From Particle Physics to Cosmology

Advances in High Energy Physics

Perspectives on Decay and Time Evolution of Metastable States: From Particle Physics to Cosmology

Lead Guest Editor: Krzysztof Urbanowski Guest Editors: Neelima G. Kelkar, Marek Nowakowski, and Marek Szydlowski Perspectives on Decay and Time Evolution of Metastable States: From Particle Physics to Cosmology Advances in High Energy Physics

Perspectives on Decay and Time Evolution of Metastable States: From Particle Physics to Cosmology

This is a special issue published in “Advances in High Energy Physics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Antonio J. Accioly, Brazil Ricardo G. Felipe, Portugal Anastasios Petkou, Greece Giovanni Amelino-Camelia, Italy Chao-Qiang Geng, Taiwan Alexey A. Petrov, USA Luis A. Anchordoqui, USA Philippe Gras, France Thomas Rössler, Sweden Michele Arzano, Italy Xiaochun He, USA Diego Saez-Chillon Gomez, Spain T. Asselmeyer-Maluga, Germany Luis Herrera, Spain Takao Sakaguchi, USA Alessandro Baldini, Italy Filipe R. Joaquim, Portugal Juan José Sanz-Cillero, Spain Marco Battaglia, Switzerland Aurelio Juste, Spain Edward Sarkisyan-Grinbaum, USA Lorenzo Bianchini, Switzerland Theocharis Kosmas, Greece Sally Seidel, USA Roelof Bijker, Mexico Ming Liu, USA George Siopsis, USA Burak Bilki, USA Enrico Lunghi, USA Luca Stanco, Italy Adrian Buzatu, UK Salvatore Mignemi, Italy Jouni Suhonen, Finland Rong-Gen Cai, China Omar G. Miranda, Mexico Mariam Tórtola, Spain Anna Cimmino, France Grégory Moreau, France Smarajit Triambak, South Africa Osvaldo Civitarese, Argentina Piero Nicolini, Germany Jose M. Udías, Spain Andrea Coccaro, Italy Carlos Pajares, Spain Elias C. Vagenas, Kuwait Shi-Hai Dong, Mexico Sergio Palomares-Ruiz, Spain Sunny Vagnozzi, Sweden Lorenzo Fortunato, Italy Giovanni Pauletta, Italy YauW.Wah,USA Mariana Frank, Canada Yvonne Peters, UK Contents

Perspectives on Decay and Time Evolution of Metastable States: From Particle Physics to Cosmology K. Urbanowski ,N.G.Kelkar ,M.Nowakowski ,andM.Szydłowski Editorial (2 pages), Article ID 6274317, Volume 2018 (2018)

Feasibility Study of the Time Reversal Symmetry Tests in Decay of Metastable Positronium Atoms with the J-PET Detector A. Gajos ,C.Curceanu,E.Czerwiński,K.Dulski,M.Gorgol,N.Gupta-Sharma,B.C.Hiesmayr, B.Jasińska,K.Kacprzak,Ł.Kapłon,D.Kisielewska,G.Korcyl,P.Kowalski,T.Kozik,W.Krzemień, E.Kubicz,M.Mohammed,Sz.Niedńwiecki,M.Pałka,M.Pawlik-Niedńwiecka,L.Raczyński,J.Raj,Z.Rudy, S.Sharma,Shivani,R.Shopa,M.Silarski,M.Skurzok,W.Wiślicki,B.Zgardzińska,M.Zieliński, and P. Moskal Research Article (10 pages), Article ID 8271280, Volume 2018 (2018)

From Quantum Unstable Systems to the Decaying Dark Energy: Cosmological Implications Aleksander Stachowski ,MarekSzydłowski , and Krzysztof Urbanowski Research Article (8 pages), Article ID 7080232, Volume 2018 (2018)

A Probability Distribution for Quantum Tunneling Times José T. Lunardi and Luiz A. Manzoni Research Article (11 pages), Article ID 1372359, Volume 2018 (2018)

From de Sitter to de Sitter: Decaying Vacuum Models as a Possible Solution to the Main Cosmological Problems G. J. M. Zilioti, R. C. Santos, and J. A. S. Lima Research Article (7 pages), Article ID 6980486, Volume 2018 (2018)

Statistical Physics of the Inflaton Decaying in an Inhomogeneous Random Environment Z. Haba Research Article (9 pages), Article ID 7204952, Volume 2018 (2018)

QFT Derivation of the Decay Law of an Unstable Particle with Nonzero Momentum Francesco Giacosa Research Article (7 pages), Article ID 4672051, Volume 2018 (2018)

Time Dilation in Relativistic Quantum Decay Laws of Moving Unstable Particles Filippo Giraldi Research Article (10 pages), Article ID 7308935, Volume 2018 (2018)

A New Inflationary Universe Scenario with Inhomogeneous Quantum Vacuum Yilin Chen and Jin Wang Research Article (15 pages), Article ID 3916727, Volume 2018 (2018)

Searches for Massive Graviton Resonances at the LHC T. V. Obikhod and I. A. Petrenko Research Article (9 pages), Article ID 3471023, Volume 2018 (2018) Electroweak Phase Transitions in Einstein’s Static Universe M. Gogberashvili Research Article (5 pages), Article ID 4653202, Volume 2018 (2018)

Moving Unstable Particles and Special Relativity Eugene V. Stefanovich Research Article (9 pages), Article ID 4657079, Volume 2018 (2018) Hindawi Advances in High Energy Physics Volume 2018, Article ID 6274317, 2 pages https://doi.org/10.1155/2018/6274317

Editorial Perspectives on Decay and Time Evolution of Metastable States: From Particle Physics to Cosmology

K. Urbanowski ,1 N. G. Kelkar ,2 M. Nowakowski ,2 and M. SzydBowski 3

1 Institute of Physics, University of Zielona Gora,´ Prof. Z. Szafrana 4a, 65-516 Zielona Gora,´ Poland 2Departamento de Fisica, Universidad de los Andes, Cra 1E, 18A-10, Bogota,´ Colombia 3Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Krakow,´ Poland

Correspondence should be addressed to K. Urbanowski; [email protected]

Received 26 August 2018; Accepted 26 August 2018; Published 12 December 2018

Copyright © K. Urbanowski et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP.

Many, if not most of the quantum states which consti- effect which loosely speaking states that “watched states decay tute the building blocks of nature, are unstable and decay differently,” a fact which can be even applied in cosmology. spontaneously. As such they require a special treatment in is behavior is followed by the exponential decay law. At quantum mechanics which displays new effects as compared large times, the latter again gets replaced by a new power to a classical theory. Examples of such unstable states can law preceded by a transition region in which the survival be found in the Standard Model of particle theory includ- probability can grow locally. is also finds applications in ing quarks, heavy gauge bosons, leptons heavier than the cosmology. e deviations from the exponential decay are a electron, baryons heavier than the proton, and all mesons genuine quantum effect. made up from a quark and an antiquark. In nuclear physics e ramifications of the spontaneous decay process in the word “radioactivity” is a synonym for the instability of understanding nature and its applications are widely spread. the nuclei which either de-excite emitting a photon, decay It is the photon emission of excited atoms which gives via a cluster emission (of which the alpha decay is the most information of the matter far away from our sun. It is the beta famous and the best studied example of a quantum tunneling decay which plays a decisive role in the nucleosynthesis of process) or undergo the weak transition known as beta decay elements heavier than iron. It is the alpha decay of thorium or inverse beta decay. In atomic physics, excited states are and uranium which in part heats up the inner core of the indeed considered as unstable and in a similar way we could earth, making it fluid, allowing for the continental dri and treat excited molecules. Finally, the whole universe can tunnel the magnetic field. It is the inverse beta decay supplying us from a false vacuum into a lower lying energy state which with the positron needed in medical tomography whose one from the point of view of quantum mechanics is understood version relies on the decay of the positronium. as a spontaneous decay. Indeed, in quantum mechanics a e present special issue consists of articles which deal decay process is quite natural as any state will decay unless with instability from a scale as small as that of elementary we have a conservation law (symmetry) which forbids it. particles to that of the universe itself. On the way, the authors Quantum mechanics allows also a unified treatment of the discover interesting phenomena related to the effects of spontaneous decay which can be applied to all unstable relativity, violation of symmetries, and the bizarre behaviour states and exhibits new phenomena (“new” as compared to of the universe under certain conditions. Chen and Wang, for the classical “exponential decay”) at short and large times. example, consider the tunneling of the universe within the At small times the exponential decay law is replaced by a scenario of an inhomogeneous quantum vacuum and, calcu- power law and is closely related to the Zeno and anti-Zeno lating the tunneling amplitude of the universe from nothing, Advances in High Energy Physics they find that the inhomogeneity leads to a faster tunneling. In contrast to this approach which uses the Friedmann- Lemaitre-Robertson-Walker metric, M. Gogberashvili considers a different approach using Einstein’s static universe metric and investigates the effects of the strong static gravitational field. A. Stachowski et al. bring in a new player, the metastable dark energy, in the investigation of the evolution of the universe. G. J. M. Zilioti et al. attempt to resolve some cosmological puzzles within decaying vacuum models. A completely different approach using concepts from statistical physics is introduced by Z. Haba to study the evolution of the expanding universe. Going over to smaller scales, the paper by T. V. Obikhod and I. A. Petrenko studies the properties of new particles predicted by the theories of extra dimensions. e survival probabilities of moving unstable particles are considered by E. V. Stefanovich, F. Giraldi, and F. Giacosa in three different papers. e feasibility of testing the time reversal symmetry in a purely leptonic system is reported by the Jagellonian-PET team from their pilot measurement involving the three photon decay of a positronium atom. Finally, the probability distribution of tunneling times of particles in connection with the recent laser induced tunnel ionization experiments is presented by J. T. Lunardi and L. A. Manzoni.

Conflicts of Interest e editors declare that there are no conﬂicts of interest regarding the publication of this special issue. K. Urbanowski N. G. Kelkar M. Nowakowski M. Szydłowski Hindawi Advances in High Energy Physics Volume 2018, Article ID 8271280, 10 pages https://doi.org/10.1155/2018/8271280

Research Article Feasibility Study of the Time Reversal Symmetry Tests in Decay of Metastable Positronium Atoms with the J-PET Detector

A. Gajos ,1 C. Curceanu,2 E. CzerwiNski,1 K. Dulski,1 M. Gorgol,3 N. Gupta-Sharma,1 B. C. Hiesmayr,4 B. JasiNska,3 K. Kacprzak,1 A.KapBon,1,5 D. Kisielewska,1 G. Korcyl,1 P. Kowalski,6 T. Kozik,1 W. KrzemieN,7 E. Kubicz,1 M. Mohammed,1,8 Sz. Niedfwiecki,1 M. PaBka,1 M. Pawlik-Niedfwiecka,1 L. RaczyNski,6 J. Raj,1 Z. Rudy,1 S. Sharma,1 Shivani,1 R. Shopa,6 M. Silarski,1 M. Skurzok,1 W. WiVlicki,6 B. ZgardziNska,3 M. ZieliNski,1 and P. Moskal1

1 Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Cracow, Poland 2INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy 3Department of Nuclear Methods, Institute of Physics, Maria Curie-Skłodowska University, 20-031 Lublin, Poland 4Faculty of Physics, University of Vienna, 1090 Vienna, Austria 5Institute of Metallurgy and Materials Science of Polish Academy of Sciences, Cracow, Poland 6Swierk´ Department of Complex Systems, National Centre for Nuclear Research, 05-400 Otwock-Swierk,´ Poland 7High Energy Department, National Centre for Nuclear Research, 05-400 Otwock-Swierk,´ Poland 8Department of Physics, College of Education for Pure Sciences, University of Mosul, Mosul, Iraq

Correspondence should be addressed to A. Gajos; [email protected]

Received 20 April 2018; Accepted 29 July 2018; Published 12 December 2018

Academic Editor: Krzysztof Urbanowski

Copyright © 2018 A. Gajos et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

Tis article reports on the feasibility of testing of the symmetry under reversal in time in a purely leptonic system constituted by positronium atoms using the J-PET detector. Te present state of T symmetry tests is discussed with an emphasis on the scarcely explored sector of leptonic systems. Two possible strategies of searching for manifestations of T violation in nonvanishing angular correlations of fnal state observables in the decay of metastable triplet states of positronium available with J-PET are proposed and discussed. Results of a pilot measurement with J-PET and assessment of its performance in reconstruction of three-photon decays are shown along with an analysis of its impact on the sensitivity of the detector for the determination of T-violation sensitive observables.

1. Introduction physical system under reversal in time required over 50 years more and was fnally performed in the system of entangled Te concept of symmetry of Nature under discrete transfor- neutral B mesons in 2012 [4]. Although many experiments mations has been exposed to numerous experimental tests proved violation of the combined CP symmetry, leading to ever since its introduction by E. Wigner in 1931 [1]. Te T violation expected on the ground of the CPT theorem, frst evidence of violation of the supposed symmetries under experimental evidence for noninvariance under time reversal spatial (P)andcharge(C) parity transformations in the weak remains scarce to date. interactions has been found already in 1956 and 1958, respec- Te Jagiellonian PET (J-PET) experiment aims at per- tively [2, 3]. However, observation of noninvariance of a forming a test of the symmetry under reversal in time 2 Advances in High Energy Physics in a purely leptonic system constituted by orthopositron- contrast to the unitary P and C operators, T can be shown ium (o-Ps) with a precision unprecedented in this sector. to be antiunitary. As a consequence, no conserved quantities Te increased sensitivity of J-PET with respect to previous may be attributed to the T operation [15] excluding symmetry discrete symmetry tests with o-Ps�→ 3� is achieved by a tests by means of, e.g., testing selection rules. large geometrical acceptance and angular resolution of the Feasibility of T tests based on a comparison between detector as well as by improved control of the positronium time evolution of a physical system in two directions, i.e., atoms polarization. In this work, we report on the results of |�(�)⟩ �→ |�(� + ��)⟩ and |�(� + ��)⟩ �→ |�(�)⟩, is also feasibility studies for the planned T violation searches by limited as most of the processes which could be used involve determination of angular correlations in the o-Ps�→ 3� decay a decaying state making it impractical to obtain a reverse basedonatestrunoftheJ-PETdetector. process with the same conditions in an experiment. Te Tis article is structured as follows: next section briefy only exception exploited to date is constituted by transitions discusses the properties of time and time reversal in quantum of neutral mesons between their favour-defnite states and systems. Subsequently, Section 3 provides an overview of CP eigenstates [16, 17]. A comparison of such reversible the present status and available techniques of testing of the transitions in a neutral B meson system with quantum 0 symmetry under reversal in time and points out the goals of entanglement of B B0 pairs produced in a decay of Υ(4�) the J-PET experiment in this feld. A brief description of the yielded the only direct experimental evidence of violation of detector and details of the setup used for a test measurement the symmetry under reversal in time obtained to date [4]. are given in Section 4. Section 5 discusses possible strategies While a similar concept of T violation searches is currently to test the time reversal symmetry with J-PET. Results of the pursued with the neutral kaon system [17–19], no direct tests feasibility studies are presented in Section 6 and their impact of this symmetry have been proposed outside the systems of on the perspectives for a T test with J-PET is discussed in neutral mesons. Section 7. In the absence of conserved quantities and with the difculties of comparing mutually reverse time evolution 2. Time and Reversal of Physical processes in decaying systems, manifestations of T violation Systems in Time may still be sought in nonvanishing expectation values of certain operators odd under the T transformation [20]. It Although the advent of special relativity made it common follows from the antiunitarity of the T operator that for any equate time with spatial coordinates, time remains a distinct operator O concept. Its treatment as an external parameter used in � � � † † � � � ∗ classical mechanics still cannot be consistently avoided in ⟨� � O ��⟩ = ⟨�� O� � ��⟩ = ⟨� � O ��⟩ , (1) today’s quantum theories [5]. As opposed to position and momentum, time lacks a corresponding operator in standard where the � subscript denotes states and operators trans- quantum mechanics and thus, countering the intuition, formed by the operator of reversal in time. Terefore, an cannot be an observable. Moreover, a careful insight into the operator odd with respect to the T transformation (i.e., O = time evolution of unstable quantum systems reveals a number −O)mustsatisfy of surprising phenomena such as deviations from exponential � � � � ∗ decay law [6, 7] or emission of electromagnetic radiation ⟨�� ⟩ = − ⟨� � � ��⟩ . (2) at late times [8]. Te decay process, inevitably involved in measurements of unstable systems is also a factor restrict- For stationary states or in systems where conditions on ing possible studies of the symmetry under time reversal interaction dynamics such as absence of signifcant fnal state [9]. interactions are satisfed [21], the mean value of a T-odd While eforts are taken to defne a time operator, obser- and Hermitian operator must therefore vanish in case of T vation of CP violation in the decaying meson systems invariance: disproves certain approaches [10]. Alternatively, concepts of ⟨�⟩ =−⟨�⟩ , time intervals not defned through an external parameter (3) may be considered using tunneling and dwell times [11, 12]. T However, also in this case invariance under time reversal is and violation of the symmetry may thus be manifested as an important factor [13]. a nonzero expectation value of such an operator. It is important to stress that all considerations made herein are only valid if gravitational efects are not considered. 3. Status and Strategies of T In the framework of general relativity with a generic curved Symmetry Testing spacetime, the concept of inversion of time (as well as the P transformation) loses its interpretation specifc only to the A number of experiments based on the property of T operator linear afne structure of spacetime [14]. demonstrated in (1)-(3) have been conducted to date. Te Te peculiar properties of time extend as well to the electric dipole moment of elementary systems, constituting operation of reversing physical systems in time (the T a convenient T-odd operator, has been sought for neutrons −26 operator), which results in grave experimental challenges and electrons in experiments reaching a precision of 10 −28 limiting the possibilities of T violation measurements. In and 10 , respectively [22, 23]. However, none of such Advances in High Energy Physics 3

+ experiments has observed T violation to date despite their laboratory conditions using typical sources of � radia- excellent sensitivity. In another class of experiments, a T- tion [38], giving positronium-based experiments a technical odd operator is constructed out of fnal state observables in + 0 + advantage over those using, e.g., aforementioned neutrino a decay process, such as the weak decay � �→ � � ] oscillations. However, few results on the discrete symmetries studied by the KEK-E246 experiment [24] in which the in the positronium system have been reported to date. Te P = muon polarization transverse to the decay plane ( most precise measurements studied the angular correlation P ⋅(p × p )/|p × p | ) was determined as an observable operators in the decay of orthopositronium states into three T whose nonzero mean value would manifest violation. photons and determined mean values of fnal state operators However, neither this measurement nor similar studies using 8 odd under the CP and CPT conjugations, fnding no decay of polarized Li nuclei [25] and of free neutrons [26] −3 violation signal at the sensitivity level of 10 [39, 40]. have observed signifcant mean values of T-odd fnal state Although the aforementioned studies sought for violation of observables. CP and CPT, it should be emphasized that the operators Notably, although the property of reversal in time shown used therein were odd under the T operation as well, leading in (1)-(3) is not limited to any particular system nor interac- to an implicit probe also for the symmetry under reversal in tion, it has been mostly exploited to test the T symmetry in weak interactions. Whereas the latter is the most promising time. candidate due to well proven CP violation, evidence for Te results obtained to date, showing no sign of viola- T noninvariance may be sought in other physical systems tion, were limited in precision by technical factors such as and phenomena using the same scheme of a symmetry detector geometrical acceptance and resolution, uncertainty test. Systems constituted by purely leptonic matter are an of positronium polarization, and data sample size. In terms example of a sector where experimental results related to of physical restrictions, sensitivity of such discrete symmetry the time reversal symmetry—and to discrete symmetries tests with orthopositronium decay is only limited by possible in general—remain rare. Several measurements of neutrino false asymmetries arising from photon-photon fnal state −9 oscillations are being conducted by the NO]AandT2K interactions at the precision level of 10 [41, 42]. Te J- CP ] �→ experiments searching for violation in the PET experiment thus sets its goal to explore the T-violating ] ] �→ ] and channels [27, 28], which may provide observables at precision beyond the presently established T −3 indirect information on the symmetry. Another notable 10 level [43]. test of discrete symmetries in the leptonic sector is the search for the violation of Lorentz and CPT invariance based on the Standard Model Extension framework [29] and 4. The J-PET Detector anti-CPT theorem [30] which has also been performed by Te J-PET (Jagiellonian Positron Emission Tomograph) is T2K [31, 32]. Other possible tests of these symmetries with a photon detector constructed entirely with plastic scintil- the positronium system include spectroscopy of the 1S-2S transition [33] and measuring the free fall acceleration of lators. Along with constituting the frst prototype of plastic positronium [34]. However, the question of the T, CP, scintillator-based cost-efective PET scanner with a large feld and CPT symmetries in the leptonic systems remains open of view [44, 45], it may be used to detect photons in the sub- as the aforementioned experiments have not observed a MeV range such as products of annihilation of positronium signifcant signal of a violation. atoms, thus allowing for a range of studies related to discrete Few systems exist which allow for discrete symmetry tests symmetries and quantum entanglement [43]. in a purely leptonic sector. However, a candidate competitive J-PET consists of three concentric cylindrical layers of with respect to neutrino oscillations is constituted by the axially arranged � detection modules based on strips of EJ- electromagnetic decay of positronium atoms, exotic bound 230 plastic scintillator as shown schematically in Figure 1. states of an electron and a positron. With a reduced mass only Each scintillator strip is 50 cm long with a rectangular cross- 2 twice smaller than that of a hydrogen atom, positronium is section of 7 × 19 mm . Within a detection module, both ends characterized by a similar energy level structure. At the same of a scintillator strip are optically coupled to photomultiplier time, it is a metastable state with a lifetime strongly dependent tubes. Due to low atomic number of the elements constituting on the spin confguration. Te singlet state referred to as plastic scintillators, � quanta interact mostly through Comp- parapositronium, may only decay into an even number of ton scattering in the strips, depositing a part of their energy photons due to charge parity conservation, and has a lifetime dependent on the scattering angle. Te lack of exact photon (in vacuum) of 0.125 ns. Te triplet state (orthopositronium, energy determination in J-PET is compensated by fast decay o-Ps) is limited to decay into an odd number of photons and time of plastic scintillators resulting in high time resolution lives in vacuum over three orders of magnitude longer than and allowing for use of radioactive sources with activity as the singlet state (�− = 142 ns) [35–37]. high as 10 MBq. Te energy deposited by photons scattered in Being an eigenstate of the parity operator alike atoms, a scintillator is converted to optical photons which travel to positronium is also characterized by symmetry under charge both ends of a strip undergoing multiple internal refections. conjugation typical for particle-antiparticle systems. Positro- Consequently, the position of � interaction along a detection nium atoms are thus a useful system for discrete symme- module is determined using the diference between efective try studies. Moreover, they may be copiously produced in light propagation times to the two photomultiplier tubes 4 Advances in High Energy Physics

→ S

→ k3 → k o-Ps 1 X → k2 Z Figure 2: Vectors describing the fnal state of an o-Ps�→ 3� annihi- Y �→ lation in the o-Ps frame of reference. � denotes orthopositronium �→ spin and � 1,2,3 are the momentum vectors of the annihilation photons, lying in a single plane. Te operator defned in (4) is a measure of angular correlation between the positronium spin and the decay plane normal vector.

Figure 1: Schematic view of the J-PET detector consisting of 192 �→ plastic scintillator strips arranged in three concentric layers with in the decay � 1,2,3 (ordered according to their descending radii ranging from 42.5 cm to 57.5 cm. Te strips are oriented along �→ �→ �→ the Z axis of the detector barrel. magnitude, i.e., | � 1|>|� 2|>|� 3|) allow for construction of an angular correlation operator odd under reversal in time:

�→ �→ �→ � = �⋅(� 1 × � 2), (4) attached to a scintillator strip [46]. In the transverse plane of the detector, � interactions are localized up to the position of a single module, resulting in an azimuthal angle resolution of which corresponds to an angular correlation between the ∘ about 1 . positronium spin direction and the decay plane as illustrated Although the J-PET � detection modules do not allow in Figure 2. for a direct measurement of total photon energy, recording Such an approach which requires estimation of the spin interactions of all photons from a 3� annihilation allows for direction of decaying positronia was used by both previous an indirect reconstruction of photons’ momenta based on discrete symmetry tests conducted with orthopositronium event geometry and 4-momentum conservation [47]. decay [39, 40]. Tese two experiments, however, adopted As J-PET is intended for a broad range of studies from diferent techniques to control the o-Ps spin polarization. In CP medical imaging [48] through quantum entanglement [49, the violation search, positronium atoms were produced 50] to tests of discrete symmetries [43], its data acquisition is in strong external magnetic feld resulting in their polariza- operating in a triggerless mode [51] in order to avoid any bias tion along a thus imposed direction [39]. A setup required in the recorded sample of events. Electric signals produced by to provide the magnetic feld, however, was associated with the photomultipliers are sampled in the time domain at four a limitation of the geometrical acceptance of the detectors CPT predefned voltage thresholds allowing for an estimation of used. Te second measurement, testing the symmetry the deposited energy using the time over threshold technique using the Gammasphere detector which covered almost a full [52]. Further reconstruction of photon interactions as well as solid angle, did not therefore rely on external magnetic feld. data preselection and handling is performed with dedicated Instead, positronium polarization was evaluated statistically sofware [53, 54]. Several extensions of the detector are by allowing for o-Ps atoms formation only in a single presently in preparation such as improvements of the J-PET hemisphere around a point-like positron source, resulting in geometrical acceptance by inclusion of additional detector estimation of the polarization along a fxed quantization axis layers [47] as well as enhanced scintillator readout with with an accuracy limited by a geometrical factor of 0.5 [40]. silicon photomultipliers [55] and new front-end electronics Neither of the previous experiments attempted to reconstruct �→ � [52]. the position of o-Ps 3 decay, instead limiting the volume of o-Ps creation and assuming the same origin point for all annihilations. 5. Discrete Symmetry Tests with Te J-PET experiment attempts to improve on the latter the J-PET Detector approach which does not require the use of external magnetic feld. Te statistical knowledge of spin polarization of the 5.1. Measurements Involving Orthopositronium Spin. Te positrons forming o-Ps atoms can be signifcantly increased symmetry under reversal in time can be put to test in the with a positronium production setup depicted in Figure 3, o-Ps�→ 3� decay by using the properties of T conjugation �→ where polarization is estimated on an event-by-event basis demonstrated by (1)-(3). Spin � of the decaying orthopositro- instead of assuming a fxed quantization axis throughout nium atom and momenta of the three photons produced the measurement. A trilateration-based technique of recon- Advances in High Energy Physics 5

scintillator strips

+ source holder → S

x x vacuum + system aluminium y z chamber y scintillator strips Figure 4: Determination of positron polarization axis using + its momentum direction (black arrow) estimated using the � Figure 3: Scheme of the positronium production setup devised for source position and reconstructed origin of the 3� annihilation of positron polarization determination in J-PET experiment. Positrons orthopositronium in the chamber wall (dark gray band). Te shaded �+ are produced in a source mounted in the center of a cylindrical region represents the angular uncertainty of positron fight direction vacuum chamber coaxial with the detector. Positronium atoms are resulting from achievable resolution of the 3� annihilation point. formed by the interaction of positrons in a porous medium covering the chamber walls. Determination of an o-Ps�→ 3� annihilation position in the cylinder provides an estimate of positron momentum direction. →  1

→   �→ � → k1 structing the position of o-Ps 3 decay created for J-PET k allows for estimation of the direction of positron propagation 3 o-Ps → in a single event with a vector spanned by a point-like k + 1 � source location and the reconstructed orthopositronium annihilation point [56]. → k Te dependence of average spin polarization of positrons 2 (largely preserved during formation of orthopositronium Figure 5: Scheme of estimation of polarization vector for a photon �→ [57]) on the angular accuracy of the polarization axis deter- produced in o-Ps�→ 3� at J-PET. Photon of momentum � 1 is scat- mination is given by (1/2)(1 + cos �) where � is the opening tered in one of the detection modules and a secondary interaction of angle of a cone representing the uncertainty of polarization �→ the scattering product � is recorded in a diferent scintillator strip. axis direction [58]. Tis uncertainty in J-PET results pre- 1 �→ Te most probable angle � between the polarization vector �1 and dominantly from the resolution of determination of the 3� �→ �→ � � 90∘ annihilation point as depicted in Figure 4 and amounts to the scattering plane spanned by 1 and 1 amounts to . ∘ about 15 [56], resulting in a polarization decrease smaller than 2%. By contrast, in the previous measurement with Gammasphere [40] where the polarization axis was fxed, the same geometrical factor accounted for a 50% polarization the J-PET detector enables a measurement of ⟨�⟩ thanks to loss. the ability to record secondary interactions of once scattered photons from the o-Ps�→ 3� annihilation as depicted in 5.2. Measurements Using Polarization of Photons. Te scheme Figure 5. of measurement without external magnetic feld for positronium polarization may be further simplifed with modifed choice of the measured T-odd operator. Tis novel approach 6. Test Measurement with the J-PET Detector of testing the T symmetry may be pursued by J-PET with �→ � Te setup presented in Figure 3 was constructed and fully a spin-independent operator constructed for the o-Ps 3 commissioned in 2017 [59]. One of the frst test mea- annihilations if the polarization vector of one of the fnal state surements was dedicated to evaluation of the feasibility of photons is included [43]: identifcation and reconstruction of three-photon events. 22 �+ �→ �→ A Na source was mounted inside a cylindrical vac- � = � ⋅ � , (5) 2 1 uum chamber of 14 cm radius. Te positronium formation- �→ enhancing medium, presently under elaboration, was not where � 1 denotes the electric polarization vector of the �→ included in the measurement. Terefore, the test of 3� most energetic � quantum and � 2 is the momentum of event reconstruction was based on direct 3� annihilation of the second most energetic one. Such angular correlation positrons with electrons of the aluminium chamber walls, operators involving photon electric polarization have never with a yield smaller by more than an order of magnitude been studied in the decay of orthopositronium. Geometry of than the rate of o-Ps�→ 3� annihilations expected in the 6 Advances in High Energy Physics

0.006 interactions recorded in the detector in cases of three interac- 1 hit in event tions observed in close time coincidence. Te tests performed with Monte Carlo simulations have shown that annihilations 2 hits in event into two and three photons can be well separated using such correlations [60]. An exemplary relative distribution of 0.004 3+ hits in event values constructed using these correlations, obtained with the test measurement, is presented in Figure 7(a). For a comparison, the same distribution obtained with a point-like annihilation medium used in another test measurement of J- 0.002 PET is presented in Figure 7(b).

counts [normalized] counts ∘ A sharp vertical band at ��2 +��3 ≈ 180 seen in Figure 7(b) originates from events corresponding to annihilations into two back-to-back photons. Broadening of this 2� band in case of the extensive chamber is a result of the 0.000 increased discrepancy between relative azimuthal angles of 0 10 20 30 40 50 60 detection module locations used for the calculation and the TOT [ns] actual relative angles in events originating in the walls of the Figure 6: Distributions of time over threshold (TOT) values, cylindrical chamber as depicted schematically in Figure 8. for � interactions observed in groups of 1, 2 and more within a Te distributions presented in Figure 7 are in good agree- coincidence time window of 2.5 ns. In the samples with increasing ment with the simulation-based expectations [60]. Selection � ∘ number of coincident photons, the contribution of quanta from of events with values of ��2 +��3 signifcantly larger than 180 3� annihilations increases with respect to other processes. Gray lines allows for identifcation of three-photon annihilations. 3� and arrow denote the region used to identify annihilation photon Te aforementioned event selection techniques allowed candidates. to extract 1164 3� event candidates from the two-day test + measurement with a � source activity of about 10 MBq placed in the center of the aluminium cylinder as depicted in fnal measurements with a porous medium for positronium Figure 3. Terefore, a quantitative estimation of the achiev- production. able resolution of three-photon event origin points and its Te capabilities of J-PET to select 3� events and discrimi- impact on the positronium polarization control capabilities + − nate background arising from two-photon � � annihilations requires a measurement including a medium enhancing the as well as from accidental coincidences are based primarily on positronium production. two factors: Resolution of the detector and its feld of view was validated with a benchmark analysis of the test data performed (i) a measure of energy deposited by a photon in Comp- using the abundant 2� annihilation events. Figure 9 presents ton scattering, provided by the time over threshold the images of the annihilation chamber obtained using 2� (TOT) values determined by the J-PET front-end events whose selection and reconstruction was performed electronics, with the same techniques as applied to medical imaging (ii) angular dependencies between relative azimuthal tests performed with J-PET [48]. Although a large part of angles of recorded � interaction points, specifc to recorded annihilations originate already in the setup holding + topology of the event [60]. the � source, a considerable fraction of positrons reach the chamber walls. Te efective longitudinal feld of view of Distributions of the TOT values, afer equalization of J-PET for 2� events which can be directly extended to 3� responses of each detection module, are presented in Figure 6. annihilations due to similar geometrical constraints spans the Separate study of TOT distributions for � quanta observed range of approximately |�| < 8 cm. in groups of 1, 2 and more recorded � hits in scintillators within a short time window reveals the diferent composition of photons from 3� annihilations with respect to those 7. Summary and Perspectives originating from background processes such as deexcitation Te J-PET group attempts to perform the frst search for �+ 22 of the decay products from a Na source (1270 keV) signs of violation of the symmetry under reversal in time and cosmic radiation. Two Compton edges corresponding to in the decay of positronium atoms. One of the available 511 1270 keV and keV photons are clearly discernible in TOT techniques is based on evaluation of mean values of fnal distributions, allowing identifying candidates for interactions state observables constructed from photons’ momenta and � of 3 annihilation products by TOT values located below positroniumspininano-Ps�→ 3� annihilation with a pre- the 511 keV Compton edge as marked with dashed lines in cision enhanced with respect to the previous realization Figure 6. of similar measurements by determination of positronium Te second event selection criterion is based on the spin distinctly for each recorded event. Moreover, the J-PET correlations between relative azimuthal angles of photon detector enables a novel test by determination of a T-odd Advances in High Energy Physics 7

250 103

4 200 10 200

2 103 10 3 150 3 150   - - 2 2 2  100 10  100 10

50 10 50

0 1 0 1 0 50 100 150 200 0 50 100 150 200 250 2 +3 2 +3 (a) (b)

Figure 7: Relations between the sum and diference of two smallest relative azimuthal angles (��2 and ��3, respectively) between � interaction points in events with three recorded interactions. (a) Distribution obtained in the test measurement with the aluminium chamber presented in Figure 3. For reference, the same spectrum obtained with a point-like annihilation medium located in the detector center [59] is displayed ∘ in (b). Te vertical band around ��2 +��3 ≈ 180 arises from two-photon annihilations and is broadened in the frst case due to extensive dimensions of the annihilation chamber used. 3� annihilation events are expected in the region located at the right side of the 2� band [60].

 

∘ Figure 8: Explanation of the broadening of the 2� annihilation band present in Figures 7(a) and 7(b) at ��2 +��3 ≈ 180 .Lef:when2� annihilations originate in a small region in the detector center, the calculated relative azimuthal angles of detection modules which registered the photons correspond closely to actual relative angles between photons’ momenta. Right: with 2� annihilations taking place in the walls of ∘ an extensive-size annihilation chamber (gray band), the broadening of the band at 180 is caused by a discrepancy between the calculated and actual relative angles. Te detector scheme and proportions are not preserved for clarity.

observable constructed using the momenta and polarization proposed for positron spin determination was validated with of photons from annihilation. a benchmark reconstruction of two-photon annihilations. Te pilot measurement conducted with the J-PET detec- Results obtained from the test measurement confrm the tor demonstrated the possibility of identifying candidates of feasibility of a test of symmetry under reversal in time by annihilation photons interactions in the plastic scintillator measurement of the angular correlation operator defned strips by means of the time over threshold measure of in (4) without external magnetic feld once a positronium deposited energy and angular dependencies between relative production medium is used. azimuthal angles of � interaction points specifc to event spatial topology. A preliminary selection of three-photon Data Availability annihilation events yielded 1164 event candidates from a two-day test measurement with a yield reduced by more As this article covers a feasibility study of time reversal than an order of magnitude with respect to the planned symmetry tests with the J-PET detector in the view of the experiments with a porous positronium production target future measurements, the data with a physical relevance will and a centrally located 10 MBq source. Te annihilation only be collected in the future and the presently used data of reconstruction resolution and performance of the setup a test measurement are mostly of technical importance only. 8 Advances in High Energy Physics

30 4000 30 35000 3500 20 20 30000 3000 10 10 25000 2500 20000 0 2000 0 X [cm]

Y [cm] 15000 1500 −10 −10 1000 10000 −20 −20 500 5000

−30 0 −30 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 X [cm] Z [cm]

(a) (b)

+ − Figure 9: Tomographic images of the cylindrical chamber used in the test run of J-PET,obtained using reconstructed � � �→ 2 � annihilation events. (a) Transverse view of the chamber (the central longitudinal region of |�| < 4 cm was excluded where the image is dominated by + annihilation events originating in the setup of the � source.). (b) Longitudinal view of the imaged chamber. In the central region, a strong image of the positron source and its mounting setup is visible.

Disclosure beta decay,” Physical Review A: Atomic, Molecular and Optical Physics,vol.105,no.4,pp.1413–1415,1957. Te opinions expressed in this publication are those of the [3] P. C. Macq, K. M. Crowe, and R. P. Haddock, “Helicity of authors and do not necessarily refect the views of the John the electron and positron in muon decay,” Physical Review A: Templeton Foundation. Atomic, Molecular and Optical Physics,vol.112,no.6,pp.2061– 2071, 1958. Conflicts of Interest [4] J. P. Lees, V. Poireau, V. Tisserand et al., “Observation of Time ReversalViolationintheB0MesonSystem,”Physical Review Te authors declare that there are no conficts of interest Letters,vol.109,p.211801,2012. regarding the publication of this paper. [5] J. G. Muga, R. Sala Mayato, and I. L. Egusquiza, Time in Quantum Mechanics,Springer-Verlag,Berlin,Germany,2nd edition, 2008. Acknowledgments [6] K. Urbanowski, “Decay law of relativistic particles: quantum theory meets special relativity,” Physics Letters B,vol.737,pp. Te authors acknowledge technical and administrative sup- 346–351, 2014. portofA.Heczko,M.Kajetanowicz,andW.Migdał.Tis [7] F. Giacosa, “Time evolution of an unstable quantum system,” work was supported by the Polish National Center for Acta Physica Polonica B,vol.48,no.10,pp.1831–1836,2017. Research and Development through Grant INNOTECH-K1/ [8] K. Urbanowski and K. Raczy´nska, “Possible emission of cosmic IN1/64/159174/NCBR/12, the Foundation for Polish Science X-and�-rays by unstable particles at late times,” Physics Letters through the MPD and TEAM/2017-4/39 programmes and B,vol.731,pp.236–241,2014. Grants nos. 2016/21/B/ST2/01222 and 2017/25/N/NZ1/00861, [9] J. Bernabeu and F. Martinez-Vidal, “Colloquium: Time-reversal the Ministry for Science and Higher Education through violation with quantum-entangled B mesons,” Reviews of Mod- Grants nos. 6673/IA/SP/2016, 7150/E-338/SPUB/2017/1, and ern Physics,vol.87,no.1,pp.165–182,2015. 7150/E-338/M/2017, and the EU and MSHE Grant no. [10]T.Durt,A.DiDomenico,andB.Hiesmayr,Falsifcation of POIG.02.03.00-161 00013/09. B. C. Hiesmayr acknowledges a Time Operator Model based on Charge-Conjugation-Parity support by the Austrian Science Fund (FWF-P26783). C. violation of neutral K-mesons,2015. Curceanu acknowledges a grant from the John Templeton [11] N. G. Kelkar, “Electron tunneling times,” Acta Physica Polonica Foundation (ID 58158). B,vol.48,no.10,pp.1825–1830,2017. [12] N. G. Kelkar, “Quantum Refection and Dwell Times of References Metastable States,” Physical Review Letters,vol.99,no.21,2007. [13] N. G. Kelkar and M. Nowakowski, “Analysis of averaged [1] E. P. Wigner, Group theory and its application to the quantum multichannel delay times,” Physical Review A: Atomic, Molecular mechanics of atomic spectra, Academic Press, New York, NY, and Optical Physics,vol.78,no.1,2008. USA, 1959. [14] L. M. Sokołowski, “Discrete Lorentz symmetries in gravitational [2]C.S.Wu,E.Ambler,R.W.Hayward,D.D.Hoppes,and felds,” Acta Physica Polonica B,vol.48,no.10,pp.1947–1953, R. P. Hudson, “Experimental test of parity conservation in 2017. Advances in High Energy Physics 9

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Research Article From Quantum Unstable Systems to the Decaying Dark Energy: Cosmological Implications

Aleksander Stachowski ,1 Marek SzydBowski ,1,2 and Krzysztof Urbanowski 3

1 Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Krakow,´ Poland 2Mark Kac Complex Systems Research Centre, Jagiellonian University, Łojasiewicza 11, 30-348 Krakow,´ Poland 3Institute of Physics, University of Zielona Gora,´ Prof. Z. Szafrana 4a, 65-516 Zielona Gora,´ Poland

Correspondence should be addressed to Marek Szydłowski; [email protected]

Received 9 April 2018; Revised 25 June 2018; Accepted 31 July 2018; Published 19 August 2018

Academic Editor: Ricardo G. Felipe

Copyright © 2018 Aleksander Stachowski et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

We consider a cosmology with decaying metastable dark energy and assume that a decay process of this metastable dark energy is a quantum decay process. Such an assumption implies among others that the evolution of the Universe is irreversible and violates the time reversal symmetry. We show that if we replace the cosmological time � appearing in the equation describing the evolution of the Universe by the Hubble cosmological scale time, then we obtain time dependent Λ(�) in the form of the series of even powers of theHubbleparameter�: Λ(�) = Λ(�). Our special attention is focused on radioactive-like exponential form of the decay process of the dark energy and on the consequences of this type decay.

