Institute),
Kashirskoe shosse 31, Moscow 115409, Russia
d N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal
University,
Kremlyovskaya ulitsa 18, Kazan 420008, Russia)

###### Abstract

We discuss the way of solving the hierarchy problem. We show that starting at
the Planck scale, the three energy scales — inflationary, electroweak and the
cosmological ones can be restored. The formation of small parameters is
proposed that leads to a successful solution of the problem. The tools
involved in the process are $f(R)$ gravity and inhomogeneous extra dimensions.
Slow rolling of a space domain from the Planck scale down to the inflationary
one gives rise to three consequences: an infinite set of causally disconnected
domains (pocket universes) are nucleated; quantum fluctuations in each domain
produce a variety of different fields and an extra-dimensional metric
distribution; these distributions are stabilized at a sufficiently low energy
scale.

11footnotetext: e-mail<EMAIL_ADDRESS>e-mail:
<EMAIL_ADDRESS>e-mail<EMAIL_ADDRESS>
##  1 Introduction

Assuming that the Universe has been formed at the Planck scale, it is
naturally implied that its initially formed parameters are of the order of the
same scale. The essence of the Hierarchy problem is the question: Why are the
observable low-energy physical parameters so small as compared to those of the
Planck scale? How did Nature manages to decrease the parameter values so
substantially?

There are at least four important energy scales during evolution of the
Universe: the Planck scale ($\sim 10^{19}$ GeV) at which our Universe cannot
be described by classical laws; the inflationary scale ($\sim 10^{13}$ GeV)
where our horizon has appeared, the electroweak scale ($\sim 10^{2}$ GeV), and
the cosmological scale specified by the cosmological constant ($\sim
10^{-123}$ GeV4) (CC).

According to the inflationary paradigm, the physical laws are formed at high
energies [1, 2], where the Lagrangian structure is yet unknown. The physics
has been established at the energy scale $M$ higher than the inflationary one,
$E_{I}\sim 10^{13}$ GeV, see [3, 4] in this context. We study the way of
substantially decreasing the physical parameters at the three scales mentioned
above assuming natural values of the initial parameters of the order of $M$.

In this paper, we invoke the idea of multidimensional gravity which is a
widely used tool for obtaining new theoretical results [5, 6, 7, 8, 9]. The
paper [10] uses warped geometry to solve the small cosmological constant
problem. Multidimensional inflation is discussed in [11, 12, 13] where it was
supposed that an extra-dimensional metric $g_{\text{n}}$ is stabilized at a
high-energy scale. Stabilization of extra space as a pure gravitational effect
has been studied in [14, 15], see also [16].

The present research is also based on nonlinear $f(R)$ gravity. The interest
in $f(R)$ theories is motivated by inflationary scenarios starting with
Starobinsky’s paper [17]. At present, $f(R)$ gravity is widely discussed,
leading to a variety of consequences, in particular, the existence of dark
matter [18, 19]. Including a function of the Ricci scalar, $f(R)$, is the
simplest extension of general relativity. In the framework of such an
extension, many interesting results have been obtained. Some viable $f(R)$
models in 4D space that satisfy the observational constraints are proposed in
[20, 21, 22, 23, 24].

The idea that the Lagrangian parameters can be considered as some functions of
a field has been widely used since Schwinger’s paper [25]. Such fields can be
involved in the classical equations of motion together with the ‘‘main’’
fields or treated as background fields. The latter were applied for fermion
localization on branes [26, 27, 28], gauge field localization [29], extensions
of gravity in a scalar-tensor form (with $f(\phi)R$) [30] and so on. In this
paper, we show that a self-gravitating scalar field can serve as a reason for
the emergence of small parameters.

As a mathematical tool, we use the Wilsonian approach technique, a well-known
method for theoretical studies of the energy dependence of physical parameters
[31]. In this approach, the physical parameters $\lambda_{i}(M)$ of the Wilson
action are fixed at a high energy scale $M$. The renormalization flow used to
descend to low energies (the top-down approach) is discussed in [32, 33, 34,
35, 36]. In particular, quantum corrections to the Starobinsky model were
discussed in [37].

In our approach, we add extra dimensions and study their role in different
scales, with a hope that it should make the renormalization procedure much
more efficient. We make use of the idea of flexible (inhomogeneous) extra
space that has been developed in [18, 38, 39]. Our preliminary study of
inhomogeneous extra metrics concerns such parameters as the cosmological
constant [38], those of the Starobinsky inflationary model and baryon
asymmetry of the Universe [40, 41]. It has been shown there that inhomogeneous
metrics can be tuned to explain the smallness of the appropriate effective
parameters. For example, encouraging results for explaining the smallness of
the cosmological constant was obtained in [38, 42]. The effect of quantum
corrections in this context was discussed in [43].

Here we continue this research by including the Higgs sector of the Standard
Model. There are three energy scales which we intend to describe —
inflationary stage, the electroweak scale, and the cosmological one. Each of
them is characterized by a specific small parameter. The initial parameters
and the Lagrangian of our model are fixed at a sub-Planckian scale and do not
vary during the Universe evolution. Special attention is paid to the emergence
mechanism small values, specific for each of the three scales.

##  2 The model

Consider $f(R)$ gravity with a minimally coupled scalar field $\zeta$ in a
$\text{D}=4+n$-dimensional manifold $M_{\text{D}}$:

$\displaystyle
S=\frac{m_{\text{D}}^{\text{D}-2}}{2}\int_{M_{\text{D}}}d^{\text{D}}X\sqrt{|g_{\text{D}}|}\,\Bigl{(}f(R)+\partial^{\text{M}}\zeta\,\partial_{\text{M}}\zeta-2V(\zeta)\Bigr{)}+S_{H_{P}}\,,$
(1)

where $g_{\text{D}}\equiv\mathop{\rm det}\nolimits g_{\text{M}\text{N}}$,
$\text{M},\text{N}=\overline{1,\text{D}}$, the $n$-dimensional manifold
$M_{n}$ is assumed to be closed, $f(R)$ is a function of the D-dimensional
Ricci scalar $R$, and $m_{\text{D}}$ is the D-dimensional Planck mass. Below,
we will work in the units $m_{D}=1$. The term $S_{H_{P}}$ denotes the Higgs
action (32) considered in Sec. 5, and it is assumed to be small as compared to
the gravitational part of the action. It is also postulated that the scalar
field $\zeta$ is very massive and hence unobservable. Nevertheless, this field
plays a key role being responsible for the emergence of small parameter(s),
see a discussion at the beginning of Sec. 4.

Variation of the action (1) with respect to the metric
$g^{\text{M}\text{N}}_{\text{D}}$ and the scalar field leads to the known
equations

$\displaystyle-\frac{1}{2}{f}(R)\delta_{\text{N}}^{\text{M}}+\Bigl{(}R_{\text{N}}^{\text{M}}+\nabla^{\text{M}}\nabla_{\text{N}}-\delta_{\text{N}}^{\text{M}}\Box_{\text{D}}\Bigr{)}{f}_{R}=-T_{\text{N}}^{\text{M}},$
(2) $\displaystyle\Box_{\text{D}}\,\zeta+V_{\zeta}=0,$ (3)

with $f_{R}={df(R)}/{dR}$,
$\Box_{\text{D}}=\nabla^{\text{M}}\nabla_{\text{M}}$, and
$V_{\zeta}={dV(\zeta)}/{d\zeta}$.

Equation (3) is known to be a consequence of equations (2). The corresponding
stress-energy tensor of the scalar field $\zeta$ is

$T_{\text{N}}^{\text{M}}=\frac{\partial L_{\rm
matter}}{\partial\bigl{(}\partial_{\text{M}}\zeta\bigr{)}}\partial_{\text{N}}\zeta-\frac{\delta_{\text{N}}^{\text{M}}}{2}L_{\rm
matter}=\partial^{\text{M}}\zeta\,\partial_{\text{N}}\zeta-\frac{\delta_{\text{N}}^{\text{M}}}{2}\,\partial^{\text{K}}\zeta\,\partial_{\text{K}}\zeta+\delta_{\text{N}}^{\text{M}}V\bigl{(}\zeta\bigr{)}\,.$
(4)

Here the Higgs field contribution is omitted. We use the conventions for the
curvature tensor $R_{\
\text{M}\text{N}\text{K}}^{\text{L}}=\partial_{\text{K}}\Gamma_{\text{M}\text{N}}^{\text{L}}-\partial_{\text{N}}\Gamma_{\text{M}\text{K}}^{\text{L}}+\Gamma_{\text{C}\text{K}}^{\text{L}}\Gamma_{\text{N}\text{M}}^{\text{C}}-\Gamma_{\text{C}\text{N}}^{\text{L}}\Gamma_{\text{M}\text{K}}^{\text{C}}$
and the Ricci tensor $R_{\text{M}\text{N}}=R^{\text{K}}_{\
\text{M}\text{K}\text{N}}$.

The metric is supposed in the form

$ds^{2}=\text{e}^{2\gamma(u)}\left(dt^{2}-\text{e}^{2Ht}(dx^{2}+dy^{2}+dz^{2})\right)-du^{2}-r(u)^{2}d\Omega_{n-1}^{2},$
(5)

where $d\Omega_{n-1}^{2}$ is the metric on a unit $n-1$-dimensional sphere.
The metric ansatz used in this paper has been widely studied in the framework
of linear gravity [44, 45, 46, 47], applying, in particular, to solving the
Hierarchy problem [48, 49, 50].

The field equations for the metric (5) and $\zeta=\zeta(u)$ read

$\displaystyle{R^{\prime}}^{2}f_{RRR}+\left[R^{\prime\prime}+\left(3\gamma^{\prime}+(n-1)\frac{r^{\prime}}{r}\right)R^{\prime}\right]f_{RR}-\left(\gamma^{\prime\prime}+4{\gamma^{\prime}}^{2}+(n-1)\frac{\gamma^{\prime}r^{\prime}}{r}-\frac{3H^{2}}{\text{e}^{2\gamma}}\right)f_{R}$
$\displaystyle\hskip
72.26999pt-\dfrac{f(R)}{2}=-\dfrac{{\zeta^{\prime}}^{2}}{2}-V\bigl{(}\zeta\bigr{)},$
(6)
$\displaystyle\left(4\gamma^{\prime}R^{\prime}+(n-1)\dfrac{r^{\prime}}{r}R^{\prime}\right)\,f_{RR}-\left(4\gamma^{\prime\prime}+4{\gamma^{\prime}}^{2}+(n-1)\dfrac{r^{\prime\prime}}{r}\right)\,f_{R}-\,\dfrac{f(R)}{2}=\dfrac{{\zeta^{\prime}}^{2}}{2}-V\bigl{(}\zeta\bigr{)}\,,$
(7)
$\displaystyle{R^{\prime}}^{2}f_{RRR}\,+\bigg{(}R^{\prime\prime}+4\gamma^{\prime}R^{\prime}+(n-2)\dfrac{r^{\prime}}{r}\,R^{\prime}\bigg{)}f_{RR}-\biggl{(}\dfrac{r^{\prime\prime}}{r}+\frac{4\gamma^{\prime}r^{\prime}}{r}+(n-2)\dfrac{{r^{\prime}}^{2}}{r^{2}}-\dfrac{(n-2)}{r^{2}}\biggr{)}f_{R}$
$\displaystyle\hskip
72.26999pt-\,\dfrac{f(R)}{2}=-\dfrac{{\zeta^{\prime}}^{2}}{2}-V(\zeta),$ (8)
$\displaystyle\zeta^{\prime\prime}+\left(4\gamma^{\prime}+(n-1)\dfrac{r^{\prime}}{r}\right)\,\zeta^{\prime}-V_{\zeta}=0,$
(9)

where the prime denotes $d/du$. Also, we will use the expression for the Ricci
scalar

$R(u)=\frac{12H^{2}}{\text{e}^{2\gamma}}-8\gamma^{\prime\prime}-20{\gamma^{\prime}}^{2}-(n-1)\left(\dfrac{2r^{\prime\prime}}{r}+\frac{8\gamma^{\prime}r^{\prime}}{r}+(n-2)\left(\dfrac{r^{\prime}}{r}\right)^{2}\,-\dfrac{(n-2)}{r^{2}}\right)$
(10)

as an additional equation and $R(u)$ will be treated as a new unknown function
to avoid 3rd and 4th order derivatives in Eqs. (2)–(2). It can be shown that
one of the equations (2)–(9) is a consequence of the others. The combination
$2\times$(7)$-f_{R}\times$(10) is the constraint equation

$\displaystyle{}\biggr{(}8\gamma^{\prime}+2(n-1)\dfrac{r^{\prime}}{r}\biggl{)}R^{\prime}f_{RR}+\Biggl{(}12{\gamma^{\prime}}^{2}+(n-1)\biggl{(}\dfrac{8\gamma^{\prime}r^{\prime}}{r}+(n-2)\dfrac{\bigl{(}{r^{\prime}}^{2}-1\bigr{)}}{r^{2}}\biggr{)}+R\Biggr{)}f_{R}$
$\displaystyle{}\hskip
72.26999pt-\frac{12H^{2}}{\text{e}^{-\gamma(u)}}f_{R}\,-f(R)={\zeta^{\prime}}^{2}-2V\bigl{(}\zeta\bigr{)}$
(11)

containing only first-order derivatives. It plays the role of a restriction on
the solutions of the coupled second-order differential equations (2)–(10).

As a result, we use three independent equations (2), (2), (10) and the
constraint (2) to fix three functions $r(u),\gamma(u),R(u)$ and the unknown
metric parameter $H$.

One of the possible numerical solutions to this system is shown in Fig. 1. We
note that the warp factor $e^{\gamma(u)}\to 0$ at the boundaries, which are
singular ends of the range of $u$ and can be imagined as a kind of poles in a
closed $n$-dimensional manifold since there $r\to 0$. The qualitative behavior
of the solution shown in Fig. 1 is quite generic. The particular form of
solutions, including the field distribution and the extra metric, depends on
the Lagrangian parameters postulated from the beginning. It also depends on
the boundary conditions at $u=0$ that are necessary for solving the second-
order differential equations. Unlike the Lagrangian parameters, the boundary
conditions ultimately depend on random initial fluctuations within a pocket
universe. Inflation produces a continuum set of such universes with different
initial conditions and therefore with different metric functions and field
distributions in the extra space.

Figure 1: Solution of Eqs. (2)–(10) for the following parameters $n=3$,
$f(R)=300R^{2}+R+0.002$, $H=0.003$, $V(\zeta)=0.01\,\zeta^{2}/2$ and boundary
conditions $r(0)=50$, $\gamma(0)=0$, $R(0)\simeq 0.00415$, $\zeta(0)=0$,
$r^{\prime}(0)=\gamma^{\prime}(0)=R^{\prime}(0)=0$, $\zeta^{\prime}(0)=6\times
10^{-8}$.

##  3 Matter localization around a singularity

In general, it is assumed here that matter is distributed throughout the extra
dimensions like in the Universal Extra Dimensional approach [51, 52]. At the
same time, there is another direction that deserves discussion. Indeed, we see
from the figures that there are two points where the metric is singular or has
sharp peaks. They could indicate the formation of branes if the extra space is
large enough and if matter is concentrated in a close neighborhood of these
peaks (certainly assuming that the formal infinities are somehow suppressed by
quantum effects). This opportunity is briefly discussed in this section.

As a rough approximation, consider the motion of classical particles near such
a singular point, see Fig. 1, bearing in mind the metric

$ds^{2}=\text{e}^{2\gamma(u)}(dt^{2}-dx^{2}-dy^{2}-dz^{2})-du^{2}-r(u)^{2}(d\xi^{2}+\sin^{2}\xi\,d\psi^{2}).$
(12)

The geodesic equations have the form

$\displaystyle{}t_{ss}+2\,t_{s}\,\gamma_{u}\,u_{s}=0,$ (13)
$\displaystyle{}x_{ss}+2\,x_{s}\,\gamma_{u}\,u_{s}=0,\quad
y_{ss}+2\,y_{s}\,\gamma_{u}\,u_{s}=0,\quad
z_{ss}+2\,z_{s}\,\gamma_{u}\,u_{s}=0,\quad$ (14)
$\displaystyle{}u_{ss}+\text{e}^{2\gamma}\,\gamma_{u}\,({t_{s}}^{2}-{x_{s}}^{2}-{y_{s}}^{2}-{z_{s}}^{2})-r\,r_{u}\,{\xi_{s}}^{2}-r\,r_{u}\sin^{2}\xi\,{\psi_{s}}^{2}=0,$
(15)
$\displaystyle{}\xi_{ss}+2\,\xi_{s}\,\frac{r_{u}}{r}\,u_{s}-\sin\xi\,\cos\xi\,{\psi_{s}}^{2}=0,$
(16)
$\displaystyle{}\psi_{ss}+2\,\psi_{s}\,\frac{r_{u}}{r}\,u_{s}+2\cot\xi\,\xi_{s}\,\psi_{s}=0,$
(17)

where the index $s$ denotes the derivative with respect to $s$, and the index
$u$ denotes the derivative with respect to $u$. These equations admit
solutions when $x,y,z,\xi$, and $\psi$ are constant. Let us assume, for
simplicity, that $\xi=\pi/2$, then

$\displaystyle{}0=t_{ss}+2\,t_{s}\,\gamma_{u}\,u_{s}=t_{ss}+2\,t_{s}\,\gamma_{s}\
\ \Rightarrow\ \ t_{s}=\text{e}^{C_{1}-2\gamma},$ (18)
$\displaystyle{}0=u_{ss}+\text{e}^{2\gamma}\,\gamma_{u}\,{t_{s}}^{2}=u_{ss}+\gamma_{u}\,\text{e}^{2C_{1}-2\gamma}=\frac{1}{u_{s}}\left(u_{ss}\,u_{s}+\gamma_{s}\,\text{e}^{2C_{1}-2\gamma}\right)\
\ \Rightarrow\ \ {u_{s}}^{2}=\text{e}^{2C_{1}-2\gamma}+C_{2},$ (19)

with integration constants $C_{i}$. The normalization relation gives

$1=\text{e}^{2\gamma}{t_{s}}^{2}-{u_{s}}^{2}=-C_{2}.$ (20)

Then

${u_{s}}^{2}=\text{e}^{2C_{1}-2\gamma}-1={u_{t}}^{2}\text{e}^{2C_{1}-4\gamma}\
\Rightarrow\
{u_{t}}^{2}=\text{e}^{2\gamma}\left(1-\text{e}^{2\gamma-2C_{1}}\right).$ (21)

Figure 2: $u\Big{|}_{t=0}=u\Big{|}_{s=0}=1,\
du/dt\Big{|}_{t=0}=du/ds\Big{|}_{s=0}=0$, in the background of the solution
shown in Fig.1.