1. Introduction of dark energy. A detailed analysis of results of recent observations shows that there is a tension between local and In the explanation of the Universe, we encounter the old primordial measurements of cosmological parameters [3]. It problem of the cosmological constant, which is related to appears that this tension may be connected with dark energy understanding why the measured value of the vacuum energy evolving in time [4]. Tis paper is a contribution to the is so small in comparison with the value calculated using discussion of the nature of the dark energy. We consider quantum feld theory methods [1]. Because of a cosmological the hypothesis that dark energy depends on time, �de = origin of the cosmological constant one must also address �de(�), and it is metastable: We assume that it decays with another problem. Namely, it is connected with our under- the increasing time � to �bare: �de(�) �→ �bare =0̸ as ��→ standing, with a question of not only why the vacuum energy ∞. Te idea that vacuum energy decays was considered in is not only small, but also, as current Type Ia supernova many papers (see, e.g., [5, 6]). Shafeloo et al. [7] assumed observations to indicate, why the present mass density of the that �de(�) decays according to the radioactive exponential Universe has the same order of magnitude [2]. decay law. Unfortunately, such an assumption is not able to Both mentioned cosmological constant problems can be refect all the subtleties of evolution in the time of the dark considered in the framework of the extension of the standard energy and its decay process. It is because the creation of the cosmological ΛCDM model in which the cosmological con- Universe is a quantum process. Hence the metastable dark stant (naturally interpreted as related to the vacuum energy energy can be considered as the value of the scalar feld at density) is running and its value is changing during the the false vacuum state and therefore the decay of the dark cosmic evolution. energy should be considered as a quantum decay process. Te Results of many recent observations lead to the conclu- radioactive exponential decay law does not refect correctly sion that our Universe is in an accelerated expansion phase all phases of the quantum decay process. In general, analysing [3]. Tis acceleration can be explained as a result of a presence quantum decay processes one can distinguish the following 2 Advances in High Energy Physics phases [8, 9]: (i) the early time initial phase, (ii) the canonical or equivalently or exponential phase (when the decay law has the exponential form), and (iii) the late time nonexponential phase. Te frst A (0) =1. (4) phase and the third one are missed when one considers H the radioactive decay law only. Simply they are invisible In (2) denotes the complete (full), self-adjoint Hamiltonian |�(�)⟩ = [−(�/ℏ)�H]|�⟩ to the radioactive exponential decay law. For example, the of the system. We have exp .Itisnot � theoretical analysis of quantum decay processes shows that difcult to see that this property and hermiticity of imply at late times the survival probability of the system considered that in its initial state (i.e., the decay law) should tend to zero as (A (�))∗ = A (−�) . ��→∞much more slowly than any exponential function (5) of time and that as a function of time it has the inverse Terefore, the decay probability of an unstable state (usually power-like form at this regime of time [8, 10, 11]. So, all called the decay law), i.e., the probability for a quantum implications of the assumption that the decay process of system to remain at time � in its initial state |�(0)⟩ ≡ |�⟩, the dark energy is a quantum decay process can be found only if we apply a quantum decay law to describe decaying def P (�) �� |A (�)|2 ≡ A (�)(A (�))∗ , (6) metastable dark energy. Tis idea was used in [12], where the assumption made in [7] that �de(�) decays according to the radioactive exponential decay law was improved by replacing must be an even function of time [8]: that radioactive decay law by the survival probability P(�), P (�) = P (−�) . (7) that is, by the decay law derived assuming that the decay processisaquantumprocess. Tis last property suggests that, in the case of the unstable Tis is the place where one has to emphasize that the states prepared at some instant �0,say�0 =0, initial condition use of the assumption that dark energy depends on time and (3) for evolution equation (2) should be formulated more is decaying during time evolution leads to the conclusion precisely. Namely, from (7), it follows that the probabilities that such a process is irreversible and violates a time reversal of fnding the system in the decaying state |�⟩ at the instant, symmetry. (Consequences of this efect will be analysed say �=�≫�0 ≡0, and at the instant �=−�are the same. in next sections of this paper) Note that the picture of Of course, this can never occur. In almost all experiments in the evolving Universe, which results from the solutions of which the decay law of a given unstable subsystem system is the Einstein equations completed with quantum corrections investigated this particle is created at some instant of time, say appearing as the efect of treating the false vacuum decay as a �0, and this instant of time is usually considered as the initial quantum decay process, is consistent with the observational instant for the problem. From property (7) it follows that the data. Te evolution starts from the early time epoch with the Λ(�) instantaneous creation of the unstable subsystem system (e.g., running and then it goes to the fnal accelerating phase a particle or an excited quantum level and so on) is practically expansion of the Universe. In such a scenario the standard Λ impossible. For the observer, the creation of this object (i.e., cosmological CDM model emerges from the quantum false the preparation of the state, |�⟩,representingthedecaying vacuum state of the Universe. subsystem system) is practically instantaneous. What is more, Te paper is organised as follows: In Section 2 one fnds using suitable detectors he/she is usually able to prove that it a short introduction of formalism necessary for considering did not exist at times �<�0. Terefore, if one looks for the decaying dark energy as a quantum decay process. Cosmo- solutions of Schrodinger¨ equation (2) describing properties logical implications of a decaying dark energy are considered of the unstable states prepared at some initial instant �0 in the in Section 3. Section 4 contains conclusions. system and if one requires these solutions to refect situations described above, one should complete initial conditions (3), 2. Decay of a Dark Energy as a Quantum (4) for (2) by assuming additionally that Decay Process � ��(�<�0)⟩ = 0 In the quantum decay theory of unstable systems, properties (8) A (�) (� < � )=0. of the survival amplitudes or 0 � A (�) =⟨�� (�)⟩ (1) Equivalently, within the problem considered, one can use |�⟩ initial conditions (3), (4) and assume that time � may vary are usually analysed. Here a vector represents the unstable �=� >−∞ �=+∞ �∈R+ state of the system considered and |�(�)⟩ is the solution of the from 0 to only, that is, that . Schrodinger¨ equation Note that canonical (that is a classical radioactive) decay law P�(�) = exp[−�/�0] (where �0 is a lifetime) does not � � � �ℏ �� (�)⟩=H �� (�)⟩. (2) satisfy property (7), which is valid only for the quantum decay �� law P(�). What is more, from (5) and (6) it follows that at very Te initial condition for (2) in the case considered is usually early times, i.e., at the Zeno times (see [8, 13]), assumed to be � �P (�)� � def � � =0, (9) ��(�=�0 ≡0)⟩�� ⟩ , (3) �� =0 Advances in High Energy Physics 3

def 2 2 which implies that �BW(�) �� (�/2�)Θ(� − �min)(Γ0/((� − �0) +(Γ0/2) )), � � def where is a normalization constant. Te parameters 0 and −�/�0 P (�) >� �� P� (�) for ��→0. (10) Γ0 correspond to the energy of the system in the unstable state and its decay rate at the exponential (or canonical) regime of So at the Zeno time region the quantum decay process the decay process. �min is the minimal (the lowest) energy is much slower than any decay process described by the of the system. Inserting �BW(�) into formula (12) for the P (�) canonical (or classical) decay law � . amplitude A(�) and assuming for simplicity that �0 =0,afer Now let us focus the attention on the survival amplitude some algebra, one fnds that A(�). An unstable state |�⟩ can be modeled as wave packets � Γ � using solutions of the following eigenvalue equation H|�⟩ = −(�/ℏ)�0� 0 A (�) = � �� ( ), (13) �|�⟩,where�∈��(H),and��(H) denotes a continuum 2� ℏ spectrum of H.Eigenvectors|�⟩ are normalized as usual: � � where ⟨�|� ⟩=�(�−�).Usingvectors|�⟩ we can model an def ∞ 1 unstable state as the following wave-packet: � (�) ��∫ �−��. � 2 (14) −� � +1/4 � � ∞ ��⟩ ≡ ��⟩ = ∫ � � � ��, � � ( ) | ⟩ (11) �=Γ�/ℏ ≡ �/� � � =ℏ/Γ �min Here 0 0, 0 is the lifetime, 0 0,and �=(�0 −��)/Γ0 >0.Teintegral��(�) has the following where expansion coefcients �(�) are functions of the energy structure: � and �min is the lower bound of the spectrum ��(H) of H. � (�) =�pole (�) +�� (�) , Te state |�⟩ is normalized ⟨�|�⟩ = 1, which means that it � � � (15) ∞ ∫ |�(�)|2�� = 1 has to be � . Now using the defnition of the min where survival amplitude A(�) and the expansion (11) we can fnd ∞ 1 A(�), which takes the following form within the formalism �pole (�) =∫ �−�� ≡ 2��−�/2, � 2 (16) considered: −∞ � +1/4 ∞ −��(�−�0) and A (�) ≡ A (� − �0)=∫ � (�) � ��, (12) ∞ �min 1 �� (�) =−∫ �+��. � 2 (17) 2 +� � +1/4 where �(�) ≡ |�(�)| >0and �(�)�� is the probability to fnd the energy of the system in the state |�⟩ between � and � � (Te integral � (�) can be expressed in terms of the integral- + ��. Te last relation (12) means that the survival amplitude � exponential function [23–26] (for a defnition, see [27, 28])) A(�) is a Fourier transform of an absolute integrable function Te result (15) means that there is a natural decomposition of �(�). If we apply the Riemann-Lebesgue Lemma to integral the survival amplitude A(�) into two parts: (12) then one concludes that there must be A(�) �→ 0 as ��→∞ . Tis property and relation (12) are an essence of A (�) = A� (�) + A� (�) , (18) the Fock–Krylov theory of unstable states [14, 15]. As it is seen from (12), the amplitude A(�) and thus where the decay law P(�) of the unstable state |�⟩ are completely � −(�/ℏ)� � pole Γ � −(�/ℏ)� � −Γ �/2 determined by the density of the energy distribution �(�) for A (�) = � 0 � ( 0 )≡�� 0 � 0 , (19) � 2� � ℏ the system in this state [14, 15] (see also [8, 10, 11, 16–21]). In the general case the density �(�) possesses properties and analogous to the scattering amplitude; i.e., it can be decom- � −(�/ℏ)� � � Γ � posed into a threshold factor, a pole-function �(�) with a A (�) = � 0 � ( 0 ), (20) � 2� � ℏ simple pole, and a smooth form factor �(�).Tereis�(�) = � Θ(� − � )(� − � ) � �(�)�(�) � min min ,where � depends on the and A�(t) is the canonical part of the amplitude A(�) describ- � � =�+� angular momentum through � [8] (see equation ing the pole contribution into A(�) and A�(�) represents the 0≤�<1 Θ(�) Θ(�) = 0 (6.1) in [8]), )and is a step function: remaining part of A(�). �≤0 Θ(�) = 1 �>0 for and for . Te simplest choice is to From decomposition (18) it follows that in the general �=0�=0�(�) = 1 �(�) take , , and to assume that has a case within the model considered the survival probability (6) Breit–Wigner (BW) form of the energy distribution density. contains the following parts: (Te mentioned Breit–Wigner distribution was found when 2 � �2 the cross section of slow neutrons was analysed [22]) It turns P (�) = |A (�)| ≡ �A� (�) + A� (�)� out that the decay curves obtained in this simplest case are (21) � �2 ∗ � �2 very similar in form to the curves calculated for the above = �A� (�)� +2R [A� (�) (A� (�)) ]+�A� (�)� . described more general �(�) (see [16] and analysis in [8]). So to fnd the most typical properties of the decay process it is Tislastrelationisespeciallyusefulwhenonelooksfora sufcient to make the relevant calculations for �(�) modeled contribution of late time properties of the quantum unstable by the Breit–Wigner distribution of the energy density �(�) ≡ system to the survival amplitude. 4 Advances in High Energy Physics

� Te late time form of the integral �� (�) and thus the late Te above described approach is self-consistent if we � (� ) � time form of the amplitude A�(�) can be relatively easy to fnd identify de 0 with the energy 0 of the unstable system � � using analytical expression for A�(�) in terms of the integral- divided by the volume 0 (where 0 is the volume of the def exponential functions or simply performing the integration �=� � (� )≡�qft ��0 =�/� ��→∞ ��→∞ system at 0): de 0 de de 0 0 and by parts in (17). One fnds for (or )that � =� /� �qft the leading term of the late time asymptotic expansion of the bare min 0.Here de is the vacuum energy density � calculated using quantum feld theory methods. In such a case integral �� (�) has the following form: 0 �� 0 −�min �de −�bare � � � �= ≡ >0, (27) � (�) ≃− +..., (��→∞) . (22) Γ � � � �2 +1/4 0 0 � =Γ/� Γ /� ≡(�0 −� )/� Tus inserting (22) into (20) one can fnd late time form of (where 0 0 0), or equivalently 0 0 de bare . A�(�). As was mentioned we consider the hypothesis that a dark 3. Cosmological Implications of � =� (�) energy depends on time, de de , and decays with the Decaying Vacuum increasing time � to �bare: �de(�) �→ �bare =0̸ as ��→∞.We assume that it is a quantum decay process. Te consequence Let us consider cosmological implications of the parameter of this assumption is that we should consider �de (�0) (where �0 Λ with the time parameterized decaying part, derived in the is the initial instant) as the energy of an excited quantum level previous section, in the form (e.g., corresponding to the false vacuum state) and the energy density �bare as the energy corresponding to the true lowest Λ≡Λef (�) =Λbare +�Λ(�) , (28) energystate(thetruevacuum)ofthesystemconsidered.Our �Λ(�) hypothesis means that (�de(�) − �bare)�→0as ��→∞. where describes quantum corrections and it is given by As it was said we assumed that the decay process of the a series with respect to 1/�; i.e., dark energy is a quantum decay process: From the point of ∞ 1 2� view of the quantum theory of decay processes this means �Λ (�) = ∑� ( ) , (� (�) − � )=0 2� (29) that lim��→∞ de bare according to the quantum �=1 � mechanical decay law. Terefore if we defne where � is the cosmological scale time and the functions def Λ (�) �Λ(�) �̃de (�) ��de (�) −�bare, (23) ef and have a refection symmetry with respect to the cosmological time �Λ(−�) = �Λ(�).Tenextstep our assumption means that the decay law for �̃de(�) has the in deriving dynamical equations for the evolution of the following form (see [12]): Universe is to consider this parameter as a source of gravity which contributes to the efective energy density; i.e., �̃de (�) = �̃de (�0) P (�) ≡ �̃de (�0) 2 (24) 3� (�) =� (�) +� (�) , (30) � �2 ∗ � �2 m de ⋅(�A� (�)� +2R [A� (�) (A� (�)) ]+�A� (�)� ), where �de(�) is identifed as the energy density of the quantum where P(�) is given by relation (6), or, equivalently, our decay process of vacuum assumption means that the decay law for �̃de(�) has the following form (compare [12]): �de (�) =Λbare +�Λ(�) . (31)

�de (�) ≡�bare + �̃de (�0) In this paper, we assume that �=8��=1. Te Einstein feld (25) equation for the FRW metric reduces to � �2 ∗ � �2 ⋅(�A� (�)� +2R [A� (�) (A� (�)) ]+�A� (�)� ), �� (�) 1 =− (�ef (�) +�ef (�)) where �̃de(�0)=(�de(�0)−�bare) and P(�) is replaced by (21). �� 2 � (32) Taking into account the standard relation between de and the 1 cosmological constant Λ we can write =− (� (�) +0+� (�) −� (�)), 2 m de de ̃ Λ ef (�) ≡Λbare + Λ(�0) where �ef =�m +�de, �ef =0+�de,or � �2 ∗ � �2 (26) ⋅(�A (�)� +2R [A (�) (A (�)) ]+�A (�)� ), � � � � � � � � �� (�) 1 1 2 =− �m (�) =− (3� (�) −Λbare −�Λ(�)) . (33) ̃ ̃ �� 2 2 where Λ(�0)≡Λ 0 =(Λ(�0)−Λbare). Tus within the considered case using defnition (6) or relation (21) we can Szydlowski et al. [12] considered the radioactive-like determine changes in time of the dark energy density �de(�) decay of metastable dark energy. For the late time, this decay (or running Λ(�)) knowing the general properties of survival process has three consecutive phases: the phase of radioactive amplitude A(�). decay, the phase of damping oscillations, and fnally the phase Advances in High Energy Physics 5

2 2 of power law decaying. When �>0for �>(ℏ/Γ0)(2�/(� + 1/4)),darkenergy canbedescribed inthefollowing form(see (25) and [12]): 1.5

2 −(Γ0/ℏ)� 1 �de (�) ≈�bare +�(4� � 1

0.5 −(Γ0/2ℏ)� 4�� sin (� (Γ0/ℏ) �) + (34) (1/4 + �2)(Γ /ℏ) � 0 0 y 1 + ), −0.5 2 2 ((1/4 + � )(Γ0/ℏ) �) −1 2 where �, Γ0,and� are model parameters. Equation (34) results directly from (25): One only needs to insert (22) into formula −1.5 for A�(�) and result (19) instead of A�(�) into (25). In this paper, we consider the frst phase of decay process, in other words, the phase of radioactive (exponential) decay. 0 0.2 0.4 0.6 0.8 1 Te model with the radioactive (exponential) decay of x dark energy was investigated by Shafeloo et al. [7]. During Figure 1: Te phase portrait of system (39). Critical point 1 (� = √ the phase of the exponential decay of the vacuum 0, � = √Λ bare/ 3�0) is the stable node and critical point 2 (� = 0, � = −√Λ /√3� ) ��Λ (�) bare 0 is the saddle. Tese critical points represent =��Λ(�) , (35) � �� the de Sitter universes. Here, 0 isthepresentvalueoftheHubble function. Te value of � is assumed as −�0. Note that the phase where �=const <0(�Λ(�) is decaying). portrait is not symmetric under refection ��→−�. While critical Te set of equations (33) and (35) constitute a two- point 1 is a global attractor, only a unique separatrix reaches critical dimensional closed autonomous dynamical system in the point 2. form �� (�) 1 =− (3� (�)2 −Λ −�Λ(�)), �� 2 bare Szydlowski et al. [12] demonstrated that the contribution (36) ��Λ (�) of the energy density of the decaying quantum vacuum =��Λ(�) . possesses three disjoint phases during the cosmic evolution. �� Te phase of exponential decay like in the radioactive decay System (36) has the time dependent frst integral in the processesislongphaseinthepastandfutureevolution.Our form estimation of model parameter shows that we are living in the 2 Universe with the radioactive decay of the quantum vacuum. �m (�) =3�(�) −Λbare −�Λ(�) . (37) It is interesting that, during this phase, the Universe violates the refection symmetry of the time: ��→−�.In At the fnite domain, system (36) possesses only one critical cosmology and generally in physics there is a fundamental point representing the standard cosmological model (the problem of the origin of irreversibility in the Universe [29]. running part of Λ vanishes, i.e., �Λ(�) = 0). Note that in our model irreversibility is a consequence of the System (36) can be rewritten in variables radioactive decay of the quantum vacuum. �Λ (�) If we considered radioactivity (in which the time reversal �= , 2 symmetry is broken) then a direction of decaying vacuum is 3�0 (38) in accordance with the thermodynamical arrow of time. First, � (�) �= note that it is in some sense very natural that the dynamics �0 of the Universe is in fact irreversible when the full quantum evolution is taken into account. Terefore, radioactive decay � where 0 isthepresentvalueoftheHubblefunction.Ten of vacuum irreversibility has a thermodynamic interpretation �� as far as the evolution of the Universe is concerned: in horizon = � �� thermodynamics the area of the cosmological horizon is 0 interpreted as (beginning proportional to) the entropy, i.e., (39) Λ �� 1 2 the Hawking entropy. In a system where decays as a result =− (3� −3ΩΛ −3�), �� 2 bare of irreversible quantum processes we obtain the very natural conclusion that the entropy of the Universe grows, in many Ω =Λ /3�2 �=�� where Λ bare bare 0 and 0 are a new cases without an upper bound [30, 31]. reparametrized time. Te phase portrait of system (39) is In the general parameterization (29), of course, the sym- shown in Figure 1. metry of changing ��→−�is present and this symmetry is 6 Advances in High Energy Physics also in a one-dimensional nonautonomous dynamical system H 600 describing the evolution of the Universe:

∞ 500 �� (�) 1 2 −2� =− (3� (�) −Λbare − ∑�2�� ). (40) �� 2 �=1 400

In cosmology, especially in quantum cosmology, the 300 analysis of the concept of time seems to be the key for the construction of an adequate quantum gravity theory, which 200 we would like to apply to the description of early Universe. Te good approximation of (40) is to replace in it the 100 cosmological time by the Hubble cosmological scale time t 1 0.002 0.004 0.006 0.008 0.010 0.012 0.014 �H = . (41) � Figure 2: Te diagram of the evolution of the Hubble function with respect to the cosmological time �,whichisdescribedby(43) In consequence, parameterization (29) can be rewritten in the with �21 =0̸ and �2� =0for every �>1. For illustration, two new form example values of the parameter �21 = are chosen: −1 and −2.Te ∞ top blue curve describes the evolution of the Hubble function in the 2� ΛCDM model. Te middle curve describes one for �21 =−1and the �Λ (�) =�Λ(� (�)) = ∑�2�� (�) . (42) �=1 bottom red curve describes one for �21 =−2. Te Hubble function is expressed in km/sMpcandthecosmologicaltime� is expressed Afer putting this form into (40), we obtain dynamical system in s Mpc/km. in an autonomous form with the preserved symmetry of time ��→−�, ��→−�. In Figure 2 presents a diagram of the H evolution of the Hubble function obtained from the following 600 one-dimensional dynamical system: 500 ∞ �� (�) 1 2 2� =− (3� (�) −Λbare − ∑�2�� (�) ). (43) 400 �� 2 �=1 300 For comparison, the evolution of the Hubble functions Λ derived in the CDM model, model (40), and model (43) are 200 presented in Figure 3. For the existence of the de Sitter global attractor as ��→∞asymptotically a contribution coming 100 ∞ 2� from the decaying part of �Λ(�(�)) = ∑�=1 �2��(�) should be vanishing. t Tis condition guarantees for us a consistency of our 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 model with astronomical observations of the accelerating Figure 3: Te diagram of the evolution of the Hubble function with phase of the Universe [3]. respect to the cosmological time �,whichisdescribedby(40)and If all parameters �2� for �>1equal zero then the Hubble (43) with �21 =0̸ and �2� =0for every �>1. For illustration, parameter is described by the following formula: the value of the parameter �21 = is chosen as −0.3.Tetopblue curve describes the evolution of the Hubble function in the ΛCDM �21−3 model. Te middle curve describes one for (43) and the bottom red �m,0� +Λbare � (�) =±√ , (44) curve describes one for (40). Te Hubble function is expressed in 3−�21 km/sMpcandthecosmologicaltime� is expressed in s Mpc/km. or Note that these models are not qualitatively diferent.

3−� � (1+�) 21 +Λ � (�) =±√ m,0 bare , (45) 3−� Figure 4 presents the evolution of the scale factor, which 21 is described by (46). Eq. (46) gives us the following formula: �=�−1 −1 where isredshif.From(44)wecanobtainthe Λ 1 bare √ following formula for the expanding Universe: � (�) = √ coth ( Λ bare (3 − �21)�) . (47) 3−�21 2 � (�) In the extension of Friedmann equation (37) matter 2/(3−� ) 21 is contributed as well as dark energy. Te total energy- � √(3 − � )Λ (46) �] �] �] m,0 21 bare momentum tensor � =� +� is of course conserved. =( sinh ( �)) . m de Λ bare 2 However, between the matter and dark energy sectors exist an interaction—the energy density is transferred between Advances in High Energy Physics 7

a In the special case of radioactive decay of vacuum (48) reduces to 1.5 �� (�) m +3�(�) � (�) =−�� =−��Λ(�) , �� m (49) 1.0 �� (�) de =��Λ(�) �� or 0.5 1 � (� (�)3 � (�))=−�� =−��Λ(�) �⇒ � (�)3 �� m

t 3 3 �� 3 0.005 0.010 0.015 0.020 �m (�) � (�) =�m,0�0 −∫�� (�) ��, (50)

Figure 4: Te diagram of the evolution of the scale factor with �� (�) � de =��Λ(�) . respect to the cosmological time ,whichisdescribedby(46).For �� illustration, two example values of the parameter �21 = are chosen: −1 and −2.Tebottombluecurvedescribestheevolutionofthe In the case of the interaction between matter and decay- scale factor in the ΛCDM model. Te middle curve describes one ing dark energy, the natural consequence of conservation of � =−1 � =−2 �] for 21 and the top red curve describes one for 21 .Te the total energy-momentum tensor � is a modifcation of cosmological time � is expressed in s Mpc/km. the standard formula for scaling matter. −3+�(�) Let �m(�) = ��,0� ,where�(�) is a deviation from the canonical scaling of dust matter [32, 33]. Ten we have  ln (�m (�) /�m,0) 0.00006 � (�) = +3. (51) ln � (�) 0.00005

0.00004 4. Conclusions

0.00003 From our investigation of cosmological implications of efects of the quantum decay of metastable dark energy, one can 0.00002 derive following results: 0.00001 (i) Te cosmological models with the running cosmological parameter can be included in the framework t 0.02 0.04 0.06 0.08 0.10 of some extension of Friedmann equation. Te new ingredient in the comparison with the standard cos- Figure 5: Te diagram of the evolution of the parameter � with mological model (ΛCDM model) is that the total � respect to the cosmological time . For illustration, two example energy-momentum tensor is conserved and the inter- � � = −100 / values of the parameter are chosen: km sMpc(the action takes place between the matter and dark energy top green curve) and � = −200km/s Mpc (the middle red curve). Λ �=0 sectors. In consequence the canonical scaling law For comparison the CDM model with the parameter � ∝�−3 Λ(�) is represented by the bottom blue curve. Here, the value of the m is modifed. Because is decaying � � (�Λ/�� < 0) energy of matter in the comoving volume parameter isequalto1.Tecosmologicaltime is expressed in 3 sMpc/km. ∝� is growing with time. (ii) We have found that the appearance of the universal exponential contribution in energy density of the decaying vacuum can explain the irreversibility of the these sectors. Tis process can be described by the system of cosmic evolution. While the reversibility ��→−�is equations still present in the dynamical equation describing the evolutional scenario, in the frst phase of radioactive decay, this symmetry is violated. �� (�) �� (�) �Λ (�) m +3�(�) � (�) =− de =− ef , (iii) We have also compared the time evolution of the �� m �� (48) Hubble function in the model under consideration �� (�) �Λ (�) (where Λ(�) is parameterized by the cosmological de = ef , �� time) with Sola et al. [34] parameterization by the Hubble function. Note that both parameterizations coincide if time � is replaced by the Hubble scale time where it is assumed that pressure of matter �m =0and �de = �� =1/�. If the evolution of the Universe is invariant −�de. Te time variability of the matter and energy density of in the scale, i.e., the scale factor � is changing in power decaying vacuum are demonstrated in Figure 5. law, then this correspondence is exact. 8 Advances in High Energy Physics

Data Availability [16] N. G. Kelkar and M. Nowakowski, “No classical limit of quantum decay for broad states,” Journal Of Physics A,vol.43, No data were used to support this study. p. 385308, 2010. [17] J. Martorell, J. G. Muga, and D. W. L. Sprung, “Quantum post- Conflicts of Interest exponential decay,” in Time in Quantum Mechanics - Vol. 2,J. G.Muga,A.Ruschhaupt,andA.delCampo,Eds.,vol.789of Te authors declare that they have no conficts of interest. Lecture Notes in Physics, pp. 239–275, Springer-Verlag Berlin Heidelberg, Germany, 2009. [18] E. Torrontegui, J. Muga, J. Martorell, and D. Sprung, “Quantum Acknowledgments decay at long times,” in Unstable States in the Continuous Spectra, Part I: Analysis, Concepts, Methods, and Results,C.A. Te authors are very grateful to Dr. Tommi Markkanen NicolaidesandE.Brandas,Eds.,vol.60ofAdvances in Quantum for suggestion of conclusion of their paper and indication Chemistry, pp. 485–535, Elsevier, 2010. of important references in the context of irreversibility of [19] G. Garcia-Calderon, R. Romo, and J. Villavicencio, “Survival quantum decay process and thermodynamic arrow of time. probability of multibarrier resonance systems: exact analytical approach,” Physical Review B,vol.76,p.035340,2007. References [20] F. Giraldi, “Logarithmic decays of unstable states,” European Physical Journal D,vol.69,p.5,2015. [1] S. Weinberg, Reviews of Modern Physics,vol.61,pp.1–23,1989. [21] F. Giraldi, “Logarithmic decays of unstable states II,” European [2] S. Weinberg, “Te cosmological constant problems,” in Sources Physical Journal D,vol.70,p.229,2016. and Detection of Dark Matter and Dark Energy in the Universe, [22] G. Breit and E. 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Research Article A Probability Distribution for Quantum Tunneling Times

José T. Lunardi 1 and Luiz A. Manzoni2

1 Department of Mathematics & Statistics, State University of Ponta Grossa, Avenida Carlos Cavalcanti 4748, 84030-900 Ponta Grossa, PR, Brazil 2Department of Physics, Concordia College, 901 8th St. S., Moorhead, MN 56562, USA

Correspondence should be addressed to Jos´eT.Lunardi;[email protected]

Received 29 June 2018; Accepted 3 August 2018; Published 19 August 2018

Academic Editor: Neelima G. Kelkar

Copyright © 2018 Jos´e T. Lunardi and Luiz A. Manzoni. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

We propose a general expression for the probability distribution of real-valued tunneling times of a localized particle, as measured by the Salecker-Wigner-Peres quantum clock. Tis general expression is used to obtain the distribution of times for the scattering of a particle through a static rectangular barrier and for the tunneling decay of an initially bound state afer the sudden deformation of the potential, the latter case being relevant to understand tunneling times in recent attosecond experiments involving strong feld ionization.

1. Introduction Te conceptual difculty in obtaining an unambiguous and well-defned tunneling time is associated with the impos- Te search for a proper defnition of quantum tunneling sibility of obtaining a self-adjoint time operator in quantum times for massive particles, having well-behaved properties mechanics [17], therefore leading to the need for operational for a wide range of parameters, has remained an important defnitions of time. Several such defnitions exist, such as and open theoretical problem since, essentially, the incep- phase time [18], dwell time [19], the Larmor times [20–23], tion of quantum mechanics (see, e.g., [1, 2] and references and the Salecker-Wigner-Peres (SWP) time [17, 24], and in therein). However, such tunneling times were beyond the some situations these lead to diferent, or even contradictory, experimental reach until recent advances in ultrafast physics results. Tis is not surprising, since by their own nature have made possible measurements of time in the attosecond operational defnitions can only describe limited aspects of scale, opening up the experimental possibility of measuring the phenomena of tunneling, and it is unlikely that any one electronic tunneling times through a classically forbidden defnition will be able to provide a unifed description of region [3–6] and reigniting the discussion of tunneling times. the quantum tunneling times in a broad range of situations. Still, the intrinsic experimental difculties associated with Nevertheless, it remains an important task to obtain a well- both the measurements and the interpretation of the results defned and real time scale that accurately describes the have,sofar,preventedanelucidationoftheproblemand, recent experiments [3–6, 25–27]. in fact, contradictory results persist, with some experiments It is important to notice that the time-independent obtaining a fnite nonzero result [3, 6] and others compatible approach to tunneling times (i.e., for incident particles with with instantaneous tunneling [4]. It should be noticed that sharply defned energy), which comprises the vast majority of the similarity between Schrodinger¨ and Helmholtz equations the literature, is ill-suited to accomplish the above-mentioned allows for analogies between quantum tunneling of massive goal, since it ignores the essential role of localizability in particles and photons [7], and a noninstantaneous tunneling defning a time scale [23, 28]; see, however, [29], which time is supported by this analogy and experiments measuring applies the time defned in [30] to investigate the half-life photonic tunneling times [8], as well as by many theoretical of �-decaying nuclei. A few works (e.g., [23, 28, 31, 32]) calculations based on both the Schrodinger¨ (for reviews see, address the issue of localizability and, consequently, arrive e.g., [1, 2]) and the Dirac equations (e.g., [9–16]). at a probabilistic defnition of tunneling times (that is, an 2 Advances in High Energy Physics average time). In particular, in [28] the SWP clock was used to angular frequency, with � being a nonnegative integer or half- obtain an average tunneling time of transmission (refection) integer giving the clock’s total angular momentum, and � is for an incident wave packet, and such time was employed to the clock’s resolution. Te weak coupling condition amounts investigate the Hartman efect [33] for a particle scattered of to assume that � is large, in such a way that the clock’s energy a square barrier and it was shown that it does not saturate in eigenvalues, �� ≡��(−� < � < �), are very small compared theopaqueregime[28,34]. to the barrier height and the particle’s energy. It is assumed Te tunneling time scales considered in [23, 28, 31] that, at �=0, well before it reaches the barrier, the particle involve taking an average over the spectral components of is well-localized far to the lef of the barrier and the wave the transmitted wave packet and, thus, obscure the interpre- function of the system is a product state of the form tation of the resulting average time. In this paper, we take Φ (�, �, � = 0) =�(�) V (�) , as a starting point the real-valued average tunneling time 0 (2) obtained in [28], using the SWP quantum clock, and obtain where �(�) is the particle’s initial state, represented by a wave a probability distribution of transmission times,byusing packet centered around an energy �0, and the clock initial a standard transformation between random variables. In state is assumed to be “in the zero-th hour” [17] addition to providing a more accurate time characterization of the tunneling process, this should provide a clearer con- 1 � V0 (�) = ∑ �� (�) , (3) nection with the experiments (which measure a distribution √2� + 1 of tunneling times; see, e.g., Figure 4 in [3]). It is worth noting �=−� that some approaches using Feynman’s path integrals address �� √ where ��(�) = e / 2� are the clock’s eigenfunctions the problem of obtaining a probabilistic distribution of the corresponding to the energy eigenvalues ��. tunneling times (see, e.g., [35]). However, these methods in Te state V0(�) is strongly peaked at �=0, thus allowing general result in a complex time (or, equivalently, multiple the interpretation of the angle � as the clock’s hand, since for time scales), and some arbitrary procedure is needed to select a freely running clock the peak evolves to ��,where�� is the the physically meaningful real time aposteriori. time measured by the clock [17]. Since here clock and particle Afer obtaining a general formula for the distribution of are coupled according to (1), when the particle passes through tunneling times, which is the main result of this work, we the region �(�) =0̸ it becomes entangled with the clock, with apply it to two specifc cases. First, to illustrate the formalism the wave function for the entire system given by in a simple scenario, we consider the situation of a particle tunneling through a rectangular barrier. Ten, we consider 1 � a slight modifcation of the model proposed in [36] for the Φ (�, �, �) = ∑ Ψ(�) (�, �) � (�) , √2� + 1 � tunneling decay of an initially bound state, afer the sudden �=−� (4) deformation of the binding potential by the application of ∞ (�) (�) −�� a strong external feld; the modifcation considered here Ψ (�, �) =∫ �� (�) �� (�) e , allows us to investigate the whole range of possibilities for 0 the tunneling times, without having an “upper cutof”, as where � is the incident particle’s energy, �=√�,and�(�) is is the case in the original model. Finally, some additional the Fourier spectral decomposition of the initial wave packet comments on the results are reserved for the last section. �(�) in terms of the free particle eigenfunctions (we are assuming delta-normalized eigenfunctions). Te functions �(�) 2. The SWP Clock’s Average Tunneling Time � (�) satisfy a time-independent Schrodinger¨ equation with a constant potential �� in the barrier region. Outside We start by briefy reviewing the time-dependent application the potential barrier region and for a particle incident from of the SWP clock to the scattering of a massive particle of a �(�)(�) localized static potential barrier in one dimension (for details the lef, the (unnormalized) solution � of the time- see [28]) which is appropriate, since it follows from the three- independent Schrodinger¨ equation is given by [28] dimensional Schrodinger¨ equation for this problem that the �� (�) −�� {e +� (�) e ,�≤−� dynamics is essentially one-dimensional [3]. (�) �� (�) = { (5) �(�) � ��,�≥�, Te SWP clock is a quantum rotor weakly coupled to { ( ) e the tunneling particle and that runs only when the particle (�) (�) is within the region in which �(�) =0̸ ,where�(�) is where � (�) [� (�)] stands for the transmission (refec- the potential energy. Te Hamiltonian of the particle-clock tion) coefcient, and it is assumed, without loss of generality, system is given by (we use ℏ=2�=1,where� is the particle’s that the potential is located in the region −� < � < mass) [17] �. Considering only the transmitted solution in (5) and substituting it into the time-dependent solution (4), it can be �2 �=− +�(�) + P (�) � , (1) shown that for weak coupling ��2 � Φ�� (�, �, �) P(�) = 1 �(�) =0̸ where if and zero otherwise. Te clock’s ∞ �(��−��) � (6) Hamiltonian is �� =−��(�/��),wheretheangle�∈[0,2�) =∫ �� (�) � (�) e V0 (� − �� (�)), is the clock’s coordinate and � = 2�/(2� + 1)� is the clock’s 0 Advances in High Energy Physics 3

� where where {��(�)} is the set of zeros of the function �(�) ≡ �� (�)−� �� (�) and �� is the derivative of �� (�) with respect to �. � �� (�) =−( ) (7) A similar defnition of the distribution of tunneling �� =0 times given in (10)-(11) can be obtained for any time scale which is probabilistic in nature, that is, of the form (8). is the stationary transmission clock time corresponding to Although several other probabilistic tunneling times exist in thewavenumbercomponent� [17, 37]. Te transmission the literature (e.g., [23, 31, 32, 35]), the SWP clock has proven coefcient �(�) corresponds to the stationary problem in the to yield well-behaved real times both in the time-independent absence of the clock. [17, 37, 38] and time-dependent approaches [28, 34, 39] and For tunneling times one is interested only in the clock’s it provides a simple procedure to derive the probabilistic reading for the postselected asymptotically transmitted wave expression (8). In addition, the role exerted by circularly packet. Tus, tracing out the particle’s degrees of freedom, the polarized light in attoclock experiments [3, 25] seems to expectation value of the clock’s measurement can be defned, provide a natural possibility for interpretation in terms of the resulting in the average tunneling time [28] SWP clock. As will be illustrated below, for the simple application of � 2 this formalism to the problem of a wave packet scattered of ⟨�� ⟩=∫��(�) �� (�) ,�(�) =�|� (�) � (�)| , (8) a rectangular potential barrier, the distribution of times (10)- (11) cannot, in general, be obtained analytically even for the � = 1/ ∫ ��|�(�)�(�)|2 where is a normalization constant simplest cases, except in trivial cases such as for a single Dirac �(�) � and is the probability density of fnding the component delta potential barrier [40–42], in which case � (�) = 0 and � in the transmitted wave packet. Similar expressions can be � ��(�) = �(�) ∫ ��(�). obtained for the refection time. It should also be noticed that, despite the fact that the derivation of the previous section leading to (8) and, thus 3. The Tunneling Times Distribution (10)-(11), assumed a scattering situation, these expressions can be shown to be valid for any situation involving prese- An important aspect of the average tunneling time considered lection of an initial state localized to the lef of a potential in the previous section is that it emphasizes the probabilistic “barrier” followed by postselection of an asymptotic trans- nature of the tunneling process. However, since the average in mitted wave packet. Tis allows us to obtain the distribution (8) is over the time taken by the spectral components of the of times for a model that simulates the tunneling decay of an wave packet, it does not lend itself to an easy interpretation, initially bound particle by ionization induced by the sudden given the spectral components of the wave packet tunnel with application of a strong external feld; the model considered diferent times. Tus, instead of (8), one would rather obtain below is a variant of that introduced in [36]. an average over (real) times of the form ∞ ⟨��⟩=∫ �� (�) , (9) 4. The Distribution of Tunneling Times for 0 a Rectangular Barrier where ��(�) stands for the probability density for observing a As a frst illustration of the formalism developed above, let particular tunneling time � for the asymptotically transmitted us consider a rectangular barrier of height �0 located in the wave packet. Tis can easily be achieved by noticing that region �∈(−�,�). Te particle’s initial state �0(�) ≡ �(�, � = in probability theory (8) and (9), which must be equal, 0) is assumed to be a Gaussian wave packet are related by a standard transformation between the two � random variables � and � through a function �� (�).Itfollows 2 that the probability distribution of times is given by 1 (� − �0) �0 (�) = exp [��0�− ], (12) 1/4 √ 4�2 � (2�) � �� (�) =∫�(�) �(�−�� (�))�� (10) � � � which, in essence, is the statement that all the �-components where the parameters 0, ,and 0 are chosen such that the � wave packet is sharply peaked in a tunneling wave number in the transmitted packet for which �� (�) = � must contribute �0 = √�0 < √�0 and is initially well-localized around �=�0, to the value of ��(�) with a weight �(�).Finally,usingthe far to the lef of the barrier; in the calculations that follow we properties of the Dirac delta function (specifcally, we use � take �0 =−8�,suchthatat�=0the probability of fnding the the fact that �(�(�)) = ∑ (�(� − ��)/|� (��)|),where{��} is � particle within or to the right of the barrier is negligible. Te the set of zeros of the function �(�) and the prime indicates transmission coefcient �(�) and the spectral function �(�) a derivative with respect to the independent variable), we are well-known and given by obtain �(� (�)) � 2��−2�� (�) =∑� �, (11) � (�) = �� (� (�))� 2 2 (13) � � � � � (� −� ) sinh (2��) + 2�� cosh (2��) 4 Advances in High Energy Physics

2 1/4 density, time � (�) =( ) √� exp [4�� (�0�+4�) � (14) 1.0 −�(�+� )(��+� �+8�)], 0 0 0.8 �=√� −�2 where 0 . Te stationary transmission clock time 0.6 (7) is [22, 28] � 0.4 �� (�) = � � 0.2 2 2 2 2 2 (15) (� +� ) tanh (2��) + 2�� (� −� ) sech (2��) ⋅ , k 2 2 4�2�2 +(�2 −�2) tanh (2��) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 with tunneling times corresponding to real values of � (i.e., |！(Ｅ)|2 2 (Ｅ) �0 >�). Figure 1 shows a plot for the stationary transmission � Ｎ＝(Ｅ) times �� (�), the distribution of wave numbers �(�) in the 2 transmitted wave packet, and the distribution |�(�)| of density, time wave numbers (momenta) in the incident packet, for two 1.0 values of the barrier width. For the chosen parameters and barrier widths both the incident and the transmitted wave 0.8 packets have an energy distribution very strongly peaked in a tunneling component (in the bottom plot of Figure 1 the 0.6 barrier is much more opaque than that in the top plot and we can observe that—despite being with a negligible probability 0.4 for the parameters chosen for this plot—in this situation some above-the-barrier components start to appear in the 0.2 distribution of the transmitted wave packet. So, in order to consider mainly transmission by tunneling we must restrict k the barrier widths to not too large ones). We also observe 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 the very well-known fact that the transmitted wave packet “speeds up” when compared to the incident particle [28]. |！(Ｅ)|2 As a general rule, the larger is the barrier width (i.e., the (Ｅ) more opaque is the barrier), the greater is the translation Ｎ＝(Ｅ) of the central component towards higher momenta. In what � Figure 1: Stationary transmission clock time �� (�) (green) and the 2 concerns the of-resonance stationary transmission time, it distributions �(�) (orange and arbitrary scale) and |�(�)| (blue, initially grows with the barrier width, and saturates for very dashed, and arbitrary scale) for the transmitted and incident wave opaque barriers (the Hartman efect); on the other hand, packets, respectively. Rydberg atomic units ℏ=2�=1are used it presents peaks at resonant wave numbers that grow and in all plots; the tunneling energies correspond to 0<�<√7, narrow with the barrier width; for a detailed discussion see corresponding to a barrier height �0 =7(the maximum tunneling √ [28]). wave number 7 is shown by a vertical grey line in the plots). In both � =1.5,�=5,� = Figure 2 shows plots of the probability distribution ��(�) plots the incident wave packet parameters are 0 0 −8� 2� = 2 2� = 16 of the tunneling times according to (10)-(11), corresponding . Top: barrier width . Bottom: barrier width . to both the barrier widths shown in Figure 1 [to obtain these plots we used a Monte Carlo procedure to generate alargenumberof� outcomes from the distribution �(�), in the bottom plot of Figure 2 is “closer” to the light time which aferwards were transformed into � values by using than the distribution shown in the top plot; [28] already � the function �=�� (�)]. Te vertical grey lines in these observed that for intermediate values of barrier widths the plots correspond to the time the light takes to cross the average transmission time—corresponding to the mean of barrier distance. It is observed that for the two distributions the distribution ��—reaches a plateau. shown in Figure 2 the probability to observe superluminal tunneling times is negligible. It is also observed that these 5. Distribution of Ionization Tunneling Times distributions have a shape that resembles that of the � distribution, albeit with a more pronounced skewness. Tis In this section we obtain a distribution for tunneling times for shape could be inferred from Figure 1 and from (11), since a particle that is initially in a bound state of a given binding T� �� (�) grows very smoothly in the region were �(�) is potential. Te potential is then suddenly deformed in such a nonvanishing. Furthermore, a comparison between the two way that the particle can escape from the initially confning plots in Figure 2 shows that the tunneling times do not grow region by tunneling. Te model considered here is a slight linearly with the barrier width and, therefore, the distribution modifcation of that proposed by Ban et al. [36] to simulate, Advances in High Energy Physics 5

density such that the particle can now tunnel through the potential 40 barrier; it is assumed that the wave function does not change during the sudden change of the potential. Finally, afer a fnite time �0 the potential returns to its original 30 confguration, �1(�), and tunneling terminates. Te cutof time �0 mimics the natural upper bound for tunneling times 20 measured in recent attoclock experiments (see, e.g., [3, 6] and references therein), since the opening and closing of the tunneling channel in these experiments occur in intervals of 10 half the laser feld’s period. Here, we deviate from [36] by setting �0 �→ ∞ ; i.e., once  deformed the potential does not return to its original form 0.059 0.101 0.167 and, afer a long enough time, the particle will be transmitted density with unit probability; thus, by eliminating the cutof (which 25 is just an experimental limitation) we are able to explore the whole range of possibilities for the ionization tunneling time. 20 In addition, for �≥0, the particle is assumed to be coupled to a SWP quantum clock running only in the region (�, �),so 15 that the clock’s readings for the asymptotic transmitted wave packet give the time the particle spent within the barrier afer 10 �=0.Following[36],weassumethatfor�<0the particle is in the ground state of the potential �1(�),whosestationary 5 wave function is given by

{sin (�0�) , 0 < � ≤ �  � (�) =� 0.058 0.077 0.132 0 { � (�−�) (18) � 0 ,�>�, {sin 0e Figure 2: Probability distribution ��(�) for the tunneling times �, � obtained by Monte Carlo samplings of -values from the distribu- where � is a normalization constant, �0 =√�0, �0 is the tion �(�) and then transforming these to time values through �= � � = √� −�2 �� (�). Te parameters are the same as in the corresponding plots in ground state energy, and 0 0 0. It is also assumed, as Figure 1 and are all expressed in Rydberg atomic units. Top: 2� = 2. in [36], that immediately afer the sudden deformation of the 2� = 16 Bottom: . Tese plots are in the same range and scale and potential from �1(�) to �2(�),at�=0,thewavefunctiondoes can be compared. Te vertical grey line in both plots corresponds not change. However, for �≥0the particle state, which is no to the time the light takes to traverse the barrier distance. Te ticks longer an energy eigenstate, is given by a superposition of the in the horizontal axes correspond to the light time, the minimum, � (�) �=√� � (�) the median and the maximum values of � in the histogram (in the energy eigenstates � ( )ofthepotential 2 , i.e., bottom plot the maximum � is out of the plot’s range). [36] ∞ � (�, � = 0) =�0 (�) = ∫ � (�) �� (�) ��, (19) 0 in a simple scenario, key features of the decay of a localized state by tunneling ionization induced by the application of a where strong external feld with a fnite duration. ∞ � � =∫ � � �∗ � ��, In [36], for �<0, the particle is in an eigenstate of a semi- ( ) 0 ( ) � ( ) (20) 0 infnite square-well potential �1(�), with {+∞ � < 0 { {� (�) sin (��) , 0<�≤� � (�) = 0 0≤�≤� { 1 { (16) { �� −�� { � (�) e +�(�) e , �<�≤� �� (�) = { (21) {�0 �>�, { { 2 √ cos [� (�−�) +Ω(�)] ,�>�, and, therefore, it cannot decay by tunneling. At �=0the { � potential is suddenly deformed to �2(�), 2 where �=√�0 −� and the coefcients �(�), �(�), �(�) +∞ � < 0 { and the phase Ω(�) are determined by the usual boundary { { conditions at �=�and �=�and are such that the {0 0≤�≤� � ⟨� (�), � � (�)⟩ = �(� − � ) �2 (�) = { (17) normalization � � holds [36]. From {� �<�<� { 0 the above expressions it follows that, without any loss of { �(�) 0�≥�, generality, we can take and all the eigenfunctions (21) { to be real. 6 Advances in High Energy Physics

In order to consider the coupling with the SWP clock 10 for times �≥0we proceed as follows. At �=0the system particle+clock is described by the product state �(�, 0)V0(�), 8 where �(�, 0) is the state (19) and V0(�) is the initial clock state given by (3). Afer �=0the particle and the clock states 6 become entangled. For the procedure of postselection of the asymptotically transmitted wave function we notice that the 4 role of the transmission coefcient for the wave function �(−��+Ω(�)(�)) (21) is played by √2/� e ,wherethesuperscript � indicates the weak coupling with the clock. Te right 2 moving asymptotic wave packet representing the coupled system formed by the transmitted particle and the clock is k 2.0 2.2 2.4 2.6 2.8 3.0 3.2 ∞ �[�(�−�)+Ω(�)(�)−��] Φ�� (�, �, �) = ∫ �� (�) e (k) 0 (22) (k) � × V0 [� − �� (�)], 0.25

��(�) = −(�Ω(�)(�)/�� ) =−(1/ 0.20 where, as before, � � ��=0 2�)(�Ω/��) [with quantities without the subscript “(�)” representing the limit �� → 0 ]. By following the same steps 0.15 described in [28], we trace out the clock’s degree of freedom in the asymptotic transmitted wave packet in order to obtain the 0.10 distribution �(�) ofthewavenumbersfortheasymptotically transmitted wave packet, which in this case is simply given by 0.05 2 � (�) = |� (�)| ; (23) k 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 i.e., the probability to fnd a wave number � in the asymptotic transmitted wave packet is the same as in the initial state, (k) which is as expected, since afer a long enough time the (k) initial wave packet will be transmitted with probability unit, � Figure 3: Top: the stationary transmission clock time �� (�) (blue) as mentioned earlier. 2 � and the wave number distribution �(�) = |�(�)| (orange, dashed, Te general behavior of �� (�) and �(�) is illustrated in and arbitrary scale), for �0 =7, �=1, �=3,and�0 ≈ 2.175932), Figures 3 and 4, corresponding to two barriers with diferent with the initial state given by (18). Bottom: close view of the above �−�=2 opacities ( and 4, respectively). Tese plots show, as plot for small times. Te vertical grey lines in the plots correspond �(�) �� expected, that the distribution is strongly peaked at the to �=�0 and �=√�0. Te regions in which �� (�) ≈ 0 (around the � � wave number 0, corresponding to the energy of the initially local maximum and minimum of �� (�)) correspond to times �≈ � bound state and is negligible for nontunneling components. �� (�0) and � ≈ 0.105 �.�. Rydberg atomic units were used in all the For tunneling wave numbers (�<√�0)thefunction plots. � �� (�) is also strongly peaked at the same wave number �0, which corresponds to a local maximum (for nontunneling wave numbers there are several other resonance peaks). � (�) From (11) we would expect that the peaks in the tunneling � coming from local maxima (resonances) and minima � associated with nontunneling values of � are suppressed, times distribution ��(�) would occur for times �=�� (�) �� since �(�) ≈ 0 in these cases. Figure 5 confrm these corresponding to values of � for which � (�) ≈ 0—which � claims. For both barrier widths considered, the distribution occur at points of local maxima and minima of the function ��(�) �(�) of tunneling times is “U” shaped, having peaks at the times � —and corresponding to nonnegligible . Terefore, corresponding to the local maxima and minima of the from the plots in Figures 3 and 4 one could expect the � stationary time �� (�) inside the tunneling region. It should frst peak of the tunneling time distribution ��(�) at �≈ 0.105 �.�. ��(�) be observed that the larger is the barrier width, the broader is (the local minimum of � ,whichissimilarfor the tunneling time distribution, due to the strong increase of ��(�) both barrier widths, since nonresonant times � change the resonant tunneling time with the barrier width. little with the barrier width for opaque barriers, as is the Figures 6 and 7 show close views of the tunneling � (�) case in Figures 3 and 4); a second peak in � is expected time distributions ��(�) for small and large tunneling times ��(�) to occur around the local maximum of � , which corre- (Figure 6 corresponds to the plot at the top of Figure 5, while � sponds to �≈�� (�0) (this local maximum—corresponding Figure 7 corresponds to the plot at the bottom of Figure 5). to resonant wave numbers—changes signifcantly with the InthetopplotsoftheseFigureswecanclearlyobservethe barrier widths; see, e.g., [28]). On the other hand, peaks in frst peak around the local minimum of ��(�) in the tunneling Advances in High Energy Physics 7

10 density

8 0.8

6 0.6

4 0.4

2 0.2

k 2.0 2.2 2.4 2.6 2.8 3.0 3.2 0.0  0 10 20 30 40 (k) density (k) 0.0007 0.20 0.0006

0.0005 0.15 0.0004

0.0003 0.10 0.0002

0.0001 0.05 0.0000  0 5000 10 000 15 000 20 000 k 2.0 2.2 2.4 2.6 2.8 Figure 5: Distributions of tunneling decay times ��(�) through the barrier of the potential �2(�) for the initial bound state �0(�) (k) given by (18). Te histograms were built by using the Monte Carlo (k) procedure described in Figure 2 and in the main text. Te vertical grey lines indicate percentiles of the distribution (the frst and the ��(�) Figure 4: Top: the stationary transmission clock time � (blue) last correspond to 1%and99%, the remaining ones range from 5% �(�) = |�(�)|2 and the wave number distribution (orange, dashed, to 95%, in steps of 5%); the three thick vertical lines indicate the � =7�=1�=5 � ≈ 2.175932 and arbitrary scale), for 0 , , ,and 0 ), frst quartile (percentile 25%), the median, and the third quartile with the initial state given by (18). Bottom: close view of the above (percentile 75%).Rydbergatomicunitswereusedinalltheplots. plot for small times. Te vertical grey lines in the plots correspond Top: barrier width �−� = 2and bin length ≈ 0.0031�.�. (≈0.15 �=� �=√� to 0 and 0. Te region of relatively slow growth of attoseconds). Bottom: barrier width �−� = 4and bin length ≈ 40�.�. �� the derivative � (�) correspond to times around 0.105 �.�. Rydberg (≈ 1,935 attoseconds). atomicunitswereusedinalltheplots.

emergence of very small times in the clock readings. We note region, which in both plots corresponds to almost the same that the introduction of the cutof �0, as in [36], would result value � ≈ 0.105 �.� ≈ 5.1 attoseconds. Te top plot of in a time distribution similar to the truncated distributions Figure 6 shows that for the less opaque barrier there exists a shown in the top plots of Figures 6 and 7. (very small) probability to observe a superluminal tunneling It is also worth observing that, for small times, the time. Even if this possibility cannot be precluded in principle distributions obtained here resemble qualitatively those in (see, e.g., [16]), in the present case the possibility of emergence Figure 4 of [3], except for the presence of several peaks at of such small times was expected, since at �=0there discrete values of the time in the latter. Te considerations was a signifcant portion of the wave packet (roughly 27%) above, relating the peaks of the distribution of clock times penetrating the whole distance of the barrier, and this has an ��(�) to the local maxima and minima of the stationary � important contribution to the emergence of small times in time �� (�) and the magnitude of distribution �(�) in the the clock’s readings associated with the transmitted particle. neighborhood of these points, suggest a scenario in which On the other hand, the top plot of Figure 7 shows that for such multiple peaks at discrete values of time can appear in the thicker barrier the probability for superluminal times is the distribution ��(�) of transmission times. Indeed, if above- negligible; the portion of the wave packet already inside the the-barrier wave numbers had a signifcant contribution to barrier at �=0isthesame(∼27%),butthewavepacket the initial wave packet, then the several local maxima and penetrates proportionally a smaller distance inside the barrier minima present in the vicinities of the resonant nontunneling and, thus, it does not contribute in a signifcant way to the components will also contribute in a signifcant way to build 8 Advances in High Energy Physics

density density 0.8 0.04

0.6 0.03

0.4 0.02

0.2 0.01

  0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 density density

0.0015 0.6

0.0010 0.4

0.0005 0.2

 19 900 19 920 19 940 19 960 19 980  47.60 47.65 47.70 47.75 47.80 47.85 Figure 7: Close views of the plot at the bottom of Figure 5, corresponding to the barrier width �−�=4. Top: small tunneling Figure 6: Close views of the plot atthe top of Figure5, correspond- times and bin length ≈ 0.0031�.�. (≈0.15attoseconds). Te vertical ing to the barrier width �−�=2,withthebinlength≈ 0.0031�.�. ≈0.15 grey line in the lef of this plot corresponds to the time the light takes ( attoseconds). Top: small tunneling times. Te vertical grey to travel the barrier distance. Te percentile 1%(correspondingto line in the lef of this plot corresponds to the time the light takes to ≈ 5.1 �.�. ≈ 247 attoseconds) is out of the range of this plot. Bottom: travel the barrier distance. Te second grey vertical line corresponds ≈ 2 �.� ≈ 100 1 large tunneling times, with bin length attoseconds. to the percentile %. Bottom: large tunneling times. Te vertical grey Te vertical grey line corresponds to the percentile 99%. lines correspond to the percentiles 95%and99%, respectively. multiple peaks in the distribution of transmission times; these peaks in the bottom plot are associated with nontunneling peaks, however, could not be associated with the tunneling components). process. We can consider such a scenario by choosing as the initial state a tightly localized state given by �(�, 0) = √ 6. Conclusions 2 sin �0�,with�0 =�and the barrier parameters �=1, �=2,and�0 =11, in Rydberg atomic units. In this situation Taking as a starting point the probabilistic (average) tunneling the initial wave function is perfectly confned to the lef of time obtained in [28] with the use of a SWP clock [17, 24, 37], the barrier (0<�<1), and above-the-barrier components we obtained a probability distribution of times (10)-(11). An contribute in a signifcant way to build the wave packet, as important advantage of using the SWP clock, in addition to can be seen from �(�) in the top plot of Figure 8 (in this those already mentioned, is that by running only when the case the probability of fnding a nontunneling � component particle is inside the barrier it allows us to address the concept inthewavepacketisapproximately75%). In this plot we can of tunneling time in a proper way, since the time spent by � also observe that all the local maxima and minima of �� (�) the particle standing in the well before penetrating the barrier shown occur in neighborhoods of wave numbers � for which is not computed. A clear advantage of having a probability �(�) is nonnegligible; therefore, all these local maxima and distribution of transmission (tunneling) times is that, in minima contribute signifcantly to build multiple peaks in addition to the usual expectation value, we can obtain all the the distribution of transmission times ��(�).Temiddleand statistical properties of this time, such as its most probable the bottom plots of Figure 8 confrm this statement: all the values (peaks of the distribution), the dispersion around the peaks of the distribution of transmission times correspond mean value, and the probability to observe extreme outcomes very closely to the local maxima and minima of ��(�),ascan (superluminal times, for instance). be seen by comparing the plots in the top and the bottom As an initial test, the distribution of times (10)-(11) was of this fgure (except for the frst, all the other signifcant applied to the simple problem of a particle tunneling through Advances in High Energy Physics 9

times, density 1.0 just the average tunneling time. But, given its probabilistic nature, an answer based only on the average time may not 0.8 be satisfactory, especially if the dispersion of the distribution 0.6 of tunneling times is large, which is ofen the case when one deals with well-localized particles, as suggested by the two 0.4 situations addressed in the present work. 0.2 As a main application of (10)-(11), we considered a slight modifcation of the problem considered in [36] to k model strong feld ionization by tunneling. Te modifcation 2 3 4 5 6 7 considered here was the elimination of the cutof time that (k) was introduced in [36] to simulate the upper bound that (k) arises in attoclock experiments [3, 6] due to the opening and closing of the tunneling channel, naturally associated density with the oscillations in the laser feld intensity. Tis cutof is not a fundamental requirement, but rather it is associated 4 with the experimental methods employed—in any case, its implementation is rather trivial, since it just truncates the 3 distribution of times. Te consideration of the full range of the distribution of times allowed us to show that an 2 important contribution to ��(�) comes from very large times associated with the resonance peaks in the tunneling region; 1 these very long tunneling times occur with a probability comparable to very short ones, thus having an important 0  4 impact on the average tunneling times and, therefore, cause 0 1 2 3 difculties when comparing theoretical predictions based on density an average time with the outcomes of experiments presenting a natural cutof in the possible time measurements. In 4 particular, in the attoclock experiments the relevant measure is ofen associated with the peak of the tunneling time, which 3 maybepromptlyidentifedonceoneknowstheprobability distribution for all possible times. A remark is in place; the distribution of times proposed here, built on the SWP clock 2 readings, refers to the time the particle dwells within the barrier, while the tunneling times ofen measured in recent 1 attoclock experiments actually refer to exit times [43]. In sum, the approach introduced above and resulting in  (10)-(11) builds upon the already (conceptually) well tested 0.1 0.2 0.3 0.4 SWP clock to provide a real-valued distribution of times ��(�) Figure 8: Top:thestationarytime � and the wave number that, in the simple models considered here, was demonstrated �(�) � =11 �−� = 36 � =� distribution ,for 0 , barrier width , 0 , to have physically sound properties and, in fact, (rough) �(�, 0) = √2 � � � =� and an initial state sin 0 ,with 0 .Tevertical similarities with the time distribution obtained in recent lines correspond to �=�0 and �=√�0. Middle: distribution � (�) experiments [3], therefore warrantying further investigation � for the transmission times, with the histogram built by the with more realistic potentials. Monte Carlo procedure described in the text. Vertical lines indicate the percentiles, as in the Figure 5. Bottom: close view of the above histogram for the range of small times. Te frst vertical line at the Data Availability lef indicates the time the light takes to cross the barrier distance. In both the histograms we used a bin length ≈ 0.0031 �.�. ≈ 0.15 Te numerical data used in the construction of the his- attoseconds. Rydberg atomic units were used in all these plots. togramsinthisarticleweregeneratedbyaMonteCarlo procedure, whose details were described in the main text. a rectangular barrier. Unsurprisingly, it revealed behavior Conflicts of Interest similar to that already known from previous works using a distribution of wave numbers (momentum)—see, e.g., Te authors declare that they have no conficts of interest. [28]—although, using ��(�) these conclusions are much more transparent. For example, one could answer the question References about the possibility of superluminal tunneling by direct calculation from the probability distribution ��(�).Inthe [1] H. G. Winful, “Tunneling time, the Hartman efect, and super- nonstationary case—which is the correct to address this luminality: a proposed resolution of an old paradox,” Physics question—this problem is usually answered by considering Reports,vol.436,no.1-2,pp.1–69,2006. 10 Advances in High Energy Physics

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[42] M. A. Lee, J. T. Lunardi, L. A. Manzoni, and E. A. Nyquist, “Double General Point Interactions: Symmetry and Tunneling Times,” Frontiers in Physics,vol.4,ArticleID10,2016. [43] N. Teeny, E. Yakaboylu, H. Bauke, and C. H. Keitel, “Ionization time and exit momentum in strong-feld tunnel ionization,” Physical Review Letters, vol. 116, no. 6, Article ID 063003, 2016. Hindawi Advances in High Energy Physics Volume 2018, Article ID 6980486, 7 pages https://doi.org/10.1155/2018/6980486

Research Article From de Sitter to de Sitter: Decaying Vacuum Models as a Possible Solution to the Main Cosmological Problems

G. J. M. Zilioti,1 R. C. Santos,2 andJ.A.S.Lima 3

1 Universidade Federal do ABC (UFABC), Santo Andre,´ 09210-580 Sao˜ Paulo, Brazil 2Departamento de F´ısica, Universidade Federal de Sao˜ Paulo (UNIFESP), 09972-270 Diadema, SP, Brazil 3Departamento de Astronomia, Universidade de SaoPaulo(IAGUSP),RuadoMat˜ ao˜ 1226, 05508-900 Sao˜ Paulo, Brazil

Correspondence should be addressed to J. A. S. Lima; [email protected]

Received 11 April 2018; Accepted 19 June 2018; Published 16 August 2018

Academic Editor: Marek Szydlowski

Copyright © 2018 G. J. M. Zilioti et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

Decaying vacuum cosmological models evolving smoothly between two extreme (very early and late time) de Sitter phases are able to solve or at least to alleviate some cosmological puzzles; among them we have (i) the singularity, (ii) horizon, (iii) graceful- exit from infation, and (iv) the baryogenesis problem. Our basic aim here is to discuss how the coincidence problem based on a large class of running vacuum cosmologies evolving from de Sitter to de Sitter can also be mollifed. It is also argued that even the cosmological constant problem becomes less severe provided that the characteristic scales of the two limiting de Sitter manifolds are predicted from frst principles.

1. Introduction more plausible than the current view of a strict constant Λ [3–13]. Te present astronomical observations are being successfully Te cosmic concordance model suggests strongly that we explainedbytheso-calledcosmicconcordancemodelor live in a fat, accelerating Universe composed of ∼1/3of Λ 0CDM cosmology [1]. However, such a scenario can hardly matter (baryons + dark matter) and ∼2/3of a constant vac- provide by itself a defnite explanation for the complete uum energy density. Te current accelerating period (�>0̈ ) cosmic evolution involving two unconnected accelerating started at a redshif �� ∼0.69or equivalently when 2�Λ = infationary regimes separated by many aeons. Unsolved ��. Tus, it is remarkable that the constant vacuum and the mysteries include the predicted existence of a space-time sin- time-varying matter-energy density are of the same order of gularity in the very beginning of the Universe, the “graceful- magnitude just by now thereby suggesting that we live in a exit” from primordial infation, the baryogenesis problem, very special moment of the cosmic history. Tis puzzle (“why that is, the matter-antimatter asymmetry, and the cosmic now”?) has been dubbed by the cosmic coincidence problem coincidence problem. Last but not least, the scenario is also (CCP) because of the present ratio Ω�/ΩΛ ∼ O(1),butitwas plagued with the so-called cosmological constant problem almost infnite at early times [14, 15]. Tere are many attempts [2]. intheliteraturetosolvesuchamystery,someofthemclosely One possibility for solving such evolutionary puzzles is to related to interacting dark energy models [16–18]. incorporate energy transfer among the cosmic components, Recently, a large class of fat nonsingular FRW type as what happens in decaying or running vacuum models or, cosmologies, where the vacuum energy density evolves like more generally, in the interacting dark energy cosmologies. a truncated power-series in the Hubble parameter �,has Here we are interested in the frst class of models because the been discussed in the literature [19–22] (its dominant term �+2 idea of a time-varying vacuum energy density or Λ(�)-models behaves like �Λ(�) ∝ � ,� > 0). Such models has (�Λ ≡ Λ(�)/8� �) in the expanding Universe is physically some interesting features; among them, there are (i) a new 2 Advances in High Energy Physics mechanism for infation with no “graceful-exit” problem, (ii) suggested by the renormalization group approach techniques thelatetimeexpansionhistorywhichisveryclosetothe applied to quantum feld theories in curved space-time [24]. cosmic concordance model, and (iii) a smooth link between It is given by the initial and fnal de Sitter stages through the radiation and �+2 matter dominated phases. 2 � Λ (�) ≡8��Λ =�0 +3]� +� � , (5) In this article we will show in detail how the coincidence �� problem is also alleviated in the context of this class of � decaying vacuum models. In addition, partially based on where � is an arbitrary time scale describing the primordial ] previous works, we also advocate here that a generic running de Sitter era (the upper limit of the Hubble parameter), and � are dimensionless constants, and �0 is a constant with vacuum cosmology providing a complete cosmic history 2 evolving between two extreme de Sitter phases is potentially dimension of [�] . able to mitigate several cosmological problems. In a point of fact, the constant � above does not represent a new degree of freedom. It can be determined with the proviso that, for large values of �,themodelstartsfroma 2. The Model: Basic Equations 2 de Sitter phase with �=0and Λ � =3�� .Inthiscase,from �] �] �] �=3(1−]) Te Einstein equations, � =8�� [�(Λ) +�(�)], for an inter- (5) one fnds because the frst two terms there acting vacuum-matter mixture in the FRW geometry read arenegligibleinthislimit[seeEq.(1) in [9] for the case �=1 � � [19, 20] and [11] for a general ]. Te constant 0 can be fxed by the timescaleofthefnaldeSitterphase.For�<<�� we also see 8�� +Λ � =3�2, 2 � ( ) (1) from (4) that �0 =3(1−])��,where�� characterizes the fnal ̇ 2 de Sitter stage (see (6) and (8)). Hence, the phenomenological 8�� −Λ(�) =−2�−3� , (2) law (5) assumes the fnal form: where �� =�� +�� and �=�� +�� are the total ��+2 Λ (�) =3(1−]) �2 +3]�2 +3(1−]) . energy density and pressure of the material medium formed � � � (6) by nonrelativistic matter and radiation. Note that the bare Λ � appearing in the geometric side was absorbed on the matter- Tisisaninteresting3-parameterphenomenologicalexpres- energy side in order to describe the efective vacuum with sion. It depends on the arbitrary dimensionless constant ] and � =−� ≡ Λ(�)/8�� energy density Λ Λ .Naturally,thetime also the two extreme Hubble parameters (��, ��) describing dependence of Λ is provoked by the vacuum energy transfer the primordial and late time infationary phases, respectively. to the fuid component. In this context, the total energy ] �] �] Current observations imply that the value of is very small, � [� +� ] =0 −6 −3 conservation law, � (Λ) (�) ;] , assumes the following |]|∼10 −10 [25, 26]. More interestingly, the analytical form: results discussed below remain valid even for ] =0.Inthis Λ̇ case, we obtain a sort of minimal model defned only by �̇ +3�(� +� )=−�̇ ≡− . (3) � � � � � Λ 8�� a pair of physical time scales, � and �, determining the entire evolution of the Universe. As we shall see, the possible What about the behavior of Λ̇? Assuming that the created existence of these two extreme de Sitter regimes suggests a particles have zero chemical potential and that the vacuum diferent perspective to the cosmological constant problem. fuid behaves like a condensate carrying no entropy, as By inserting the above expression into (3) we obtain the what happens in the Landau-Tisza two-fuid description equation of motion: employed in helium superfuid dynamics[23], it has been Λ<0̇ 3 (1+�)(1−]) �2 �� shown that as a consequence of the second law of �+̇ �2 [1 − � − ]=0. 2 � (7) thermodynamics [10], that is, the vacuum energy density 2 � �� diminishes in the course of the evolution. Terefore, in what follows we consider that the coupled vacuum is continuously In principle, all possible de Sitter phases here are simply transferring energy to the dominant component (radiation or characterized by a constant Hubble parameter (��) satisfying ̇ 2 nonrelativistic matter components). Such a property defnes the conditions �=�=�=0and Λ=3��. For all physically precisely the physical meaning of decaying or running vac- relevant values of ] and � in the present context, we see that uum cosmologies in this work. the condition �=0̇ is satisfed whether the possible values Now, by combining the above feld equation it is readily of �� are constrained by the algebraic equation involving the checked that arbitrary (initial and fnal) de Sitter vacuum scales �� and ��:

3 (1+�) 2 1+� �+2 � 2 2 � �̇ + � − Λ (�) =0, (4) � −� � +� � =0. (8) 2 2 � � � � � �=2 where the equation of state �� =�� (�≥0)wasused.Te In particular, for ,thevaluepreferredfromthe above equations are solvable only if we know the functional covariance of the action, the exact solution is given by form of Λ(�). �2 �2 4�2 Te decaying vacuum law adopted here was frst proposed �2 = � ± � √1− � , � 2 (9) based on phenomenological grounds [7–9, 11] and later on 2 2 �� Advances in High Energy Physics 3

and since �� << �� we see that the two extreme scaling solutions for �=2are �1� =�� and �2� =��.However, 1.0 we also see directly from (8) that the condition �� << ��, also guarantees that such solutions are valid regardless of the 0.8 values of �. In certain sense, since �0 is only the present day expansion rate, characterizing a quite casual stage of the

２ 0.6 recent evolving Universe, probably, it is not the interesting +Ω

scale to be a priori predicted. In what follows we consider that － � ,� 0.4 the pair of extreme de Sitter scales ( � �)arethephysically ,Ω Λ relevant quantities. Tis occurs because diferent from �0, Ω the expanding de Sitter rates are associated with very specifc 0.2 limiting manifolds. For instance, it is widely known that de Sitter spaces are static when written in a suitable coordinate 0.0 system. Besides the discussion on the coincidence problem (see next section), a new idea to be advocated here is that the −2 −1 01234 75 76 77 78 79 80 81 82 prediction of such scales, at least in principle, should be an ln (1+z) interesting theoretical target. Teir frst principles prediction Ω－ +Ω２ would open a new and interesting route to investigate the ΩΛ cosmological constant problem. Figure 1: Standard coincidence problem in the cosmic concordance Te solutions for the Hubble parameter describing analyt- model (Λ 0CDM). Solid and dashed lines represent the evolution of ically the transitions vacuum-radiation (�=1/3)andmatter- the vacuum (ΩΛ ) and total matter-radiation (Ω� +Ω�)density �=0 0 vacuum ( ) can be expressed in terms of the scale factor, parameters. Te circle marks the low (and unique!) redshif present- � ,� ],� the couple of scales ( � �), and free parameters ( ): ing the extreme coincidence between the density parameter of the vacuum and material medium. Note that the model discussed here �� = , is fully equivalent to Λ 0CDM when the time dependent corrections 1/� (10) [1 + ��2�(1−])] in the decay Λ(�) expression are neglected [see (5)].

−3(1−]) 1/2 �=�� [�� +1] . (11) In Figure 1 we display the standard view of the coin- We remark that the transition radiation-matter is like that in cidence problem in terms of the corresponding density the standard cosmic concordance model. Te only diference parameters: Ω� =Ω� +Ω�� (baryons + cold dark matter) ] is due to the small parameter that can be fxed to be zero and Ω� =Ω� +Ω] (CMB photons + relic neutrinos). As (minimal model). Indeed, if one fxes ] =0,thematter- one may conclude from the fgure, the ratio was practically vacuum transition is exactly the same one appearing in the infnite at very high redshifs, that is, at the early Universe fat ΛCDM model. As we shall see below, the fnal scale �� (say, roughly at the Planck time). However, both densities are can be expressed as a simple function of �0, ],andΩΛ. (Ω +Ω )/Ω ∼1 nearly coincident at present. Te ratio � � Λ 0 ) Naturally, the existence of such an expression is needed in is at low redshifs. Note also that, in the far future, that is, order to compare with the present observations. However, it very deep in the de Sitter stage, the ratio approaches zero or cannot be used to hide the special meaning played by �� in a equivalently the inverse ratio is almost infnite because the possible solution of the cosmological constant problem. vacuum component becomes fully dominant. A natural way to solve this puzzle is to assume that the 3. Alleviating the Coincidence Problem vacuum energy density must vary in the course of the expansion. As shown in the previous section, the characteristic Te so-called coincidence problem is very well known. scales of the Λ(�) model specify the evolution during the It comes from the fact that the matter-energy density of extreme de Sitter phases: the primordial vacuum solution 2�(1−]) the nonrelativistic components (baryons + dark matter) with �� << 1 and �=�� behaves like a “repeller” in decreases as the Universe expands while the vacuum energy the distant past, while the fnal vacuum solution for �>>1, � ��3(1−]) ��→ 0 �=� density ( Λ 0 ) is always constant in the cosmic concordance that is, and �,isanattractorinthe model (Λ 0CDM). Tis happens also because the energy distant future. densities of the radiation �� (CMB photons) and neutrinos Te arbitrary integration constants C and D are also easily (�]) are negligible today. Tus, in a broader perspective, one determined. Te constant C can be fxed by the end of the may also say that the ratio (�� +��)/�Λ ,where�� =�� +�], primordial infation (�=0̈ )orequivalently�Λ =��.Tis 0 −2�(1−]) was almost infnite at early times, but it is nearly of the order means that �=�(��) /(1 − 2]) [�(��) corresponds to the unity today. value of the scale factor at vacuum-radiation equality]. In Te current fne-tuning behind the coincidence problem terms of the present day observable quantities we also fnd √ can also be readily defned in terms of the corresponding �=Ω�0/(ΩΛ0 − ]) and �� =�0√ΩΛ0 − ]/ 1−].For Ω ∼0.7 Ω +Ω ∼0.3 ] =0 Ω ∼0.7 � ∼ 0.83� density parameters, since ( Λ 0 and � � ), so and Λ0 one fnds � 0,asexpected that the ratio is of the order unity some 14 billion years later. a little smaller than �0. Te small observable parameter 4 Advances in High Energy Physics

−3 ] <10 quantifes the diference between the late time decaying vacuum model and the cosmic concordance cosmol- 1.0 ogy; namely, 0.8 �0 −3(1−]) 1/2 �= [Ω�0� +1−Ω�0 − ]] . (12) √1−]

２ 0.6

As remarked above, the �(�) expression of the standard +Ω ΛCDM model is fully recovered for ] =0. － 0.4 ,Ω Te solution of the coincidence problem in the present Λ Ω framework can be demonstrated as follows. Te density 0.2 parameters of the vacuum and material medium are given by

Λ (�) �2 �� 0.0 Ω ≡ = ] + (1−]) � + (1−]) , Λ 3�2 �2 �� (13) � −2 −1 01234 75 76 77 78 79 80 81 82 �2 �� ln (1+z) Ω ≡1−Ω =1−] − (1−]) � − (1−]) . Ω +Ω � Λ �2 �� (14) －２ � ΩΛ

Such results are a simple consequence of expression (6) Figure 2: Solution of the coincidence problem in running vacuum for Λ(�) and constraint Friedman equation (1). Note that cosmologies. Te right graphic is our model; the lef is ΛCDM. Ω Ω� ≡Ω�+Ω� is always describing the dominant component, Solid and dashed lines represent the evolution of the vacuum ( Λ) Ω +Ω either the nonrelativistic matter (�=0)orradiation(�= and total matter-radiation ( � �) density parameters for n=2, ] =10−3 � /� =1060 1/3). ,and � 0 . Te late time coincidence between the Te density parameters of the vacuum and material density parameter of the vacuum and material medium (lef circle) has already occurred at very early times (right circle). Note also that medium are equal in two diferent epochs specifying the the values 5 and 75 in the horizontal axis were glued in order to dynamic transition between the distinct dominant compo- show the complete evolution (the suppressed part presents exactly nents. Tese specifc moments of time will be characterized the same behavior). Diferent values of � change slightly the value of �� here by Hubble parameters 1 and 2 .Tefrstequality the redshif for which ΩΛ =Ω� +Ω� at the very early Universe (see (vacuum-radiation, �Λ =��) occurs just at the end of the �� also discussion in the text). frst accelerating stage (�=0̈ ), that is, when �1 =[(1− 1/� 2])/2(1 − ])] ��, while the second one is at low redshifs �� 1/2 �=� when �2 =[2(1−])/(1 − 2])] ��. Note that such results a pure unstable vacuum de Sitter phase with � (in the are also valid for the minimal model by taking ] =0.In beginning there is no matter or radiation, ΩΛ =1,Ω� +Ω� = particular, inserting ] =0in the frst expression above we 0). Te vacuum decays and the model evolves smoothly to a �� 1/� �� ] fnd � =��/2 .Tescale� can also be determined in quasi-radiation phase parametrized by the small -parameter. 1 2 Ω =Ω +Ω terms of �0. By adding the result �� ∼0.83�0 we fnd for Te circles show the redshifs for which Λ � �. �� Of course, the existence of two equality solutions alleviates ] =0that �2 ∼ 1.18�0, which is higher than �0,asshould be expected for the matter-vacuum transition. the cosmic coincidence problem. Naturally, the existence of two subsequent equalities on Te robustness of the solution must also be commented �>0 the density parameter suggests a solution to the coincidence on. It holds not only for any value of but also for −120 −60� ] =0 problem. Neglecting terms of the order of 10 and 10 . In the latter case, the primordial nonsingular vacuum in above expressions, it is easy to demonstrate the following state defates directly to the standard FRW radiation phase. results: Later on, the transition from radiation to matter-vacuum dominated phase also occurs, thereby reproducing exactly the Ω =1 Ω =0 Λ (1) lim��→�� Λ and lim��→�� , matter-vacuum transition of the standard 0CDM model. Te “irreversible entropic cycle” from initial Sitter (��) (2) lim��→� ΩΛ =1and lim��→� Ω� =0. � � to the late time de Sitter stage is completed when the Hubble Te meaning of the above results is quite clear. Te density parameter approaches its small fnal value (� ��→ � �). Te parameters of the vacuum and material components (radia- de Sitter space-time that was a “repeller” (unstable solution) tion + matter) perform a cycle, that is, ΩΛ,andΩ� +Ω� are at very early times (��→∞) becomes an attractor in the periodic in the long run. distant future (��→−1) driven by the incredibly low energy In Figure 2, we show the complete evolution of the vac- scale �� which is associated with the late time vacuum energy 2 uum and matter-energy density parameters for this class of density, �� → 0 , � Λ� ∝��. decaying vacuum model. Diferent from Figure 1 we observe Like the above solution to the coincidence problem, some that the values of ΩΛ and Ω� +Ω� are cyclic in the long run. cosmological puzzles can also be resolved along the same Tese parameters start and fnish the evolution satisfying lines because the time behavior of the present scenario even the above limits. Te physical meaning of such evolution is fxing �=1−] has been proven here to be exactly the one also remarkable. For any value of �>0,themodelstartsas discussed in [20] (see also [9] for the case �=1). Advances in High Energy Physics 5

4. Final Comments and Conclusion decayingvacuumprocess(see[21]fortheentropy produced in the case �=2). For an arbitrary �> Λ(�) As we have seen, the phenomenological -term provided 0, the exit of infation and the entropy production a possible solution to the coincidence problem because the had also already been discussed [22]. Some possible Ω /Ω ratio � Λ is periodic in long run (see Figure 2). In curvature efects were also analyzed [33]. other words, the coincidence is not a novelty exclusive of the current epoch (low redshifs) since it also happened in (iv) Baryogenesis problem: recently, it was shown that the very early Universe at extremely high redshifs. In this the matter-antimatter asymmetry can also be induced framework, such a result seems to be robust because it is not by a derivative coupling between the running vac- altered even to the minimal model, that is, for ] =0. uum and a nonconserving baryon current [34, 35]. It should also be stressed that the alternative complete Such an ingredient breaks dynamically CPT thereby cosmological scenario (from de Sitter to de Sitter) is not a triggering baryogenesis through an efective chemical singular attribute of decaying vacuum models. For instance, it potential (for a diferent but related approach see was recently proved that at the background level such models [36]). Naturally, baryogenesis induced by a running are equivalent to gravitationally induced particle production vacuum process has at least two interesting features: cosmologies [27, 28] by identifying Λ(�) ≡ �Γ/3�,where (i) the variable vacuum energy density is the same Γ is the gravitational particle production rate. In a series of ingredient driving the early accelerating phase of the papers [29, 30], the dynamical equivalence of such scenario Universe and it also controls the baryogenesis process; at late times with the cosmic concordance model was also (ii) the running vacuum is always accompanied by discussed. It is also interesting that such a reduction of particle production and entropy generation [8, 10, 22]. the dark sector can mimic the cosmic concordance model TisnonisentropicprocessisanextrasourceofT- violation (beyond the freeze-out of the B-operator) (Λ 0CDM) at both the background and perturbative levels [31, 32]. In principle, this means that alternative scenarios which as frst emphasized by Sakharov [37] is a basic evolving smoothly between two extreme de Sitter phases ingredient for successful baryogenesis. In particular, for ] =0it was found that the observed B-asymmetry are also potentially able to provide viable solutions of the � main cosmological puzzles. However, diferent from Λ(�)- ordinarily quantifed by the parameter cosmologies, such alternatives are unable to explain the −10 −10 5.7 × 10 <�<6.7×10 (15) cosmological constant problem with this extreme puzzle becoming restricted to the realm of quantum feld theory. can be obtained for a large range of the relevant At this point, in order to compare our results with � ,� alternative models also evolving between two extreme de parameters ( � ) of the present model [34, 35]. Sitter stages, it is interesting to review briefy how the main Tus, as remarked before, the proposed running cosmological problems are solved (or alleviated) within this vacuum cosmology may also provide a successful classofmodelsdrivenbyapuredecayingvacuuminitialstate: baryogenesis mechanism. (i) Singularity: the space-time in the distant past is (v) de Sitter Instability and the future of the Universe: another interesting aspect associated with the pres- a nonsingular de Sitter geometry with an arbitrary ence of two extreme Sitter phases as discussed here is energy scale ��. In order to agree with the semiclassi- the intrinsic instability of such space-time. Long time cal description of gravity, the arbitrary scale �� must 19 ago, Hawking showed that the space-time of a static be constrained by the upper limit �� ≤10 GeV black hole is thermodynamically unstable to macro- (Planck energy) in natural units or equivalently based scopic fuctuation in the temperature of the horizon on general relativity is valid only for times greater �−1 ≥10−43 [38]. Later on, it was also demonstrated by Mottola than the Plank time, � sec. [39] based on the validity of the generalized second (ii) Horizon problem: the ansatz (6) can mathematically law of thermodynamics that the same arguments used be considered as the simplest decaying vacuum law by Hawking in the case of black holes remain valid for which destabilizes the initial de Sitter confguration. the de Sitter space-time. In the case of the primordial Actually, in such a model the Universe begins as a de Sitter phase, described here by the characteristic �� steady-state cosmology, �∼� .Sincethemodel scale ��, such an instability is dynamically described is nonsingular, it is easy to show that the horizon by solution (10) for �(�).Asweknow,itbehaveslike problem is naturally solved in this context (see, for a “repeller” driving the model to the radiation phase. instance, [22]). However, the instability result in principle must also (iii) “Graceful-Exit” from infation: the transition from be valid to the fnal de Sitter stage which behaves the early de Sitter to the radiation phase is smooth like an attractor. In this way, once the fnal de Sitter and driven by (10). Te frst coincidence of density phase is reached, the space-time would evolve to an �� parameters happens for �=�1 ,�Λ =��,and energy scale smaller than � thereby starting a new �=0̈ , that is, when the frst infationary period evolutionary “cycle” in the long run. ends (see Figure 2). All the radiation entropy (�0 ∼ (vi) Cosmological constant problem: it is known that phe- 88 10 , in dimensionless units) and matter-radiation nomenological decaying vacuum models are unable content now observed were generated during the early to solve this conundrum [22, 34]. Te basic reason 6 Advances in High Energy Physics

seems to be related to the clear impossibility to [2] S. Weinberg, “Te cosmological constant problem,” Reviews of predict the present day value of the vacuum energy Modern Physics,vol.61,no.1,pp.1–23,1989. density (or equivalently the value of �0)fromfrst [3] M. Bronstein, “On the expanding universe,” Physikalische principles. However, the present phenomenological Zeitschrif der Sowjetunion,vol.3,pp.73–82,1933. approach can provide a new line of inquiry in the [4] M. Ozer¨ and M. Taha, “A possible solution to the main search for alternative (frst principle) solutions for this cosmological problems,” Physics Letters B,vol.171,no.4,pp. remarkable puzzle. In this concern, we notice that the 363–365, 1986. minimal model discussed here depends only on two [5]M.OzerandM.O.Taha,“Amodeloftheuniversefreeof relevant physical scales (��,��)whichareassociated cosmological problems,” Nuclear Physics,vol.287,p.776,1987. with the extreme de Sitter phases. 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Urbanowski, “Quantum ever, although justifed from diferent viewpoints, the main mechanical look at the radioactive-like decay of metastable dark difculty of such models seems to be a clear-cut covariant energy,” Te European Physical Journal C, vol. 77, no. 12, 2017. Lagrangian description. [14] P. J. Steinhardt, L. Wang, and I. Zlatev, “Cosmological tracking solutions,” Physical Review D: Particles, Fields, Gravitation and Data Availability Cosmology,vol.59,no.12,ArticleID123504,1999. [15] P.J. Steinhardt, Critical Problems in Physics,V.L.FitchandD.R. Te data used to support the fndings of this study are Marlow, Eds., Princeton University Press, Princeton, N. J, 1999. available from the corresponding author upon request. [16] S. Dodelson, M. Kaplinghat, and E. Stewart, “Solving the Coincidence Problem: Tracking Oscillating Energy,” Physical Conflicts of Interest Review Letters, vol. 85, no. 25, pp. 5276–5279, 2000. Te authors declare that there are no conficts of interest [17] W. Zimdahl, D. Pavón, and L. P. Chimento, “Interacting quintessence,” Physics Letters B,vol.521,no.3-4,pp.133–138, regarding the publication of this paper. 2001. [18] A. Barreira and P.P.Avelino, Physical Review D: Particles, Fields, Acknowledgments Gravitation and Cosmology,vol.84,no.8,2011. G. J. M. Zilioti is grateful for a fellowship from CAPES [19]J.A.S.Lima,S.Basilakos,andJ.Solá, “Expansion History (Brazilian Research Agency), R. C. Santos acknowledges the with Decaying Vacuum: A Complete Cosmological Scenario,” INCT-Aproject,andJ.A.S.Limaispartiallysupportedby Monthly Notices of the Royal Astronomical Society,vol.431,p. CNPq, FAPESP (LLAMA project), and CAPES (PROCAD 923, 2013. 2013). Te authors are grateful to Spyros Basilakos, Joan Sola,` [20]E.L.D.Perico,J.A.S.Lima,S.Basilakos,andJ.Solà, and Douglas Singleton for helpful discussions. “Complete cosmic history with a dynamical Λ=Λ(H) term,” Physical Review D: Particles, Fields, Gravitation and Cosmology, References vol. 88, Article ID 063531, 2013. [21]J.A.Lima,S.Basilakos,andJ.Solà, “Nonsingular decaying [1] P.A.R.Adeetal.,Planck Collaboration A&A,vol.594,pp.1502– vacuum cosmology and entropy production,” General Relativity 1589, 2016. and Gravitation,vol.47,no.4,article40,2015. Advances in High Energy Physics 7