For nonrelativistic particles, only the second equation matters. It can be
approximated as

$u_{ss}\simeq-2e^{2\gamma}\gamma_{u}.$ (22)

We see that the acceleration of a particle is directed to a singular point,
which should ultimately lead to concentration of matter at such a point. As a
result, matter is localized around both ‘‘poles,’’ as should be the case in a
brane world (this time consisting of two branes on the two ‘‘poles’’). It
opens a door for developing a mechanism of strong reduction of the initial
parameter values. For example, an interaction term of the form

$\kappa\int d^{D}Z\sqrt{|g_{D}|}\chi(z)\bar{\psi}(z)\psi(z)$

contains overlapping integral

$I_{\rm overlap}\equiv\int d^{n}y\sqrt{|g_{n}|}\chi(y)\bar{\psi}(y)\psi(y)$

over the extra dimensions which could be arbitrarily small if the fields
$\chi(y)$ and $\psi(y)$ are localized near different branes. It leads to the
coupling constant renormalization

$\kappa\to\kappa^{\prime}=\kappa I_{\rm overlap}\ll\kappa.$

We will leave this idea for future studies and return to our main discussion.

##  4 Intermediate energies. The Starobinsky model

The second energy scale relates to the inflationary stage with the small
parameter $H/m_{\text{Pl}}\sim 10^{-6}$. Different inflationary models use
different parameters of this order. It could be the inflaton mass in the
simplest model of inflation with a quadratic potential or a constant factor of
$R^{2}$ term in the Starobinsky model. Let us restore the latter. To this end,
we should solve the system (2)–(9) and obtain the necessary values of its
parameters.

The scalar field $\zeta$ affects the extra-space metric through the Einstein
equations, but here we are interested in small amplitude solutions of this
field, i.e., $\zeta(X)\ll 1$. Therefore, its role in the metric formation is
negligible, and it can be considered as a test field acting in the background
metric. This approximation makes the analysis easier but is not very
significant for our reasoning.

#### The emergence of small parameters

A successful solution of the Hierarchy problem implies the presence of small
parameters, and we have enough tools to create them. Indeed, in our picture,
there is an infinite set of different universes created during inflation [53,
54, 13] which contain the independently fluctuating field $\zeta$. These
fluctuations decay with time and lead to static field distributions in each
universe. There is an infinite set $\aleph$ of such static distributions that
form a continuum set. The situation is similar to the boson star formation
model [55] where a self-gravitating scalar field forms a variety of dense
stable clumps. The set $\aleph$ contains a subset of small-amplitude
distributions $\zeta(u)$ like those presented in Fig. 1, right panel. Their
values averaged over the extra dimensions represent a set of small parameters
to be widely used below.

To proceed, let us restore some formulas from our previous paper [56] for a
relation between the D-dimensional Planck mass and the 4-dimensional one,
which is needed to convert units $m_{\text{D}}=1$ into the physical units. To
this end, define

$R_{4}\equiv 12H^{2},\qquad R_{n}\equiv R(u)-\text{e}^{-2\gamma(u)}R_{4},$
(23)

see (10). Substitution of the Taylor series

$f(R)\simeq
f(R_{n})+f_{R}(R_{n})\text{e}^{-2\gamma(u)}R_{4}+\frac{1}{2}f_{RR}(R_{n})\text{e}^{-4\gamma(u)}R_{4}^{2}+\ldots$
(24)

into the gravitational part of the action (1) leads to an effective theory
after integration over the extra coordinates:

$S_{\text{eff}}=\dfrac{m_{\text{Pl}}^{2}}{2}\int\limits_{M_{4}}d^{4}x\sqrt{|g_{4}|}\Bigl{(}a_{\text{eff}}R_{4}^{2}+R_{4}+c_{\text{eff}}\Bigr{)}.$
(25)

Here $g_{4}$ is the determinant of the 4D metric

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dt^{2}-\text{e}^{2Ht}\delta_{ij}dx^{i}dx^{j}\,,$
(26)

and

$\displaystyle{}m^{2}_{\text{Pl}}=\mathcal{V}_{n-1}\int_{u_{\min}}^{u_{\max}}f_{R}(R_{n})\,\text{e}^{2\gamma}\,r^{n-1}\,du,$
(27)
$\displaystyle{}a_{\text{eff}}=\frac{\mathcal{V}_{\text{n}-1}}{2m_{\text{Pl}}^{2}}\int_{u_{\min}}^{u_{\max}}f_{RR}(R_{n})\,\text{e}^{4\gamma}\,r^{n-1}\,du,$
(28)
$\displaystyle{}c_{\text{eff}}=\frac{\mathcal{V}_{n-1}}{m_{\text{Pl}}^{2}}\int_{u_{\min}}^{u_{\max}}\Bigl{(}f(R_{n})-(\zeta^{\prime})^{2}-2V(\zeta)\Bigr{)}\,\text{e}^{4\gamma}\,r^{n-1}\,du.$
(29)

where $\mathcal{V}_{n-1}=\int
d^{n-1}x\sqrt{|{g}_{n-1}|}=\dfrac{2\pi^{n/2}}{\Gamma(n/2)}$. The r.h.s. of Eq.
(27) is written in units $m_{D}=1$. This relation is used to express the
D-dimensional Planck mass in terms of the 4D Planck mass $m_{\text{Pl}}$. Here
we suppose that the functions $\gamma(u),\ r(u),\ \zeta(u),\ R(u)$ form a
particular solution to the system (2)-(9) for a specific value of $H$.
Therefore the values of $a_{\text{eff}}(H)$ and $c_{\text{eff}}(H)$ are
functions of the Hubble parameter. They are approximately constants during
inflation and at the present times, being different in these two periods. The
parameter $c_{\text{eff}}(H)$ is fixed at the present epoch when $H\lll
m_{\text{Pl}}$, while the parameter $a_{\text{eff}}(H)$ is determined by the
appropriate inflation rate. The parameter $a_{\text{eff}}$ must be
approximately equal to the observable value obtained from the COBE
normalization [57], as

$a_{\text{Starob}}\simeq 1.12\cdot
10^{9}\left(\dfrac{N_{\text{e}}}{60}\right)^{2}m^{-2}_{\text{Pl}}.$ (30)

For the solution shown in Fig. 1, $a_{\text{eff}}\simeq 7.2\cdot
10^{8}\,m_{\text{Pl}}^{-2}$, and the Hubble parameter is $H\simeq 1\cdot
10^{-6}m_{\text{Pl}}$. One can see that the Starobinsky inflationary model has
been restored. The parameter values $a=300,\,c=0.002$ lead to the following
values of the dimensionless parameters:

$a^{\prime}=\sqrt{am_{D}^{2}}\simeq 17,\qquad
c^{\prime}=\sqrt{c/m_{D}^{2}}\simeq 0.045$ (31)

which looks natural.

The other parameter, $c_{\text{eff}}(H)$, needs a separate discussion.
Equation (29) for $c_{\text{eff}}(H)$ is derived under the assumption that all
functions in Eqs. (2)–(9) are stationary, which takes place for a 4D de Sitter
metric in which the Hubble parameter $H={\rm const}$. This approximation is
valid at slow-roll inflation with the small parameter $|\dot{H}|/H^{2}\ll 1$.
Fortunately, this inequality holds for a wide range of the parameter
$c_{\text{eff}}$, in particular for $c_{\text{eff}}=0$ [17]. However, Eq. (29)
has a practical meaning only for a pure de Sitter metric but not for slow
rolling. This formula could be valid at the present times, for example, if we
suppose that the cosmological constant $\Lambda=-c_{\text{eff}}/2$ is really
constant. It is a subject of detailed discussion, see Sec. 6.

##  5 The electroweak scale. Restoration of the Higgs parameters

In this section, the reasoning is in the spirit of our previous paper [56],
but without introducing an external scalar field.