[22]J.A.S.Lima,S.Basilakos,andJ.Solà, “Termodynamical [41]T.Harko,F.S.N.Lobo,S.Nojiri,andS.D.Odintsov,“F(R, aspects of running vacuum models,” Te European Physical T)gravity,”Physical Review D: Particles, Fields, Gravitation and Journal C,vol.76,no.4,article228,2016. Cosmology,vol.84,no.2,ArticleID024020,2011. [23] L. Landau and E. Lifshitz, Journal of Fluid Mechanics,Pergamon Press, 1959. [24] I. L. Shapiro and J. Solà, “Scaling behavior of the cosmological constant and the possible existence of new forces and new light degrees of freedom,” Physics Letters B,vol.475,no.3-4,pp.236– 246, 2000. [25] S. Basilakos, D. Polarski, and J. Solà, “Generalizing the running vacuum energy model and comparing with the entropic-force models,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 86, Article ID 043010, 2012. [26] A. Gomez-Valent and J. Solà, “Vacuum models with a linear and a quadratic term in H: structure formation and number counts analysis,” Monthly Notices of the Royal Astronomical Society,vol. 448, p. 2810, 2015. [27] I. Prigogine, J. Geheniau, E. Gunzig, and P. Nardone, “Termo- dynamics and cosmology,” General Relativity and Gravitation, vol. 21, p. 767, 1989. [28] M. O. Calvão, J. A. S. Lima, and I. Waga, “On the thermodynamics of matter creation in cosmology,” Physics Letters A,vol. 162, no. 3, pp. 223–226, 1992. [29]J.A.S.Lima,J.F.Jesus,andF.A.Oliveira,“CDMaccelerating cosmology as an alternative to ΛCDM model,” Journal of Cosmology and Astroparticle Physics, vol. 11, article 027, 2010. [30] J. A. S. Lima, S. Basilakos, and F. E. M. Costa, “New cosmic accelerating scenario without dark energy,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.86,ArticleID 103534, 2012. [31] R. O. Ramos, M. Vargas dos Santos, and I. Waga, “Matter creation and cosmic acceleration,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.89,no.8,ArticleID 083524, 2014. [32] R. O. Ramos, M. Vargas dos Santos, and I. Waga, “Degeneracy between CCDM and ΛCDM cosmologies,” Physical Review D, vol. 90, Article ID 127301, 2014. [33] P. Pedram, M. Amirfakhrian, and H. Shababi, “On the (2+1)- dimensional Dirac equation in a constant magnetic feld with a minimal length uncertainty,” International Journal of Modern Physics D,vol.24,no.2,ArticleID1550016,8pages,2015. [34] J. A. S. Lima and D. Singleton, “Matter-Antimatter Asymmetry Induced by a Running Vacuum Coupling,” Te European Physical Journal C,vol.77,p.855,2017. [35] J. A. S. Lima and D. Singleton, “Matter-antimatter asymmetry and other cosmological puzzles via running vacuum cosmologies,” International Journal of Modern Physics D,2018. [36] V. K. Oikonomou, S. Pan, and R. C. Nunes, “Gravitational Baryogenesis in Running Vacuum models,” International Jour- nal of Modern Physics A, vol. 32, no. 22, Article ID 1750129, 2017. [37] A. Sakharov, “Violation of CP invariance, b asymmetry, and baryon asymmetry of the universe,” JETP Letters,vol.5,p.24, 1967. [38] S. W. Hawking, “Black holes and thermodynamics,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.13, no. 2, pp. 191–197, 1976. [39] E. Mottola, “Termodynamic instability of de Sitter space,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 33, no. 6, pp. 1616–1621, 1986. [40] T. P. Sotiriou and V. Faraoni, “f(R) theories of gravity,” Reviews of Modern Physics,vol.82,no.1,article451,2010. Hindawi Advances in High Energy Physics Volume 2018, Article ID 7204952, 9 pages https://doi.org/10.1155/2018/7204952

Research Article Statistical Physics of the Inflaton Decaying in an Inhomogeneous Random Environment

Z. Haba

Institute of Teoretical Physics, University of Wroclaw, 50-204 Wroclaw, Plac Maxa Borna 9, Poland

Correspondence should be addressed to Z. Haba; [email protected]

Received 8 April 2018; Accepted 21 May 2018; Published 9 July 2018

Academic Editor: Marek Nowakowski

Copyright © 2018 Z. Haba. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

We derive a stochastic wave equation for an infaton in an environment of an infnite number of felds. We study solutions of the linearized stochastic evolution equation in an expanding universe. Te Fokker-Planck equation for the infaton probability distribution is derived. Te relative entropy (free energy) of the stochastic wave is defned. Te second law of thermodynamics for the difusive system is obtained. Gaussian probability distributions are studied in detail.

1. Introduction the quantum Big Bang. Te quantum fuctuations expand forming the observed galactic structure. Te transition to Te ΛCDM model became the standard cosmological model classicality requires a decoherence. Te decoherence can since the discovery of the universe acceleration [1, 2]. It be obtained through an interaction with an environment. describes very well the large-scale structure of the universe. Te environment may consist of any unobservable degrees Te formation of an early universe is explained by the of freedom. In [11, 12] these unobservable variables are infationary models involving scalar felds (infatons) [3, 4]. the high energy modes of the felds present in the initial Such models raise the questions concerning the dynamics theory. We assume that the environment consists of an from their early stages till the present day. Te origin of the infnite set of scalar felds interacting with the infaton. cosmological constant as a manifestation of “dark energy” Te model is built in close analogy to the well-known couldalsobeexplored.Inmodelsofthedarksectorwehope infnite oscillator model [13–16] of Brownian motion. As the to explain the “coincidence problem”: why the densities of the model involves an infnite set of unobservable degrees of dark matter and the dark energy are of the same order today freedom the statistical description is unavoidable. We have as well as “the cosmological constant problem” [5]; why the an environment of an infnite set of scalar � felds which have cosmological constant is so small. We assume that the dark an arbitrary initial distributions. We begin their evolution matter and dark energy consist of some unknown particles from an equilibrium thermal state. In such a case we obtain and felds. Tey interact in an unknown way with baryons and arandomphysicalsystemdrivenbythermalfuctuationsof the infaton. Te result of the interaction could be seen in a the environment. We could also take into account quantum dissipative and difusive behaviour of the observed luminous fuctuations extracting them as high momentum modes of matter. Te difusion approximation does not depend on the infaton as is done in the Starobinsky stochastic infation the details of the interaction but only on its strength and [17]. In such a framework a description of a quantum random “short memory” (Markovian approximation). In this paper evolution is reduced to a classical stochastic process. We we follow an approach appearing in many papers (see [6– can apply the thermodynamic formalism to the study of 10] and references quoted there) describing the dark matter evolution of stochastic systems. In such a framework we and the felds responsible for infation (infatons) by scalar can calculate the probability of a transition from one state felds. In the ΛCDM model the universe originates from to another. In particular, a vacuum decay can be treated 2 Advances in High Energy Physics as a stochastic process leading to a production of radiation We write [18]. �=V +�−3/2�.̃ Te plan of this paper is the following. In Section 2 (7) we derive the stochastic wave equation for an infaton Ten interacting with an infnite set of scalar felds in a homoge- 2 � −2 � 2 � 3/2 −3/2 2 neous expanding metric. In Section 3 we briefy discuss a �� ̃ −� ��̃ +��̃ =−�� +� � (�̃ ) (8) generalization to inhomogeneous perturbations of the metric satisfying Einstein-Klein-Gordon equations. In Section 4 we where approximate the nonlinear system by a linear inhomogeneous 3 9 �2 =�2 +8� V2 − � �− �2. stochastic wave equation with a space-time-dependent mass. � � � � 2 � 4 (9) Te Fokker-Planck equation for the probability distribution We consider large �→∞so that for a large time we may of the infaton is derived. A solution of the Fokker-Planck −2 2 equation in the form of a Gibbs state with a time-dependent neglect � term. Moreover, we assume that �� >0and that temperature is obtained. In Section 5 a general linear system is �� is approximately constant (this is exactly so for the de Sitter space and approximate for power-law expansion when the � discussed. In Section 6 we obtain partial diferential equations −2 for the correlation functions of this system. In Section 7 dependent term decays as � ). Ten, the solution of (8) is Gaussian solutions of the Fokker-Planck equation and their �̃� = (� �) �̃� + (� �) �−1�̃� relation to the equations for correlation functions are studied. cos � 0 sin � � 0 In Section 8 we discuss thermodynamics of time-dependent � −1 3/2 (10) (nonequilibrium) difusive systems based on the notion of −�� ∫ sin (�� (�−�))�� (�) �� . � therelativeentropy(freeenergy).IntheAppendixwetreat 0 a simple system of stochastic oscillators in order to show that Inserting the solution of (3) in (2) we obtain an equation of the formalism works well for this system. the following form: −1/2 1/2 � � 2. Scalar Fields Interacting Linearly � �� (� � )�+� (�) with an Environment � (11) = ∫ K (�, ��)�(��)�� +�−3/2�(�(0) , �),̃ Te CMB observations show that the universe was once in an �0 equilibrium state. Te Hamiltonian dynamics of scalar felds usually discussed in the model of infation do not equilibrate. where We can achieve an equilibration if the scalar feld interacts K (�, �) =�(�)−3/2 ∑�2 (� (�−�))�−1� (�)3/2 � sin � � (12) with an environment. We suggest a feld theoretic model � which is an extension of the well-known oscillator model � discussed in [13–16]. We consider the Lagrangian and the noise depends linearly on the initial conditions (�(0),̃ �(0))̃ : 1 L =�√�+ � ��−�(�) 2 � � =−∑� (� �) �̃� +� (� �) �−1�̃�. � � cos � 0 � sin � � 0 (13) 1 1 (1) � + ∑ ( � �� − �2�� −� �� −� (��)) . 2 � 2 � � � � If the correlation function of the noise is to be stationary (depend on time diference) then we need Equations of motion for the scalar felds read � � −2 � � ⟨�̃ �̃ ⟩ = ⟨� �̃ �̃ ⟩ . (14) �−1/2� (�1/2��)�=−�� − ∑� ��, 0 0 � 0 0 � � (2) � Ten, (assuming that ⟨�̃�⟩=0̃ ,trueintheclassicalGibbs �� state) we get �−1/2� (�1/2��) �� +�2�� =−� �− � , � � � � (3) �� ⟨�� (x) �� (y)⟩ where ��] is the metric tensor and �=|det[��]]|. −3 2 −2 � � (15) =� ∑�� ⟨(�� cos (�� (�−�)) �̃0 (x) , �̃0 (y))⟩ . We consider the fat expanding metric: � ��2 =��2 −�2�x2, (4) We assume a certain probability distribution for initial values. Relation (14) is satisfed for classical as well as quantum Gibbs In the metric (4), (3) reads ̃ 2 2 2 distribution with the Hamiltonian �� = (1/2)(�̃ +��̃ ). �� 2�� +3�� −�−2�� +�2�� =−� �− � . In the classical feld theory in the Gibbs state the covariance � � � � �� (5) −2 2 −1 −2 of the felds in (14) is (−� �+��) .If� �=0then this �−1�−2�(x−y) We may choose covariance is approximated by [19] � .Wechoose � 2 2 2 � ≃ √��−1/2� �� (� ) =�� (�� − V�) . (6) � � (16) Advances in High Energy Physics 3

Under assumption (16) and a continuous spectrum of �� in and (11) we shall have 2 −2 2 �� + (3� + Γ) �� −� �� +�� (1+2�) � ∫ ��K (�, �) � (�) �� (22) =−� � (1+2�) − � (1+2�) �0 � �� 3 =−�2� � (�) − �2� (�) � (�) +�2� (0) � (�) (17) � 2 where 2 −3/2 3/2 −Γ = � �+3��−�−2�(�2� �−��) −� �(�−�0)�(�0)�(�) �(�0) � � � (23)

−1 2 −2 Let Here, �(�) comes from � ∑� �� cos(��);the�(0) term is (an infnite) mass renormalization which appears already in � = (�) �̃ + V 2 � exp � � (24) the Caldeira-Leggett model [15]; it could be included in � . Te last term can be neglected when �0 tends to −∞.Weshall with omit these terms in further discussion. 1 In an expanding metric, (11) takes the following form: � �=− (3� + Γ) � 2 (25) 3 �2�−�−2�� + (3� + �2)��+ �2�� + �� (�) �̃ � � 2 Ten, in the approximation neglecting higher powers of in (18) � −3/2 =�� , 2 −2 2 �� ̃� −� ��̃� +Ω�� =−�� exp (−�)(1+2�) � (26)

where where ⟨� (�, x) �(��, x�)⟩ = � (� − ��)� (x, x�). 1 � (19) Ω2 =(�2 +8V2� ) (1+2�) − (3� + Γ)2 � � � � 4 � � (27) Here, �(x, x )=�(x−x ) comes from the expectation value of 1 −2 − � (3� + Γ) the initial values �̃ and �̃ with the neglect of � �.Ifwedonot 2 � −2 neglect � � in (8) then the form of (19) would be much more 2 2 2 complicated (we would obtain a nonlocal equation). We make We again assume that Ω ≃��+8V��. Ten, in the derivation theapproximation(19)whichpreservestheMarkovproperty of (18) for � the only change comes from the diferentiation and provides stochastic felds which are regular functions of of exp(�) inside the integral (11) and the factor (1 + 2�) x ( it can be considered as a cutof ignoring high momenta multiplying the felds. So, components of �). Equation (18) has been derived earlier in 3 1 [19]. It is applied in the model of warm infation [20]. �2�� → �2 (3� + Γ) � (1+2�)2 2 2 (28)

3. Stochastic Equations in a Perturbed and

Inhomogeneous Metric 2 2 � ��→� (1+2�) �� ((1+2�) �) (29) We did not write yet equations for the metric which result from Lagrangian (1). Te power spectrum of infaton pertur- Hence, our fnal equation is (for a general theory of a bations depends on the metric [3, 4, 21–23]. In general, the stochastic wave equation on a Riemannian manifold see [24] equations for the metric are difcult to solve. Tey are solved and references cited there) in perturbation theory. We consider only scalar perturbations 1 of the homogenous metric using the scalar felds �, �, �, �. �2�+(3�+�2 +Γ)��−�−2�� + �2 (3� + Γ) � � � 2 Ten, the metric is expressed in the following form [3, 23]: +6�2�� + 2�2� �� + 4�2�� (30) 2 2 � � � �� == − (1+2�) �� +2�� (20) +�� (�) (1+2�) =�(1+2�) �−3/2� 2 � � +� ((1 − 2�) �� +2��) �� 4. A Simplified System of a Decaying Inflaton Inserting the metric in (2)-(3) we obtain Te metric (�,�,�,�)can be expressed by � from Einstein 2 −2 � �� +(3� + Γ) ��−� �� + � (�) (1+2�) equations resulting from the Lagrangian (1). We expand � (21) infaton equation with a potential around the classical =−∑�� (1+2�) solution of Klein-Gordon-Einstein equation. Te linearized � version of the equation for fuctuations takes the form of the 4 Advances in High Energy Physics

3 Klein-Gordon equation with a space-time-dependent mass where √�=�.Itcanbecheckedthat [3,4,23,25,26]: � −1 2 3 2 3 2 �=exp (2� �) � (�+� ∫ �� (�) exp (2� �)) (38) ��=Π 0 3 �Π + (3� + �2)Π��+ �2�� + ]�� + �−2�� and 2 (31) 1 �−1� �=− �2�−3�� (��) + (3� + �2)�(0) ∫ �x. � 2 (39) =��−3/2�� Tis normalization factor is infnite (needs renormalization) where we write �=��/��and treat (31) as Ito stochastic but the value of � does not appear in the expectation values −1 diferential equation [27]. Te function ] depends on the ⟨�⟩ = (∫ �) ∫��. potential � in (18) and on the choice of coordinates (�, �) (the � has the meaning of the inverse temperature. Te choice of gauge [28]). We do not discuss ] in this paper. We dependence (38) of the temperature of the difusing system consider in this section the simplifed version of (31) without on the scale factor � has been derived (for �=0and arbitrary −3 � terms: �)in[31–33]foranysystemwith�=1and the � correction �=1 2 −3/2 (35) to the cosmological term (in eq. (38) )(fortime- �Π + (3� + � )Π��=�� (32) dependent cosmological term see [5, 34, 35]).

We defne the energy density: 5. The Linearized Wave Equation 1 �= Π2 2 (33) We can rewrite (4)-(5) in a way that they do not contain frst- order time derivatives of felds. Let 1 Ten, from (32) applying the stochastic calculus [27, 29] and �=�−3/2 (− �2�) Φ (32), we obtain exp 2 (40) �Π2 = 2Π�Π + �Π�Π Ten, the linearized version of the infaton equation expanded around the classical solution (with an account of 2 2 −3/2 = −2 (3� + � )Π �� + 2�� Π�� (34) metric perturbations ) reads ��Φ=Π (41) +�2�−3� (�, �) �� 2 3 9 2 1 2 � ��Π+� Φ− ��Φ − � Φ− � Φ+]Φ We may frst integrate this equation and use ⟨∫ ��⟩ = 0 for 2 4 4 0 (42) the Ito integral. Diferentiating the expectation value over � 1 =�Π+�2Φ+̃]Φ=��3/2 ( �2�) � we obtain � exp 2 � ⟨�⟩ + 6� ⟨�⟩ �� = −2�2 ⟨�⟩ �� where 2 −2 2 1 (35) � =−� �+� (43) + �2� (�, �) �−3�� 2 For the stochastic system (42) the Fokker-Planck equation reads Equation (35) describes the infaton density with �=�/�=1 −3 �2 �2 and a cosmological term varying with the speed � .Te � �= ∫ �x�x�G (x, x�) � 2 � � � term −2� ⟨�⟩ violates the energy conservation of the infaton. 2 �Π (x) �Π (x ) � It describes a decay of the infaton into the felds. If we � couple the � felds to radiation then if � felds are invisible the + ∫ �x (�2Φ+̃]Φ) � (44) �Π x observable efect will be detected as a production of radiation ( ) from the decay of the infaton [18, 30]. � − ∫ �xΠ (x) �≡A . For the stochastic system (32) the Fokker-Planck equation �Φ (x) P reads where �2 �2 � �= ∫ �x�x�G (x, x�) � G (x, x�)=�3 (�2�) � (x, x�) � 2 � �Π (x) �Π (x�) � exp � (45) (36) ] � � and depends on the classical solution in the potential [23, + ∫ �x (3� + �2)Π�. 25]. �Π (x) We may write the noise in the Fourier momentum space: � � � � Ten, the Gaussian solution is ⟨� (�, k) �(� , k )⟩ = G� (k) �(k + k )�(�−� ) (46) 1 �=�exp (− ∫ √�Π�Π) (37) Ten, (42) is rewritten as an ordinary (instead of partial) 2 diferential equation. Advances in High Energy Physics 5

2 6. A Differential Equation for Correlations �� (x, y)=�� (x, y)+�� (y, x)+� G� (x, y) (54)

Let us consider (42) expressed in the following form: �� (y, x) =�� (x, y) +�� (x, y) (55)

��Φ=Π (47) If the system is translation invariant then we can Fourier 1 � Π=�Φ+��3/2 ( �2�) � transform these equations obtaining a system of ordinary � exp 2 (48) diferential equations for Fourier transforms: where �� (�) =�� (�) +�� (−�) (56) 2 −2 2 −� = � + ̃] =−� �+� + ̃] (49) −2 2 2 �� (�) =−(� � +� + ̃])(�� (�) +�� (−�)) (57) Let us denote 2 +� G� (�)

⟨Φ� (x) Φ� (y)⟩ =�� (x, y) (50) −2 2 2 �� (�) =−(� � +� + ̃])�� (�) +�� (�) (58) ⟨Φ� (x) Π� (y)⟩ =�� (x, y) (51) where G�(�) is defned in (46). ⟨Π� (x) Π� (y)⟩ = �� (x, y) (52)

Using the stochastic calculus [27, 29] and taking the expec- 7. Gaussian Solutions of tation value we get a system of diferential equations for the the Fokker-Planck Equation correlation functions: We look for a solution of the Fokker-Planck equation (44) in �� (x, y)=�� (x, y)+�� (y, x) (53) the following form:

1 1 �� =�(�) (−�−2 ∫ �x�x� ( Π� (�, x, x�)Π+Π� (�, x, x�)Φ+ Φ� (�, x, x�)Φ)+∫ �x�Φ + ∫ �x�Π) . � exp 2 1 2 2 3 (59)

or in the momentum space

1 1 �� =�(�) (−�−2 ∫ �k ( Π� (�, k) Π+Π� (�, k) Φ+ Φ� (�, k) Φ) + ∫ �k� (k) Φ (−k) + ∫ �k� (k) Π (−k)). � exp 2 1 2 2 3 (60)

In the confguration space � is an operator and in the It is useful to introduce instead of �� the variables (defned momentum space a function of k. �(�) is determined by by Fourier transforms of �): normalization or directly from the Fokker-Planck equation: 2 �2 �=�3 − (66) −1 1 � � � � � � �=− ∫ �x�x G (x, x )� (x, x ) 1 � 2 � 1 (61) �2 �= (67) Te equations for � read �1

�3 ��1 =−�1G��1 −2�2 (62) �= (68) �1 2 � � =−�G � +� � −� (63) � 2 2 � 1 1 3 We can invert these relations: 2 � � =−�G � +2� � (64) � � 3 2 � 2 2 � = 1 �−�2 (69) where �� = 2 2 (70) 2 −2 2 2 �−� � =� � +� + ̃] (65) �� 3 = (71) We skip the equations for � and �. �−�2 6 Advances in High Energy Physics

� �� can be expressed as Hence,

� � � � � � � � � � � ∫ �� Π �0 (� ,Π )⟨�(�� (� ,Π ),Π� (� ,Π ))⟩ �� =�(�) exp (− ∫ �x�x

(72) � � � � � � � 1 = ∫ �� Π � (� ,Π ) ∫ ��Π�� (� ,Π ;�,Π) (84) ⋅ �−2 ((Π+�Φ) � (Π+�Φ) +Φ�Φ)). 0 2 1 � ⋅�(�,Π)=∫ ��Π�� (�, Π) � (�, Π) From(72)itcanbeseenthattheprobabilitydistribution is diagonal in the variables Φ and Π+�Φ.So,weobtain � � � the expectation value ⟨(Π + �Φ)(x)(Π + �Φ)(x )⟩ = Note that the initial value 0 in (84) according to (59) is 2 −1 � determined by the initial values of ��. A possible choice for � �1 (x, x ). Assume that we calculate the expectation values at time �: the initial value is the thermal Gibbs distribution: � 1 2 2 2 2 �0 = exp (− ∫ �x (Π +(∇�) +� � )) (85) 2 2 2 −1 2 −1 2� �=⟨Π� ⟩=��3 (�1�3 −�2) =�� (73)

−1 which corresponds to the initial condition �2(� = 0) = 0, �=⟨Φ2⟩ =�2� (� � −�2) =�2�−1 � � � � 1 1 3 2 (74) �1(� = 0, x, x ) = (1/�)�(x − x ) and �3(� = 0, x, x )= � (1/�)��(x−x ). We could also consider the initial probability 2 2 −1 2 −1 �=⟨Φ�Π�⟩=−��2 (�1�3 −�2) =−�� (75) distribution:

� 1 2 � = exp (− ∫ �x (� (x) − V) ) (86) �, �, � can be expressed by �, �, � as 0 2�2

−1 describing the feld concentrated at V.Ten,in(59)�(� = �=�� −2 (76) 0, x)=� V. In such a case, the probability distribution (59) −1 describes the probability of the transition from V to � (see [36] �=−�� (77) for such calculations in quantum mechanics). 2 −1 �=�� (78) 8. The Relative Entropy �, �, � If we know then we can express Assume we have a functional equation of the form (like (44)):

−1 2 2 1 � � � �1 =�(��−�) (79) � �= ∫ �x�x D (x, x ) � � 2 � �Π (x) �Π (x�) 2 −1 �2 =−�(�� − � ) (80) � + ∫ �x �1 (� (x) ,Π(x))� (87) 2 −1 �Π (x) �3 =�(��−�) (81) � + ∫ �x � (Φ (x) ,Π(x)) �≡A� �Φ (x) 2 Note that �2(� = 0) = 0 means �(� = 0) = 0. Te relations (67)-(81) allow relating the solutions of the Let us assume that we have two solutions �1 and �2 of this stochastic equation (42) with the solutions of the diferential equation. Defne the relative entropy equations (62)-(64) and the solution of the Fokker-Planck � equation (44). In fact, the solution � canbeexpressedbythe �=∫ �Φ�Π�−1� (� �−1� �−1) Fokker-Planck transition function �� (which is defned by the 1 1 ln 2 1 1 2 (88) solution of the stochastic equation (42) [27]) as follows: where �� (�, Π) = ∫ ��Π�� (��,Π�)� (��,Π�;�,Π) � 0 � (82) �1 = ∫ �Φ�Π�1 (89)

Te expectation values of the solution of the stochastic � = ∫ �Φ�Π� � � 2 2 (90) equation with the initial condition (� ,Π ) is From the defnition of � it follows that [37] � � � � ⟨� (�� (� ,Π ),Π� (� ,Π ))⟩ �≥0 (91) (83) = ∫ ��Π� (��,Π�;�,Π)�(�,Π) � Calculation of the time derivative of � gives Advances in High Energy Physics 7

1 2 � � � �=− ∫��Π� (� �−1) �x�x�D (x, x�) (� �−1) (� �−1) ≤0 � 2 1 2 1 � �Π (x) 1 2 �Π (x�) 1 2 (92)

� We choose �2 =� (then �2 =1because �(�) is the normal- and observations of galaxies evolution (including galaxies ization factor). Ten, we defne the entropy (the entropy of distribution). Te CMB spectrum and its fuctuations are infaton and gravitational perturbations has been discussed the test ground for models involving quantum and thermal earlier in [38, 39]): fuctuations. A simplifed description of an interaction of a relativistic system with an environment leads to a stochastic −1 −1 �=−� ∫ �Φ�Π� ln (� �) (93) wave equation for an infaton generating the expansion (infation) of the universe. We considered a linearization of the wave equation. We discussed the Fokker-Planck equation for Using (87) we calculate the time derivative: the probability distribution of the infaton. Gaussian solutions −1 −1 of the Fokker-Planck equation for linearized systems can be ��=−� ∫ �Φ�ΠA� ln �−� ∫ �Φ�ΠA� (94) treated as Gibbs states with a time-dependent temperature. Te model leads to a formula for density and temperature Te second term is zero, whereas the frst term is equal to evolution. We have derived the density evolution law in (35) and the temperature evolution in (38) in a simplifed model. �� In order to obtain the results in the complete model we would have to solve (numerically) equations of Section 7. 1 −1 � � � � = ∫ �Φ�Π� �x�x D� (x, x ) � � Te comparison of density evolution (38) with observations 2 �Π (x) �Π (x�) (95) is discussed in [32] and in similar models with the decaying �� vacuum ( see [34, 35] and references cited there). Te + ∫ ��Π ∫ �x (�1 +�2 ) model allows calculating (and compare with observations) �Π (x) �Φ (x) the power spectrum resulting from thermal fuctuations which may go beyond the approximations applied in the In(95)thefrsttermispositivewhereasthesecondterm warminfationof[20].Wehaveintroducedathermodynamic depends on the dynamics (it is vanishing for Hamiltonian description of the expanding difusive systems in terms of � =�� dynamics). Using formula (59) for 2 we obtain the relative entropy (free energy) and entropy. Te state of 1 a stochastic system can be identifed with its probability �+�=�−1 ∫ �Φ�Π� ( �−2Π� Π+�−2Φ� Π 2 1 2 distribution. Te relative entropy allows comparing the evo- (96) lution of the probability distributions with diferent initial 1 + �−2Φ� Φ) − � (�) conditions. In this sense relative entropy can be treated as a 2 3 ln quantitative measure of a decay of one state into another state (as an alternative to a quantum description of vacuum decay Formula (96) has a thermodynamic meaning relating the in cosmology [40, 41]). sum of free energy � and entropy � to the internal energy expressedbytherhsof(96).Attheinitialtime(withtheinitial conditions discussed at the end of Section 7) the rhs of (96) Appendix isthemeanvalueoftheenergy A. Statistical Physics of a Static Finite 1 � = ∫ �x (Π2 + (∇Φ)2 +�2Φ2) Dimensional Model 0 2 (97) � A fnite dimensional analog of the wave equation is (x ∈�): In the static universe we would have an equilibrium distribution as �2.Insuchacasethethermodynamicrelation(95) �� would describe the standard version of the second law of =�� (A.1) thermodynamics of difusing systems. � with ��≤0in (92) �� would show the approach to equilibrium. In the expanding �� universe the relation (96) can serve for a comparison of =−Γ�� −�2�� +�� (A.2) various probability measures starting from diferent initial �� conditions. Te Fokker-Planck equation reads 9. Summary 2 2 � � � � 2 � � � � Te main source of observational data [1, 2] comes from � �=−� + (Γ� +� � )�+ (A.3) � �� 2 �� measurements of the cosmic microwave background (CMB) 8 Advances in High Energy Physics

Te stationary solution is Acknowledgments

Γ 2 2 2 E Interesting discussions with Zdzislaw Brzezniak on stochastic �∞ = exp (− (p +� x )) = exp (− ) �2 � (A.4) wave equations during my stay at York University are grate- fully acknowledged where E is the energy of the oscillator. It describes a Gibbs state with the temperature References �2 �= (A.5) [1] P. A. R. Ade, N. Aghanim, M. Arnaud et al., https://arxiv.org/ 2Γ abs/1502.01589. We look for a solution of (A.3) in the following form: [2]K.T.Story,C.L.Reichardt,Z.Houetal.,“Ameasurementof the cosmic microwave background damping tail from the 2500- 1 1 square-degree SPT-SZ survey,” Te Astrophysical Journal,vol. � =�(�) (− � p2 −� xp − � x2) � exp 2 1 2 2 3 (A.6) 779,no.1,p.86,2013. [3] V.F. Mukhanov and G. V.Chibisov, “Quantum fuctuations and Ten a non-singular universe,” JETP Letters, vol. 33, p. 532, 1981. � �−1� �=− �2� +Γ� [4] M. Sasaki, “Large scale quantum fuctuations in the infationary � 2 1 (A.7) universe,” Progress of Teoretical and Experimental Physics,vol. 76,p.1036,1986. 2 2 ��1 =−2�2 −� �1 +2Γ�1 (A.8) [5] S. Weinberg, “Te cosmological constant problem,” Reviews of Modern Physics,vol.61,no.1,pp.1–23,1989. 2 2 ��2 =−�3 −� �1�2 +Γ�2 +� �1 (A.9) [6] P. J. E. Peebles and A. Vilenkin, “Noninteracting dark matter,” 2 2 2 Physical Review D: Particles, Fields, Gravitation and Cosmology, ��3 =−��2 +2� �2 (A.10) vol. 60, no. 10, Article ID 103506, 1999. [7]P.J.E.PeeblesandB.Ratra,“Cosmologywithatime-variable Let us write c o s m o l o g i c a l ’c ons t ant’,” Astrophysical Journal,vol.325,p.L17, Γ 1988. �� = (− �) �� exp 2 (A.11) [8] Y. L. Bolotin, A. Kostenko, O. A. Lemets, and A. D. Yerokhin, “Cosmological evolution with interaction between dark energy Ten, the stochastic equation for � reads and dark matter,” International Journal of Modern Physics D,vol. 24, Article ID 1530007, 2014. �2�� Γ =−Ω2�� +� ( �) �� (A.12) [9]M.C.Bento,O.Bertolami,andN.C.Santos,“Atwo-feld ��2 exp 2 quintessence model,” Physical Review D: Particles, Fields, Grav- itation and Cosmology,vol.65,no.6,2002. where [10] F. Bezrukov, D. Gorbunov, and M. Shaposhnikov, “On initial 2 conditions for the hot big bang,” JournalofCosmologyand 2 2 Γ Ω =� − (A.13) Astroparticle Physics,vol.2009,no.06,pp.029-029,2009. 4 [11] E. Calzetta and B. L. Hu, “Quantum fuctuations, decoherence Te solution of (A.12) is of the mean feld, and structure formation in the early Universe,” Physical Review D: Particles, Fields, Gravitation and Cosmology, �� (�) vol.52,p.6770,1995. [12] D. Polarski and A. A. Starobinsky, “Semiclassicality and deco- � −1 � = cos (Ω�) � (0) + sin (Ω�) Ω �� (0) herence of cosmological perturbations,” Classical and Quantum (A.14) Gravity,vol.13,no.3,pp.377–391,1996. � −1 Γ [13] G.W.Ford,J.T.Lewis,andR.F.O’Connell,“QuantumLangevin +�∫ sin (Ω (�−�)) Ω exp ( �) � (�) �� 0 2 equation,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 37, no. 11, pp. 4419–4428, 1988. � We can easily calculate the correlation functions of � (�) [14] G. W. Ford and M. Kac, “On the quantum Langevin equation,” � and � (�) in two ways: either from the stochastic equations Journal of Statistical Physics,vol.46,no.5-6,pp.803–810,1987. or using the probability distribution �� resulting from the [15] A. O. Caldeira and A. J. Leggett, “Path integral approach to solution of the diferential equations (A.8)-(A.10). quantum Brownian motion,” Physica A,vol.121,p.587,1983. [16]H.KleinertandS.V.Shabanov,“QuantumLangevinequation from forward-backward path integral,” Physics Letters A,vol. Data Availability 200, Article ID 9503004, p. 171, 1995. [17] A. A. Starobinsky, “Stochastic de Sitter (infationary) stage in Tedatausedtosupportthefndingsofthisstudyare the early universe,” in Current Topics in Field Teory,Quantum available from the corresponding author upon request. Gravity and Strings, H. J. Vega and N. Sanchez, Eds., vol. 246 of Lecture Notes in Physics, pp. 107–126, Springer, 1986. Conflicts of Interest [18] P. J. Steinhardt and M. S. Turner, “Prescription for successful new infation,” Physical Review D: Particles, Fields, Gravitation Te author declares that there are no conficts of interest. and Cosmology,vol.29,no.10,pp.2162–2171,1984. Advances in High Energy Physics 9

[19] A. Berera, “Termal properties of an infationary universe,” [39] R. Brandenberger, V. Mukhanov, and T. Prokopec, “Entropy Physical Review D: Particles, Fields, Gravitation and Cosmology, of the gravitational feld,” Physical Review D: Particles, Fields, vol. 54, no. 4, pp. 2519–2534, 1996. Gravitation and Cosmology,vol.48,no.6,pp.2443–2455,1993. [20]A.Berera,I.G.Moss,andR.O.Ramos,“Warminfationand [40] L. M. Krauss and J. Dent, “Late time behavior of false vacuum its microphysical basis,” ReportsonProgressinPhysics,vol.72, decay: possible implications for cosmology and metastable Article ID 026901, 2009. infating states,” Physical Review Letters,vol.100,no.17,Article [21] V.F. Mukhanov, “Quantum theory of gauge invariant cosmolog- ID 171301, 4 pages, 2008. ical perturbations,” Soviet Physics—JETP, vol. 67, p. 1297, 1988. [41] A. Stachowski, M. Szydlowski, and K. Urbanowski, “Cosmolog- [22] A. A. Starobinsky, “Dynamics of phase transition in the new ical implications of the transition from the false vacuum to the infationary universe scenario and generation of perturbations,” true vacuum state,” Te European Physical Journal C,vol.77,p. Physics Letters B,vol.117,no.3-4,pp.175–178,1982. 357, 2017. [23] B. A. Bassett, S. Tsujikawa, and D. Wands, “Infation dynamics and reheating,” Reviews of Modern Physics,vol.78,no.2,pp. 537–589, 2006. [24] Z. Brzezniak and M. Ondrejat, “Strong solutions to stochastic wave equations with values in Riemannian manifolds,” Journal of Functional Analysis,vol.253,no.2,pp.449–481,2007. [25] J. Hwang, “Curved space quantum scalar feld theory with accompanying metric fuctuations,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.48,no.8,pp.3544–3556, 1993. [26]J.E.Lidsey,A.R.Liddle,E.W.Kolb,E.J.Copeland,T.Barreiro, and M. Abney, “Reconstructing the infaton potential—an overview,” Reviews of Modern Physics,vol.69,no.2,pp.373–410, 1997. [27] N. Ikeda and S. Watanabe, Stochastic Diferential Equations and Difusion Processes, North-Holland, Amsterdam, Te Nether- lands, 1981. [28] J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, “Sponta- neous creation of almost scale free density perturbations in an infationary universe,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.28,no.4,p.678,1985. [29] B. Simon, Functional Integration and Quantum Mechaics,Aca- demic Press, New York, 1979. [30] A. Berera and L.-Z. Fang, “Termally induced density perturbations in the infation era,” Physical Review Letters,vol.74,no. 11, pp. 1912–1915, 1995. [31] Z. Haba, “A relation between difusion, temperature and the cosmological constant,” Modern Physics Letters A,vol.31,no.24, p. 1650146, 2016. [32]Z.Haba,A.Stachowski,andM.Szydłowski,“Dynamicsofthe difusive DM-DE interaction—Dynamical system approach,” Journal of Cosmology and Astroparticle Physics,vol.2016,no.07, pp. 024-024, 2016. [33] Z. Haba, “Termodynamics of difusive DM/DE systems,” General Relativity and Gravitation,vol.49,no.4,Art.58,21 pages, 2017. [34] J. M. Overduin and F. I. Cooperstock, “Evolution of the scale factor with a variable cosmological term,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.58,no.4,1998. [35] M. Szydłowski and A. Stachowski, “Cosmology with decaying cosmological constant—exact solutions and model testing,” JournalofCosmologyandAstroparticlePhysics,vol.2015,no.10, pp. 066-066, 2015. [36] A. H. Guth and S. Pi, “Fluctuations in the new infationary universe,” Physical Review Letters, vol. 49, no. 15, pp. 1110–1113, 1982. [37] H. Risken, Te FokkerPlanck Equation: Method of Solution and Applications, Springer, Berlin, Germany, 1989. [38] R. Brandenberger, V. Mukhanov, and T. Prokopec, “Entropy of a classical stochastic feld and cosmological perturbations,” Physical Review Letters,vol.69,no.25,pp.3606–3609,1992. Hindawi Advances in High Energy Physics Volume 2018, Article ID 4672051, 7 pages https://doi.org/10.1155/2018/4672051

Research Article QFT Derivation of the Decay Law of an Unstable Particle with Nonzero Momentum

Francesco Giacosa 1,2

1 Institute of Physics, Jan-Kochanowski University, Ul. Swietokrzyska 15, 25-406 Kielce, Poland 2Institute for Teoretical Physics, J. W. Goethe University, Max-von-Laue-Str. 1, 60438 Frankfurt, Germany

Correspondence should be addressed to Francesco Giacosa; [email protected]

Received 7 April 2018; Accepted 3 June 2018; Published 26 June 2018

Academic Editor: Neelima G. Kelkar

Copyright © 2018 Francesco Giacosa. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

We present a quantum feld theoretical derivation of the nondecay probability of an unstable particle with nonzero three- momentum p. To this end, we use the (fully resummed) propagator of the unstable particle, denoted as �, to obtain the energy p probability distribution, called � (�), as the imaginary part of the propagator. Te nondecay probability amplitude of the particle � ∞ p �p(�) = ∫ ��p(�)�− � with momentum turns out to be, as usual, its Fourier transform: 2 +p2 ( ℎ is the lowest energy threshold �ℎ in the rest frame of � and corresponds to the sum of masses of the decay products). Upon a variable transformation, one can p ∞ −2 +p2 0 �ℎ 0 rewrite it as � (�) = ∫ ��(�)� [here, �(�) ≡ �(�) is the usual spectral function (or mass distribution) in the rest �ℎ frame]. Hence, the latter expression, previously obtained by diferent approaches, is here confrmed in an independent and, most importantly, covariant QFT-based approach. Its consequences are not yet fully explored but appear to be quite surprising (such as the fact that the usual time-dilatation formula does not apply); thus its frm understanding and investigation can be a fruitful subject of future research.