###  5.1 Analytical formulas

In the previous section, we have reproduced the Starobinsky model of inflation
at the scale of $10^{13}$ GeV. The appropriate values of the initial
Lagrangian parameters $a,c$ are fixed. These parameters must be the same at
low scales where the Hubble parameter is equal to zero, $H\simeq 0$, as
compared to the Planck scale. On the contrary, the extra space metric depends
on the energy scale, the Hubble parameter in our case.

In this section we discuss the Hierarchy problem at the electroweak scale
using the Higgs field as an example. Within the framework of our approach
outlined in the Introduction, we assume that the physics of the Higgs field is
formed at the Planck scale.

Suppose that the form of the Higgs action at the Planck scale is the same as
at the electroweak scale,

$S_{\text{H}_{\text{P}}}=\frac{1}{2}\int
d^{\text{D}}X\sqrt{|g_{\text{D}}|}\,\Bigl{(}\partial^{\text{M}}{H_{\text{P}}}^{\dagger}\partial_{\text{M}}H_{\text{P}}+\nu{H_{\text{P}}}^{\dagger}H_{\text{P}}-\lambda\bigl{(}{H_{\text{P}}}^{\dagger}H_{\text{P}}\bigr{)}^{2}\Bigr{)},$
(32)

where the symbol $\dagger$ means Hermitian conjugation, $\nu$ and $\lambda>0$
are arbitrary numbers and $H_{\text{P}}$ is a proto-Higgs field.

We have managed to avoid large/small initial dimensionless parameter values
$a^{\prime},c^{\prime}$, see (31) when describing the Starobinsky model acting
at energies $\sim 10^{13}$ GeV. Our intention is to repeat this success at the
electroweak energies. To do that, we need to show that the initial parameters
can be reduced by many orders of magnitude. All numerical values in the
Lagrangian (32) are of the order of unity in $m_{\text{D}}$ units. More
definitely, let us express the dimensionful parameters $\nu,\lambda$ in terms
of the dimensionless ones $\nu^{\prime},\lambda^{\prime}$:

$\nu\to(\nu^{\prime}m_{D})^{2},\qquad\lambda\to(\lambda^{\prime}/m_{D})^{D-4}.$
(33)

It is these dimensionless parameters $\nu^{\prime}$ and $\lambda^{\prime}$
that should vary around unity.

The classical equations of motion are obtained by varying the action (32) with
respect to $H_{\text{P}}$, which gives

$\square_{\text{D}}H_{\text{P}}=\nu
H_{\text{P}}-2\lambda\bigl{(}{H_{\text{P}}}^{\dagger}H_{\text{P}}\bigr{)}H_{\text{P}}.$
(34)

The proto-Higgs field can be presented as

$H_{\text{P}}=h(x)\ {U}(u)+\delta H_{\text{P}},\qquad\delta
H_{\text{P}}=\sum_{k}h_{k}(x)Y_{k}(u)$ (35)

where $h(x)$ and $h_{k}(x)$ are 2-component columns acting in the fundamental
representation of $SU(2)$. In what follows, we will consider the case

$H_{\text{P}}\simeq h(x)\,U(u),\qquad\delta H_{\text{P}}\ll h(x)\,U(u).$ (36)

The dimensionality of the proto-Higgs field is $[H_{P}]=m_{D}^{(D-2)/2}$,
$[h]=[h_{k}]=m_{D}$, $[U]=[Y_{k}]=m_{D}^{n/2}$.

Our immediate aim is to find the distribution of the field $H_{\text{P}}$ over
the extra coordinates governed by the scalar function $U(u)$ by solving Eqs.
(34), (36). The inhomogeneities of the field $h(x)$ are important at low
energies, but they are exponentially stretched during the first de Sitter-like
stage so that $h(x)={\rm const}$ with great accuracy. It means that

$h(x)=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v_{0}+\rho(x)\end{pmatrix}\simeq\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v\end{pmatrix}.$ (37)

Therefore, the approximation (37) transforms Eq. (34) in the following way:

$\square_{n}U(u)=\nu U(u)-\lambda\,v^{2}U^{3}(u),$ (38)

with a yet unknown parameter $v$. We suppose further on that the metric
functions as well as the Lagrangian parameters remains the same as those
considered above with one exception: the Hubble parameter is extremely small
at the present epoch as compared to the inflationary epoch, and below we put
$H\approx 0$.

The knowledge of solutions to Eq. (38) permits us to integrate out the
internal coordinates and to reduce the action (32) to the 4D form

$\displaystyle{}S_{\text{H}}=\frac{\mathcal{V}_{\text{n}-1}}{2}\int
d^{4}x\sqrt{|\tilde{g}_{4}|}\int_{u_{\min}}^{u_{\max}}\biggl{[}\text{e}^{-2\gamma(u)}U^{2}(u)\tilde{g}^{ij}\partial_{i}h^{\dagger}\partial_{j}h$
$\displaystyle{}\hskip
72.26999pt+\Bigl{(}-(\partial_{u}U)^{2}+\nu\,U^{2}(u)\Bigr{)}h^{\dagger}h-\lambda\,U^{4}(u)\bigl{(}h^{\dagger}h\bigr{)}^{2}\biggr{]}\text{e}^{4\gamma(u)}r^{n-1}(u)\,du$

after substitution of (36) into (32). To study this action at low energies, we
choose the Minkowski metric

$\tilde{g}_{4,ij}=\eta_{ij}$ (39)

and define the following parameters by integration over $u$:

$\displaystyle{}K_{h}=\mathcal{V}_{\text{n}-1}\int_{u_{\min}}^{u_{\max}}U^{2}(u)\,\text{e}^{2\gamma(u)}r^{\text{n}-1}(u)\,du,$
(40)
$\displaystyle{}m^{2}_{h}=\mathcal{V}_{\text{n}-1}\int_{u_{\min}}^{u_{\max}}\Bigl{(}-(\partial_{u}U)^{2}+\nu\,U^{2}(u)\Bigr{)}\text{e}^{4\gamma(u)}r^{\text{n}-1}(u)\,du,$
(41)
$\displaystyle{}\lambda_{h}=\mathcal{V}_{\text{n}-1}\int_{u_{\min}}^{u_{\max}}\lambda\,U^{4}(u)\,\text{e}^{4\gamma(u)}r^{\text{n}-1}(u)\,du.$
(42)

###  5.2 Comparison with the Higgs parameters

Recall that a natural range for the dimensionless parameters
$\lambda^{\prime}$ and $\nu^{\prime}$ is $10^{-2}$ to $10^{2}$. It means that
acceptable ranges of the ‘‘physical’’ parameters are $(10^{-6}\div 10^{6})$
for $\lambda$ and $(10^{-4}\div 10^{4})$ for $\nu$ according to the
definitions (33).

The substitution

$H_{0}(x)={h(x)\sqrt{K_{h}}}$ (43)

leads to the 4D effective Higgs Lagrangian

$\displaystyle{}S_{\text{H}}=\frac{1}{2}\int
d^{4}x\sqrt{|\tilde{g}_{4}|}\biggl{(}\partial_{i}H_{0}^{\dagger}\partial^{i}H_{0}+m_{H}^{2}H_{0}^{\dagger}H_{0}-\lambda_{H}\bigl{(}H_{0}^{\dagger}H_{0}\bigr{)}^{2}\biggr{)}\,,$
(44) $\displaystyle{}\hskip
28.45274ptm_{H}^{2}\equiv\frac{m_{h}^{2}}{K_{h}},\qquad\lambda_{H}\equiv\frac{\lambda_{h}}{K_{h}^{2}}.$
(45)

Here $H_{0}$ is the observable Higgs field at zero energy. The experimentally
measured parameters are the Higgs boson mass and its vacuum average,

$m_{\rm Higgs}=125\,\text{GeV},\quad v_{\rm Higgs}=246\,\text{GeV}\,.$ (46)

according to [58]. They are related to the parameters $m_{H}$ and
$\lambda_{H}$ of the effective Higgs action (44) as follows:

$m_{H}=m_{\rm Higgs}/\sqrt{2}=88.6\,\text{GeV}\simeq 10^{-17}m_{\text{Pl}},$
(47)

and

$\lambda_{H}=({m_{H}}/v_{\rm Higgs})^{2}/2\simeq 0.13,$ (48)

The vacuum energy of the Higgs field is

$V_{\min}=-\frac{1}{2}m_{H}^{2}v_{\rm Higgs},$

so that the parameter $c$ in the function $f(R)$ should be corrected, $c\to
c+V_{\min}$. Note, however, that $V_{\min}$ is very small as compared to the
D-dimensional Planck scale and may be neglected.

The above formulas contain the function $U(u)$, the solution to Eq. (38) with
a yet unknown constant $v$. It is of interest that the Lagrangian structure
(32) allows us to avoid the determination of this constant. Indeed, a solution
to Eq. (38) can be found for the function

$\tilde{U}(u)=vU(u)$

because Eq. (38) $\tilde{U}(u)$ does not contain the unknown parameter $v$ in
this case. Moreover, substitution of $U=\tilde{U}(u)/v$ into the expressions
(40), (41) and (42) gives

$K_{h}[U]=\frac{K_{h}[\tilde{U}]}{v^{2}},\qquad
m_{h}^{2}[U]=\frac{m_{h}^{2}[\tilde{U}]}{v^{2}},\qquad\lambda_{h}[U]=\frac{\lambda_{h}[\tilde{U}]}{v^{4}},$
(49)

hence the observable parameters $m_{H}$ and $\lambda_{H}$ in (45) do not
depend on $v$. This quantity also appears in the relation

$v\simeq v_{\rm Higgs}/\sqrt{K_{h}[U]},$ (50)

following from (43) and the substitution $H_{0}\to v_{\rm Higgs},\,h\to v$.
Luckily, this relation does not depend on $v$ as well. After taking into
account the first equality in (49), we obtain an additional restriction for
the function $\tilde{U}$:

$1\simeq v_{\rm Higgs}/\sqrt{K_{h}[\tilde{U}]}.$ (51)

The quantity $K_{h}[\tilde{U}]$ is calculated in $m_{D}$ units. Therefore,
$v_{\rm Higgs}$ should be also expressed in $m_{D}$ units.

Figure 3: Solution of Eq. (38) for the function $\tilde{U}(u)=vU(u)$.
Parameter values: $\nu=7.5\times 10^{-6}$, $\lambda=2.9\times 10^{5}$ (the
dimensionless parameters defined by (33) are as follows: $\nu^{\prime}\simeq
0.003,\ \lambda^{\prime}\simeq 70$). Additional conditions:
$\tilde{U}(0)=4.5\times 10^{-17}$, $\tilde{U}^{\prime}(0)=2.6\times 10^{-20}$
in the background shown in Fig. 1. For this solution, $v_{\rm
Higgs}/\sqrt{K_{h}[\tilde{U}]}\simeq 0.99,\ \lambda_{H}\simeq 0.13,\
m_{H}\simeq 10^{-17}m_{\text{Pl}}$.

Figure 3 presents the Higgs field distribution $\tilde{U}$ in the extra
dimensions.

In this section, we have found the conditions at which the initial parameter
values written in Fig.3 reproduce the Higgs Lagrangian with the observed
parameters. The origin of small parameters has been discussed earlier, see the
beginning of Sec. 4. Quantum fluctuations produce a variety of field
amplitudes in the countable set of pocket universes. A small measure of them
contains (extremely) small amplitudes $\tilde{U}$. The fields values of the
order of $\tilde{U}\sim 10^{-16}$ (see Fig. 3) are suitable for the relations
(47), (48) and (51).

##  6 Low energies. The cosmological constant

Analytical formulas for 4D gravity have been obtained in Sec. 4. It is assumed
that they are approximately valid at the inflationary scale $\sim 10^{13}$
GeV. The parameter $a_{\text{eff}}$ was obtained for specific values of the
initial parameters $a,c,m_{D}$. A calculation of $c_{\text{eff}}$ representing
the effective cosmological constant (CC) is not necessary because its value
could vary in a wide range without any effect on the inflationary process.
This value is important at low energies where the Hubble parameter is $\sim
10^{-61}\approx 0$ GeV2.

Therefore, the extra metric must be found by solving the Einstein equations at
$H=0$. Luckily, the resulting metric weakly depends on the Hubble parameter if
it varies within the interval $0<H<0.01$, and Fig. 1 gives the appropriate
impression.

At the low energy scale, the Hubble parameter $H$ is small, and the curvature
squared $R_{4}^{2}$ can be neglected in (25). In this case, the 4D Einstein
equations lead to a relation between the CC and the Hubble parameter,

$c_{\text{eff}}\equiv-2\Lambda=-6\,H^{2},$ (52)

which means that $c_{\text{eff}}$ should be extremely small as well.

On the other hand, the same value was found above starting from the initial
D-dimensional action, see Eq. (29). It can be presented in the following form
(see the Appendix):

$\displaystyle c_{\text{eff}}$ $\displaystyle=$
$\displaystyle-6H^{2}+\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{\text{D}-2}}{m_{\text{Pl}}^{2}}2\int_{u_{\min}}^{u_{\max}}\Big{[}\Big{(}f_{RR}\,R^{\prime}-f_{R}\,\gamma^{\prime}\Big{)}\,\text{e}^{4\gamma}r^{\text{n}-1}\Big{]}^{\prime}\,du+O(H^{6}).$
(53)

A comparison of the expressions (52) and (53) derived with arbitrary initial
parameters and boundary conditions indicates that the integral in (53) must be
zero. It makes sense to prove this statement directly. To this end, we should
find the function
$\Phi(u)\equiv\Big{(}f_{RR}\,R^{\prime}-f_{R}\,\gamma^{\prime}\Big{)}\,\text{e}^{4\gamma}r^{\text{n}-1}$
at the boundary points $u_{\min}$ and $u_{\max}$. Numerical simulations
indicate that this function tends to zero indeed, see Fig.4. Unfortunately,
the accuracy is unsatisfactory while approaching the boundary points. To
clarify the situation, the following can be suggested: we modify the nonlinear
term in the action $R^{2}\to R^{2}e^{-\epsilon R^{2}}$ from the beginning,
$\epsilon\lll 1$ in $m_{D}$ units, and put $\epsilon=0$ finally. It does not
affect the equation of motion, but smooths out the singularities. In this
case, $\Phi\to 0$ at the boundary points, where $r=0$ by definition. Hence,
the integral as a whole equals zero.

Figure 4: The integrand
$\Phi(u)=\Big{(}f_{RR}\,R^{\prime}-f_{R}\,\gamma^{\prime}\Big{)}\,\text{e}^{4\gamma}r^{\text{n}-1}$
in the right-hand side of Eq. (53) is zero at the boundary points $u_{\max}$
and $u_{\min}$. Solution of equations Eqs.(2)–(10) is taken for parameters
$n=3$, $f(R)=300R^{2}+R+0.002$, $H=0$, $V(\zeta)=0.01\,\zeta^{2}/2$ and
boundary conditions $r(0)=50$, $\gamma(0)=0$, $R(0)\simeq 0.00396$,
$\zeta(0)=0$, $r^{\prime}(0)=\gamma^{\prime}(0)=R^{\prime}(0)=0$,
$\zeta^{\prime}(0)=3.62\times 10^{-5}$.

In this subsection, we have proved that the well-known relation
$H^{2}=\Lambda/3$ can be derived from D-dimennsional gravity. In the standard
notations, $c_{\text{eff}}=-2\Lambda$, where $\Lambda$ is the cosmological
constant. In a more general case of the the function $f(R)$, terms
proportional to $H^{6}$ and other nontrivial terms can appear in the
expression (53). In this case, the inflationary dynamic requires a separate
study.

##  7 The role of quantum fluctuations

###  7.1 The smallest scale of compact extra dimensions

It is known that our instruments ‘‘feel’’ average values of a field
$\bar{\zeta}(u)$ calculated as

$\bar{\zeta}(u)=\zeta_{\rm classical}(u)+\delta\zeta(u),$

where $\zeta_{classical}(u)$ is the classical part, and $\delta\zeta(u)$ is a
quantum correction to it. It makes sense to calculate the classical part only
if $\zeta_{\rm classical}(u)\gg\delta\zeta(u)$. This inequality holds only if
the action $S\gg 1$ (the steepest descend method). Thus the classical approach
can be applied in our approach if

$S=\int dv_{D}f(R)\gg 1,$ (54)

or in other words

$S\simeq\delta v_{D}\langle f(R)\rangle\gg 1,$ (55)

where $\delta v_{D}\simeq\delta u^{D}$ is a small volume parametrized by the
coordinate $u$, and $\langle...\rangle$ stands for averaging over this volume.
It means that the volume $\delta v_{D}$ must not be smaller than $\sim
1/\langle f(R)\rangle\to\delta u\sim\langle f(R)\rangle^{-1/D}.$ Thus for a
classical description to make sense (i.e. to approximately coincide with the
averages), the averaging range must not be smaller than

$\delta u\sim\langle f(R)\rangle^{-1/D}.$ (56)

For example, for a seven-dimensional space and $f(R)\sim 10$, we have $\delta
u\sim 10^{-1/7}\sim 1$. It means that the size of extra dimensions should be
larger than $1/m_{D}$. Also, it is dangerous to make physical conclusions
based on classical solutions in the vicinity of singular points at a distance
smaller than $1/m_{D}$.