1. Introduction energy (or mass) between � and �+dm]. A review of the derivation is presented in Section 2. Quite remarkably, there Te study of the decay law is a fundamental part of quantum are many interesting properties linked to this equation, which mechanics (QM). It is now theoretically [1–9] and experimen- include deviations from the standard dilatation formula; see tally [10, 11] established that deviations from the exponential below. decay exist, but they are usually small. Such deviations are Te purpose of this work is straightforward: we derive also present in quantum feld theory (QFT) [9, 12]. (1) in a QFT framework (see Section 3). We thus confrm An interesting question addresses the decay of an unstable its validity and, as a consequence, its peculiar features. For p. particle with nonzero momentum In [13–17], it was defniteness, an underlying Lagrangian involving scalar felds shown—by using QM-based approaches enlarged to include is introduced, but our discussion is valid for any spin of the special relativity—that the nondecay probability of an unsta- unstable particle and of the decay products. A central quantity � p ble particle with momentum is given by (in natural units) of our study is the relativistic propagator of the unstable p � p �2 particle �: the mass distribution �(�) is then obtained by the � (�) = �� (�)� imaginary part of the propagator. ∞ 2 2 (1) In this Introduction, we recall some basic and striking �p (�) =∫ � (�) �− +p , with dm features connected to (1). Te normalization �ℎ ∞ � (�) � where is the energy (or mass) distribution of in its rest ∫ dm� (�) =1 (2) frame [dm�(�) is the probability that the unstable state � has �ℎ 2 Advances in High Energy Physics is a crucial feature of the spectral function, implying that Namely, the quantity �=√p2 +�2/� = 1/√1−k2 is p � (0) = 1. It must be valid both in QM and in QFT. Here, the usual dilatation factor for a state with (defnite) energy without loss of generality, we set the lower limit of the integral �. Deviations between (8) and (9) are very small, as the to �ℎ ≥0. In fact, a minimal energy �ℎ is present in numerical discussion in [17] shows. Although not measurable each physical system; in particular, for a (relevant for us) by current experiments [20], the very fact that deviations exist relativistic system, it is given by the sum of the rest masses is very interesting and deserves further study. of the produced particles (�ℎ =�1 +�2 +⋅⋅⋅≥0).Clearly, As a last point, it should be stressed that in this work we p. for p = 0, (1) reduces to the usual expression consider unstable states with a defnite momentum Tis is a subtle point: while for a state with defnite energy, a � ∞ �2 0 � 0 �2 � −� boost and a momentum translation are equivalent, this is not � (�) =� (�) = �� (�)� = �∫ � (�) � � . (3) � � � dm � so for an unstable state, since it is not an energy eigenstate. � �ℎ � Even more surprisingly, a boost of an unstable state is a A detailed study of (1) shows that the usual time dilatation quantum state whose nondecay probability is actually zero: does not hold: it is already decayed (on the contrary, its survival probability

p � presents a peculiar time contraction [21]). In other words, � (�) =�̸ (� ), (4) √�2 + p2 a boosted muon consists of an electron and two neutrinos [15, 17]. In this sense, the boost mixes the Hilbert subspace where � is the mass of the state � defned, for instance, as of the undecayed states with the subspace of the decay the position of the peak of the distribution �(�); in general, products; see [17] for details. Tere, it is also discussed however, other defnitions are possible, such as the real part why the basis of unstable states contains states with defnite of the pole of the propagator; see Section 3. Te point is that, three-momentum. Indeed, the investigation of this paper also no matter which defnition one takes, expression (4) remains confrms this aspect: unstable states with defnite momentum an inequality. naturally follow from the study of its propagator in QFT. It is always instructive to investigate the exponential limit, Te paper is organized in this way: in Section 2 we recall in which the spectral function of the state � reads [18, 19] the QM derivation of (1), while in Section 3—the key part of this paper—we present this derivation in a QFT context. In 2 −1 Γ 2 Γ the end, in Section 4 we describe our conclusions. � (�) = [(�−�) + ] , (5) 2� 4 2. Recall of the QM-Based Derivation of (1) where � is the “mass of the unstable state” corresponding to the peak. Even if the spectral function � (�) is clearly For completeness, we report here the “standard” derivation of unphysical because there is no minimal energy (�ℎ →−∞), (1). To this end, we use the arguments presented in [17], but in many physical cases it is a good approximation for a quite similar ones can be found in [13–16]. � broad energy range. Here, the decay amplitude and the decay We consider a system described by the Hamiltonian , law in the rest frame of the decaying particle notoriously read whose eigenstates are denoted as ,0 −−Γ/2 |�, p⟩ =�p |�, 0⟩ , (10) � (�) =� �→ (6) where �p is the unitary operator associated with the trans- , −Γ � (�) =� . lation in momentum space. Standard normalization expressions are assumed: When a nonzero momentum is considered, the nondecay probability is still an exponential given by ⟨�1, p1 |�2, p2⟩=�(�1 −�2)�(p1 − p2). (11) ,p −Γp Te state |�, p⟩ has defnite energy, � (�) =� (7) where the width is [17] � |�, p⟩ = √p2 +�2 |�, p⟩ , (12) Γ p defnite momentum,

1/2 P |�, p⟩ = p |�, p⟩ , Γ2 2 Γ2 (8) (13) = √2√[(�2 − + p2) +�2Γ2] −(�2 − + p2). 4 4 and defnite velocity p/√p2 +�2. Note, assuming that the energy of |�, p⟩ is√p2 +�2,wehavearelativisticspectrum. Clearly, Γp=0 =Γ.One realizes, however, that Γp difers from the naively expected standard time-dilatation formula, Formally, the Hamiltonian can be written as according to which the decay width of an unstable state with ∞ �=∫ 3 ∫ ��√p2 +�2 |�, p⟩⟨�, p| momentum p should simply be d p �ℎ Γ� (14) =�Γ. =∫ 3 � √p2 +�2 (9) d p p Advances in High Energy Physics 3 where follows. Note that (20) is not a state with defnite velocity. Tis ∞ is due to the fact that each state |�, p⟩ in the superposition has √ 2 2 �p = ∫ �� p +� |�, p⟩⟨�, p| (15) p/√p2 +�2 a diferent velocity .TesubsetofHilbertspace �ℎ 2 given by {|�, p⟩∀p ⊂�} represents the set of all undecayed is the efective Hamiltonian in the subspace of states with quantum states of the system under study. p. defnite momentum Te form of the Hamiltonian �p in terms of the states � Let us now consider an unstable state in its rest frame. |�, p⟩ and �p|k, 0⟩=|k, p⟩ canbeinprinciplederivedby Te corresponding quantum state at rest is assumed to be using the expressions above. Together with (20), one shall also ∞ take (19) and apply �p in order to get |�, 0⟩ =∫ dm� (�) |�, 0⟩ , (16) �ℎ ∞ � |k, 0⟩ = |k, p⟩ =∫ � (�) |�, p⟩ . � (�) � p dm k (22) where is the probability amplitude that the state has �ℎ energy �. Hence, it is natural that the quantity �(�) = 2 � |�(�)| is the mass distribution: �(�)dm is the probability Ten, one should invert (20) and (22) and insert it into p of that the unstable particle � has a mass between � and �+dm. (15). However, its explicit expression is defnitely not trivial ∞ ∫ � (�) = 1 but, fortunately, also not needed in the present work. Hence, As a consequence, 0 dm , as already discussed in the Introduction. we do not attempt to write it down here. � For the states of zero momentum, the Hamiltonian �p=0 We now turn to the nondecay amplitude of the state . p = 0 can be expressed in terms of the undecayed state |�, 0⟩ and its In its rest frame ( ), upon starting from a properly decay products in the form of a Lee Hamiltonian [22] (similar normalized state with zero momentum, |�, 0⟩/√�(0),one efective Hamiltonians are used also in quantum mechanics obtains the usual expression [6, 19] and quantum feld theory [9, 23]): 0 1 � −� ∞ � (�) = ⟨�, 0 �� , 0⟩ � (0) � � �p=0 =∫ dm� |�, 0⟩⟨�, 0| �ℎ 1 ∞ � � = � −� ∫ dm1dm2 ⟨�1, 0 �� 2, 0⟩ (23) 3 � (0) =�0 |�, 0⟩⟨�, 0| +∫d k� (k) |k, 0⟩⟨k, 0| (17) �ℎ ∞ − 3 =∫ � (�) � , d k dm +∫ �� (k) [|�, 0⟩⟨k, 0| + |k, 0⟩⟨�, 0|], �ℎ (2�)3/2 in agreement with (3). Te theory of decay is discussed in |k, 0⟩ where represents a decay product with vanishing total great detail for the case p = 0 in[4,6,7,9]andreferences |k, 0⟩ momentum: in the two-body decay case, describes two therein. Note here the nondecay probability coincides with k particles, the frst with momentum and the second with the survival probability (that is, the probability that the state −k momentum ,hence did not change), but in general this is not the case [17]. Next, we consider a normalized unstable state � with � (k) = √k2 +�2 + √k2 +�2. (18) 1 2 nonzero momentum: |�, p⟩/√�(0). Te resulting nondecay Te last term in (17) represents the “mixing” between |�, 0⟩ probability amplitude and |k, 0⟩, which causes the decay of the former into the latter. p 1 � −� Moreover, � is a coupling constant and �(k) encodes the � (�) = ⟨�, p �� , p⟩ � (0) � � dependence of the mixing on the momentum of the produced |k, 0⟩ ∞ particles. Te explicit expressions connecting the states 1 � −� = ∫ ⟨� , p �� , p⟩ to |�, 0⟩ formally read � (0) dm1dm2 1 � � 2 (24) �ℎ ∞ ∞ |k, 0⟩ =∫ � (�) |�, 0⟩ 2 2 dm k (19) =∫ � (�) �− +p �ℎ dm �ℎ where �k(�) can be found by diagonalizing the Hamiltonian (17). coincides with (1), hence concluding our derivation. Let us now consider an unstable state with defnite In principle, one could also start from the Hamiltonian momentum p,whichisdenotedas|�, p⟩: �p and obtain the energy distribution associated with this p p ∞ state, denoted as � (�). Ten, � (�) should also emerge as |�, p⟩ =�p |�, 0⟩ =∫ dm� (�) |�, p⟩ . (20) the Fourier transform of the latter. Tis is hard to do here, �ℎ since the explicit expression of �p in terms of |�, p⟩ and |k, p⟩ Te normalization was not written down (as mentioned previously, this is not an easy task). Quite interestingly, in the framework of QFT, the p ⟨�, p1 |�,p2⟩ =�(p1 − p2) (21) function � (�) can be easily determined; see Section 3. 4 Advances in High Energy Physics

2 As a last comment of this section, we recall that the quantity Π(� ) is the one-particle irreducible diagram. Its general nondecay probability of an arbitrary state |Ψ⟩ reads calculation is of course nontrivial (it requires a proper

� � � �2 regularization), but it is not needed for our purposes. Te 3 � � −� � �|Ψ⟩ (�) =∫d p �⟨�, p �� Ψ⟩� , (25) imaginary part is

2 2 2 whose interpretation is straightforward: we project |Ψ⟩ onto ImΠ(� )=√� Γ(√� ) the basis of undecayed states. In general, �|Ψ⟩(0) is not unity. |k| (30) Notice also that �|Ψ⟩(�) is not the survival probability of the 2 2 = � �Λ (|k|) +⋅⋅⋅, state |Ψ⟩ (a state can change with time but still be undecayed 8�√�2 if it is a diferent superposition of |�, p⟩). When a boost �k on the state with zero momentum (and where dots refer to higher orders, which are however typically 2 2 hence with zero velocity) |�, 0⟩ is considered, the resulting very small [25]. Once ImΠ(� ) is fxed, ReΠ(� ) can be state reads [17] determined by dispersion relations (for an example of this � 2 ��k⟩=�k |�, 0⟩ technique, see, e.g., [26]). Te quantity Γ =Γ(√� =�) ∞ � (26) is the usual tree-level decay width; hence in the exponential =∫ � (�) √��3/2 ��, ��k⟩, −Γ�� dm � limit the decay law �(�) = � must be reobtained. As �ℎ mentioned in the Introduction, an unstable state has not a 2 −1/2 where �=(1−k ) . In fact, each element of the defnite mass: this is why diferent defnitions for � (which � superposition, |�, ��k⟩,hasvelocityk.Ofcourse,|�k⟩ is not is not the bare mass 0 entering in (29)) are possible: −1 2 2 an eigenstate of momentum, since each element in (26) has ReΔ (� =�)=0(zero of the real part of the p =��k. −1 a diferent momentum In this respect, the state denominator), or Re[√�],withΔ (�)=0(real part of |�, 0⟩ is special: it is the only state which has at the same the pole), or the maximum of the spectral function defned time defnite momentum and defnite velocity (both of them below. vanishing). As mentioned in the Introduction, the nondecay We also recall that probability associated with |�k⟩ vanishes: 2 �4 +(�2 −�2) −2�2 (�2 +�2) � (�) =0 ∀k ̸= 0. √ 1 2 1 2 |k⟩ (27) |k| = (31) 4�2 As soon as a nonzero velocity is considered, the state has decayed. Tis result is quite surprising but also rather “del- coincides, for the on-shell decay, with the modulus of the | ⟩ � k (�) three-momentum of one of the outgoing particles. Te vertex icate”: when a wave packet is considered, is nonzero � (|k|) (even if it is not 1 for �=0)[17]. function Λ is a proper regularization which fulflls the condition �Λ(|k|→0)=1and describes the high- 3. Covariant QFT Derivation of (1) energy behavior of the theory (its UV completion); hence the parameter Λ is some high-energy scale; �Λ(|k|) is formally Let us consider an unstable particle described by the scalar not present in (28) since it appears in the regularization feld �(�) ≡ �(�, x). For an illustrative example, one can couple procedure, but it can be included directly in the Lagrangian � with bare mass �0 to two scalar felds �1 (with mass �1)and by rendering it nonlocal [27] in a way that fulflls covariance �2 (with mass �2) via the interaction term ��1�2,leadingto [28]. In a renormalizable theory (such as the one of (28)), the the QFT Lagrangian dependence on Λ disappears in the low-energy limit. 1 2 1 2 Te properties outlined above, although in general very L = [(� � ) −�2�2]+ [(� � ) −�2�2] 2 1 1 1 2 2 2 2 important in specifc calculations, turn out to be actually (28) secondary to the proof that we present below, where only 1 2 2 2 the formal expression of the propagator of (29) is relevant. + [(��) −�0 � ]+��1�2. 2 Moreover, even when the unstable particle is not a scalar, one TisistheQFTcounterpartoftheQMsystemoftheprevious can always defne a scalar part of the propagator which looks section. Note we use scalar felds for simplicity, but our just as in (29), then the outlined properties apply, mutatis discussion is in no way limited to it. mutandis, to each QFT Lagrangian. � As a next step, upon introducing the Mandelstam variable Te (full) propagator of the state (details in [24]) reads 2 �=�,thefunction�(�) defned as 2 1 Δ (� )= 1 �2 −�2 +Π(�2)+�� (�) = [Δ (�2 =�)] 0 (29) �Im (32) 2 2 2 with � =� − p , fulflls the normalization condition: ∞ �=�0 p where is the energy and the three-momentum. ∫ ds� (�) =1, (33) 2 2 Because of covariance, Δ (� ) depends only on � .Te �ℎ Advances in High Energy Physics 5

2 † where �ℎ =�ℎ is the minimal squared energy. For the case is not simply given by �0 |0⟩,where|0⟩ is the perturbative 2 † of (28), one has obviously �ℎ =(�1+�2) . Te normalization vacuum and �p the creator operator of the noninteracting (33) is a consequence of the Kall¨ en–Lehmann´ representation feld �. Te case of neutrino oscillations shows a similar [29] situation: the state corresponding to a certain favour, such ] ∞ � (�) as the neutrino ,mustbeconstructedwithduecare Δ (�2)=∫ , ds 2 (34) by making use of Bogolyubov transformations [32]. Along � −�+�� ℎ this line, the exact and formal determination of the state |�, 0⟩ � (�) 2 , corresponding to the mass distribution ,inthe in which the propagator Δ (� ) has been rewritten as the 2 −1 context of QFT requires a generalization of Bogolyubov “sum” of free propagators [� −�+��] ,eachoneofthem transformations and was, to our knowledge, not yet explicitly weighted by �(�):ds�(�) is the probability that the squared done (it is lef for the future). Nevertheless, it is not needed mass lies between � and �+ds. Of course, the normalization for the purpose of this paper. (33) is a very important feature of our approach. For a detailed Let us now consider the particle � moving with a certain 2 2 proof of its validity, we refer to [30]. Here we recall a simple momentum p.Uponusing�=�− p , the energy version of it, which is obtained by assuming the rather strong � 2 2 2 distribution—as function of —is obtained by requirement Π(� )=0for � >Λ,whereΛ is a high-energy �p (�) = � (�) , scale (no matter how large). Under this assumption dE ds (40) 1 leading to Δ (�2)= �2 −�2 +Π(�2)+�� p 2 2 0 � (�) =2��(�=� − p ) (35) Λ2 � (�) � (41) =∫ ds . = � (√�2 − p2). �2 −�+�� 2 2 �ℎ √� − p 2 2 �p(�) � Ten, upon taking a certain value � ≫Λ,theprevious Te quantity dE is the probability that the particle with defnite momentum p has an energy between � and equation reduces to p=0 �+dE (clearly, � (�) = �(� = �)). Also in this case, 1 Λ2 � (�) =∫ �→ the normalization 2 ds 2 ∞ � � �ℎ ∫ �p (�) =1 (36) dE (42) 2 2 +p2 Λ �ℎ ∫ � (�) =1 ds � (�) �ℎ is a consequence of (33). When has a maximum at p √ 2 2 �,then� (�) has a maximum at ∼ � + p .Notethe 2 2 2 2 Te general case in which Π(� →∞)=0smoothly requires very fact that the propagator depends on � =�− p p more steps, but the fnal result of (33) still holds [30]. allows to determine the spectral function � (�) for a defnite Let us now consider the rest frame of the decaying p |�, p⟩ 2 2 2 momentum , that corresponds to the state of Section 2. particle: p = 0, �=� =� =�. Here, upon a simple Te nondecay probability’s amplitude for a state � moving variable change (� = √�), we obtain the mass distribution with momentum p is then given by p=0 (or spectral function) � (�) through the equation ∞ �p (�) = ∫ �p (�) �−, dE (43) p=0 2 +p2 dm� (�) = ds� (�) , (37) �ℎ out of which wherewehavetakenintoaccountthattheminimalenergyis √ 2 2 now given by �ℎ + p . � (�) =�p=0 (�) =2��(�=�2). (38) Tis expression can be manipulated by using (41) and via achangeofvariable: � (�) As already mentioned, dm is the probability that the ∞ particle � has a mass between � and �+dm [24, 31]. In this p p − � (�) =∫ dE� (�) � 2 +p2 context, the normalization �ℎ ∞ ∞ � ∫ � (�) =1 √ 2 2 − dm (39) =∫ dE � ( � − p )� (44) �ℎ 2 +p2 √ 2 2 �ℎ � − p � (�) ∞ follows from (33). Once the function is identifed −2+p2 as the mass distribution of the undecayed quantum state, =∫ dm� (�) � , 0 �ℎ the nondecay probability’s amplitude � (�) can be obtained by repeating the steps of Section 2. Te result coincides, as which coincides exactly with (1), as we wanted to demon- expected, with (3). Yet, it should be stressed that the unstable strate. Tus, we confrm the validity of (1) in a covariant QFT- quantum state |�, 0⟩ characterized by the distribution �(�) based framework. 6 Advances in High Energy Physics

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Letters,vol.86,no.13,pp.2699–2703,2001. [26] T. Wolkanowski, F. Giacosa, and D. H. Rischke, Physical Review [6] P. Facchi and S. Pascazio, “Spontaneous emission and lifetime D: Particles, Fields, Gravitation and Cosmology,vol.93,no.1, modifcation caused by an intense electromagnetic feld,” Phys- 2016. ical Review A, vol. 62, article 023804, 2000. [27] J. Terning, “Gauging nonlocal Lagrangians,” Physical Review D, [7] K. Urbanowski and K. Raczy´nska, “Possible emission of cosmic vol.44,p.887,1991. X-and�-rays by unstable particles at late times,” Physics Letters [28] M. Soltysiak and F. Giacosa, “A covariant nonlocal Lagrangian B,vol.731,pp.236–241,2014. for the description of the scalar kaonic sector,” Acta Physica [8] M. Peshkin, A. Volya, and V.Zelevinsky, “Non-exponential and Polonica B, 9, 2016, https://arxiv.org/abs/1607.01593. oscillatory decays in quantum mechanics,” EPL (Europhysics [29] M. E. Peskin and D. V. 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Research Article Time Dilation in Relativistic Quantum Decay Laws of Moving Unstable Particles

Filippo Giraldi

School of Chemistry and Physics, University of KwaZulu-Natal and National Institute for Teoretical Physics (NITeP), Westville Campus, Durban 4001, South Africa

Correspondence should be addressed to Filippo Giraldi; [email protected]

Received 20 March 2018; Accepted 22 April 2018; Published 29 May 2018

Academic Editor: Marek Nowakowski

Copyright © 2018 Filippo Giraldi. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

Te relativistic quantum decay laws of moving unstable particles are analyzed for a general class of mass distribution densities which behave as power laws near the (nonvanishing) lower bound �0 of the mass spectrum. Te survival probability P�(�),the instantaneous mass ��(�), and the instantaneous decay rate Γ�(�) of the moving unstable particle are evaluated over short and long times for an arbitrary value � of the (constant) linear momentum. Te ultrarelativistic and nonrelativistic limits are studied. Over long times, the survival probability P�(�) is approximately related to the survival probability at rest P0(�) by a scaling law. Te scaling law can be interpreted as the efect of the relativistic time dilation if the asymptotic value ��(∞) of the instantaneous mass isconsideredastheefectivemassoftheunstableparticleoverlongtimes.Teefectivemasshasmagnitude�0 at rest and moves 2 2 with linear momentum � or, equivalently, with constant velocity 1/√1+�0/� . Te instantaneous decay rate Γ�(�) is approximately independent of the linear momentum �,overlongtimes,and,consequently,isapproximatelyinvariantbychangingreferenceframe.

1. Introduction provides the survival probability in the reference frame where the particle is at rest, while the condition of nonvanishing Te description of the decay laws of unstable particles via linear momentum, �>0, can be referred to the laboratory quantum theory has been a central topic of research for frame of an observer where the particles move with linear decades [1, 2]. Many unstable particles which are generated in momentum �. Te efects of the relativistic time dilation in astrophysical phenomena or high-energy accelerator experi- quantum decay laws of moving unstable particles remain a ments are moving in the laboratory frame of the observer at matter of central interest. See [7–17], to name but a few. relativistic or ultrarelativistic velocity. For this reason, plenty As a continuation of the scenario described above, here, of studies have been devoted to formulate the decay laws in we evaluate the survival probability, the instantaneous energy, terms of relativistic quantum theory. See [1, 3–6], to name but and the instantaneous decay rate of a moving unstable afew. particle over short and long times for a wide variety of A fundamental subject in the description of the relativis- MDDs and for an arbitrary value of the (constant) linear tic decays of moving unstable particles is the way the decay momentum. In light of the short-time transformations of laws transform by changing the reference frame. Naturally, it the survival probability according to the relativistic time is essential to understand how the decay laws, holding in the dilation, we search for further scaling relations in the decay rest reference frame of the moving particle, are transformed laws which can relate the conditions of nonvanishing and in the laboratory frame of an observer. Te nondecay or vanishing linear momentum. In this way, we aim to fnd survivalprobabilityofanunstableparticlehasbeenevaluated further descriptions of the ways the decay laws of moving in [7, 8] for nonvanishing and vanishing values of the linear unstable particle transform by changing reference frame. momentum in terms of the mass distribution density (MDD). Te paper is organized as follows. Section 2 is devoted Te condition of vanishing linear momentum, �=0, to the relativistic quantum decay laws of a general moving 2 Advances in High Energy Physics unstable particle. In Section 3, the survival probability is Te state ket |�, �⟩ describes the moving unstable particle evaluated over short and long times for an arbitrary value with nonvanishing linear momentum �. Such state is related of the linear momentum and a general class of MDD. to the state ket |�⟩ as follows: |�, �⟩ = �(Λ)|�⟩.Teform In Section 4, the transformation of the long-time survival 2 2 �(�, �) = �� = √� +� is obtained by considering the probability is described via a scaling law. Section 5 is devoted energy-momentum 4-vector and the Lorentz invariance [6, to the evaluation of the instantaneous mass and instantaneous � (�) � (�) = ⟨�, �|�−��|�, �⟩ decayrateovershortandlongtimes.InSection6,thescaling 18]. Te quantity � is defned as � and represents the survival amplitude in the reference frame law, describing the transformation of the survival probability, � is interpreted in terms of the relativistic time dilation and where the particle has linear momentum . Te approach of the instantaneous mass of the moving unstable particle. described above leads to the following integral expression of Summary and conclusions are reported in Section 7. the survival amplitude:

∞ −�√�2+�2� 2. Relativistic Quantum Decay Laws �� (�) = ∫ � (�) � ��. (2) �0 An extended and detailed description of the relativistic quantum decay laws of moving unstable particles has been Te quantity P�(�) represents the survival probability that recently provided in [14, 17]. Following these references, a the decaying particle is in the initial state at the time � in brief summary of unstable quantum states, of the survival thereferenceframewheretheunstableparticlehaslinear probability, and of the instantaneous energy and decay rate momentum � and is given by the square modulus of the isreportedbelowforthesakeofclarity,byadoptingthe P (�) = |� (�)|2 ℏ=�=1 survival amplitude, � � .See[7,8,17]fordetails. system of units where . Te motion is assumed Te decay laws of the unstable particles are obtained from to be one-dimensional, due to the conservation of the linear theMDDvia(1)and(2).Inliterature,theMDDisusually momentum [1, 7, 8, 14–17]. represented via the Breit-Wigner function [19]: In the Hilbert space H of the quantum states which describe the unstable particle, let the state kets |�, �⟩ be the � Θ(�−� )� Γ/ (2�) common eigenstates of the linear momentum operator and � (�) = 0 BW , the Hamiltonian � self-adjoint operator, �|�, �⟩ = �|�, �⟩ BW 2 2 (3) (� − �0) + Γ /4 and �|�, �⟩ = �(�, �)|�, �⟩, for every value of the mass parameter � and of the linear momentum �.Temass � Θ(�) parameter � belongs to the spectrum of the Hamiltonian where BW is a normalization factor, is the Heaviside � whichissupposedtobecontinuouswithlowerbound�0, unit step function, 0 is the rest mass of the particle, Γ which means �≥�0. In the rest reference frame of the and is the decay rate at rest. A detailed analysis of the moving particle the linear momentum vanishes and the survival amplitude of a moving unstable particle has been mentioned state kets become |�, 0⟩, while the eigenstates performed in [8] by considering the Breit-Wigner form of the of the Hamiltonian are �(�, 0) = �.Let|�⟩ be the ket MDD [7]. Te long-time behavior of the survival amplitude of the Hilbert space H which describes the quantum state results in dominant inverse power laws, besides additional of the unstable particle. Such state can be expressed in decaying exponential terms. Refer to [8] for details. In [17, terms of the eigenstates |�, 0⟩ of the Hamiltonian as |�⟩ = 20, 21] general forms of MDD are considered with power-law ∞ ∫ �(�)|�, 0⟩��, via the expansion function �(�).Te behaviors near the lower bound of the mass spectrum. Te �0 long-time decay of the survival amplitude �0(�) is described survival amplitude �0(�) is defned in the rest reference frame � (�) = ⟨�|�−��|�⟩ by inverse power laws which are determined by the low- of the moving unstable particle as 0 and is mass profle of the MDD [20, 21]. Additional removable given by the integral expression logarithmic singularities in the low-mass form of the MDD ∞ −�� lead to logarithmic-like relaxations of the survival amplitude �0 (�) = ∫ � (�) � ��, (1) at rest �0(�) which can be arbitrarily slower or faster than �0 inverse power laws [22, 23]. where � is the imaginary unit. Te function �(�) represents Since the unstable particle decays, the initial state is 2 theMDDandreads�(�) = |�(�)| .TeprobabilityP0(�) notaneigenstateoftheHamiltonianandtheinstantaneous that the decaying particle is in the initial state at the time �, mass (energy) is not defned during the time evolution. For i.e., the survival probability, is given by the following form, this reason, instantaneous mass (energy) and decay rate are 2 P0(�) = |�0(�)| , in the rest reference frame of the moving defned in the rest reference frame of the moving particle unstable particle. in terms of an efective Hamiltonian. Such operator acts on Let Λ be the Lorentz transformation which relates the the subspace of the Hilbert space H which is spanned by reference frame where the unstable moving particle is at theinitialstate.Referto[14,17,20,21,24]fordetails.Inthe rest, to the one with velocity V =�/(��),where�� is same way, the instantaneous mass (energy) and decay rate the corresponding relativistic Lorentz factor. Let �(Λ) be an Γ�(�) of the moving unstable particle are defned in the frame unitary representation of the transformation Λ acting on the system where the particle has linear momentum �.Forboth Hilbert space H such that |�, �⟩ = �(Λ)|�, 0⟩ for every vanishing and nonvanishing values of the linear momentum value of the mass parameter � in the Hamiltonian spectrum. �,theinstantaneousmass��(�) and decay rate Γ�(�) are Advances in High Energy Physics 3

� (�) 2 2 obtained from the survival amplitude � via the following �=��, �=�/��, �=�/��,and�=√� +� .Te forms [14, 17, 20–23]: MDD is expressed in terms of the auxiliary function Ω(�) via ̇ the scaling law �(��) = Ω(�)/��,forevery�≥�0,where �� (�) � (�) =− { } �0 =�0/��.Inthisway,thesurvivalamplitude��(�) results � Im � � (4) � ( ) in the expression below: ̇ ∞ �� (�) −�� Γ (�) =−2 { }. �� (�) = ∫ Ω (�) � ��. (7) � Re (5) � �� (�) 0 Te MDDs under study are defned over the infnite ��(�) Te long-time behavior of the instantaneous mass and support [�0,∞)by auxiliary functions Ω(�) of the following Γ (�) decay rate � has been evaluated in [14, 15] for the Breit- form: Wigner form of the MDD. � Ω (�) = (�−�0) Ω0 (�) . (8) 2.1. Relativistic Time Dilation. Te interpretation of decay In order to study the long-time behavior of the decay laws of processes via the theory of special relativity suggests that themovingunstableparticle,theMDDsarerequestedtoobey the lifetime which is detected in the rest reference frame of the conditions below. Te lower bound of the mass spectrum the unstable particle increases in the reference frame where is chosen to be nonvanishing, �0 >0. Te constraints �≥0 the particle is moving. Te increase is due to the relativistic and Ω0(�0)>0are also requested. Te function Ω0(�) and dilation of times and is determined by the relativistic Lorentz (�) the derivatives Ω0 (�) are required to be summable, for every factor [25]. �=1,...,⌊�⌋+4 Te appearance of the relativistic time dilation in quan- , and continuously diferentiable in the whole support [�0,∞),forevery� = 1,...,⌊�⌋+3.Consequently, tum decays is a matter of great interest and fruitful dis- (�) (�) + Ω (�) Ω (� ) cussions. See [7–17], to name but a few. A broadly shared the limits lim�→�0 0 must exist as fnite and be 0 0 (�) opinion is that the relativistic time dilation infuences the for every � = 0,...,⌊�⌋ + 4.TefunctionsΩ0 (�) have survival probability uniquely in the short-time exponential to decay sufciently fast as �→+∞,sothattheauxiliary P (�) (�) decay. Briefy, this means that the survival probability � function Ω(�) and the derivatives Ω (�) vanish as �→+∞, and the survival probability at rest P0(�) are given by the �=0,...,⌊�⌋ −�/� −�/� for every . P (�) ∼ � � P (�) ∼ � 0 exponential forms � and 0 ,over As far as the short-time behavior of the survival amplitude short times. Under this assumption, the survival probability is concerned, let the auxiliary function decay as follows: obeys the scaling law −1−�0 Ω(�) = O(� ) for �→+∞,with�0 >5. Under this � condition, the survival amplitude evolves algebraically over P� (�) ∼ P0 ( ) , (6) short times: �� 2 3 �� (�) ∼1−��0�−�1� +��2� , (9) over short times. Te parameter �0 represents the lifetime of theparticleatrest,while�� is the lifetime which is detected in for �≪1/��.Teconstants�0, �1,and�2 are given by the the reference frame where the particle is moving with linear expressions below: momentum �. According to the relativistic time dilation, the ∞ � =�� 2 2 lifetimes are related as follows: � 0 �,where � is the �0 = ∫ � (�) √� +� ��, � corresponding relativistic Lorentz factor. Refer to [7, 8, 12– 0 17] for an extended explanation. In this context, we intend ∞ 1 2 2 to study the survival probability P�(�) and the survival �1 = ∫ � (�) (� +� )��, (10) 2 � probability at rest P0(�) over short and long times for a wide 0 ∞ variety of MDDs. We search for further ways to describe the 1 2 2 3/2 �2 = ∫ � (�) (� +� ) ��. transformations of the decay laws which occur by changing 6 � reference frame. 0 Te short-time evolution of the survival probability is derived 3. Survival Probability versus from (7) and (9) and is algebraic: 2 Linear Momentum P� (�) ∼1−�0� , (11) In the present section the short- and long-time behaviors 2 for �≪1/��,where�0 =2�1 −� . of the survival probability are studied for a general value � 0 Te long-time behavior of the survival amplitude is of the constant linear momentum of the moving unstable obtained from (7) and from the following equivalent form: particle. Te analysis performed in the whole paper is based entirely on form (2) of the survival amplitude [7, 12]. For the 2 2 ∞ �Ω (√� −� ) sake of convenience, the survival amplitude is expressed via −�� (�) = ∫ � ��, (12) the dimensionless variables �, �, �,and�. Tese variables � �0 √�2 −�2 are defned in terms of a generic scale mass �� as follows: 4 Advances in High Energy Physics

2 2  (t) where �0 = √� +�0.Noticethattheconstraintofnon- p 1.0 vanishing lower bound of the mass spectrum, �0 >0,is fundamental for the equivalence of expressions (7) and (12) ofthesurvivalamplitude.Teasymptoticanalysis[26,27]of 0.8 the integral form, appearing in (12) for �≫1, provides the expression of the survival amplitude over long times: 0.6 1+� (e) 2 2 �� −�((�/2)(1+�)+√�0 +� �) �� (�) ∼�0� ( ) , (13) 0.4 �� (b) (d) (a) for �≫1/��,where�0 = Γ(1 + �)Ω0(�0),and 0.2 (c) �2 � = √1+ . msＮ � 2 (14) 0 5 10 15 20 �0 P (�) (� �) Asymptotic form (13) of the survival amplitude holds for Figure 1: (Color online) the survival probability � versus � 0≤��≤20 �=0 � =� every value of the linear momentum �,nonvanishing,arbi- for � , MDDs given by (21), , 0 �, and diferent � (�) �= trarily large or small, or vanishing; for the variety of MDDs values of the linear momentum .Curve corresponds to 0��; (�) corresponds to �=��; (�) corresponds to �=2��; (�) under study; for every �≥0;andforeveryvalue�0 of the corresponds to �=3��; (�) corresponds to �=4��. lower bound of the mass spectrum such that �0/�� >0. Te last condition is crucial and will be interpreted in the next section in terms of the instantaneous mass at rest of the moving unstable particle. Te square modulus of asymptotic for �≫1/��. Tis is the main result of the paper. In fact, expression (13) approximates the survival probability over the above scaling law describes how the survival probability long times: at rest transforms, approximately over long times, in the � 2(1+�) reference frame where the particle moves with constant 2 � P (�) ∼� ( ) , (15) linear momentum �. Te transformation can be interpreted, � 0 � � � approximately, as the efect of a time dilation which is deter- � for �≫1/��.Noticethatthetimescale1/�� and, mined by the scaling factor �. Notice that the scaling factor, � →0+ consequently, the short or long times, �≪1/�� or �≫ given by (14), diverges in the limit 0 .InSection6, 1/��, are independent of the auxiliary function Ω(�) and are the scaling law (18) is interpreted, via the special relativity, as determined uniquely by the MDD. the efect of the relativistic time dilation. Tis interpretation holds if the asymptotic value of the instantaneous mass is 4. Scaling Law for the Survival Probability consideredastheefectivemassofthemovingunstable particle over long times. Expression (15) of the survival probability holds for every Te correction to the scaling law (18) can be estimated via (P (�) − P (�/� ))/P (�/� ) value of the constant linear momentum �. Such arbitrariness the expression � 0 � 0 � .FortheMDDs allows evaluating the survival probability in the nonrelativis- under study, such correction vanishes inversely quadratically tic and ultrarelativistic limits. For vanishing value of the linear over long times: momentum, �=0, or, equivalently, in the rest reference P (�) − P (�/� ) � frame of the unstable particle, the survival probability is � 0 � � ∼ , (19) approximated over long times as follows: 2 P0 (�/��) (��) �2 P (�) ∼ 0 , �≫1/� 0 2(1+�) (16) for �,where (��) 2 � (1+�)(2+�) �� Ω0 (�0/��) �� for �≫1/��.Instead,considerlargevaluesofthelinear � = (2 − � �3 Ω (� /� ) � �2 momentum, �≫�0. In such condition the survival 0 0 0 � 0 � probability is approximated over long times as follows: (20) 5 �2 2(1+�) ×(3+�+( +�) )) . 2 � 2 �2 P� (�) ∼�0 ( ) , (17) 0 �0�� for �≫1/��.Bycomparing(15)and(16),weobservethat, Numerical analysis of the survival probability P�(�) has for the MDDs under study, the survival probability obeys, been displayed in Figures 1, 2, 3, 4, and 5. Te computed approximately over long times, the following scaling law: MDDs are given by the following toy form of the auxiliary function: � P� (�) ∼ P0 ( ), (18) 2 2 � −�2 �� Ω (�) =��(� −�0) � , (21) Advances in High Energy Physics 5

p(t) p(Ｎ)/0(Ｎ) 1.0 25 (e) 0.8 20 (d) 0.6 15

(e) (c) 0.4 (c) 10

(a) (d) (b) 0.2 (b) 5 (a)

msＮ msＮ 0 246810 12 14 10 20 30 40 50

Figure 4: (Color online) ratio P�(�)/P0(�) versus (��) for 0≤ Figure 2: (Color online) the survival probability P�(�) versus (��) ��≤50, MDDs given by (21), �0 =��, �=0, and diferent for 0≤��≤15, MDDs given by (21), �=1, �0 =��, and diferent values of the linear momentum �.Curve(�) corresponds to �=��; values of the linear momentum �.Curve(�) corresponds to �= (�) corresponds to �=2��; (�) corresponds to �=3��; (�) 0��; (�) corresponds to �=��; (�) corresponds to �=2��; (�) corresponds to �=4��; (�) corresponds to �=5��. corresponds to �=3��; (�) corresponds to �=4��.

|ＦＩＡ(p(Ｎ))| p(Ｎ)/0(Ｎ) 10 (f) (d) 250 (h) 8 (i) (e) (c) 200 (g) (b) 6 (d) 150 (c) 4 (a) 100 (b) 2 50 (a)

m Ｎ log(mst) s −1012345 10 20 30 40 50 P (�)/P (�) (� �) 0≤ | (P (�))| (� �) Figure 5: (Color online) ratio � 0 versus � for Figure 3: (Color online) quantity log � versus log � for � �≤50 � =� �−1 ≤��≤�5 � =� � , MDDs given by (21), 0 �, and diferent values of the � , MDDs given by (21), 0 �, and diferent parameter � and of the linear momentum �.Curve(�) corresponds values of the parameter � and of the linear momentum �.Curve to �=1and �=2��; (�) corresponds to �=1and �=3��; (�) (�) corresponds to �=0and �=5��; (�) corresponds to �=0 corresponds to �=2and �=2��; (�) corresponds to �=1and and �=3��; (�) corresponds to �=0and �=0; (�) corresponds �=4��. to �=1and �=5��; (�) corresponds to �=1and �=2��; (�) corresponds to �=2and �=4��; (�) corresponds to �=2and �=2��; (ℎ) corresponds to �=1and �=0��; (�) corresponds to �=2 �=0 and . 5. Instantaneous Mass and Decay Rate versus Linear Momentum

2 �0 where �� is a normalization factor and reads �� =2� /Γ(1+ In the present section the instantaneous mass and decay �). Diferent values of the nonnegative power � and linear rate are analyzed over short and long times for every value momentum � are considered. Te corresponding MDDs of the constant linear momentum � of the particle, which belong to the class under study which is defned in Section 3 is detected in the laboratory frame of the observer. Te via (8). Te asymptotic lines appearing in Figure 3 agree instantaneous mass and decay rate are evaluated from the with the long-time inverse-power-law decays of the survival survival amplitude via (4) and (5). −1−� probability, given by (15). Te ordinates of the asymptotic Again, let the auxiliary function decay as Ω(�) = O(� 0 ) horizontal lines of Figures 4 and 5 are in accordance with for �→+∞,with�0 >5. Te short-time evolution of form (14) of the scaling factor ��. Te long-time dilation the instantaneous mass and decay rate are obtained from in the survival probability, given by the scaling law (18), behavior (9) of the survival amplitude and from (4) and (5). In is confrmed by the common horizontal asymptotic line, at this way, the following algebraic evolution results over short ordinate 1,whichappearsinFigures6and7. times: 6 Advances in High Energy Physics

p(Ｎ)/0(Ｎ/p) Te instantaneous mass tends over long times, �≫1/��, � (∞) 1.00 to the asymptotic value � ,givenby

0.95 √ 2 2 (e) �� (∞) = �0 +� , (23) 0.90 (d) (c) (b) 0.85 according to the following dominant algebraic decay: (a) 0.80 −2 �� (�) ∼�� (∞) (1 + �� (��) ). (24) 0.75 Te constant �� is defned as 0.70

msＮ � = (1+�) 10 20 30 40 50 �

P (�)/P (�/� ) (� �) 0≤ � � � �2 Ω� (� /� ) (25) Figure 6: (Color online) ratio � 0 � versus � for ⋅ � ((1 + ) � − 0 0 � ). ��≤50, MDDs given by (21), �0 =��, �=0, and diferent values 2 2 �0 2 �0 �0 +� Ω0 (�0/��) of the linear momentum �.Curve(�) corresponds to �=5��; (�) corresponds to �=4��; (�) corresponds to �=3��; (�) For large values of the linear momentum, �≫�0,the corresponds to �=2��; (�) corresponds to �=��. instantaneous mass decays over long times, �≫1/��,as follows:

 (Ｎ)/ (Ｎ/ ) −2 p 0 p �� (�) ∼�(1+�� (��) ), (26) 1.0 where (e) 0.8 � �� Ω0 (�0/��) (c) �� = (1+�) ((1 + ) − ). (27) �0 2 �0 Ω0 (�0/��) 0.6 (d) If the linear momentum vanishes, �=0,theinstanta- 0.4 (b) neous mass (at rest) tends over long times, �≫1/��,tothe (a) minimum value of the mass spectrum: 0.2 �0 (∞) =�0, (28) msＮ 10 20 30 40 50 60 with the following dominant algebraic decay: Figure 7: (Color online) ratio P�(�)/P0(�/��) for 0≤��≤60, � =� −2 MDDs given by (21), 0 �, and diferent values of the parameter �0 (�) ∼�0 (∞) (1 + �0 (��) ), (29) � and of the linear momentum �.Curve(�) corresponds to �=2 and �=4��; (�) corresponds to �=1and �=5��; (�) corresponds where to �=2and �=3��; (�) corresponds to �=1and �=2��; (�) corresponds to �=1and �=��. � �� Ω0 (�0/��) �0 =−(1+�) . (30) �0 Ω0 (�0/��) 2 Γ (�) �� (�) ∼�0 −�1� , Te instantaneous decay rate � vanishes over long (22) times, �≫1/��, according to the following dominant Γ� (�) ∼�2�, algebraic decay: 2 (1+�) 3 2 Γ� (�) ∼ . (31) for �≪1/��,where�1 =�0 +3(�2−�0�1) and �2 = 2(2�1−�0 ). � Te long-time behavior of the instantaneous mass and decayratearestudiedincasethattheMDDfulfllsthe Diferently from the survival probability and from the constraints which are reported in the second paragraph of instantaneous mass, the dominant asymptotic form of the Section 3. In addition, the functions �Ω0(�) and �Ω(�) are instantaneous decay rate is independent of the linear momen- required to obey the conditions which are requested in the tum �. Consequently, the instantaneous decay rate at rest same paragraph for the function Ω0(�) and the auxiliary remains approximately unchanged over long times in the function Ω(�), respectively. Te asymptotic analysis [26, 27] reference frame where the unstable particle moves with linear of the instantaneous mass or the instantaneous decay rate is momentum �. Notice that decay laws (29) and (31) are in obtained from the results of Section 3 and from (4) or (5), accordance with the ones which are obtained in [23] for a respectively. wider class of MDDs. Advances in High Energy Physics 7

Mp(Ｎ)/ms Γp(Ｎ)/ms 1.0 (e) 4.0 (e) 0.8 3.5 (d) (d)

3.0 0.6 (c) (c) 2.5 0.4 (b) 2.0 (b) 0.2 (a) 1.5 (a)

msＮ msＮ 0246810 12 14 0 5 10 15 20 25 30

Figure 10: (Color online) quantity Γ�(�)/�� versus (��) for 0≤ Figure 8: (Color online) quantity ��(�)/�� versus (��) for 0≤ ��≤30, MDDs given by (21), �0 =��, �=1, and diferent values ��≤15, MDDs given by (21), �0 =��, �=1, and diferent values of the linear momentum �.Curve(�) corresponds to �=4��; of the linear momentum �.Curve(�) corresponds to �=0��; (�) (�) corresponds to �=3��; (�) corresponds to �=2��; (�) corresponds to �=��; (�) corresponds to corresponds to �=2��; corresponds to �=1��; (�) corresponds to �=0��. (�) corresponds to �=3��; (�) corresponds to �=4��.