###  7.2 Fluctuations in the present epoch

The field value $H_{P}\propto 10^{-17}$ is quite a small value. Let us
estimate the probability of large fluctuations that could destroy the
solution. The cosmological probability to find a field value $\chi_{2}$ at the
instant $t_{2}=t_{1}+t$ in a spatial region of the horizon size $H^{-1}$ was
studied in [59, 60]. Based on the outcomes, it is possible to demonstrate that
the first mode amplitude fluctuation probability, $h_{1}$, can be determined
as

$\displaystyle
dP=dP_{1}=dh_{1}\cdot\sqrt{q_{1}/\pi}\exp[-q_{1}\,h_{1}^{2}],\quad t\to\infty$
(57)

where

$\displaystyle
q_{1}=\frac{\mu}{\sigma^{2}},\qquad\mu=\frac{m_{1}^{2}}{3H},\qquad\sigma=\frac{H^{3/2}}{2\pi}.$
(58)

$m_{1}\sim m_{D}$, and the present-day Hubble parameter is $H=1.2\times
10^{-61}m_{\text{Pl}}$. Their knowledge allows one to estimate the parameter
$q_{1}$. The fluctuation $h_{1}$ should be of the order of the classical part,
$h_{1}\sim\tilde{U}\sim 10^{-17}$ (see Fig. 3) or larger to destroy it. Now we
have everything to estimate the exponent,

$q_{1}\,h_{1}^{2}\sim\text{e}^{211}\left(\frac{m_{D}}{M_{\text{Pl}}}\right)^{4},\qquad
m_{D}>10^{-5}M_{\text{Pl}}.$

This estimate can be substituted to (57) to demonstrate how unlikely it is
that even such tiny classical field ($\tilde{U}\sim 10^{-17}$) will be
destroyed.

###  7.3 Quantum corrections

The essence of the Wilson approach is to fix a Lagrangian and its parameters
at the highest scale and shift down to a low energy scale. It is achieved by
sequentially integrating the Euclidean action over a small slice of the
momentum interval $\Delta k_{E}$. The renormalization group equations thus
obtained are widely used in this concern [31]. The relations between low-
energy parameter values and high-energy ones are discussed in [61]. Also,
quantum fluctuations could modify the same form of a Lagrangian, [62, 63].

The inclusion of a compact extra space into consideration complicates the
procedure. Indeed, we cannot choose an arbitrarily small momentum interval due
to the energy level discreteness. For example, if a size is quite small,
$\Delta k_{E}<1/r$, $r$ being the scale of extra dimensions, then this
momentum interval does not contain energy levels at all. A possible way to
overcome this difficulty is discussed in [43], where truncated Green functions

$G_{T}(Z,Z^{\prime})\equiv\sum_{N\in\cal{N}}\frac{Y_{N}(Z)Y_{N}(Z^{\prime})^{*}}{\lambda_{N}}$

were introduced. Here $Y_{N}(Z)$ is a subset of $n+4$-dimensional
eigenfunctions. The coordinates $Z$ describe both 4D space and a compact extra
space. It allows for approximately calculating the parameters at low energies.
As a result, quantum corrections caused by a scalar field are proportional to
its self-coupling. This means that such quantum effects cannot be responsible
for reducing the parameter values by many orders of magnitude, from the Planck
scale to the electroweak scale. The classical mechanism discussed in this
paper was elaborated just for this aim. The procedure of quantum
renormalization is a necessary and unavoidable element that leads to fine
tuning of the physical parameters at low energies.

##  8 Conclusion

In this paper, we have discussed an approach which provided the hierarchy of
three energy scales, the inflationary, electroweak and cosmological ones.
Necessary tools for small parameters formation and a successful solution of
the problem are $f(R)$ gravity and inhomogeneous extra dimensions.

The set of small parameters is formed in the following way. Slow rolling of a
spatial domain from a sub-Planckian scale down to the inflationary one gives
rise to several consequences: (1) nucleation of an infinite set of causally
disconnected domains (pocket universes), (2) quantum fluctuations in each
domain produce a variety of fields and an extra-space metric distribution, (3)
these distributions are stabilized when the energy scale is low enough. Self-
gravitating (scalar) fields do not necessarily settle at states with minimum
energy. On the contrary, e.g., the boson stars activity [55] is based on the
fact that self-gravitating scalar fields can settle at a continuum set of
static states. There are states with arbitrarily small amplitudes among them.
These states are formed in a small but finite set of universes. As a result, a
small but nonzero measure of different universes contains small effective
parameters that are applied here to solve the Hierarchy problem at three
energy scales.

The mechanism developed should be accompanied by a renormalization group
analysis aimed at correction of the initial parameter values.

## Acknowledgements

The work of SGR and KAB was funded by the Ministry of Science and Higher
Education of the Russian Federation, Project "New Phenomena in Particle
Physics and the Early Universe" FSWU-2023-0073 and the Kazan Federal
University Strategic Academic Leadership Program. The work of AAP was funded
by the development program of Volga Region Mathematical Center (agreement No.
075-02-2023-944). KAB also acknowledges support from Project No.
FSSF-2023-0003.

## Appendix

The validity Condition of Eq. (52) is not so trivial in the $4+n$-dimensional
case, and we will discuss it here.

We can exclude the terms with a scalar field in the definition of
$c_{\text{eff}}$; using the expression (2), we obtain

$\displaystyle c_{\text{eff}}$
$\displaystyle=\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{D-2}}{m_{\text{Pl}}^{2}}\int\limits_{u_{\text{min}}}^{u_{\text{max}}}\Bigl{(}f\bigl{(}R_{\text{n}}\bigr{)}-\bigl{(}\zeta^{\prime}\bigr{)}^{2}-2V\bigl{(}\zeta\bigr{)}\Bigr{)}\,\text{e}^{4\gamma}\,r^{\text{n}-1}\,du$
$\displaystyle=\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{D-2}}{m_{\text{Pl}}^{2}}\int_{u_{\text{min}}}^{u_{\text{max}}}\,\left\\{f\bigl{(}R_{\text{n}}\bigr{)}+2{R^{\prime}}^{2}f_{RRR}(R)+2\Bigl{[}R^{\prime\prime}+R^{\prime}\Big{(}3\gamma^{\prime}+(\text{n}-1)\dfrac{r^{\prime}}{r}\Big{)}\Bigr{]}f_{RR}(R)\right.$
$\displaystyle\ \ \
\left.-2\left(\gamma^{\prime\prime}+4{\gamma^{\prime}}^{2}+(\text{n}-1)\frac{\gamma^{\prime}r^{\prime}}{r}\right)f_{R}(R)+\frac{6H^{2}}{\text{e}^{2\gamma(u)}}f_{R}-f(R)\right\\}\,\text{e}^{4\gamma}r^{\text{n}-1}\,du=0.$
(59)

Part of this expression can be transformed as follows:

$\displaystyle\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{D-2}}{m_{\text{Pl}}^{2}}\int_{u_{\text{min}}}^{u_{\text{max}}}\,\left\\{\biggl{(}2{R^{\prime}}^{2}f_{RRR}+2\Bigl{[}R^{\prime\prime}+R^{\prime}\Big{(}3\gamma^{\prime}+(\text{n}-1)\dfrac{r^{\prime}}{r}\Big{)}\Bigr{]}f_{RR}-2\left(\gamma^{\prime\prime}+4{\gamma^{\prime}}^{2}\right.\right.$
$\displaystyle\left.\left.+(\text{n}-1)\frac{\gamma^{\prime}r^{\prime}}{r}\right)f_{R}\right\\}\,\text{e}^{4\gamma}r^{\text{n}-1}\,du=\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{D-2}}{m_{\text{Pl}}^{2}}2\int\Big{[}\Big{(}f_{RR}R^{\prime}-f_{R}\gamma^{\prime}\Big{)}\,\text{e}^{4\gamma}r^{\text{n}-1}\Big{]}^{\prime}\,du.$
(60)

The remaining part of the expression (Appendix) can be rewritten in a more
conventional form (by substituting the expansion (24) and the definition
(23)):

$\displaystyle\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{D-2}}{m_{\text{Pl}}^{2}}\int\left(f(R_{\text{n}})+\frac{6H^{2}}{\text{e}^{2\gamma}}f_{R}(R)-f(R)\right)\,\text{e}^{4\gamma}r^{\text{n}-1}\,du$
$\displaystyle\hskip
28.45274pt=\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{D-2}}{m_{\text{Pl}}^{2}}\int\left[f(R_{\text{n}})+\frac{6H^{2}}{\text{e}^{2\gamma(u)}}\left(f_{R}(R_{n})+\frac{R_{4}}{\text{e}^{2\gamma(u)}}f_{RR}(R_{n})\right)-\Big{(}f(R_{n})\right.$
$\displaystyle\hskip
72.26999pt\left.+\frac{R_{4}}{\text{e}^{2\gamma(u)}}f_{R}(R_{n})+\frac{R_{4}^{2}}{2\text{e}^{4\gamma(u)}}f_{RR}(R_{n})\Big{)}+O\left(\frac{R_{4}^{3}}{\text{e}^{6\gamma(u)}}f_{RRR}(R_{n})\right)\right]\,\text{e}^{4\gamma}r^{\text{n}-1}\,du$
$\displaystyle\hskip
28.45274pt=\mathcal{V}_{\text{n}-1}\frac{m_{\text{D}}^{D-2}}{m_{\text{Pl}}^{2}}\int\left[\frac{(6H^{2}-R_{4})}{\text{e}^{2\gamma}}f_{R}(R_{n})+\frac{(12H^{2}-R_{4})R_{4}}{2\text{e}^{4\gamma}}f_{RR}(R_{n})\right.$
$\displaystyle\hskip
72.26999pt\left.+O\left(\frac{R_{4}^{3}}{\text{e}^{6\gamma(u)}}f_{RRR}(R_{n})\right)\right]\,\text{e}^{4\gamma}r^{\text{n}-1}\,du.$
(61)

Since $m_{\text{Pl}}^{2}$ is defined by the expression (27) and
$R_{4}=12H^{2}$, Eq. (Appendix) can be rewritten as

$-6H^{2}+O(H^{6}).$ (62)

Thus we can present $c_{\text{eff}}$ in the form of (53) by summing the
expressions (Appendix) and (62).

## References

  * [1] Robert H. Brandenberger, Ali Nayeri, Subodh P. Patil and Cumrun Vafa  ‘‘String gas cosmology and structure formation’’  In _Int. J. Mod. Phys._ A22, 2007, pp. 3621–3642  DOI: 10.1142/S0217751X07037159
  * [2] Max Tegmark, Anthony Aguirre, Martin Rees and Frank Wilczek  ‘‘Dimensionless constants, cosmology and other dark matters’’  In _Phys. Rev._ D73, 2006, pp. 023505  DOI: 10.1103/PhysRevD.73.023505
  * [3] Abraham Loeb  ‘‘An Observational Test for the Anthropic Origin of the Cosmological Constant’’  In _J. Cosmol. Astropart. Phys._ 0605, 2006, pp. 009  DOI: 10.1088/1475-7516/2006/05/009
  * [4] Amjad Ashoorioon, Konstantinos Dimopoulos, M.. Sheikh-Jabbari and Gary Shiu  ‘‘Reconciliation of High Energy Scale Models of Inflation with Planck’’  In _J. Cosmol. Astropart. Phys._ 1402, 2014, pp. 025  DOI: 10.1088/1475-7516/2014/02/025
  * [5] Richard B. Abbott, Stephen M. Barr and Stephen D. Ellis  ‘‘Kaluza-Klein Cosmologies and Inflation’’  In _Phys. Rev._ D30, 1984, pp. 720  DOI: 10.1103/PhysRevD.30.720
  * [6] Masud Chaichian and Archil B. Kobakhidze  ‘‘Mass hierarchy and localization of gravity in extra time’’  In _Phys. Lett._ B488, 2000, pp. 117–122  DOI: 10.1016/S0370-2693(00)00874-1
  * [7] Lisa Randall and Raman Sundrum  ‘‘An Alternative to compactification’’  In _Phys. Rev. Lett._ 83, 1999, pp. 4690–4693  DOI: 10.1103/PhysRevLett.83.4690
  * [8] Adam R. Brown, Alex Dahlen and Ali Masoumi  ‘‘Compactifying de Sitter space naturally selects a small cosmological constant’’  In _Phys. Rev._ D90.12, 2014, pp. 124048  DOI: 10.1103/PhysRevD.90.124048
  * [9] K.A. Bronnikov, S.G. Rubin and I.V. Svadkovsky  ‘‘High-order multidimensional gravity and inflation’’  In _Grav. & Cosm._ 15, 2009, pp. 32–33  DOI: 10.1134/S0202289309010083
  * [10] Axel Krause  ‘‘A Small cosmological constant and back reaction of nonfinetuned parameters’’  In _J. High Energ. Phys._ 09, 2003, pp. 016  DOI: 10.1088/1126-6708/2003/09/016
  * [11] A.. Green and A. Mazumdar  ‘‘Dynamics of a large extra dimension inspired hybrid inflation model’’  In _Phys. Rev. D_ 65.10, 2002, pp. 105022  DOI: 10.1103/PhysRevD.65.105022
  * [12] K.A. Bronnikov, S.G. Rubin and I.V. Svadkovsky  ‘‘Multidimensional world, inflation and modern acceleration’’  In _Phys. Rev. D_ 81, 2010, pp. 084010  DOI: 10.1103/PhysRevD.81.084010
  * [13] Júlio C. Fabris, Arkady A. Popov and Sergey G. Rubin  ‘‘Multidimensional gravity with higher derivatives and inflation’’  In _Phys. Lett. B_ 806, 2020, pp. 135458  DOI: 10.1016/j.physletb.2020.135458
  * [14] U. Günther, P. Moniz and A. Zhuk  ‘‘Nonlinear multidimensional cosmological models with form fields: Stabilization of extra dimensions and the cosmological constant problem’’  In _prd_ 68.4, 2003, pp. 044010  DOI: 10.1103/PhysRevD.68.044010
  * [15] U. Günther, P. Moniz and A. Zhuk  ‘‘Asymptotical AdS space from nonlinear gravitational models with stabilized extra dimensions’’  In _Phys. Rev. D_ 66.4, 2002, pp. 044014  DOI: 10.1103/PhysRevD.66.044014
  * [16] A. Arbuzov, B. Latosh and A. Nikitenko  ‘‘Effective potential of scalar-tensor gravity with quartic self-interaction of scalar field’’  In _Class. Quant. Grav._ 39.5, 2022, pp. 055003  DOI: 10.1088/1361-6382/ac4827
  * [17] Alexei A. Starobinsky  ‘‘A New Type of Isotropic Cosmological Models Without Singularity’’  In _Phys. Lett._ B91, 1980, pp. 99–102  DOI: 10.1016/0370-2693(80)90670-X
  * [18] Vakhid A. Gani, Alexander E. Dmitriev and Sergei G. Rubin  ‘‘Deformed compact extra space as dark matter candidate’’  In _Int. J. Mod. Phys._ D24, 2015, pp. 1545001  DOI: 10.1142/S0218271815450017
  * [19] Elena Arbuzova, Alexander Dolgov and Rajnish Singh  ‘‘$R^{2}$-Cosmology and New Windows for Superheavy Dark Matter’’  In _Symmetry_ 13.5, 2021, pp. 877  DOI: 10.3390/sym13050877
  * [20] Antonio De Felice and Shinji Tsujikawa  ‘‘f(R) theories’’  In _Living Rev. Rel._ 13, 2010, pp. 3  DOI: 10.12942/lrr-2010-3
  * [21] K. Bamba et al.  ‘‘Bounce cosmology from F(R) gravity and F(R) bigravity’’  In _J. Cosmol. Astropart. Phys._ 1, 2014, pp. 8  DOI: 10.1088/1475-7516/2014/01/008
  * [22] Leszek M. Sokolowski  ‘‘Metric gravity theories and cosmology:II. Stability of a ground state in f(R) theories’’  In _Class. Quant. Grav._ 24, 2007, pp. 3713–3734  DOI: 10.1088/0264-9381/24/14/011
  * [23] S. Nojiri, S.. Odintsov and P.. Tretyakov  ‘‘Dark energy from modified F(R)-scalar-Gauss Bonnet gravity’’  In _Physics Letters B_ 651, 2007, pp. 224–231  DOI: 10.1016/j.physletb.2007.06.029
  * [24] S. Nojiri, S.. Odintsov and V.. Oikonomou  ‘‘Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution’’  In _Phys. Rept._ 692, 2017, pp. 1–104  DOI: 10.1016/j.physrep.2017.06.001
  * [25] Julian S. Schwinger  ‘‘On gauge invariance and vacuum polarization’’  In _Phys. Rev._ 82, 1951, pp. 664–679  DOI: 10.1103/PhysRev.82.664
  * [26] Masoumeh Moazze Sorkhi and Zahra Ghalenovi  ‘‘Fermion Localization on the Deformed Brane with the Derivative Coupling Mechanism’’  In _Acta Phys. Polon. B_ 49, 2018, pp. 123–144  DOI: 10.5506/APhysPolB.49.123
  * [27] Tao-Tao Sui, Li Zhao, Yu-Peng Zhang and Qun-Ying Xie  ‘‘Localization and mass spectra of various matter fields on Weyl thin brane’’  In _Eur. Phys. J. C_ 77.6, 2017, pp. 411  DOI: 10.1140/epjc/s10052-017-4922-6
  * [28] Masato Arai, Filip Blaschke, Minoru Eto and Norisuke Sakai  ‘‘Massless bosons on domain walls: Jackiw-Rebbi-like mechanism for bosonic fields’’  In _Phys. Rev. D_ 100.9, 2019, pp. 095014  DOI: 10.1103/PhysRevD.100.095014
  * [29] A… Chumbes, J.. Hoff da Silva and M.. Hott  ‘‘A model to localize gauge and tensor fields on thick branes’’  In _Phys. Rev. D_ 85, 2012, pp. 085003  DOI: 10.1103/PhysRevD.85.085003
  * [30] K.. Bronnikov and V.. Melnikov  ‘‘Conformal frames and D-dimensional gravity’’  In _International School of Cosmology and Gravitation: 18th Course: The Gravitational Constant: Generalized Gravitational Theories and Experiments: A NATO Advanced Study Institute Erice, Italy, April 30-May 10, 2003_ , 2003, pp. 39–64  DOI: 10.1007/978-1-4020-2242-5_2