Γp(Ｎ)/ms 1.4 Mp(Ｎ)/ms (e) 4.5 1.2 (d) 4.0 (e) 1.0

3.5 (d) 0.8 (c) 3.0 0.6 (c) (b) 2.5 0.4 (b) (a) 2.0 0.2 (a) 1.5 msＮ 0 5 10 15 20 25 30 msＮ 0246810 12 14 Figure 11: (Color online) quantity Γ�(�)/�� versus (��) for 0≤ ��≤30, MDDs given by (21), �0 =��, �=2, and diferent values of the linear momentum �.Curve(�) corresponds to �=4��; Figure 9: (Color online) quantity ��(�)/�� versus (��) for 0≤ (�) corresponds to �=3��; (�) corresponds to �=2��; (�) ��≤15, MDDs given by (21), �0 =��, �=2, and diferent values corresponds to �=1��; (�) corresponds to �=0��. of the linear momentum �.Curve(�) corresponds to �=0��; (�) corresponds to �=��; (�) corresponds to corresponds to �=2��; (�) corresponds to �=3��; (�) corresponds to �=4��. 6. Relativistic Time Dilation and Survival Probability Numerical analysis of the instantaneous mass or the instantaneous decay rate is displayed in Figures 8, 9, 12, and In Section 2.1 how the survival probability at rest, P0(�), 13 or Figures 10, 11, 14, and 15, respectively. Te computed transforms, due to the relativistic time dilation, in the refer- MDDsaregivenbythetoyform(21)oftheauxiliary ence frame where the unstable particle moves with constant function for diferent values of the nonnegative power � linear momentum � is reported. Te transformed survival and of the linear momentum �.Teasymptoticlinesof probability, P�(�), is related to the survival probability at rest, Figure 12 and the asymptotic horizontal lines of Figure 13 are P0(�), by the scaling law (6). Te scaling factor consists in the in accordance with the long-time inverse-power-law decays corresponding relativistic Lorentz factor ��. of the instantaneous mass, given by (24) and (29). Te Te analysis performed in Section 3 shows that, for the asymptotic horizontal lines of Figure 13 agree with (32) and class of MDDs under study, the survival probability P�(�) with expression (14) of the scaling factor. Te asymptotic lines and the survival probability at rest P0(�) are related by the appearing in Figure 14 and the asymptotic horizontal lines scaling law (18) over long times. Te corresponding scaling of Figure 15, at ordinate 1, agree with the long-time inverse- factor �� is given by (14). It is worth noticing that the power-law behavior of the instantaneous decay rate, given by scaling factor �� coincides with the ratio of the asymptotic (31). form of the instantaneous mass ��(�) and the asymptotic 8 Advances in High Energy Physics

Mp (t) |log(Γp(t)/ms)| |ＦＩＡ(| −1|)| 2 2 3.0 p +0

6 (i) 2.5 (f) 5 (h) (f) 2.0 4 (e) (c) 1.5 (g) (a) (c) (e) 3 (b) 1.0 2 (d) (d) 0.5 (b) 1 (a) log(mst) 1.5 2.0 2.5 3.0 log(mst) 1.5 2.0 2.5 3.0 3.5 2 2 Figure 12: (Color online) quantity |log (|��(�)/√� +�0 − 1|)| 3.3 Figure 14: (Color online) quantity |log (Γ�(�)/��)| versus log (��) versus log (��) for �≤��≤� , MDDs given by (21), �0 =��, 1.5 3.8 for � ≤��≤� , MDDs given by (21), �0 =��, and diferent and diferent values of the parameter � and of the linear momentum values of the parameter � and of the linear momentum �.Curve(�) �.Curve(�) corresponds to �=2and �=4��; (�) corresponds corresponds to �=2and �=0��; (�) corresponds to �=2and to �=2and �=2��; (�) corresponds to �=1and �=2��; (�) �=2��; (�) corresponds to �=2and �=4��; (�) corresponds corresponds to �=2and �=��; (�) corresponds to �=0and to �=1and �=0��; (�) corresponds to �=1and �=2��; (�) �=3��; (�) corresponds to �=0and �=5��. corresponds to �=1and �=5��; (�) corresponds to �=0and �=0��; (ℎ) corresponds to �=0and �=3��; (�) corresponds to �=0 �=5�� Mp(Ｎ)/M0(Ｎ) and .

4.0 Γp(t)/Γ0(t) (g) 3.5 (h) 1.0 3.0 (e) 0.8 (f) 2.5 (e) (d) (d) 0.6 (a) 2.0 (c) (c) 1.5 (b) 0.4 (a) msＮ 0246810 (b) 0.2 Figure 13: (Color online) ratio ��(�)/�0(�) versus (��) for 0≤ � �≤10 � =� m Ｎ � , MDDs given by (21), 0 �, and diferent values of the 0 5 10 15 20 25 30 s parameter � and of the linear momentum �.Curve(�) corresponds Γ (�)/Γ (�) (� �) 0≤ to �=1and �=��; (�) corresponds to �=0and �=��; (�) Figure 15: (Color online) ratio � 0 versus � for � �≤30 � =� corresponds to �=1and �=2��; (�) corresponds to �=0and � , MDDs given by (21), 0 �, and diferent values of the � � (�) �=2��; (�) corresponds to �=2and �=3��; (�) corresponds parameter and of the linear momentum .Curve corresponds �=1 �=4� (�) �=2 �=3� (�) to �=1and �=3��; (�) corresponds to �=1and �=4��; (ℎ) to and �; corresponds to and �; �=2 �=2� (�) �=0 corresponds to �=0and �=4��. corresponds to and �; corresponds to and �=2��; (�) corresponds to �=0and �=��. expression of the instantaneous mass at rest �0(�) of the moving unstable particle: particle is not defned. On the contrary, the instantaneous mass is properly defned in terms of the survival amplitude. �� (∞) See [14, 16, 17] for details. �� = . (32) �0 (∞) In light of the above observations, the long-time scaling law (18) can be interpreted as an efect of the relativistic time At this stage, consider the reference frame S where a dilation if the asymptotic value ��(∞) of the instantaneous mass at rest which is equal to the asymptotic value �0(∞) mass is considered to be the efective mass of the unstable S becomes ��(∞) due to the relativistic transformation of particle over long times. In fact, in the reference frame the the mass. According to (32), in the reference frame S the mass at rest �0(∞),whichisequaltothevalue�0,moves corresponding relativistic Lorentz factor coincides with the with linear momentum �, or, equivalently, with constant �� 2 2 scaling factor . We remind that, initially, the unstable velocity 1/√1+�0/� , and becomes the relativistic mass quantum system is not in an eigenstate of the Hamiltonian. √ 2 2 Consequently, in the present model the mass of the unstable ��(∞),whichisequaltothevalue �0 +� .Concurrently, Advances in High Energy Physics 9

in the reference frame S the survival probability at rest P0(�) While the instantaneous mass transforms by changing transforms, according to the relativistic time dilation, in the reference frame, no transformation is found, approximately, survival probability P�(�) and obeys the scaling law (6), for the instantaneous decay rate over long times. In fact, or,equivalently,(18),overlongtimes.Inthiscontext,the the instantaneous decay rate vanishes, over long times, crucial condition of nonvanishing lower bound of the mass approximately independently of the linear momentum of the spectrum, �0/�� >0, suggests that the long-time relativistic moving particle. Consequently, the long-time instantaneous dilation and the scaling law (6), or, equivalently, (18), hold decay rate is approximately invariant by changing reference uniquely for an unstable moving particle with nonvanishing frame. efective mass, �0(∞) > 0. In conclusion, the present analysis shows further ways to describe the long-time transformations of the decay laws of moving unstable particles in terms of model-independent properties of the mass spectrum. Te role of the (nonvanish- 7. Summary and Conclusions ing) mass at rest in the relativistic transformation is assumed in the present description by the (nonvanishing) lower bound Te relativistic quantum decay laws of a moving unstable of the mass spectrum. particle have been analyzed over short and long times for an arbitrary value � of the (constant) linear momentum. Te MDDs under study exhibit power-law behaviors near the Data Availability (nonvanishing) lower bound �0 ofthemassspectrum.Dueto the arbitrariness of the linear momentum, the ultrarelativistic Te results of this article are entirely theoretical and ana- and nonrelativistic limits have been obtained as particular lytical. For each result, the main steps of the corresponding cases. demonstration are reported in the text. Te article is fully Te survival probability, which is detected in the rest consistentwithoutthesupportofanyadditionaldata. reference frame of the unstable particle, transforms in the reference frame where the unstable particle moves with linear Conflicts of Interest momentum �, approximately according to a scaling law, over long times. Te scaling factor is determined by the lower Te author declares that there are no conficts of interest bound �0 of the mass spectrum and by the linear momentum regarding the publication of this paper. � of the particle. Te scaling law can be interpreted as theefectoftherelativistictimedilationiftheasymptotic References form of the instantaneous mass ��(�) is considered as the efectivemassofthemovingunstableparticleoverlongtimes. [1] L. A. Khalfn, “Quantum Teory of Unstable Particles and S In fact, consider the reference frame where a mass at Relativity,” Tech. Rep. PDMI PREPRINT-6/1997, St. Petersburg 2 2 rest of magnitude �0 moves with velocity 1/√1+�0/� ,or, Department of Stelkov Mathematical Institute, St. Ptersburg, Russia, 1997. equivalently, with linear momentum �.Temassatrest�0 coincides with the asymptotic value of the instantaneous mass [2] L. Fonda, G. C. Ghirardi, and A. Rimini, “Decay theory of at rest, �0(∞), of the moving unstable particle. In the refer- unstable quantum systems,” ReportsonProgressinPhysics,vol. ence frame S the transformed mass, which is equal to the 41, no. 4, pp. 587–631, 1978. 2 2 [3] B. Bakamjian, “Relativistic particle dynamics,” Physical Review value √� +� , coincides with the asymptotic value ��(∞) 0 A: Atomic, Molecular and Optical Physics,vol.121,no.6,pp. of the instantaneous mass of the particle. Simultaneously, 1849–1851, 1961. in the reference frame S the dilation of times, which is [4] F. Coester and W. N. Polyzou, “Relativistic quantum mechanics suggested by the special relativity, transforms the survival of particles with direct interactions,” Physical Review D: Parti- probabilityatrestaccordingtothementionedscalinglaw. cles, Fields, Gravitation and Cosmology,vol.26,no.6,pp.1348– 2 2 Te above description indicates the value 1/√1+�0/� as the 1367, 1982. asymptotic velocity of the moving unstable particle. [5] P. Exner, “Representations of the Poincaregroupassociated´ We stress that the present interpretation is an attempt with unstable particles,” Physical Review D: Particles, Fields, to ascribe the transformation laws of the long-time survival Gravitation and Cosmology, vol. 28, no. 10, pp. 2621–2627, 1983. probability to the dilation of times which is provided by the [6]E.V.Stefanovich,Relativistic Quantum Teory of Particles,vol. theory of special relativity. However, a clear scaling transfor- 1 and 2, Lambert Academic, 2015. mation of the survival probability at rest holds, approximately [7]E.V.Stefanovich,“Quantumefectsinrelativisticdecays,” over long times, if the decay is observed in the reference International Journal of Teoretical Physics,vol.35,no.12,pp. frame where the unstable particle moves with constant linear 2539–2554, 1996. momentum. Te scaling law can still be interpreted as the [8] M. Shirokov, “Decay law of moving unstable particle,” Interna- efect of a time dilation which appears by changing reference tional Journal of Teoretical Physics,vol.43,no.6,pp.1541–1553, frame. Te dilation is determined uniquely by the scaling 2004. factor which depends on the mass spectrum and on the [9] D. H. Frisch and J. H. Smith, “Measurement of the relativistic dynamics of the unstable particle. 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[10]J.Bailey,K.Borer,F.Combleyetal.,“Measurementsofrelativis- tic time dilatation for positive and negative muons in a circular orbit,” Nature,vol.268,no.5618,pp.301–305,1977. [11] F. J. M. Farley, “Te CERN (g-2) measurements,” Zeitschrif fur¨ Physik C Particles and Fields, vol. 56, supplement 1, pp. S88–S96, 1992. [12] M. I. Shirokov, “Evolution in time of moving unstable systems,” Concepts of Physics,vol.3,pp.193–205,2006. [13] M. I. Shirokov, “Moving system with speeded-up evolution,” Physics of Particles and Nuclei Letters,vol.6,no.1,pp.14–17, 2009. [14] K. Urbanowski, “Decay law of relativistic particles: quantum theory meets special relativity,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology,vol.737,pp.346–351, 2014. [15] F. Giacosa, “Decay law and time dilatation,” Acta Physica Polonica B,vol.47,no.9,pp.2135–2150,2016. [16] K. Urbanowski, “On the velocity of moving relativistic unstable quantum systems,” Advances in High Energy Physics,vol.2015, Article ID 461987, 2015. [17] K. Urbanowski, “Non-classical behavior of moving relativistic unstable particles,” Jagellonian University. Institute of Physics. Acta Physica Polonica B,vol.48,no.8,pp.1411–1432,2017. [18] W. M. Gibson and B. R. Pollard, Symmetry Principles in Elementary Particle Physics, Cambridge, UK, 1976. [19] M. L. Goldberger and K. M. Watson, Collision Teory,Wiley, New York, NY, USA, 1964. [20] K. Urbanowski, “General properties of the evolution of unstable states at long times,” Te European Physical Journal D,vol.54, no. 1, pp. 25–29, 2009. [21] K. Urbanowski, “Long time properties of the evolution of an unstable state,” Open Physics,vol.7,no.4,pp.696–703,2009. [22] F.Giraldi, “Logarithmic decays of unstable states,” Te European Physical Journal D,vol.69,no.1,2015. [23] F. Giraldi, “Logarithmic decays of unstable states II,” Te European Physical Journal D, vol. 70, no. 11, article no. 229, 2016. [24] K. Urbanowski, “Early-time properties of quantum evolution,” Physical Review A: Atomic, Molecular and Optical Physics,vol. 50, pp. 2847–2853, 1994. [25] C. Møller, Te theory of relativity, Clarendon Press, Oxford, UK, 1972. [26] A. Erdelyi,´ Asymptotic Expansions,Dover,NewYork,NY,USA, 1956. [27] R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, Mass, USA, 1989. Hindawi Advances in High Energy Physics Volume 2018, Article ID 3916727, 15 pages https://doi.org/10.1155/2018/3916727

Research Article A New Inflationary Universe Scenario with Inhomogeneous Quantum Vacuum

Yilin Chen 1 and Jin Wang 1,2,3

1 College of Physics, Jilin University, Changchun, Jilin 130021, China 2Department of Chemistry and Physics, State University of New York, Stony Brook, NY 11794, USA 3State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Changchun, Jilin 130022, China

Correspondence should be addressed to Yilin Chen; [email protected] and Jin Wang; [email protected]

Received 7 February 2018; Accepted 1 April 2018; Published 8 May 2018

Academic Editor: Marek Szydlowski

Copyright © 2018 Yilin Chen and Jin Wang. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

We investigate the quantum vacuum and fnd that the fuctuations can lead to the inhomogeneous quantum vacuum. We fnd that the vacuum fuctuations can signifcantly infuence the cosmological inhomogeneity, which is diferent from what was previously expected. By introducing the modifed Green’s function, we reach a new infationary scenario which can explain why the Universe is still expanding without slowing down. We also calculate the tunneling amplitude of the Universe based on the inhomogeneous vacuum. We fnd that the inhomogeneity can lead to the penetration of the Universe over the potential barrier faster than previously thought.

1. Introduction the vacuum energy. Despite the success, one issue remains on how the Universe quits this stage. Many proposals were Gravity governs the evolution of the Universe. A great break- suggested to resolve this issue [3, 4]. It turns out rather through in gravitational research in the last century is the difcult to complete and have a graceful exit for the old discovery of general relativity (GR). Afer Einstein laid down infationary scenario with multiple bubbles coalescing in a therelationshipbetweenthespace-timegeometrythrough Universe suggested by Guth [1]. Chaotic infationary scenario the curvature and the matter through energy momentum, has been suggested with essentially one bubble for a Universe the general relativity theory has been applied to many but forever evolving to avoid the exiting issue. felds, especially in astrophysics and cosmology. Another Te idea of the infationary scenario is to combine the great achievement of modern physics is quantum mechanics. quantum vacuum energy for describing the matter and the Basedonthat,quantumfeldtheory(QFT)emerged,which Einstein’s equation for describing the space-time evolution has been tested in many experiments. Due to the success of together. Based on the equivalence principle of GR, each form relativity in the macroscopic world and quantum mechanics of the energy infuences the space-time in the same way. in the microscopic world, it is natural to ask how we can Quantumvacuumbringsanewsourceofenergy.Naively,one combine them together. In cosmology, this issue becomes can study how the quantum vacuum infuences the space- more apparent afer the birth of the theory of infationary time evolution by simply put the energy-momentum tensor Universe [1, 2] and successful in solving the horizon and forthequantumvacuumontherighthandsideandthe fatness problems and eventually quantifying the seeds in Einstein space-time curvature tensor on the lef hand side of terms of the density fuctuations for large scale structure the Einstein equation of GR. formation and inhomogeneity for the microwave background Unfortunately,thereisanissueonceonenaivelyputs radiation. In this theory,at the very early history,the Universe these two theories of general relativity and quantum mechan- expanded exponentially, and the expansion was sustained by ics together. Tis is because there is currently no applicable 2 Advances in High Energy Physics method for quantizing GR or space-time. Tis indicates that In this study, by improving the method developed in [6], quantum mechanics and GR are at totally diferent footing we suggest going one step further beyond the conventional and do not match each other. Terefore, many existing semiclassical method for combining the GR and the QFT theories suggested certain approximate equations relating by taking into account the vacuum fuctuations and, based these two theories. However, these approaches ofen show on that, reaching a new infationary scenario. In this new great ambiguity. Firstly, let us look at Einstein’s feld equation: scenario, the cosmological constant issue, which arises when 1 tryingtocombineGRandQFT,canberesolved. �] − ��] =��]. (1) Tis paper is organized as follows: in Section 2, we 2 illustrate that the quantum vacuum is not homogeneous, but

Tisequationinthecurrentformwouldnotmakesenseif�] inhomogeneous, due to quantum fuctuation. In Section 3, represents the quantum (such as vacuum here) rather than by introducing the modifed Green’s function, we build up a classicalmatter.Tisisbecausetheenergy-momentumtensor model to quantify the fuctuations of the quantum vacuum. is an operator in quantum world. In quantum mechanics, We study the infuence of the quantum vacuum fuctuations once we attempt to observe something about a system, the and its physical interpretation. In Section 4, we consider expected observed values correspond to the average values of a simple case and solve the corresponding Einstein’s feld the corresponding operator for the observable. Terefore, it equation. In Section 5, by introducing fnite temperature feld seems natural to modify the Einstein equation by changing theory, we take into account the temperature, which is a key the energy-momentum operator for the average value of it: element in cosmological evolution. In Section 6, based on our solutions to the cases in Sections 4 and 5, we propose 1 � − �� =�⟨� ⟩. (2) a new infationary scenario, which can help to resolve the ] 2 ] ] cosmological constant problem. In Section 7, we analyze the infuence of the inhomogeneous vacuum on the tunneling Tis describes, at the average level, how the quantum matter amplitude of the Universe from nothing. infuences the space-time evolution. For quantum matter, we Te units and metric signature are set to be �=ℏ=1 know that fuctuations are unavoidable. Tis is even true for and (+,−,−,−)throughout. And in this paper, 4-vectors are the quantum vacuum. One natural question to ask is how denoted by light italic type, and 3-vectors are denoted by the quantum matter fuctuations infuence the space-time boldface type. evolution. Another way to modify the Einstein equation for taking the fuctuations into account is to take the square of both sides of the Einstein equation and then take the average 2. The Quantum Fluctuation and valueontherighthandside: Inhomogeneous Vacuum 1 2 Vacuum energy plays an very important role in the infa- (� − �� ) =�⟨� 2⟩ . (3) ] 2 ] ] tionary theory. In this theory, at the very early time, the Universe expanded exponentially. In this period, vacuum In fact, (2) and (3) are equivalent only if the fuctuation of energy dominated the expansion of the Universe. Usually, the energy-momentum tensor is zero, but this of course is not the vacuum energy density is treated as a constant; for example, case since the energy-momentum tensor even for quantum just as in (2), the average value of �00 is vacuum is not zero. Tis example illustrates that before we “totally” under- Λ4 stand quantum gravity, there could be many ways of com- ⟨� ⟩∼ , (4) 00 16�2 bining the general relativity and quantum feld theory, which arenotequivalentatthesemiclassicallevel.Furthermore, 2̇ 2 2 2 these diferences in semiclassical treatments are caused by the where �00 = (1/2)(� +(∇�) +� � ) is the energy density fuctuations of the quantum feld vacuum. Te cosmological ofafreescalarfeld,andthehighenergycutofΛ is much constant problem [5] is a good example to demonstrate greaterthanthemassinthefreescalarfeld. this issue, where the vacuum energy density predicted by By recalling the example in the introduction section, the quantum feld theory is much larger than the cosmological vacuum fuctuations are not zero. Tis is because the vacuum constant from the observations. A natural question one can |0⟩ is not the eigen state of �00,buttheeigenstateofHamil- 3 ask is whether taking the fuctuation into account can be tonian �=∫�00� � (for the detailed discussions, see [6]). an efective way to improve the quantitative descriptions of Terefore, the fuctuations in energy density should be con- the cosmological evolution driven by the quantum vacuum sidered. Due to the vacuum fuctuations, the conventional energy [6]. Te idea of considering the quantum vacuum assumption of homogeneous Universe is only approximately fuctuations was suggested [6] to solve the cosmological correct. When fuctuations are taken into account, the vac- constant problem at present. However, quantum fuctuations uum is not homogeneous. Tus, a more suitable theory exist not only in the current Universe with approximately should include the efects of the fuctuations in energy fat space-time but also in the very early history of the density. To achieve this goal, the strategy we adopt is to Universe. Terefore, it is also important to consider the modify both sides of the feld equation, in order to have impacts of quantum vacuum fuctuations on the evolution of the fne structures which are compatible with the fuctua- the Universe, in particular the early Universe. tions. Advances in High Energy Physics 3

2.1. Generalizing the FLRW Metric. To describe a homo- value (expectation value) of the tensor over the entire space- geneous, isotropic expanding Universe, we introduce Fried- time should not be taken as a constant due to the vacuum mann–Lemaˆıtre–Robertson–Walker (FLRW) metric [7]: fuctuations. By only taking the expectation values of the energy-momentum tensor, some fne structures which come ��2 ��2 =��2 −�2 (�) ( +�2��2 +�2 2��2), from the fuctuations are lost. Here, our main task is to fnd a 2 sin (5) 1−�� correct �] for (8), which describes the inhomogeneity of the vacuum fuctuations. where � can be −1, 0, or +1, which indicates the 3-dimensional � � ⟨0 �� 0⟩ = � (x,�) . space is elliptical space (closed), Euclidean space (fat), or � ]� ] (10) �(�) hyperbolic space (open), respectively. Te factor ,known Before studying how to fnd the suitable �],weintroduce � as the scale factor, depends only on . Although the FLRW the scalar feld to describe the matter feld in our toy model. metric can naturally describe an expanding Universe, it For simplicity, �-4 theory is adopted: cannot describe the inhomogeneous Universe where the 1 1 1 3�4 inhomogeneity is caused by the vacuum fuctuations. Te L = �]� �� − �2�2 − ��4 − , (11) corresponding resolution is to allow the scale factor �(�) to 2 ] 2 4! 2� have spatial dependence. 2 2 where � =−� <0. Due to the Higgs mechanism, the sym- ��2 =��2 metry is broken spontaneously at low temperatures. Te efective potential becomes ��2 (6) −�2 (�, �) ( +�2��2 +�2 2��2). �2 � 3�4 1−��2 sin �(�)=− �2 + �4 + >0. (12) 2 4! 2� To simplify our model, we only assume that the scale factor To be compatible with the modifcation of the FLRW metric, has only radius � dependence, so the rotational symmetry is we reduce a number of degrees of freedom of the feld �(�, �) preserved. For the spatial part, �=1is chosen (the reason and preserve the rotational symmetries just as what we did will be explained in Section 5). Now, the Ricci tensor for the before. metric becomes According to Noether’s theorem, the energy-momentum 3�̈ tensor for this feld is given as �00 =− , � �] =��]� 2(��̇ −��̇ ) � 1 3�4 (13) � =� = , − ] (�� −�2�2 − ��4 − ). 01 10 �2 2 12 2�

2 2 2 2(1−2� )� 2� 2� 2(�̇ +1)+��̈ Substituting (13) in (9), we reach � = + − − , (7) 11 �(�2 −1)� �2 � �2 −1 �00 =��−�(�), � � = 33 2 22 2 1−� sin � � =− � �� −�(�), (14) 11 �2 (�, �) � [(4�2 −3)� +�(�2 −1)� +��̈ 2 +2�(�2̇ +1)�] = , � �22 =�33 =−�(�). � where the dot represents the derivative with respect to �,and Now, we are ready to substitute into Einstein’s equations. At the prime represents the derivative with the respect to �.Ten frst, let us look at one of the Einstein equations: the Ricci tensor can be substituted into the Einstein’s feld 3�̈ � =− =8�� (�, �) equation: 00 � 00 (15) ̇ ̇ �] =8��], (8) =8��(⟨��⟩ − ⟨� (�)⟩) . where �] is given by the energy-momentum tensor: Next, the main challenge becomes the evaluations of ̇ ̇ ⟨�(�)⟩ and ⟨��⟩.However,itisimpossibletoobtainthe ̇ ̇ 1 correct ⟨�(�)⟩ and ⟨��⟩ in conventional methods. Te key to � =� − � �, (9) ] ] 2 ] obtain the correct expectation values of the potential is to frst fnd out ⟨0|��|0⟩. In general, this expectation value is given � � where is the trace of the energy-momentum tensor ]. as � � � � 2.2. Quantifcation of the Inhomogeneous Vacuum. Afer ⟨0 �� (�) � (�)� 0⟩ = lim ⟨0 ��(� )�(�)� 0⟩ �→ � � transforming the lef hand side of (8), now we can focus on (16) the right hand side. Following the description of the intro- = �(�,�), lim� duction section, the �] which is ofen taken as the average → 4 Advances in High Energy Physics

where �(� ,�)is the full Green’s propagator. To gain insights, (because when the scale factor � is large, the Ricci scalar −1 the free propagator should be derived at frst following for FLRW metric is proportional to � ,sothecurvature, the perturbation theory. Due to the spontaneous symmetry especially afer infation, is very small). Meanwhile, the van breaking, the free propagator cannot be obtained directly. To Vleck determinant Δ(�, � ) should also be one in this case. resolve this issue, we make a shif of the variable, �→�+�0, Tus, the approximate propagator in the curved space-time 2 2 where �0 =6�/�.Tenwehave becomes 3 2 4 � � 1 −(Δ+k⋅Δx) 1 ] 1 ��0 2 2 � 4 3� �(�,� )=∫ � , L = � � �� − ( −� )� − � − 3 (22) 2 ] 2 2 4! 2� (2�) 2�

2 where the Δ� and Δx are geodesic distances. Afer that, we 2 �� 3 +(� − 0 )� �− � � (17) can calculate the full propagator. For simplicity, the tree level 6 0 6 0 propagator is considered, and higher order corrections are 2 neglected. � � 2 2 +( − �0)�0. 2 4! 3. Modifications of the Green’s Function Now it is easy to see that the square of the efective mass of In this section, we aim to explore the fuctuations of the the new feld � is vacuum and take this into account in our model by modifying the propagator; then we can establish an efective feld theory 2 � 2 2 2 �ef = �0 −� =2�. (18) taking into account of the efects of the fuctuations by 2 introducing the modifed Green’s function. Afer shifing the variables, we can derive the solution for � 3.1. Another Approach to Obtaining the Propagator. Following and return back to the original variable: the previous section, the free propagator for the scalar feld is 3 given as −3 � � † ∗ �=�0 + (2�) ∫ [� � (�, �) +�� (�, �)], (19) √2� �3� �(�,�)=∫ �. 3 (23) where � satisfes the equation of motion for the free scalar (2�) 2� feld. For example, in the fat space-time, �(�) = exp(��), Usually, in quantum feld theory, the free propagator can andinthiscase,theGreen’sfunctionis be derived by calculating the Green’s function in momentum 2 space and then integrating over � with a suitable contour. �(� ,�)=� +�0 ⟨�⟩ + �0 ⟨� ⟩ + ⟨�� ⟩ 0 However, the two-point correlation function obtained by 3 ⟨0|��|0⟩(�) � � � (20) this way is not the expectation value, ,whichwe =�2 + ∫ �(− ), 0 (2�)3 2� expect for the energy-momentum tensor. Tis is because certain hidden structures inside the correlation functions which can cause the fuctuations are ignored. To show the fne struc- �=√�2 +�2 = √�2 +4�2 where ef .Inthecurvedspace, tures of the expectation values for the energy-momentum the propagator is not as simple as it is in the fat space. tensor, here we introduce another approach to obtaining the Following [8], by introducing Riemann normal coordinates Green’s function. We will write down the explicit solution of and expanding the metric, the propagator in the curved space � andthencalculatethecorrelationdirectly. canalsobeobtainedinmomentumspace: Here, we represent the solution of � in terms of the † creation and annihilation operators � and � for the scalar 1/2 �Δ (�, � ) � � � �(�,�)= ∫ �(− ) feld without interactions: /2 (4�) (2�) �3� � (x,�) = ∫ (� �− +�†�) . 3 (24) � � 2 (2�) √2� ⋅[1+� (�, �)(− )+� (�, �)( ) ] (21) 1 2 2 2 �� Ten the multiplication of the two �sbecomes

1 3 3 ⋅ , � �� 1 �2 +�2 � (x,�) �(x ,� )=∫ (2�)6 2√�� (�, �) � (�, �) where 1 and 2 are certain functions which are −(+��) † † (+��) ⋅[� � � � +� � � (25) related to the curvature tensor. If the curvature is not quite � large,thesecondandthirdtermsin(21)wouldbemuch † −(−��) † (−��) +� � � +� � � � ]. smaller than the frst term. In that case, we can establish a � perturbative propagator in the curved space-time. For our toymodel,weomit�1(�, � ) and �2(�, � ) terms and just Conventionally, in (25), the frst two terms do not give preserve the leading term for the approximate fat space-time contributions to the propagator. Tis is because their vacuum Advances in High Energy Physics 5

2 expectation values are zero. When we take the expectation �(0, 0) is related to Λ . In order to be independent of the high value of (25), ⟨0|�(�)�(� )|0⟩, there would be no diference momentum cutof, we can rewrite this result by subtracting in the result in (23). However, the crucial thing to consider is �(0, 0) = �0,so how to evaluate the frst two terms. To see the signifcance of the frst two terms, here by setting � →�and rewriting (25), � (x,�) =�(x,�) −�0 we have �3��3� 1 = √⟨�2 (x,�)⟩+√⟨�2 (x,�)⟩ �2 (x,�) = ∫ (31) (2�)6 2√�� − √⟨�2 (0, 0)⟩−√⟨�2 (0, 0)⟩. † † ⋅[(� � � +� � ) (� + � )� � cos † † (26) Now,thenewGreen’sfunctionistotallycomposedofthe +�(� � � −�� ) sin (�+�)� terms of the fuctuations �(x,�)and �(x,�)and starts at zero. 2 † † Meanwhile, the large constant which is proportional to Λ +(�� +� �� ) cos (�−�)� is eliminated. Tis constant is the vacuum energy which is † † +�(� � � −� � ) (�−�)�]. nonobservable, due to the QFT. � sin

Similarly, the last two terms indicate zero point energy, and 3.2. Te Interpretation of Modifed Green’s Function. To gain the frst two terms are zero when they are taken vacuum the modifed Green’s function, (28) should be computed. However, the integration is not simple. Here is a way which expectation value. However, the frst two terms are the parts 2 in (26) where the fuctuations emerge canbeusedtoestimatetheintegralsfor⟨|�(x,�)| ⟩ and 2 ⟨|�(x,�)| ⟩. At frst, divide the interval of the integration: � (x,�)

3 3 ⟨|�|2 (x,�)⟩ � �� † † = ∫ (� � � +� � ) (� + � )�, 6 � cos 2 (2�) √�� Λ Λ 3 3 0 � �� 1 (27) =(∫ + ∫ ) 2 (� + �)�, 6 2�� cos � (x,�) Λ 0 0 (2�) (32) 3 3 2 �� † † ⟨|�| (x,�)⟩ = ∫ (� � � −�� ) sin (�+�)�. 6 √ 2 (2�) �� Λ Λ 3 3 0 � �� 1 =(∫ + ∫ ) 2 (�+�)�. 2 6 2�� sin Obviously, ⟨�(�, �)⟩ = ⟨�(�, �)⟩ = 0.However,⟨� (�, �)⟩ and Λ 0 0 (2�) 2 ⟨� (�, �)⟩ arenotzero.Infact,theyarethereasonwhythe vacuum state is not the eigen state of the energy density. In For the frst part of integrals, because Λ≫2�,wehave this case, �(�, �) and �(�, �) are the key elements to show the “hidden” structure of ⟨��⟩.Here,wehave �=√2�2 +�2 ≈�. (33) 3 3 2 � � � � 1 2 ⟨|�| ⟩=∫ cos (� + � )�, (2�)6 2�� In this case, the integrals become (28) �3��3� 1 ⟨|�|2⟩=∫ 2 (�+�)�. 6 2��sin Λ �3��3� 1 (2�) ⟨�2 ⟩=∫ 2 (� + �)� 1 6 2�� cos Λ 0 (2�) A simplest way to preserve the part which quantifes the 2 fuctuation is given as ≈ ∫ �� ∫ sin � sin � �� 2 0 0 Λ 2 2 � (x,�) = + √⟨|�| (x,�)⟩+√⟨|�| (x,�)⟩. (29) Λ �� 8�2 ⋅ ∫ 2 (�+�)�, 6 cos Λ 0 2 (2�) Here, we simply preserve the latest nonzero order of (34) ⟨|�(x,�)| ⟩ and ⟨|�(�, �)| ⟩,soastoleaveallthepartsin(26) Λ �3��3� 1 ⟨�2⟩=∫ 2 (�+�)� “survived” afer taking the expectation value. Fortunately, 1 6 2�� sin Λ 0 (2�) �(x,�)is convergent when x and � go to infnity and 2 Λ �� 2 ≈ ∫ �� ∫ � �� ∫ lim � (x,�) ∼Λ sin sin 6 x,→∞ 0 0 Λ 0 2 (2�) (30) 2 � (0, 0) ∼Λ2 ⋅ sin (� + � )�. 6 Advances in High Energy Physics

To see the result of integrals in (34) more clearly, we set x = 2 2 0.20 0 and keep ⟨�1⟩ and ⟨�1⟩ as functions only with variable �. Here we have 0.15 2 2 1 4 2 2 2

⟨� (0, �)⟩= [4� (Λ −Λ ) 2 1 4 0 32�  0.10 /2 2 2 2 2 2 Λ +(1−4Λ0� ) cos (4Λ 0�) + (1 − 4Λ � ) cos (4Λ�) 0.05 2 +2(4Λ0Λ� −1)cos (2� (Λ 0 +Λ)) 1 2345 −4(Λ0 +Λ)�sin (2� (Λ 0 +Λ)) Λt

+4Λ0�sin (4Λ 0�) + 4Λ�sin (4Λ�)], Figure 1: Approximate �(0, �) in (41). (35) 1 2 ⟨�2 (0, �)⟩= [4�4 (Λ2 −Λ2 ) 1 32�4 0 0.20 2 2 2 2 +(4Λ0� −1)cos (4Λ 0�) + (4Λ � −1)cos (4Λ�) 0.15 2 +2(1−4Λ Λ�2) (2� (Λ +Λ)) 0 cos 0 2 0.10  /2 +4(Λ +Λ)� (2� (Λ +Λ)) 2

0 sin 0 Λ 0.05

−4Λ0� sin (4Λ 0�) − 4�Λ sin (4Λ�)]. 1 2345 ⟨�2 ⟩ ⟨�2⟩ Afer direct calculations, 1 and 1 have limits where Λt 2 (Λ2 −Λ2 ) 2 2 0 Figure 2: �(0, �) derived by numerical calculation. lim ⟨� ⟩= lim ⟨� ⟩= . (36) →∞ 1 →∞ 2 64�4 At the origin, we also have �=0 2 For simplicity, setting ,wecaneasilyevaluate(39) (Λ2 −Λ2 ) directly: ⟨�2 (0, 0)⟩= 0 , 1 32�4 (37) 1 Λ 6 ⟨�2 (0, �)⟩= 0 2 (2√2��) , 2 2 4 2 cos ⟨�1 (0, 0)⟩=0. 288� � (40) Ten, think about the next part of integrals. In the next 1 Λ 6 Λ ≪� ⟨�2 (0, �)⟩= 0 2 (2√2��) . part of integrals, because 0 , 2 288�4 �2 sin �=√2�2 +�2 ≈ √2�. (38) Combining (36) and (40), the modifed Green’s function � (0, �) Λ≫�≫Λ Similarly, we have is obtained. Because 0, the approximate Λ 3 3 result is given as 2 0 � �� 1 2 ⟨� ⟩=∫ cos (� + � )� 2 2 2� 6 2�� Λ 0 ( ) � (0, �) ≈ √⟨�2 (0, �)⟩+√⟨�2 (0, �)⟩− . 1 1 2 (41) 2 4√2� ≈ ∫ �� ∫ sin � sin � �� 0 0 In this case, the limit when time goes infnity becomes Λ 2 2 0 � � �� 2 ⋅ ∫ 2 (� + �)�, (2 − √2) Λ 6 2 cos 0 4 (2�) � lim � ≈ . (42) →∞ 8�2 (39) Λ 3 3 0 � �� 1 ⟨�2⟩=∫ 2 (� + �)� To check the validity of the approximate modifed Green’s 2 6 2�� sin 0 (2�) function derived here, we calculate �(0, �) numerically. 2 In Figure 2, the numerical result is seen to be close to ≈ ∫ �� ∫ sin � sin � �� the result in Figure 1 which is plotted with the approximate 0 0 Green’s function. Te amplitude of the oscillation of �(0, �) Λ 2 2 0 � � �� remains nearly a constant in Figure 1 at initial times; how- ⋅ ∫ 2 (� + �)�. 6 sin ever, when the time becomes very long, the amplitude of 0 4 (2�) �2 the oscillations of our approximate result approaches zero. Advances in High Energy Physics 7

4. Toy Model: How the Inhomogeneous Vacuum Influences the Evolution of 0.298614 the Universe 2

2 Afer deriving the modifed Green’s function, now we are  0.298614

/2 able to explore how the inhomogeneous vacuum infuences 2

Λ the evolution of the Universe. Recalling Einstein’s equation in Section 2 (15), adopting the approximation in (41), and 0.298614 substituting it into (15), we have [9] 501 502 503 504 3�̈ �2 � 3�2 Λt − =8��(⟨��⟩̇ ̇ + � + � 2 + ) . � 2 4! 2� (44) Figure 3: �(0, �) derived by numerical calculation when � goes � (0, �) � larger. Te oscillation of canbeseenclearlywhen is large. However, here, a new issue emerges, that is, how to calculate ̇ ̇ ⟨��⟩.Asamatteroffact,wehavesucharelationbetween ̇ ̇ ⟨��⟩ andthemodifedGreen’sfunction: Te maximal values of the two results derived by diferent ̇ ̇ ̇ ̇ methods are slightly diferent. In fact, this is not hard to ⟨��⟩ (�) = lim ⟨� (�) �(� )⟩ 2 2 � explain. In (41), we drop the ⟨� (0, �)⟩ and ⟨� (0, �)⟩ terms, → 2 2 (45) whichinfuencetheamplitudeandthevalueofpeak.Roughly = lim �� ⟨� (�) �(� )⟩ . speaking, comparing to the value of the Green’s function’s →� limit when �→∞, the oscillation’s amplitude is much smaller compared to the overall value. Terefore, it can be In analogy to the method that we use to derive the ⟨��⟩ omitted. Te oscillation of the Green’s function is shown modifed Green’s function for in the previous section, �(x,�) �(x,�) in Figure 3 by computing the Green’s function numeri- functions and canalsobeintroduced,anddue �(x,�) �(x,�) cally. to (45), the new and are given as Now, a problem arises naturally: how to interpret this � (x,�) result. In Figures 1 and 2, the modifed Green’s function starts at the origin and fnally reaches the peak. Meanwhile 3 3 � �� √�� † † afer reaching the peak, the function oscillates with a small = ∫ (�� +� � � ) cos (�+�)�, (2�)6 2 amplitude. To make a correspondence to the evolution of �, �=0 �=0 � (46) we notice that when ,thescalarfeld and then � (x,�) evolves with time. However, � cannot always increase. Tis is because once it starts evolving not at the true vacuum, 3 3 � �� √�� † † the global minimal point of the potential, there is always a = ∫ (� � � −�� ) sin (� + � )�. (2�)6 2 tendency towards getting back to the stable true vacuum. � (0, 0) = 0 Taking this thought into account, indicates Aferthedirectcalculations,resultsin(46)arequitesimilarto that the scalar feld � starts at the origin and reaches the that of the �. Terefore, the regulation is also necessary for true vacuum periodically. Because the energy for driving the ̇ ̇ ⟨��⟩. However, unlike the method we used for the modifed infationoftheUniverseisgraduallydissipated,theamplitude ⟨��⟩̇ ̇ of � becomes smaller and smaller. Finally, � oscillates around Green’s function, the fnal version of afer regulation the global minimal point with a small amplitude. Tis idea becomes canbetrueonlywhenweadmitsuchapostulation: ⟨��⟩̇ ̇ = √⟨�2 (x,�)⟩+√⟨�2 (x,�)⟩ (2 − √2) Λ2 6�2 (47) 2 2 the maximum of � ≈ = V = . (43) − lim (√⟨� (x,�)⟩+√⟨� (x,�)⟩) . 8�2 � →∞

Terefore, we obtained a correlation of the ultraviolet cutof Te reason that this type of regulation is adopted, which is Λ and the global minimal of the potential. Tis kind of diferent from what we used for �, is not hard to explain. It relationship does not appear in QFT. Tis is because, in is clear that the scalar feld � fnallydecaystothetruevacuum QFT, the ultraviolet cutof is set by an exterior constant which is the minimal of the potential energy. Equivalently, which is there just for regularization. However, for this case, recalling the interpretation for � in the previous section, wehopetoestablishanefectivefeldtheorywhichtakes we can discuss the correspondence to the evolution of �.We into consideration the quantum vacuum fuctuations. In this notice that when � grows larger and larger, � will be stable at case,thekeyquestionishowtheenergydensityofthis the minimal point. Meanwhile, all the derivatives of � should efective feld drives the Universe to expand. As a result, there be approximately zero at this stage in order to preserve the ̇ ̇ is an exterior constraint on the cosmological background stability. Te numerical result of ⟨��⟩ isshowninFigure4. and we cannot treat it as the usual QFT in fxed back- Te evolution of this correlation function is similar to the ground. evolution of the modifed Green’s functions. 8 Advances in High Energy Physics

400,000 1 2 3 4 5 −0.01 Λt 300,000 −0.02 2 −1

2 −0.03

 200,000

/2 −0.04 2 Λ −0.05 (a − 1)/H 100,000 −0.06 −0.07

̇ ̇ 12 12 12 12 12 12 12 Figure 4: Te correlation function ⟨��⟩ afer regulation in (47). 1×10 2×10 3×10 4×10 5×10 6×10 7×10 Λt −9 3. × 10 Figure 6: Scale factors when time is large. Te blue curve is for our 2.5 × 10−9 toy model, and the orange one is for standard scenario. Like Figure 5, −9 theverticalaxisisalsorescaled. −1 2. × 10 H 1.5 × 10−9 −9 the Lagrangian changes. If the feld is not at zero Kelvin, the

(a − 1)/ 1. × 10 efective mass becomes 5. × 10−10 2 2 � 2 � �→ � − � . (49) 4! 0.2 0.4 0.6 0.8 1.0 Λt Tus, the Lagrangian becomes 1 1 � � Figure 5: Scale factors at the beginning. Te blue curve is for our L = �]� �� + (�2 − �2)�2 − �4 toy model, and the orange one is for standard scenario. To illustrate 2 ] 2 4! 4! the behavior of scale factors clearly, the vertical axis is rescaled as (50) (� − 1)/�−1 �=√8��(0)/� 3�4 ,where . − , 2� and in this case, the minimal position of the potential is no 2 Substituting (47) into (44), we are able to solve the equa- longer at ] =6�/�,butat � tion for the evolution of the scale factor .Afernumerical 6�2 (�) 6 � �2 calculation, two fgures are plotted for the scale factor �(0, �) ] (�) = = (�2 − �2)=] (0) − . (51) � � 4! 4 compared to the standard scenario. InFigures5and6,itisnothardtoseethatatthe Recalling the postulation in (43), the ultraviolet cutof now in beginning, the scale factor in our toy model grows much fnite temperature feld theory is related to the temperature faster that the scale factor in the standard scenario. However, Λ=Λ(�). But what would happen when the temperature is when the time becomes larger, the scale factor in our toy 2 higher than the critical point �>� = √24� /�?Inthis model grows slower and fnally �∝�. Tis is because when ](�) = 0 � is large, the scalar feld � falls down from the top of the case, ,andthespontaneoussymmetrybreaking potential and oscillates around the global minimal point. In no longer exists. Meanwhile, due to the postulation in (43), thiscase,therighthandsideof(44)isapproximatelyzero,so the ultraviolet cutof should be zero. However, the behavior � � �̇ is also approximately zero [9]. of when time becomes long gives us hint that should oscillate around the origin at the minimal of the potential. 5. More Elaborated Model: When Temperature For more details, the modifed Green’s function should be discussed, which involves temperature. Similarly to the zero Is Involved temperature approach, we start with �(x,�) and �(x,�). Replacing �→�+��,wereach[11] Once the temperature is introduced in our model, there are 3 3 signifcant changes. Based on the fnite temperature quan- � �� † † − � (x,�) = ∫ (� � � � +� � � � ) tum feld theory, correlation functions should satisfy Kubo– 2 (2�)6 √�� Martin–Schwinger relations [10] ⋅ cos (�+�)�, ⟨� (�) � (� )⟩ = ⟨� (� ) � (�+��)⟩ . (52) (48) 3 3 �� † † − � (x,�) = ∫ (� � � −� � � � ) 6√ � Here, time � is also extended to the complex plane and � 2 (2�) �� =�−1 in (48) is the reciprocal for the temperature, .Not ⋅ (� + �)�. only the background becomes more complicated, but also sin Advances in High Energy Physics 9

Notice here that the square of the efective mass in � is the vacuum energy drove the Universe expanding expon- 2 2 2 �ef =��/4! − � . Following the process in previous sec- entially. Following the previous section, the symmetry was tion, √|�(x,�)|2 and √|�(x,�)|2 need to be calculated. Nev- restored because the feld was at a very high temperature ertheless, the term which depends on � is canceled out. which is much higher than the critical temperature where Terefore, fnally, the modifed Green’s function here is not phase transition of the feld occurs. Terefore, � oscillates at diferent from the previous one. To fnd the explicit form the origin with a tiny amplitude which can almost be negli- � when � is very high, considering the approximation in (39) gible. By omitting the terms containing in (56), during this 2 and Λ≪��,wereach period, the equation for the scale factor becomes 3 4 4 √2 Λ � � � � �̈ 8�� 3� 4�� 2 � (0, �) ≈ (� 2� �� + � 2� �� −1), ≈ = =�. (57) 2 �cos � �sin � (53) � 3 2� � 12� � 2 2 2 2 Taking the initial condition when �≈0,wereach�(�, �) = where � =��/4! − � ≈��/4!.Itiseasytofndoutthat −1 �(�) = � cosh(��). Tis also gives the solution to the Fried- �(0, �) oscillates at the origin and the amplitude is small. Before solving the Einstein’s feld equation for the scale mann equation for homogeneous and isotropy space-time ̇ ̇ with positive curvature [13]. In this case, we have the same factor, we notice that the term ⟨��⟩ in (15) is still unknown. solution to (57): Similarly to the previous described approach with zero � (x,�) � (x,�) −1 temperature, introducing and in (46) for the �≈� cosh (��) . (58) fnite temperature feld, we reach In fact, following the Friedmann equations, besides the 00 3 3 √ � �� † † − component of Einstein’s feld equation as (15), the other ones � (x,�) = ∫ (�� +� � � � ) (2�)6 2 arederivedfromthefrstoneandthetraceofEinstein’sfeld equations. Similarly, we can derive the second equation for ⋅ cos (�+�)�, our model. (54) �3��3� √�� 1 �00 �11 �22 �33 † † − LHS.= ( − − − ) � (x,�) = ∫ (� � � � −�� ) (2�)6 2 6 �00 �11 �22 �33 2 2 2 2 ⋅ (� + �)�. 4 ((3� −2)� +�(� −1)� )�−2�(� −1)� sin = 6��4 Using the approximation �≈�ef , in this case, we reach �2̇ +1 + , ̇ ̇ 2 �2 ⟨��⟩ = � �. (55) ef (59) 1 � � � � .= ( 00 − 11 − 22 − 33 ) Ten plugging (53) and (55) into (15), we obtain the equation RHS 6 � � � � for the scale factor: 00 11 22 33 2 4 8�� 1−� �̈ 8�� 1 2 � 2 3� = (⟨��⟩̇ ̇ + ⟨� � ⟩+2�(�)) = (− � (�) � + � + ). 2 � 3 2 ef 4! 2� (56) 6 �

8�� 2 To solve this equation, the relation between the temperature ≈ � (0) =�. 3 and the scale factor is needed. Due to the total entropy con- −1 servation, the relation is given as �∝� .In(56),because OnthelefhandsideofthesecondFriedmannequation,it the amplitude of � is negligibly small, the last term �(0) = separates into two parts. One part comes from the spatial 4 (3� )/(2�) dominates the equation. Terefore, it is not hard to dependence of the scale factor, and the other part is a com- predictthatthescalefactor� increases exponentially before monterm.Becausetherighthandsideisapproximatelya the temperature is below the critical temperature. constant,wecanalsomaketheapproximationsothat�(�, �) ≈ �(�). Ten the frst part on the lef hand side disappears. 6. A New Inflationary Scenario Hence, a much more simplifed equation is derived: 2 2 2 In this section, we mainly discuss a new infationary scenario �̇ +1≈� � . (60) which is built based on our model. At the beginning, in this new infationary scenario, we investigate the idea that It is obvious that the solution to this equation is the same as theUniverseiscreatedfromnothing[12,13].Hence,inour that of (57). model, the curvature is greater than zero or, equivalently, we At the frst stage, the Universe expanded exponentially, so take �=1in the FLRW metric. From the tunneling theory, the temperature of the entire Universe decreased rapidly, due −1 before tunneling through the barrier, the “Universe” was in to the relation of temperature and entropy [1], �∝� .Once Euclidean space. Afer penetrating the potential barrier, the the temperature becomes lower than the critical temperature, Universe was “created” and began expanding. At this stage, theUniverseswitchedtothenextstage.RecallingSection4 10 Advances in High Energy Physics

accelerationcannotbezero,but,ataverysmallvalue,wewill explain this efect next). Afer the switching period, the Universe cooled down. due to the coupling with other felds, the bosons of the scalar 1.5 feld would decay into other particles. Tus, the � = �(�) 1.0 and �=�(�)in (63) dropped down, so the speed of linear 0.5 10 expansion in the last period also decreased rapidly. Moreover, 0.0 at this stage, the vacuum energy of the scalar feld contributed 10 to the Universe’s expansion decreases, so the speed of expan- 5 t sion continued decreasing. However, this does not mean that 5 r the Universe stopped expanding. In fact, the expansion of the Universe was still accelerating because of one element that we 0 0 neglect in the previous stage, which became more important. In Section 3, the behavior of � shows that the feld � falls Figure 7: Numerical result of �(�, �) in 3D. into the minimal of the potential very fast. However, once it reaches that point, it will oscillate at that point with a tiny amplitude. Te oscillation makes a tiny shif, which ⟨��⟩̇ ̇ � and considering that goes to zero and reaches the sta- isquitesmallcomparedtotherighthandsideof(61).To blepointextremelyfast,thesetermsmakeverylimitedcontri- calculate the tiny shif caused by the oscillations of �,thetwo butions. Terefore, they can be neglected in the large scale approximations in Section 3 which are applied to calculate � space-time. However, if we want to solve the second Fried- should be both considered; when � is large, we reach mann equation, (59), for the scale factor during this period, there is still an issue: how the spatial dependence of the scale � (x,�) = √�2 (x,�) +�2 (x,�) factor infuences the result. Ideally, if the spatial factor has 1 2 no signifcant efects, we can also drop those terms involving + √�2 (x,�) +�2 (x,�) − √�2 (0, 0) +�2 (0, 0) � and � and obtain a simple equation as (60). To check 1 2 1 2 this assumption, both two variables should be considered for � √ 2 2 √ 2 . To illustrate how the spatial part infuences the modifed − �1 (0, 0) +�2 (0, 0) ≈( �1 (x,�) Green’s function, we plot a 3D fgure for �(�, �).InFigure7, � except the very little area where is small, the surface in the + √�2 (x,�) − √�2 (0, 0) − √�2 (0, 0)) fgure is almost fat. Tis means that the spatial fuctuationsof 1 1 1 (64) the variables are very small and can be omitted when � is not toosmall.Tus,itisfnetogetridofthepartwhichcontains 1 �2 (x,�) �2 (x,�) 1 � � + ( 2 + 2 )≈] (�) + or in (59) and obtain a new equation: 2 2 √�2 (x,�) √�2 (x,�) 36� 8�� 1 1 �2̇ +1≈ �2�(�,�)� � 6 3 �min Λ ⋅ 0 , (61) 2 2 8�� 2�2 ��4 Λ (�) � (�) = ( − )�2. 3 8 384 where Λ(�) is the ultraviolet cutof which is related to 2 2 2 temperature. � (�) = � −�� /4! is the square of the efect- Because the expansion of the Universe is adiabatic, the total 2 ive mass in this temperature. ](�) = 6� (�)/� indicates the entropy of the Universe is conserved and set �� = �;then� minimal point of the potential. Tus, we can defne the shif canbereplacedin(61): of the scalar feld �: 2 2 4 2 8�� 2 6 �̇ +1≈ ( − ) � . (62) 2 1 Λ 0 3 8�2 384�4 (Δ�) = lim � (x,�) − ] (�) = . (65) →∞ 36�2 Λ2 (�) �2 (�)

In this switching period, (62) could be rewritten as Ten,thetinyshifofthepotentialcanalsobeobtained

2 2 4 1/2 �� (� ) 2 �≈[(̇ −1)− ] = √�−�/�2. (63) Δ� = � (� +Δ�)−�(� )≈ 0 (Δ�) , (66) 3�2 144�2 0 0 2 � = √](�) Terefore, it is easy to fnd that when scale factor grew larger, where 0 ,whichindicatestheminimalpoint,and 2 2 2 the second term �/� became smaller and � ≈ constant, � =��/�� . Substituting (65) and the Lagrangian into which means the Universe expanded approximately in a fxed (66), here we have speed at this stage. In this case, the expansion of the Universe Λ 6 was linearly in time. Meanwhile, during this period, the Δ� = 0 . 2 2 (67) acceleration of the expansion kept on decreasing (in fact, the 36� Λ (�) Advances in High Energy Physics 11

2 2 Currently, in the case that � ≈ 2.7�,wehaveΛ (�) ≈ Λ (0) Ω =Ω =0, ΩΛ +Ω =1,andΩΛ ≈ 0.7.In(69),because 2 2 √ � ≈ (48� � )/((2 − 2)�) ≫ Λ 0, so the energy shif Δ� is a at this stage islarge,thefrsttermissmall.Tomatchthe very small constant. current statue of the Universe, an approximate formula can Besides the energy shif began playing a role in the be laid down. Universe expansion; afer the particles of the scalar feld �2�2/3 3 decayed into other particles, the density of particle � kept �−Γ/2 ≈ . (72) decreasing. According to the radiation thermodynamics, the 8 8�� 4 density of particle is proportional to [14]. Terefore, in this Tus, the curvature constant is efectively equal to zero. Ten, period, the particle density is given as by neglecting the second term on the right hand side of (69), �=��−Γ ∝�4�−Γ, we reach the equation which describes the current Universe 0 (68) as Γ 8�� where is the decay rate which is dependent on the coupling �2̇ ≈ ( +Δ�)�2. constants of � and other felds. Terefore, equivalently, we can 3 �3 (73) replace the original temperature by �→�exp(−Γ�/4).Now, we are able to establish the equation for the last stage of the In this case, the expansion of the Universe is dominated by infation evolution: vacuum energy which is from the energy shif caused by the � 2 oscillations of the scalar feld around the potential minimal �̇ +1 and nonrelativistic matter, while other efects are ruled out. Γ� Meanwhile, plugging the current observed Hubble constant 8�� 2�2 − ��4 � (69) [17] in (71), we obtain the energy density of vacuum energy ≈ ( � 2 − �−Γ + +Δ�)�2, 3 8�2 384�4 �3 (cosmological constant): −47 4 �Λ0 ≈ 2.57 × 10 (GeV) . (74) where � indicates the density of nonrelativistic matter. Now, we have established the equation which can explain the Tisresultismuchsmallerthantheenergyscalesofvarious scenariooftheUniversenowadays.Inthisequation,the theories of QFT. Since Δ� in (67) is also much smaller than −2 frst term which is proportional to � decreases rapidly. the energy that QFT can predict, this result can be used to Te next term is negligible when � is large. Te third term explain why the current energy density of the vacuum energy comes from the nonrelativistic matter which is created by the is so small [6]. In our model, the density of efective vacuum energy that the scalar feld dissipates. Te last term, which is energy caused by Δ� is approximately a constant, plays the role of vacuum energy or, −2 in other words, the cosmological constant. �Λ =Δ�∝Λ . (75) According to the observations, now, the Universe is still −1 expanding, and the expansion is accelerating. Using the In this case, the Hubble parameter �∝Λ →0.Compared observational data, one can check which factor dominates the to another resolution to the cosmological constant problem current Universe evolution. To describe the behavior of the [6], the efective Hubble parameter approaches zero, �∝ −Λ current Universe, a general evolution equation is introduced: Λ� →0. Although both two models draw a conclusion that the Hubble parameter �→0when the ultraviolet cutof �2̇ is taken as infnity, Λ→∞, the results in the two models � 2 � 3 � 4 (70) are still diferent. Tis is because, in our model, the quantum =�2 [Ω +Ω ( 0 ) +Ω ( 0 ) +Ω ( 0 ) ], vacuum fuctuations have direct impacts on the scalar feld �. 0 Λ � � � Tus we obtain the efective density of vacuum energy Δ�.In [6], the vacuum fuctuations have direct impacts on the space- where �0 is the current Hubble’s parameter and �0 is the time structure, but not on the feld itself. To summarize, there current scale factor. Meanwhile, the energy densities of are three main stages of this new infation scenario (shown in nonrelative matter, radiation, and vacuum are, respectively, Figure 8): At the frst stage (curve (I) in Figure 8), the scalar 3�2Ω feldhadnosymmetrybreakingattheveryhightemperature � = 0 , and the Universe expanded exponentially. Meanwhile, the 0 8�� temperaturekeptdecreasingrapidlyandonceitdropped 3�2Ω below the critical point, the scenario switched to the next � = 0 , (71) 0 8�� stage. At this transient period (curve (II) in Figure 8), the expansion was no longer exponential, but the acceleration 3�2Ω of the expansion decreased, and fnally the expansion was � = 0 Λ . Λ0 8�� approximately linearly in time. Next, along the decrease of the temperature, the � and � in (63) can no longer be constants Byfttingwiththeobserveddataofthesupernova[15,16],the but went down quickly. Terefore, in this stage (curve (III) ratio of each component which sustains the expansion of the in Figure 8), the speed of the expansion became slower and Universecanbeobtained.Oneoftheresultsobtainedisthat slower, and fnally �(�) ≈ 0.However,oneefectburied 12 Advances in High Energy Physics

2 2 2 By setting �=sin�, the metric becomes �� =�� −�(�, �)[��2 + 2�(��2 + 2��2)] = ��2 −�2(�, �)�Ω2 (III) sin sin 3.Ten afer direct calculation, the Ricci scalar in our model is given as

2 2 2 �=− [� −2�(2cot (�) � +� )+3(1+�̇ ) a (II) �4 (78) +3�3�]̈ ,

(I) where �̇ = ��/�� and � = ��/��.Substitutingthepotential H−1 and Ricci scalar into action and writing out the action t explicitly, we have

Figure 8: Te scale factor of the new scenario. Notice that there are ̇ �=∫ �� (�, �,̇ � ;�,�, � ), (79) three stages of the evolution of the scale factor. where the Lagrangian reads in previous stages became important now. Afer �(�) being 1 2 �=∫ [ (� −2�(2cot (�) � +� ) approximately zero, the efective potential in (67) leads to the 8��4 acceleration of the expansion of the current Universe. �2 Compared to the standard infationary scenario, the new +3(1+�2̇ )�2 +3�3�)̈ + �2̇ − −�(�)] (80) scenario proposed here has no issue of how the Universe �2 quits the infationary stage from the exponential expansion. 3 2 Te acceleration of expansion decreased smoothly from the ⋅� sin � sin ��. frst stage to the second stage, and fnally the expansion was almost linearly in time. Tis smooth transition is guaranteed Tereisanissueinvolvingthedifcultyinevaluating bythebehaviorofthemodifedGreen’sfunctionwhichshows the integrations in (80). To evaluate the integral, we make � � that the scalar feld at each point of the whole Universe can the approximation that and are approximately constants � reach the local minimum on the efective potential landscape. when integrating over . In fact, this approximation is � Terefore, the new infationary scenario does not have to quit acceptable. Recalling Figure 7, the spatial variation of is � � through bubble collisions. small, and because is determined by , the spatial variation of � is also small. Ten we have the approximate Lagrangian 1 7. Tunneling Amplitude: How the �=2�2 [ (�2 +3(1−�2̇ )�2)+�3�2̇ −��2 Inhomogeneous Vacuum Influences the 8�� Creation of the Universe (81) −�3�(�)]. Afer establishing the model describing the evolution of the Universe, let us go back to the moment where the Universe Ten, following the same idea, we can make further approx- has not been created. Due to the tunneling theory [12, 13], imations that neglect � . Terefore, now we have a simpler the Universe can be created from “nothing” by tunneling Lagrangian through the potential barrier. It is therefore useful to estimate 2 3 2 3 2 2 3 thetunnelingamplitudeinourmodelandtheinfuenceof �=2� [ (1 − �̇ )�+� �̇ −�� −� �(�)]. (82) the inhomogeneous vacuum on the tunneling amplitude. 8�� To compute the amplitude, we followed the following step. Now, we can derive the WD equation. Firstly, the canonical First fnd the minimal coupling action. Ten derive the momenta should be obtained: Wheeler–DeWitt (WD) equation. Finally solve the WD �� 3��̇ equation, then fnd the outgoing wavefunction, and calculate � = = , the tunneling amplitude. �� 2� (83) Te minimal coupling of Einstein–Hilbert action and the �� = =2�2�3�.̇ scalar feld action is given as ��

� 1 ] 4 By Legendre transform and substituting the canonical mo- �=∫ [− + � ��]�−�(�)] √−�� , (76) 16�� 2 menta into (82), we have the Hamiltonian: � 1 where the potential �(�), based on our model, is given as �=− � 2 + � 2 3�� 4�2�3 (84) 1 1 3�4 3� 8�� (�)= �2�2 + ��4 + . (77) − �[1− (�2 +�2�(�))]. 2 4! 2� 4� 3 Advances in High Energy Physics 13

To obtain Wheeler–DeWitt equation, introduce canonical quantization by replacing � →−��/��, � →−��/��;then we have the WD equation: �2 � � 1 �2 [ + − −�(�, �)] �=0, ��2 � �� 2 ̃2 (85) �� a ̃2 2 and here, � =4��/3,theparameter� represents the H−1 ambiguity: U 1 � 2 ∼�2�2̇ = (��)̇ � (��)̇ . � (86) 2 2 2 �(�) �= In this case, it is not appropriate to just write � as � /�� , Figure 9: Potential for WD equation. Notice the step at �−1 but . 2 2 1 � � � � � � =− (� )= + . (87) � �� 2 � �� andthisisalsotheinitialconditionthatweusedforour model. � In the WD (85), the potential is given as Following the same approach by which we obtained the �<�−1 3� 2 8�� WD equation for Lorentz region, in ,theWD � (�, �) = ( ) �2 [1− (�2 +�2� (�))] . (88) equation for Euclidean region is simpler: 2� 3 �2 � � 1 �2 Basedonthetunnelingapproach,beforetheUniversewas [ + − −� (�, �)] �=0, 2 2 ̃2 0 (93) created, everything was formulated in Euclidean space. To �� knowthebehavioroftheUniversebeforetunnelingthrough the potential barrier, gravitational instanton solution should where the potential �0 is � be considered. At this stage, is assumed not to change with 2 time. Terefore, it is possible to set �=0. Tis is because, 3� 2 2 2 �0 (�, �) = ( ) � [1 − � � ]. (94) in the infationary scenario that we discussed in the pre- 2� vious section, the scalar feld � starts at the extremum, �= 0 Afer the WD equation is obtained, we can calculate the . We assume that before tunneling through the barrier, in �(�−1)/�(0) Euclidean space, the Universe preserved the highest sym- tunneling amplitude, which is proportional to . metry. Terefore, we assume that the Universe was homo- WKB approximation can be used to derive the amplitude: geneous and the scale factor was not related to �. Terefore, �(�−1) −1 in Euclidean region, �=0. Now we have Freedman equation = exp [− ∫ √�0 (�)��] in Euclidean space as � (0) 0 (95) −�2̇ +1=�2�2, 3 (89) = [− ]. exp 16�2� (0) 1/2 4 where � = (8��(0)/3) ,V(0) = 3� /2�.Tesolutionto 2 this simple equation is Because of � ,thepotential�(�) is a little bit diferent from before.InFigure9,wecanseeclearlythatthereisaclear −1 −1 � (�) =� cos (��) . (90) sharp step at �=� , which means the tunneling amplitude should be greater than the conventional result exp(−3/ Tis solution shows that, before the Universe “penetration” 2 [16� �(0)]).Tisrefectstheinfuencecausedbytheinho- through the potential barrier in Euclidean space, it expanded mogeneous vacuum. Meanwhile, the sharp step caused by and contracted periodically. In this case, when we analyze 2 −1 � and � is also a spatial function. Terefore, the size of the the wave function of the WD equation, the region �<� −1 sharp step at � is also related to spatial variable �.Tis is in Euclidean space representation. Meanwhile, the region −1 means that the tunneling probabilities vary at diferent spatial �>� is in Lorentz space representation. At the very early locations. Terefore, in our model, each spatial point of the time afer the Universe has been created, �≪1,followingthe Universe is not expected to tunnel through the barrier at the Lorentz space representation, we have same time due to the diferent tunneling probability. Te WD 2 2 2 �=−1 �̇ +1≈� � . (91) equationcanbesolvedexactlywiththechoiceof .Te choice of the factor-ordering factor � dose not infuence the Te solution is probabilities. Now introducing a new variable,

−1 2 2 � (�, �) ≈� cosh (��) , (92) �=−(1−�� ). (96) 14 Advances in High Energy Physics

√3 √3 −1 Tus, the potential �0 becomes {�1Ai (− ��) + �2Bi (− ��) , � < � �={ (101) 2 � (−√3 ��)+� (−√3 ��), �>�−1, 3� 2 { 1Ai 1Bi � =( ) � �. (97) 0 2� 2 where �=(3�/4��).Next,weshouldfndthecoefcients Setting �=−1,wehave in the solution. Te continuities of � and ��/�� should be considered. Meanwhile, based on the tunneling theory, only 2 � 3� 2 � an outgoing wave should be considered outside the barrier, [ + ( ) ] �=0. (98) −1 −1 ��2 2� 4�2 which means �� /�� > 0 for �>� .Forlarge�,we reach the asymptotic formulas for Ai and Bi: Afer introducing the new variable, we can rewrite the WD −1 3/2 equation in �<� . Similarly, we can also introduce another sin ((2/3) � +�/4) −1 Ai (−�) ∼ , variable for the WD equation in the region �>� [12]: √��1/4 (102) 3/2 8�� 2 2 2 cos ((2/3) � +�/4) � =−(1− � −� � ) . (99) (−�) ∼ . 3 Bi √��1/4 At the very early times, �≈0. Terefore, here, 8��(�)/3 ≈ 2 2 Due to the asymptotic formulas and Euler’s formula, it is easy � � � −1 and are irrelevant to .Teequationcanberewritten to fnd out that the proper form of � for �>� is once again as �=�(Ai (−� )+�Bi (−� )) . (103) �2 3� 2 � [ +( ) ]�=0. ��2 2� 4�2 (100) Ten we can obtain the coefcients �1 and �2 related to � by two continuity equations. Now, we are able to calculate the Te solution to these equations are Airy functions: tunneling amplitude:

−1 �(� ) 2⋅33/2� = , 3 3 (104) � (0) (34/3Γ (2/3) �−3Γ(1/3) �) Ai (√�) + (35/6Γ (2/3) �+31/2Γ (1/3) �) Bi (√�)

where � and � inside (104) are nothing. We found that the spatial fuctuations caused by the inhomogeneous vacuum lead to faster tunneling while 8��√3 ��2 8��√3 ��2 the tunneling amplitude is dependent on the spatial loca- �= (− )+� (− ), Ai 3 Bi 3 tions. (105) Tere are still some issues that we have not addressed 8��√3 ��2 8��√3 ��2 with our model in this paper. First, in our model, we take the �= (− )+� (− ). Ai 3 Bi 3 simplest potential for the scalar feld. However, our method which describes the inhomogeneity by modifying Green’s function is general. Terefore, an improvement on our Compared to the conventional solution (95), the result in model is to consider more complicated cases, such as grand (104) is much more complicated, and we can see that not only � � unifed theory. Second, no higher order correlations, such but also plays an important role in tunneling process, as one-loop correlations, are considered here in our model. which means the quantum vacuum fuctuation also afects the Tus,theloopcorrectionshouldbeinvolvedinourmodel creation of the Universe. andthenmoreaccurateefectivepotentialcanbeobtained. Tird, it would be interesting to consider the cosmological 8. Conclusion consequences of the new infation scenario suggested here, In this paper, we analyzed the inhomogeneous vacuum in the especially the density fuctuations, as the seed of the large Universe and come up with a new method for introducing the scale structure formation and the related fuctuation power inhomogeneity by modifying Green’s function. Meanwhile, spectrum. Tese are closely associated with the current substituting the modifed Green’s function in Friedmann observations of the microwave background radiation and equation, we obtained a new infationary scenario which can large scale density map. explain why the Universe is still expanding and where the vacuum energy comes from which leads to the accelerated Conflicts of Interest expansion. At last, we also applied our inhomogeneous model to fnd the tunneling amplitude of the Universe from Te authors declare that they have no conficts of interest. Advances in High Energy Physics 15

Acknowledgments Jin Wang would like to thank NSF PHYS 76066 for partial support.

References

[1] A. H. Guth, “Infationary universe: a possible solution to the horizon and fatness problems,” Physical Review D: Particles, Fields,GravitationandCosmology,vol.23,no.2,pp.347–356, 1981. [2] A. Linde, “Chaotic infation,” Physics Letters B,vol.129,no.3-4, pp. 177–181, 1983. [3]S.W.Hawking,I.G.Moss,andJ.M.Stewart,“Bubblecollisions in the very early universe,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.26,no.10,pp.2681–2693,1982. [4] A. H. Guth and E. J. Weinberg, “Could the universe have recovered from a slow frst-order phase transition?” Nuclear Physics B,vol.212,no.2,pp.321–364,1983. [5] S. Weinberg, “Te cosmological constant problem,” Reviews of Modern Physics,vol.61,no.1,pp.1–23,1989. [6]Q.Wang,Z.Zhu,andW.G.Unruh,“Howthehugeenergy of quantum vacuum gravitates to drive the slow accelerating expansion of the Universe,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.95,no.10,ArticleID103504, 2017. [7]R.M.Wald,General Relativity, University of Chicago Press, 1984. [8] T. S. Bunch and L. Parker, “Feynman propagator in curved spacetime: A momentum-space representation,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.20, no. 10, pp. 2499–2510, 1979. [9] In fact, here, �4 should be replaced by 4-point correlation function, which includes at least one four-point interaction. How- ever, once O(�) and higher order diagrams are omitted, GR2 is still good enough to be used for the approximation. [10] D. Ashok, Finite Temperature Field Teory, World Scientifc Publishing Company, 1997. [11] Strictly speaking, the imaginary time method should be applied to discuss the modifcation of A and B, but it makes no diference to our result. For details of the imaginary time, see Ref. [10]. [12] A. Vilenkin, “Birth of infationary universes,” Physical Review D: Particles, Fields, Gravitation and Cosmology,vol.27,no.12, pp. 2848–2855, 1983. [13] A. Vilenkin, “Creation of universes from nothing,” Physics Let- ters B,vol.117,no.1-2,pp.25–28,1982. [14] S. Weinberg, Cosmology, Oxford University Press, New York, NY, USA, 2008. [15] A. G. Riess, P. E. Nugent, R. L. Gilliland et al., “Te farthest known supernova: Support for an accelerating universe and a glimpse of the epoch of deceleration,” Te Astrophysical Journal ,vol.560,no.1,pp.49–71,2001. [16] A. G. Riess, “Type Ia supernova discoveries at �>1from the Hubble Space Telescope: evidence for past deceleration and constraints on dark energy evolution,” Te Astrophysical Journal, vol. 607, pp. 665–687, 2004. [17] Planck Collaboration, Ade, P. A. R., et al. Planck 2015 results - xiii. cosmological parameters. AA, 594:A13, 2016. Hindawi Advances in High Energy Physics Volume 2018, Article ID 3471023, 9 pages https://doi.org/10.1155/2018/3471023

Research Article Searches for Massive Graviton Resonances at the LHC

T. V. Obikhod andI.A.Petrenko

InstituteforNuclearResearch,NationalAcademyofSciencesofUkraine,47Prosp.Nauki,Kiev03028,Ukraine

Correspondence should be addressed to T. V. Obikhod; [email protected]

Received 17 January 2018; Accepted 21 February 2018; Published 30 April 2018

Academic Editor: Marek Nowakowski

Copyright © 2018 T. V. Obikhod and I. A. Petrenko. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

Te Standard Model problems lead to the new theories of extra dimensions: Randall-Sundrum model, Arkani-Hamed- −1 Dimopoulos-Dvali model, and TeV model. In the framework of these models, with the help of computer program Pythia8.2, the production cross sections for Kaluza-Klein particles at various energies at the LHC were calculated. Te generation of monojet events from scalar graviton emission was considered for number of extra dimensions (�=2,4,and6)fortheenergyattheLHC 14 TeV.Te graviton production processes through the gluon-gluon, quark-gluon, and quark-quark fusion processes are also studied and some periodicity was found in the behavior of the graviton mass spectrum. Production cross sections multiplied by branching fractions were calculated for the massive graviton, G, within Randall-Sundrum scenario and the most probable processes of graviton decay at 13 TeV, 14 TeV, and 100 TeV were counted.

1. Introduction Universe formation is deeply connected with the conception of the space-time. According to hierarchy formula [1], Plank Te problems with theoretical explanation of vacuum energy energy can be reduced to the energy of about 10 TeV that as well as dark energy, dark matter, and cosmological constant is achieved at the LHC. So, the phenomena of the Universe problems are only the tip of the iceberg of problems in the formation at the early stages and the accompanying processes modern theoretical physics. Some of them are of particle physics could be studied at modern colliders. (i) ordinary matter accounting for about 5% of mass In spite of the fact that no new physics beyond the SM is energy in the Universe and no dark matter candidate discovered at the LHC at 13 TeV, the planned upgrading of in the Standard Model (SM), theLHCtohighluminositiesandenergiesupto100TeV (ii) hierarchy problem, gives the possibility for the discovery of new physics. Among such searches for new physics, the most popular are the (iii) fne tuning of SM Higgs mass, experimental searches for the Kaluza-Klein (KK) particles. (iv) no explanation for fermion masses and mixings and Historically, KK theory appeared as the unifcation of three family structures, gravitational and electromagnetic interactions due to the (v) unifcation of strong, electroweak, and gravitational proposition of a ffh dimension in addition to the usual forces, four-dimensional space-time [2–4], which leads to the consideration of the concept of deformation of Riemannian (vi) compositeness of leptons and quarks, geometry defned by extrinsic curvature of the space-time. It is an experimental fact that there is something we cannot Te conclusions of this result are based, in particular, on the explain within the SM. fve-dimensional space from the paper [5]. Arkani-Hamed et As is known, vacuum is produced in the processes of al. proposed the solution to the hierarchy problem on the phase transitions in Early Universe and the space-time struc- basis of the existence of new compact spatial dimensions. ture of the physical vacuum exhibits the connection to the KK excitations in this extra dimensional space through the structure formation in our Universe. So, the understanding of combined efect of all the gravitons became observable. 2 Advances in High Energy Physics

Weak brane

Gravity brane

Gravity production function

Figure 1: RS theory presented by the gravity and weak branes as the 4-dimensional boundaries of the extra dimension (from [12]).

Today, the idea of additional space as the instrumentation calculations of KK particle properties. In the framework of of the unifcation of all four interactions is of interest not only M-theory [13], the metric of ADD model is as follows: in theoretical physics [6–8] but also in experimental searches 2 � (�, �) = � + ℎ (�, �) , at the LHC for exotic matter that deviates from normal matter �� 1+�/2 �� (1) [9]. Our paper is devoted to the searches for KK parti- where ��(�, �) is the metric of (4 + �)-dimensional space- cles in three models of extra dimensions: Arkani-Hamed- time with compact extra dimensions, where the gravitational Dimopoulos-Dvali (ADD) model, [6], Randall-Sundrum feld propagates and the SM localized on a 3-brane embedded −1 � � � (RS) model [7, 8], and TeV model [10]. Using computer into the (4 + )-dimensional space-time, �� is (4 + )- ℎ (�, �) program Pythia8.2 [11], within these three extra dimensional dimensional Minkowski background and �� is the � models, we have calculated deviation of Minkowski metrics, is the fundamental mass scale, and � is the number of extra dimensions. Masses of KK (i) the production cross section of KK modes of massive modes for ADD model are given by gravitons and gauge bosons at energies from 14 TeV to 1 |�| planned 100 TeV, √ 2 2 2 �� = �1 +�2 +⋅⋅⋅+�� = . (2) (ii) the graviton mass spectrum for three graviton, G, � � →�� →�� emission processes: (a) gg ,(b)qg ,and Five-dimensional metric of RS model with one extra dimen- ��→�� 1 (c) at 14 TeV at the LHC, sion compactifed to the orbifold, S /Z2, is with nonfactoriz- (iii) the graviton mass spectrum at 14 TeV at the LHC for able geometry: numbers of extra dimensions (� =2,4,and6)(for ��2 =�−2�(�)� ��] +��2. simplicity and brevity). Since the maximum of � is �] (3) equal to 6 for ADD model, it was of interest to look at Two 3-branes are located at points �=0and �=��of the thebehaviorofgravitonmassspectrumattheextreme � 1 ��, � = 0,1,2,3 � values of �,from� =2to� =6, orbifold with radius, ,ofS. ,and �] are four-dimensional coordinates and Minkowski metrics; the (iv) the production cross section of graviton, �� → �, �(�) −�� < � < �� (� → ��) function inside the interval is equal multiplied by branching ratios, Br (gluon- to �(�) = �|�|. (� > 0, dimensional parameter). In Figure 1 gluon (gg) pair), Br(� → ��) (leptons, � are of any �, �, � (� → ℎℎ) ℎ a nonfactorizable geometry with one spatial extra dimension type, ), and Br ( , Higgs boson) of the is presented as a line segment between two four-dimensional most probable processes of decay within RS model at branes, known as Planck and TeV brane. 13 TeV, 14 TeV, and 100 TeV. Masses of KK particles for RS model are given by −�� 2. Models of Extra Dimensions �� =�� , (4)

In this section, we will observe three models of extra dimen- where �� = 3.83, 7.02, 10.17, 13.32, . . . for � = 1, 2, 3, 4, . . . . −1 −1 sions, ADD, RS, and TeV , which are the base for our further Te metric of TeV model for ten-dimensional string theory Advances in High Energy Physics 3

q l+ g l+

q G l− g G l− (a) (b)

Figure 2: Processes for the graviton resonance production through (a) quark-quark and (b) gluon-gluon fusion.

is determined by the conditions on Calabi-Yau manifold: of these processes in the Monte Carlo generator Pythia8.2 Ricci-fatness of metric, vanishing frst Chern class, and was presented in this paper. Te considered processes are SU(�) holonomy. Low-energy efective action has much connected with monojet, diphoton, and dilepton fnal states. smallerscalethanPlanckmassrelatedtoaninternalcom- It is also possible to generate monojet events from scalar pactifcation radius. In these scenarios, the SM felds as well graviton emission as described in [17]. −1 as ZKK and WKK resonancesareallowedtopropagateinthe TeV sized extra dimensional KK production processes bulk, but gravity is not included in the model. Masses of KK involve the production of electroweak KK gauge bosons, ZKK −1 −1 modes for TeV model are given by and WKK,inoneTeV sized extra dimension. Te processes are described in [18, 19] and allow the specifcation of fnal �⋅� 1/2 � =(�2 + ) , (5) states. In this article, the observation of a certain KK hard � 0 �2 process in pp interactions at the LHC was presented within � 1 −1 the S /Z2,TeV extra dimensional theoretical framework. �⋅� �=(�,� ,...) where is scalar production of 1 2 which labels Te analytic form for the hard process cross section has been � KK excitation levels, and 0 is the mass of gauge zero-mode, calculated and has been implemented within the Pythia8.2 which corresponds to SM gauge feld. Monte Carlo generator. Te advantage of the presented models of extra dimensions lies in the possibility of the observation of the physics 3.1. Te Production Cross Section of KK Particles at Diferent of Planck scales, �Pl, at the energies achievable at modern Energies. With the help of computer program Pythia8.2 for colliders, �, due to the presence of extra dimensions, �. three models of extra dimensions, we have calculated the Tisresultispossibleduetothehierarchyformulasforthe production cross sections of KK particles at energies varying presented models: from 14 to 100 TeV. According to the latest experimental �2 ∼��2+� data presented in [15], we reconstructed jets with the anti- ADD model: Pl jet 2 3 2�� k� clustering algorithm, transverse momenta, � ≻20 GeV, RS model: � =(�/�)(� −1) � Pl and rapidity |�| ≺ 2.4 of jets to drop data connected with −1 TeV model: hierarchy formula is determined by the missing transverse momentum. Events were produced with low-energy efective action Monte Carlo event sample using the NNPDF23LO parton distribution function (PDF) for the proton beam and scales of 3. Results of Computer Modeling for renormalizations and factorizations. Te results are presented Properties of KK Particles in Figure 3. FromFigure3(a),wecanseethesignifcantpredom- RS resonances, connected with production of KK graviton, inance of production for dijet fnal states above monojet G, are described in [14]. Experimental searches for massive fnal states within LED model. Figure 3(b) shows that RS KK Graviton production are carried out at the LHC by ATLAS particles are produced at much higher values than ZKK and −1 and CMS Collaborations and the latest data on the lower limit WKK bosons in TeV extra dimensional model. −1 on graviton mass are presented in [15]. Te production of Within TeV model with parameters �=10and ∗ narrow graviton resonances in the TeV range at the LHC as � =2–10TeV,wecalculatedtheproductioncrosssections wellasthedecaysintofermionandbosonpairswasstudiedin at 20 TeV, 60 TeV, and 100 TeV at the center of mass energies this paper. For the discovering of graviton resonance, G, the asafunctionofKKmass,MZKK , presented in Figure 4. As the parton showering formalism was used, which agrees with the decay of ZKK to muon pair is the dominant one, we decided NLO matrix element calculations. Te partonic subprocesses to calculate, namely, this process of KK particle decay. are demonstrated in Figure 2. From Figure 4, we can see the sharp drop in the curve Te ADD graviton emission and virtual graviton for20TeVcomparedwithothercurvesat60TeVand exchange processes are described in [16]. Within model 100TeVatthecenterofmassenergies.Tisresultemphasizes with large extra dimensions (LED), the processes that could the most important result for the further searches of new give rise to new phenomena at LHC due to emission or physics at high energies. As the last two curves (60 TeV and exchange of gravitons were considered. Te implementation 100 TeV) are almost parallel, it is preferable to search for new 4 Advances in High Energy Physics

ExtraDimensionsLED: dijet processes ExtraDimensionsLED: monojet processes 0 20 40 60 80 100 120 20,000 40,000 60,000 80,000 100,000 1e +11 1e +11

10,000 10,000

+10 +10 (pb) 1e 1e (pb) 100 100  

1 1 1e +09 1e +09 0 20 40 60 80 100 120 20,000 40,000 60,000 80,000 100,000

E＝Ｇ (TeV) E＝Ｇ (TeV) ∗ ∗ gg → (LED_＇ ) → gg fbar → (LED_＇ ) → gammagamma ∗ ∗ qg → (LED_＇ ) → qg gg → (LED_＇ ) → gammagamma  ∗  ∗ qq(＜；Ｌ) → (LED_＇ ) → qq(＜；Ｌ) fbar → (LED_＇ ) →ll ∗ gg → (LED_＇ ) →ll (a) ∗ ExtraDimensions ＇ processes NewGaugeBoson processes 0 20 40 60 80 100 120 20 40 60 80 100 1,000 1,000 12 12

100 100 10 10 8 8 10 10 6 6 (pb) (pb)   1 1 4 4

0,1 0,1 2 2

0,01 0,01 0 0 02040 60 80 100 120 20 40 60 80 100

E＝Ｇ (TeV) E＝Ｇ (TeV) ∗  qg →＇ q f＜；Ｌ → W+− ∗ ∗ gg →＇ g fbar →Ａ；ＧＧ； /Z0/Zprime0 ∗ gg →＇ (b)

Figure 3: Te production cross section at the center of mass energies varying from 14 to 100 TeV for (a) LED model: lef panel, dijet fnal −1 state; right panel, monojet fnal state; (b) lef panel, RS model; right panel, TeV model.