  * [31] Michael E. Peskin and Daniel V. Schroeder  ‘‘An Introduction to quantum field theory’’  In _Reading, USA: Addison-Wesley (1995) 842 p_ , 1995  URL: http://www.slac.stanford.edu/spires/find/books/www?cl=QC174.45%3AP4
  * [32] C.. Burgess  ‘‘The Cosmological Constant Problem: Why it’s hard to get Dark Energy from Micro-physics’’  In _Proceedings, 100th Les Houches Summer School: Post-Planck Cosmology: Les Houches, France, July 8 - August 2, 2013_ , 2015, pp. 149–197  DOI: 10.1093/acprof:oso/9780198728856.003.0004
  * [33] Mark P. Hertzberg and Ali Masoumi  ‘‘Can Compactifications Solve the Cosmological Constant Problem?’’  In _J. Cosmol. Astropart. Phys._ 1606.06, 2016, pp. 053  DOI: 10.1088/1475-7516/2016/06/053
  * [34] A. Babic, B. Guberina, R. Horvat and H. Stefancic  ‘‘Renormalization group running of the cosmological constant and its implication for the Higgs boson mass in the standard model’’  In _Phys. Rev._ D65, 2002, pp. 085002  DOI: 10.1103/PhysRevD.65.085002
  * [35] E. Dudas, C. Papineau and V.. Rubakov  ‘‘Flowing to four dimensions’’  In _J. High Energ. Phys._ 03, 2006, pp. 085  DOI: 10.1088/1126-6708/2006/03/085
  * [36] Christof Wetterich  ‘‘Effective average action in statistical physics and quantum field theory’’  In _Int. J. Mod. Phys. A_ 16, 2001, pp. 1951–1982  DOI: 10.1142/S0217751X01004591
  * [37] Ana R. Romero Castellanos, Flavia Sobreira, Ilya L. Shapiro and Alexei A. Starobinsky  ‘‘On higher derivative corrections to the $R+R^{2}$ inflationary model’’  In _JCAP_ 12, 2018, pp. 007  DOI: 10.1088/1475-7516/2018/12/007
  * [38] Sergey G. Rubin  ‘‘Scalar field localization on deformed extra space’’  In _Eur. Phys. J._ C75.7, 2015, pp. 333  DOI: 10.1140/epjc/s10052-015-3553-z
  * [39] S.. Rubin  ‘‘The role of initial conditions in the universe formation’’  In _Grav. & Cosm._ 21, 2015, pp. 143–151  DOI: 10.1134/S0202289315020103
  * [40] Valery V. Nikulin, Polina M. Petriakova and Sergey G. Rubin  ‘‘Formation of Conserved Charge at the de Sitter Space’’  In _Particles_ 3.2, 2020, pp. 355–363  DOI: 10.3390/particles3020027
  * [41] Valery V. Nikulin and Sergey G. Rubin  ‘‘Cosmological baryon/lepton asymmetry in terms of Kaluza–Klein extra dimensions’’  In _Int. J. Mod. Phys. D_ 30.16, 2021, pp. 2140004  DOI: 10.1142/S0218271821400046
  * [42] K.. Bronnikov et al.  ‘‘Inhomogeneous compact extra dimensions’’  In _J. Cosmol. Astropart. Phys._ 10, 2017, pp. 001  DOI: 10.1088/1475-7516/2017/10/001
  * [43] Sergey G. Rubin  ‘‘How to make the physical parameters small’’  In _Adv. High Energy Phys._ 2020, 2020, pp. 1048585  DOI: 10.1155/2020/1048585.
  * [44] Itsaso Olasagasti and Alexander Vilenkin  ‘‘Gravity of higher-dimensional global defects’’  In _Phys. Rev. D_ 62.4, 2000, pp. 044014  DOI: 10.1103/PhysRevD.62.044014
  * [45] Inyong Cho and Alexander Vilenkin  ‘‘Gravity of superheavy higher-dimensional global defects’’  In _Phys. Rev. D_ 68.2, 2003, pp. 025013  DOI: 10.1103/PhysRevD.68.025013
  * [46] Satsuki Shimono and Takeshi Chiba  ‘‘Numerical solutions of inflating higher dimensional global defects’’  In _Phys. Rev. D_ 71.8, 2005, pp. 084002  DOI: 10.1103/PhysRevD.71.084002
  * [47] Christophe Ringeval, Patrick Peter and Jean-Philippe Uzan  ‘‘Stability of six-dimensional hyperstring braneworlds’’  In _Phys. Rev. D_ 71.10, 2005, pp. 104018  DOI: 10.1103/PhysRevD.71.104018
  * [48] Ruth Gregory  ‘‘Nonsingular Global String Compactifications’’  In _Phys. Rev. Lett._ 84.12, 2000, pp. 2564–2567  DOI: 10.1103/PhysRevLett.84.2564
  * [49] Tony Gherghetta and Mikhail Shaposhnikov  ‘‘Localizing Gravity on a Stringlike Defect in Six Dimensions’’  In _Phys. Rev. Lett._ 85.2, 2000, pp. 240–243  DOI: 10.1103/PhysRevLett.85.240
  * [50] K.. Bronnikov and B.. Meierovich  ‘‘Global strings in extra dimensions: A Full map of solutions, matter trapping and the hierarchy problem’’  In _J. Exp. Theor. Phys._ 106, 2008, pp. 247–264  DOI: 10.1007/s11447-008-2005-0
  * [51] Nima Arkani-Hamed, Savas Dimopoulos and G.. Dvali  ‘‘The Hierarchy problem and new dimensions at a millimeter’’  In _Phys. Lett._ B429, 1998, pp. 263–272  DOI: 10.1016/S0370-2693(98)00466-3
  * [52] T. Bringmann, M. Eriksson and M. Gustafsson  ‘‘Cosmological evolution of homogeneous universal extra dimensions’’  In _Phys. Rev. D_ 68.6, 2003, pp. 063516  DOI: 10.1103/PhysRevD.68.063516
  * [53] Alan H. Guth  ‘‘The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems’’  In _Phys. Rev. D_ 23, 1981, pp. 347–356  DOI: 10.1103/PhysRevD.23.347
  * [54] A.. Linde  ‘‘Particle Physics and Inflationary Cosmology’’  Harwood Academic Publishers, Switzerland, 1990
  * [55] Steven L. Liebling and Carlos Palenzuela  ‘‘Dynamical boson stars’’  In _Living Rev. Rel._ 15, 2012, pp. 6  DOI: 10.1007/s41114-023-00043-4
  * [56] Polina Petriakova, Arkady A. Popov and Sergey G. Rubin  ‘‘Flexible extra dimensions’’  In _Eur. Phys. J. C_ 83.5, 2023, pp. 371  DOI: 10.1140/epjc/s10052-023-11542-7
  * [57] Y. Akrami  ‘‘Planck 2018 results. X. Constraints on inflation’’  In _Astron. Astrophys._ 641, 2020, pp. A10  DOI: 10.1051/0004-6361/201833887
  * [58] R.. Workman  ‘‘Review of Particle Physics’’  In _PTEP_ 2022, 2022, pp. 083C01  DOI: 10.1093/ptep/ptac097
  * [59] M.. Khlopov and S.. Rubin  ‘‘Cosmological Pattern of Microphysics in the Inflationary Universe’’  P.O. Box 17, 3300 AA Dordrecht, The Netherlands: Kluwer Academic Publishers, 2004
  * [60] Sergey G. Rubin and Julio C. Fabris  ‘‘Distortion of extra dimensions in the inflationary Multiverse’’  In _arXiv e-prints_ , 2021, pp. arXiv:2109.08373  arXiv:2109.08373 [gr-qc]
  * [61] I.. Marian et al.  ‘‘Vacuum energy and renormalization of the field-independent term’’  In _JCAP_ 03.03, 2022, pp. 062  DOI: 10.1088/1475-7516/2022/03/062
  * [62] Lei-Hua Liu, Tomislav Prokopec and Alexei A. Starobinsky  ‘‘Inflation in an effective gravitational model and asymptotic safety’’  In _Phys. Rev. D_ 98.4, 2018, pp. 043505  DOI: 10.1103/PhysRevD.98.043505
  * [63] Vsevolod R. Ivanov, Sergei V. Ketov, Ekaterina O. Pozdeeva and Sergey Yu. Vernov  ‘‘Analytic extensions of Starobinsky model of inflation’’  In _J. Cosmol. Astropart. Phys._ 2022.3, 2022, pp. 058  DOI: 10.1088/1475-7516/2022/03/058