1,000 2,000 3,000 4,000 5,000 6,000 10 10

1 1

0,1 0,1 (pb)

 0,01 0,01

0,001 0,001

0,0001 0,0001 1,000 2,000 3,000 4,000 5,000 6,000

M：ＥＥ (TeV) 100 TeV f far → mu+ mu− 60 TeV f far → mu+ mu− 20 TeV f far → mu+ mu−

−1 Figure 4: Production cross sections at 20 TeV, 60 TeV, and 100 TeV at the center of mass energies as a function of KK mass for TeV model. Advances in High Energy Physics 5

dN/m: graviton mass spectrum dN/m: graviton mass spectrum dN_m dN_m 250 Entries 10000 240 Entries 10000 Mean 3295 Mean 3354 Std Dev 1203 220 Std Dev 1223 200 200 180 160 150 140 120 100 100 80 60 50 40 20 0 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 (a) (b) dN/m: graviton mass spectrum dN_m 220 Entries 10000 Mean 4103 200 Std Dev 1296 180 160 140 120 100 80 60 40 20 0 0 1000 2000 3000 4000 5000 6000 7000 (c)

Figure 5: Graviton mass spectrum within monojet LED model for three graviton, G, emission processes: (a) gg →��,(b)qg→��,and(c) ��→��atthecenterofmassenergiesof14TeV.

phenomena at energies up to 30 TeV, when the production As LED model depends on the number of extra dimen- cross sections of KK particles can be varied in the wide sions, �,itwasimportanttostudythegravitonmassspectrum range. distributions for �=2, 4, and 6. Te results of our calculations of G jet emission process, gg →��,atthecenter 3.2. Te Graviton Mass Spectrum at 14 TeV at the LHC. We of mass energies of 14 TeV is presented in Figure 6. will apply large-extra-dimensional (ADD-type) models in From Figure 6, substantial dependence of G jet emission production processes for virtual extra dimensional scalars process in the LED model on the number of extra dimensions of graviscalar type. With the help of Pythia8.2, it is possible is seen. Moreover, we can see clear periodicity of dependence, to generate monojet events from scalar graviton emission as when peaks are shifed by 1 TeV with an increase of the described in [17]. number of extra dimensions by 2. According to the hierarchy Te group of lowest-order G jet emission processes within formula of ADD model, this shifing is connected with the monojet model was considered with the following parame- change in the radius of compactifcation, �.As� increases, � ters: ExtraDimensionsLED: � = 6; ExtraDimensionsLED: � must decrease. Terefore, the mass shif to the right of 1 TeV = 10000. Tree graviton, G, mass spectra for G jet emission indicates the decreasing of compactifcation radius. processes, gg →��(gluon-gluon fusion (gg) with emission of graviton, G, and gluon, g), qg →��(quark-gluon fusion 3.3. Te Production Cross Section of Graviton Emission Mul- (qg) with emission of graviton, G, and quark, q), and ��→ tiplied by Branching Ratio in RS Model. As is known, the �� (quark-antiquark fusion (��) with emission of graviton, discovery of a Higgs boson, ℎ,attheLHCmotivatesthe G, and gluon, g) at the center of mass energies of 14 TeV at searches for physics beyond SM in channels involving Higgs theLHC,arepresentedinFigure5. boson. Higgs pair production is predicted by RS model with From Figure 5, the peak of the graviton mass spectrum KK graviton, GKK, emission that may decay to a pair of distribution is viewed depending on the process of monojet Higgs bosons. Such experimental searches were performed by emission. Although this dependence is insignifcant, never- ATLAS Collaboration, [20] at 13 TeV, presented in Figure 7. theless, for the process ��→��of monojet graviton, G, From [12], branching fractions of graviton decay to SM emission, it is shifed by almost 1 TeV. particles are taken, presented in Figure 8. Te predictions 6 Advances in High Energy Physics

dN/m: graviton mass spectrum dN/m: graviton mass spectrum dN_m dN_m Entries 10000 Entries 10000 450 Mean 1148 Mean 2290 Std Dev 811.6 250 Std Dev 1100 400 350 200 300 250 150 200 100 150 100 50 50 0 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 (a) (b)

dN/m: graviton mass spectrum dN_m 250 Entries 10000 Mean 3295 Std Dev 1203 200

150

100

0 0 1000 2000 3000 4000 5000 6000 7000 (c)

Figure 6: Graviton mass spectrum within monojet LED model for G jet emission process gg →��for (a) �=2,(b)�=4,and(c)�=6at thecenterofmassenergiesof14TeV.

104 ATLAS Preliminary −1 s=13TeV, 13.3 ＠＜ ) (f) 103

2 10 Resolved Boosted →ＢＢ→＜＜＜＜＋＋ ∗

10 ( ＪＪ → ＇ 

1 500 1000 1500 2000 2500 3000

m ∗ ＇＋＋ (GeV) k/M = 1.0 Bulk RS, ０l ％ＲＪ？＝Ｎ？＞ ± 1 Observed Limit (95% CL) ％ＲＪ？＝Ｎ？＞ ± 2 Expected Limit (95% CL)

Figure 7: Te expected and observed upper limit for pp →�KK →ℎℎ→�� in the bulk RS model with �/� = 1 at the 95% confdence level. Advances in High Energy Physics 7

0.100

BR 0.010

0.001

4 500 1000 5000 10 MGr (GeV) RS1 gg WW qq  ZZ tt ll hh

Figure 8: Branching fractions of graviton, G (from [12]).

for decay were updated using state-of-the-art computation character but also of experimental one, which is confrmed tools with the highest branching fraction to dijet fnal by modern experiments on the Higgs boson. Our work is states. dedicated to the studying of the properties of the new par- We will consider three processes of graviton decay, G → ticles predicted by the theories of extra dimensions. Within −1 ��,G →��,andG →ℎℎ,forfurther�×��calculations three models, ADD, RS, and TeV ,wehavecalculated within RS scenario for graviton production process gg → the production cross sections of massive graviton formation � at 13TeV, 14TeV, and 100TeV at the center of mass as well as the production of KK modes of gauge bosons energies. In Figure 9, our calculations performed with the depending on the energies of the modern and future colliders. help of Pythia8.2 are presented, with parameters �/� =1and Within LED model, the behavior of graviton mass spectrum 0.1. for G jet emission processes for diferent numbers of extra From Figure 9, we see the dependence of the �×�� dimension (�=2, 4, and 6) was studied and clear periodicity calculations on the parameter �/� and on the energy at of peaks shifed by 1 TeV was seen with an increase in the center of mass. Comparison of Figure 9(a) and 9(b) the number of extra dimensions by 2. Te graviton mass shows that the resonance peak shifs from 5TeV to 7TeV spectrum for three graviton, G, emission processes was also with increasing of energy at the collider. Tis mass shif of investigated: (a) gg →��,(b)qg →��, and (c) ��→ 2 TeV for case (b) indicates decreasing of the compactifcation �� at 14 TeV at the LHC. Te experimental searches for radius of RS model according to hierarchy formula. In KK graviton emission and decay to a pair of Higgs bosons, the case of Figure 9(c), we can see that there is no peak performed by ATLAS Collaboration at 13TeV, stimulated us at 100 TeV and that the cross section for the formation to perform calculations at diferent parameters and energies of the resonance as a function of energy is observed to within RS model. Our calculations of �×��shows that the decrease. resonance peak shifs from 5 TeV to 7 TeV with increasing of energy at the colliders from 13 TeV to 14 TeV as well as the 4. Conclusion absence of peak at energy of 100 TeV at the center of mass energies. Te modern high energy physics is connected with experimental searches of new physics beyond the SM. Tese Conflicts of Interest searches are connected not only with new possibilities of modern accelerating technics but also with problems of Te authors declare that there are no conficts of interest SM physics. Te SM problems are not only of theoretical regarding the publication of this paper. 8 Advances in High Energy Physics

RS-model. E＝Ｇ = 13 TeV. =1 RS-model. E＝Ｇ = 14 TeV.  = 0.1 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 100,000 100,000 1 1 1 1 1e −05 1e −05 1e −05 1e −05 1e −10 1e −10 1e −10 1e −10 (pb) (pb) 1e −15 1e −15 1e −15 1e −15 −20 −20 ×＂Ｌ 1e 1e ×＂Ｌ 1e −20 1e −20 1e −25 1e −25 1e −25 1e −25 1e −30 1e −30 1e −35 1e −35 1e −30 1e −30 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000

M ∗ M＇∗ ＇ ∗ ∗ gg2＇ hh gg2＇ hh ∗ ∗ gg2＇ ll gg2＇ ll ∗ ∗ gg2＇ gg gg2＇ gg (a) (b) RS-model. E＝Ｇ = 100 TeV 0 10,000 20,000 30,000 40,000 50,000 100 100

1 1

0,01 0,01

0,0001 0,0001 (pb)

1e −06 1e −06 ×＂Ｌ 1e −08 1e −08

1e −10 1e −10

1e −12 1e −12 0 10,000 20,000 30,000 40,000 50,000

M＇∗ ∗ gg2＇ hh ∗ gg2＇ ll ∗ gg2＇ gg (c)

Figure 9: �×��for graviton production and decay as the function of graviton mass, ��,at(a)13TeV,�/� =1;(b)14TeV,�/� = 0.1; and (c) 100 TeV, �/� = 0.1.

References [6] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “Phenom- enology, astrophysics and cosmology of theories with sub- [1] Y. A. Kubyshin, Models with Extra Dimensions and Teir millimeter dimensions and TeV scale quantum gravity,” Physical Phenomenology,2001. Review D,vol.59,article086004,1999. [2] T.Kaluza, “Zum Unitatsproblem in der Physik,” Sitzungsberichte [7] L. Randall and R. Sundrum, “A large mass hierarchy from a der Koniglich Preußischen Akademie der Wissenschafen (Math- small extra dimension,” Physical Review Letters,vol.83,no.17, ematical Physics), pp. 966–972, 1921. pp. 3370–3373, 1999. [3] O. Klein, “Quantentheorie und funfdimensionale¨ Relativitats-¨ [8] L. Randall and R. Sundrum, “An alternative to compactifca- theorie,” Zeitschrif fur¨ Physik,vol.37,no.12,pp.895–906,1926. tion,” Physical Review Letters,vol.83,no.23,pp.4690–4693, [4] M. D. Maia, “Te Poincare conjecture and the cosmological 1999. constant,” International Journal of Modern Physics: Conference [9] S. Maruyama, for the ATLAS, and CMS Collaborations, Series, vol. 03, pp. 195–202, 2011. “Searches for Exotic Phenomena at ATLAS and CMS,” High [5] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “Te hierarchy Energy Physics - Experiment,2014. problem and new dimensions at a millimeter,” Physics Letters B, [10] I. Antoniadis, “A possible new dimension at a few TeV,” Physics vol. 429, no. 3-4, pp. 263–272, 1998. Letters B,vol.246,no.3-4,pp.377–384,1990. Advances in High Energy Physics 9

[11] T. Sjostrand,¨ S. Ask, J. R. Christiansen et al., “An introduction to PYTHIA 8.2,” Computer Physics Communications,vol.191,no.1, pp.159–177,2015. [12] A. Carvalho, Gravity Particles from Warped Extra Dimensions, Predictions for LHC,2017. [13] P.Hoˇrava and E. Witten, “Heterotic and Type I String Dynamics from Eleven Dimensions,” Nuclear Physics B,vol.460,no.3,pp. 506–524, 1996. [14] J. Bijnens, P. Eerola, M. Maul, A. Mansson,˚ and T. Sjostrand,¨ “QCD signatures of narrow graviton resonances in hadron colliders,” Physics Letters B,vol.503,no.3-4,pp.341–348,2001. [15] Te ATLAS Collaboration, “Search for diboson resonances with boson-tagged jets in pp collisions at s = 13 TeV with the ATLAS detector,” Physics Letters B,vol.777,no.91,2017. [16] S. Ask, I. V. Akin, L. Benucci, A. De Roeck, M. Goebel, and J. Haller, “Real emission and virtual exchange of gravitons and unparticles in Pythia8,” Computer Physics Communications,vol. 181, no. 9, pp. 1593–1604, 2010. [17] G. Azuelos, P.-H.Beauchemin, and C. P.Burgess, “Phenomeno- logical constraints on extra-dimensional scalars,” Journal of Physics G: Nuclear and Particle Physics,vol.31,pp.1–20,2005. [18] G. Bella, E. Etzion, N. Hod, and M. Sutton, “Introduction to the MCnet Moses project and Heavy gauge bosons search at the LHC,” High Energy Physics - Experiment,2010. [19] G. Bella, E. Etzion, N. Hod, Y. Oz, Y. Silver, and M. Sutton, “A search for heavy Kaluza-Klein electroweak gauge bosons at the LHC,” Journal of High Energy Physics,vol.2010,no.9,articleno. 025, 2010. [20] Te ATLAS Collaboration, “Search for pair production of Higgs bosons in the bbbb fnal state using proton-proton collisions at s =13 TeV with the ATLAS detector,” ATLAS-CONF-049,2016. Hindawi Advances in High Energy Physics Volume 2018, Article ID 4653202, 5 pages https://doi.org/10.1155/2018/4653202

Research Article Electroweak Phase Transitions in Einstein’s Static Universe

M. Gogberashvili 1,2

1 Javakhishvili Tbilisi State University, 3 Chavchavadze Avenue, 0179 Tbilisi, Georgia 2Andronikashvili Institute of Physics, 6 Tamarashvili Street, 0177 Tbilisi, Georgia

Correspondence should be addressed to M. Gogberashvili; [email protected]

Received 11 November 2017; Accepted 14 March 2018; Published 19 April 2018

Academic Editor: Marek Szydlowski

Copyright © 2018 M. Gogberashvili. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

We suggest using Einstein’s static universe metric for the metastable state afer reheating, instead of the Friedman-Robertson- Walker spacetime. In this case, strong static gravitational potential leads to the efective reduction of the Higgs vacuum expectation value, which is found to be compatible with the Standard Model frst-order electroweak phase transition conditions. Gravity could also increase the CP-violating efects for particles that cross the new phase bubble walls and thus is able to lead to the successful electroweak baryogenesis scenario.

According to standard cosmology at the electroweak scale, leads to the very small value for the Higgs self-coupling param- our universe is in radiation dominated phase where all Stan- eter, dard Model particles are massless [1]. Once the temperature drops below the critical value, � ∼ 170 GeV, the electroweak � ≈ 2� ≈ 0.02, (3) phase transition has occurred and the Higgs boson, gauge bosons, and fermions (except neutrinos) acquire masses which according to (1) is incompatible with the observed through the Higgs mechanism. Te order of this phase Higgs boson mass, transition depends on the details of the Higgs potential with � = √2�V ≃ 125 . the temperature dependent terms [2–5]. To have a frst-order � GeV (4) phase transition efective Higgs potential of the model should have several minima. Of special importance is the cubic term Tis means that within the minimal Standard Model the elec- in efective potential, which is essential to generate a potential troweak phase transition is a smooth crossover, or of the barrier between the symmetric and broken phases and thus second order [4, 5]. can provide the phase transition to be of the frst order. In On the other hand, the frst-order electroweak phase tran- 3 the Standard Model the cubic term, �� , is contributed only sitions may solve some cosmological problems, like the gen- by the electroweak gauge bosons. If at zero temperature the eration of the baryon asymmetry of the universe (see recent Higgsfeldattheminimumofthepotentialhasthevalue reviews [6, 7]). A frst-order cosmological phase transition proceeds through the formation and expansion of cosmic V ≈ 246 GeV, (1) bubbles. In this scenario spacetime is separated into two manifolds with their own distinct metrics, which are typically � the parameter isthecubictermoftheefectivepotentialof joined across a thin wall (domain wall). Te dynamics of such order of objects can be very complicated, depending on the matter 2�3 +�3 content of the interior (true vacuum) and exterior (false �≈ � � ≈ 0.01, (2) 4�V3 vacuum)regionsaswellasthetensiononthebubblewallsand how they interact with the surrounding plasma [8]. In bubble where �� and �� are the gauge bosons masses. Ten the models matter-antimatter asymmetry can be generated at condition that the Higgs efective potential has two minima the electroweak scale, because all three Sakharov conditions 2 Advances in High Energy Physics

(baryon number violation, � and CP-violation, and departure cosmological models usually is considered only as the initial from the thermal equilibrium) are fulflled. Baryon produc- state for the infationary phase [20]. tionisabubblesurfaceefectbutcanbeusedtoexplain In our opinion, the short metastable period with �=̈ �=̇ the observed matter-antimatter asymmetry, since the entire 0, in course of transition of the infation (�>0̈ )intothedecel- universe is swept out during the frst-order phase transition eration stage (�<0̈ ), that is, during or afer the reheating andwemayliveinonebubbletoday[9,10].However,intra- epoch, when our universe was a spherule with the homoge- ditional approach not only is the electroweak phase transition nous cosmic fuid of all kinds of ultrarelativistic particles, is not of frst order, but the CP-violation from the Cabibbo- natural to be described by Einstein’s static solution (5), with Kobayashi-Maskawa matrix is too small, and to obtain the the later transition to the nonstationary radiation dominated observed baryon asymmetry various extensions of the Stan- phase with the Friedmann-Robertson-Walker metric. Tis dard Model have been proposed [4–7]. canbeachievedbyintroducingadarkenergylikesubstance, We want to emphasize that the conventional scenario of presence of which at early times is a quite generic feature of electroweak baryogenesis does not take into account grav- several dynamical dark energy models [21, 22], for example, itationalefects.Itisassumedthatattheelectroweakscale �(�) modifcation of gravity [23, 24], or the decaying scalar the universe was radiation dominated (flled with relativis- feld (quintessence) [25], rather than a large fxed value classi- tic particles in thermal equilibrium) and the Friedmann- cal cosmological constant. One can also assume that the high- Robertson-Walker scale factor was unimportant for the par- er-energy degrees of freedom of the quantum vacuum during ticle reactions [1]. However, details of the anticipated periods reheating do not cancel the contribution of the zero-point (infation and reheating) and conditions at the moment of motion of the quantum felds and the nullifcation of vacuum transition to the radiation dominated phase, with the normal energy in the equilibrium vacuum is not acquired at this expansion of spacetime, are still poorly understood. For stage. instance, if one assumes the existence of black holes in the Introduction of Einstein’s static metric (5) for our spher- early universe, the gravitational efects in their static felds are ule-universe can radically change the common view that for able to modify parameters of the efective Higgs potential and the observed large Higgs mass the cosmological electroweak lead to the frst-order phase transitions [11–14]. phase transition is of the second order [4, 5]. In a static island It is known that the Cosmological Principle for the ofspacewecanexpectappearanceoflargegravitational uniform matter distribution not only leads to the Friedmann- potential in the Standard Model Lagrangian, similar to the Robertson-Walker nonstatic solution, but gives the Einstein’s case with a static black hole background [11–14]. Indeed, static universe as well [15]. Einstein’s static universe refers introducing the density function, Ω=�/��,where to the homogenous and isotropic universe with the positive 3 � = , cosmological constant and positive spatial curvature. In the � 2 (9) framework of General Relativity this model has been widely 8�� investigated for several kinds of matter sources (see [16, 17] is the critical density parameter and � is the horizon distance and references therein). Te metric of Einstein’s universe can at the electroweak scale, and the gravitational potential (6) be written in the form: can be written in the form: 2 2 2 2 �� 2 2 2 2 � �� =�� − −� (�� + sin �� ), (5) Φ (�) =Ω . (10) 1−Φ(�) �2 where A preferred location in the universe is absent and for uniform 8� matter distribution we expect the existence of a constant aver- Φ (�) = ��2 ⟨Φ(�)⟩ 3 (6) age gravitational potential, , and the radial dependence will disappear. Te total matter density of our universe at all denotes the gravitational potential and � is the uniform stagesofitsevolutionisassumedtobeclosetounity[26], � =1 cosmic fuid density. In (5), it has been set �� , since in Ω ≲ 1.005. cosmological case there must be a universal proper time for (11) all fundamental observers. Introducing the radial coordinate Tentheaveragegravitationalpotential(10)intheEinstein transformation, static universe (5) is estimated to reach the value: � �= , 1 3Ω � 2 (7) ⟨Φ (�)⟩ = ∫ ��Φ (�) = ∫ ��4 ≈0.6. 1+2�� /3 5 (12) � � 0 the metric (5) can be written also in terms of Cartesian coor- Tus the factor which would multiply spatial components of dinates [18, 19]: the metric in matter Lagrangian will be of the order of 2 2 1 2 2 2 �� =�� − (�� +�� +�� ). 2 1 2 (8) � = ≈ 2.5. (1 + 2��2/3) 1−⟨Φ (�)⟩ (13)

It was found that Einstein’s static universe (5) is critically Unlike the case of Friedmann-Robertson-Walker scale factor, unstable to gravitational collapse or expansion and in modern one cannot hide the function � in the defnitions of the spatial Advances in High Energy Physics 3 coordinates, which are already set by the assumption to have were investigated in the infnite universe at the one-loop level the asymptotical Minkowski (or de Sitter) metric. [33, 34]. In the case of static, isotropic metrics (5), one can apply Since a solution of the Higgs equation in the metric ��] the well-established efective potential technique to particle is the solution of the similar equation with an efective mass models. Te optical-mechanical analogy leads to a remark- (for simplicity, we consider minimal coupling of the Higgs able simplifcation of the equations of motion of particles and feld to gravity) in the conformal metric, then under the general-relativistic problems become formally identical to the transformation (15) the Standard Model Lagrangian obtains classical ones with the efective index of refraction [27, 28], theordinaryform,butwiththemodifedHiggsmass, 1 �� 1 � �→ � ≈0.6� . � (�) = √ = ≈ 0.6. (14) � � � � (17) �� Tis means that in the early universe at the electroweak scale In addition to the gravitational refraction index (14), in the (in the symmetric phase) the efective vacuum expectation Standard Model Lagrangian one can take into account the value (1) probably was smaller than in the present broken constant gravitational factor (13) by introduction of the new phase, time parameter, �→��, and conducting the conformal transformation of the Minkowski metric, V �→ 0.6V ≈ 156 GeV. (18) 1 � �→ � , Under the conformal rescaling (15), other parameters of �] �2 �] (15) the Standard Model are unchanged (including gauge boson 1 masses), and in the perturbative analysis on the static back- √−� �→ √−�. �4 ground the modifcation of the vacuum expectation value (18) leads to the alteration of the parameter (2), Itisknownthatitispossibletobringtheconformallyequiv- � alentmatterLagrangiantotheMinkowskianformbythere- ��→ ≈ 0.05. (19) scaling, 0.2 �] �→ � ], Ten from the condition to have the frst-order electroweak phase transition in the early universe (3), we obtain the ��→��, (16) acceptable value for the Higgs self-coupling constant, 3/2 ��→� �, � ≃ 0.1. (20) ofthegauge,scalar,andspinorfelds,respectively[29,30]. Tis means that for the minimal Standard Model in the Ein- In general, the gravitational feld afects strongly the stein static universe background (5), the electroweak phase symmetry behaviors of all quantum-feld models, including transitions can be of the frst order. Afer the expansion of new scalar feld Lagrangian [29, 30]. Currently, we do not have phase bubbles and the passage to the Friedman-Robertson- a standard theory of massive scalar bosons in curved space- Walker expansion of spacetime in the broken phase, the time, and several models exist with the minimal or conformal gravitational potential in the universe (6) will tend to zero couplings with curvature. Nonminimal couplings usually are and all parameters of the Standard Model will get the present supposed in infation models [31]. Investigations of the min- values. imal case are important as well, since massive vector mesons Note that, together with the allowance of electroweak and gravitons satisfy equations of this type [29], and also for phase transition to be of the frst order, gravitational efects nonminimal couplings in general it is impossible to conserve are able to solve another problem of the Standard Model conformal invariance not only of the efective action (the baryogenesis—the smallness of CP-violating parameters. conformal anomaly) but of the action itself [30]. It is known that, in general, gravitation can induce CP- To explore properties of cosmological phase transitions in violation processes [35–44]. Since antimatter can be inter- the presence of static external gravitational feld, one should preted as an ordinary matter propagating backward in time, evaluate the expectation value of the Higgs feld over the for nonstationary spacetime (like an expanding new phase lowest energy state. To fnd an energy spectrum it is impor- bubble) the time and thus CP-violation could be accrued in tant construction of the Hamiltonian of the system, which a CPT preserving framework [45]. To explore gravitational in general is difcult problem in the presence of gravitational CP-violations for static backgrounds as well, one can use the feld. However, in stationary conformal metric there exist fact that some quantum mechanical efects depend upon the some eligible models, even for massive scalar felds [32]. It was gravitational potential themselves, not only to its gradient. found that, in most situations, in regions where the conformal Equations of quantum particles with a gravitational interac- factor is almost constant, the conformal transformations tion terms contain inertial and gravitational mass separately. fnally amount to rescaling of a scalar boson mass in this If within a model inertial and gravitational masses are not region [29, 30, 32]. Our situation with the static gravitational equal then in fermions wavefunctions there can appear the feld in some fnite region of bulk spacetime difers with the mass dependent gravitationally induced CP-violating phases cases where the parameters of cosmological phase transitions [46, 47]. 4 Advances in High Energy Physics

It is known that topological defects could violate the Week bubble walls, which are appearing in electroweak baryogene- Equivalence Principle and exhibit nontrivial gravitational sis scenarios. features [48]. Since it is impossible to surround any topological object by a spatial boundary, one cannot defne their Conflicts of Interest gravitational mass, �,bytheintegralfromthezero-zero � component of energy-momentum tensor, 00, but needs to Te author declares that there are no conficts of interest. use the Tolman formula, 0 1 2 3 Acknowledgments �=∫ √−�� (�0 −�1 −�2 −�3 ). (21) Te author would like to thank the University of Bonn for From this formula it follows that, in spite of having large ten- hospitality. Tis research partially is supported by Volkswa- sions, cosmic strings do not produce any gravitational force genstifung (Contract no. 86260). on the surrounding matter locally, while global monopoles, global strings, and planar domain walls exhibited repulsive nature [48]. 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Research Article Moving Unstable Particles and Special Relativity

Eugene V. Stefanovich

1763 Braddock Court, San Jose, CA 95125, USA

Correspondence should be addressed to Eugene V. Stefanovich; [email protected]

Received 5 December 2017; Accepted 12 March 2018; Published 19 April 2018

Academic Editor: Krzysztof Urbanowski

Copyright © 2018 Eugene V. Stefanovich. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

In Poincare-Wigner-Dirac´ theory of relativistic interactions, boosts are dynamical. Tis means that, just like time translations, boost transformations have a nontrivial efect on internal variables of interacting systems. In this respect, boosts are diferent from space translations and rotations, whose actions are always universal, trivial, and interaction-independent. Applying this theory to unstable particles viewed from a moving reference frame, we prove that the decay probability cannot be invariant with respect to boosts. Diferent moving observers may see diferent internal compositions of the same unstable particle. Unfortunately, this efect is too small to be noticeable in modern experiments.

1. Introduction However, such a quantum clock as an unstable particle cannot be at rest (i.e., cannot have zero Time dilation is one of the most spectacular predictions velocity or zero momentum) and simultaneously of special relativity. Tis theory predicts that any time- be at a defnite point (due to the quantum uncer- dependent process slows down by the universal factor of tainty relation). So, the standard derivation of 2 2 1/√1−V /� ≡ cosh � when viewed from a reference frame the moving clock dilation is inapplicable for the moving with the speed V (and rapidity �). Te textbook quantum clock. Te related quantum-mechanical example of such a time-dependent process is the decay law derivation must contain some reservations and Υ(0, �) of an unstable particle at rest. Te function Υ(0, �) is corrections. Shirokov [10] the probability of fnding the unstable particle at time �,if it was prepared with 100% certainty at time �=0.Ten, Indeed, detailed quantum-mechanical calculations [8, according to special relativity, the decay law of a moving 10–12] suggest that (1) is not accurate, and that corrections particle should be exactly cosh � times slower: to this formula should be expected, especially at large times, exceeding multiple lifetimes. Although these corrections are too small to be observed in modern experiments, their �� Υ (�, �) =Υ(0, ). (1) presence casts doubt on the limits of applicability of Einstein’s cosh � special relativity. Unfortunately, the corrections to (1) were derived in [8, Indeed, this prediction was confrmed in numerous mea- 10–12] under certain assumptions and approximations. So, surements [1–4]. Te best accuracy of 0.1% was achieved in the question remains whether one can design a relativistic experiments with relativistic muons [5, 6]. model in which the decay slowdown will acquire exactly the However, the exact validity of (1) is still a subject of form (1) demanded by special relativity [13, 14]? controversy. One point of view [7–9] is that special-relativistic In order to answer this question we will analyze the status time dilation was derived in the framework of classical theory of interactions in special relativity from a more general point andmaynotbedirectlyapplicabletounstableparticles,which of view. We are going to prove that under no circumstances are fundamentally quantum systems without well-defned the decay law transforms with respect to boosts exactly as in masses, velocities, positions, and so on. (1). 2 Advances in High Energy Physics

Table 1: Inertial transformations. Transformation Type Parameter Generator Meaning of generator

Space translation Kinematical Distance aP= P0 Total momentum

Rotation Kinematical Angle � J = J0 Total angular momentum

Time translation Dynamical Time ��=�0 +� Total energy (Hamiltonian)

Boost Dynamical Rapidity � K = K0 + Z Boost operator

2. Materials and Methods Consider the light clock shown in Figure 1(a). It consists of two parallel mirrors and the light pulse refecting back and 2.1. Inertial Transformations. Te theory of relativity tries to forth between them. Te period of the clock at rest is equal connect views of diferent inertial observers. Te principle of to �=2�=2�/�. If the clock is moving, as in Figure 1(b), the � � relativity says that all such observers are equivalent; that is, distance traveled by the light pulse increases to � =2�� = two inertial observers performing the same experiment will 2√�2 +(V��)2. Solving this system of equations with respect obtain the same results. to the clock period, we obtain Tere are four classes of inertial transformations, space translations, time translations, rotations, and boosts, and � � � � =2� = =�cosh �. their actions on observed systems difer very much (see √1−V2/�2 (3) Table 1). For example, it is easy to describe the results of space translations and rotations. An observer displaced by So, the moving clock runs cosh � times slower than the clock the 3-vector a sees all atoms in the Universe simply shifed at rest. Tis is the time dilation efect that we used in (1). in the opposite direction −a. Tis shif is absolutely exact and Let us now consider the same clock oriented parallel to its universal. It applies to all systems, however complicated. Te velocity, as in Figure 2. Te clock’s rate should not depend on samecanbesaidaboutrotations.Onecanswitchtothepoint its orientation, so we already know the period of this clock in of view of the rotated observer by simply rotating all atoms motion (3). Taking into account the invariance of the speed of in the Universe. For example, rotation through the angle � light, this result can be achieved only if the distance between about the �-axis results in the transformation of coordinates the two mirrors decreases. Te corresponding system of � � =�cos �−�sin � equations is

� � � � =� �+� � (2) (� + V�1) (� − V�2) cos sin �� =� +� = + . (4) 1 2 � � �� =� � Solving with respect to � , we obtain the familiar length which is independent on the composition of the observed contraction formula: system and on its interactions. Due to this exact universality, we can regard space translations and rotations as purely V2 � geometrical or kinematical transformations. �� =�√1− = . (5) 2 Time translation is also an inertial transformation, � cosh � because repeating the same experiment at diferent times will not change the outcome. However, this transformation Formulas (3) and (5) already imply that no material object is by no means kinematical. Time evolutions of interacting can move faster than the speed of light. Otherwise, the factor systems can be very complicated. Teir description requires √1−V2/�2 wouldbecomeimaginary,whichisabsurd. intimate knowledge of the system’s composition, state, and We can also make a clock, in which, instead of the light interactions acting between system’s parts. We will say that pulse, we have a massive steel ball bouncing between the time translations are dynamical inertial transformations. two mirrors. Te ball’s speed � is less than �, and the speed Now, what about boosts? Are they kinematical or dynam- invariance postulate does not apply to �. Nevertheless, we ical? In nonrelativistic classical physics boosts are defnitely expect this clock to obey the same time dilation and length regarded as kinematical, they simply change velocities of contraction rules as derived above. Ten, for consistency, all atoms in the Universe. However, things become more we have to modify the classical velocity transformation law. complicated in relativistic physics, as we shall see below. For example, if the resting clock in Figure 2(a) had ball’s velocities ±�, then the moving clock in Figure 2(b) should 2.2. Boosts in Special Relativity. Description of boost trans- have velocities formations is the central subject of special relativity. Einstein �+V based his approach on the already mentioned relativity � = 1 2 (6) postulate and on his second postulate about the invariance 1+�V/� of the speed of light. It is remarkable how all results of −� + V special relativity can be derived from these two simple and � = . 2 2 (7) undeniable statements. 1−�V/� Advances in High Energy Physics 3

t

t l

(a) (b) Figure 1: Light clock: (a) at rest and (b) in motion perpendicular to the clock’s axis.

l 

t time time t2

(a) (b) Figure 2: Light clock: (a) at rest and (b) in motion parallel to the clock’s axis. Te time evolution is shown in three frames stacked up vertically.

Indeed, it is not difcult to verify that these values solve the 4-coordinates (�,�,�,�)in the rest frame will have other 4- system of equations coordinates � � � � =�cosh �−( ) sinh � �1�1 =� + V�1 � � � � =�cosh �−��sinh � −� � =� − V� (10) 2 2 2 (8) �� =� 2� � +� =( ) � � 1 2 � cosh � =� in the frame moving with velocity V =�tanh � along the �- that describe the movement of the ball during one clock axis. Te linear character and the exact universality of these period. formulas are similar to transformations of 3-coordinates As a consistency check, by applying the velocity addition under rotations (2). So, it is tempting to continue this analogy (� = �) law (6) to the photon moving with the speed of light , andtointroducetheideaofthe4DMinkowskispace-time, we return back to Einstein’s light speed invariance postulate whose points constitute physical events, and where boosts are represented by purely geometrical (kinematical) pseudo- �+V rotations. �1 = =�. (9) 1+V/� However, it is important to note that all the above derivations used model systems without interactions. Te Specialrelativityexplainshowalltheseparticularresults second Einstein postulate is formally applicable only to light can be generalized into Lorentz transformations for the pulsesandeventsassociatedwiththem.So,strictlyspeaking, times and positions of events. For example, any event having we are not allowed to extend results of special relativity 4 Advances in High Energy Physics

3 beyond corpuscular optics. In addition, one can show [15] [� ,� ]=�ℏ∑� � that Lorentz transformations (10) can be extended to special � � �� (14) �=1 events, such as intersections of particle trajectories, involving massive noninteracting particles, for example, our steel ball [��,��]=[��,�]=[��,�]=0 (15) and mirrors. (Of course, refections of light pulses or steel balls from the mirrors do involve interactions, but in our �ℏ 3 [� ,� ]=− ∑� � idealized thought experiments we can assume that these � � �2 �� (16) processes take negligibly short times.) �=1 How can we be confdent that the same conclusions apply �ℏ [� ,�]=− �� (17) to interacting systems? For example, what if the steel ball is � � �2 �� bouncing between plates of a charged capacitor? Can we be sure that Lorentz formulas (10) still apply? [��,�]=−�ℏ��. (18) Here we meet the following fork in the road. On the one hand, we can choose to postulate that the laws of 2.4. Hilbert Space of Unstable Particle. According to Wigner special relativity established above are valid independent of (�) and Weinberg [24, 26], the Hilbert space H of each stable interactions. Ten boosts should be rigorously kinematical, elementary particle carries an unitary irreducible represen- just as space translations and rotations. Tis nonobvious �(�) postulate is tacitly assumed in all textbooks. In particular, tation � of the Poincare´ group. Te Hilbert space of an � it was used in numerous attempts [16–22] to derive Lorentz -particle system is constructed as a tensor product (with transformations from the frst Einstein postulate only. proper (anti)symmetrization) of one-particle spaces

Alternatively, we can assume that, similar to time transla- � (1) (2) (�) tions, boosts are dynamical; that is, they involve interactions, H = H ⊗ H ⊗⋅⋅⋅⊗H . (19) and their actions cannot be expressed by simple universal formulas, like (10). We will discuss these two possibilities In the formalism with varied numbers of particles (e.g., in in Section 3. However, before doing that, in the rest of quantum feld theory), one builds the Fock space as a direct this section, we are going to recall the fundamentals of sum of spaces (19). For example, in a good approximation one � relativistic quantum theory pioneered by Wigner and Dirac. can describe the unstable particle with one decay channel �→�+� Tistheoryisbasedontheextremelyimportantfactthat , in the part of the Fock space, which includes the � �+� inertial transformations form the Poincaregroup.´ particle itself and its decay products (�) (�) (�) 2.3.RepresentationsofthePoincareGroupinQuantum´ H = H ⊕ (H ⊗ H ) . (20) Mechanics. Weareinterestedinapplicationofinertialtrans- formations to quantum systems. Properties of such systems Each normalized state vector |Ψ⟩ in the Hilbert space H are described by objects in the Hilbert space H,suchasstate canberepresentedasasumoftwoorthogonalcomponents vectors and Hermitian operators of observables. So, we have to defne the action (a.k.a. representation) of inertial transfor- |Ψ⟩ = |Ψ⟩� + |Ψ⟩�� , (21) mations in H. Operators of this representation �� must preserve quantum-mechanical probabilities, so these operators where the component have to be unitary [23]. Tis brings us to the classical math- (�) ematical problem of constructing unitary representations of |Ψ⟩� ≡�|Ψ⟩ ∈ H (22) the Poincare´ group in the given Hilbert space H [24]. Important role is played by the so-called “infnitesimal lies entirely in the subspace of the unstable particle, while the transformations” or generators. Tey are represented by Her- other component mitian operators in H (see Table 1). Unitary representatives (�) (�) �� of fnite transformations can be expressed by exponential |Ψ⟩�� ≡ (1−�) |Ψ⟩ ∈ H ⊗ H (23) functions of the Hermitian generators. For example, a general inertial transformation � is a composition of (boost �) × is in the subspace of decay products. Here we denoted by � (�) (rotation �) × (space translation a) × (time translation �). It is the Hermitian projection on the subspace H represented by the following product of unitary exponents: (�) (�) −(��/ℏ)K⋅� −(�/ℏ)J⋅� −(�/ℏ)P⋅a (�/ℏ)�� H = H . (24) �� =� � � � . (11) According to basic rules of quantum mechanics, the norms of Commutators of the Hermitian generators are fully deter- these two components have simple physical interpretations: minedbythestructureofthePoincaregroup[25,26]´ 3 � �2 Υ≡�|Ψ⟩�� (25) [��,��]=�ℏ∑�� (12) �=1 � 3 is the probability of fnding the unstable particle in the state |Ψ⟩ ‖|Ψ⟩ ‖2 =1−‖|Ψ⟩‖2 [��,��]=�ℏ∑�� (13) ,and �� is the probability of fnding �=1 its decay products. Advances in High Energy Physics 5

Observationsoftheunstableparticlecanbealso kinematical/dynamical character of inertial transformations described in the quantum-logical language of yes-no ques- from Section 2.1, we can immediately conclude that gener- tions,like“Doweseetheunstableparticle?”andanobserv- ators of space translations and rotations coincide with their able, which can take two values 1 or 0, corresponding to noninteracting counterparts the possible answers “yes” or “no.” Obviously, the Hermitian P = P operator of this observable is the projection � introduced 0 (32) (�) (�) (�) above. Its eigensubspaces H and H ⊗ H correspond J = J , to the eigenvalues 1 and 0, respectively. Ten the probability 0 (33) of fnding the particle � can be written as the expectation value of the projection �.Takingintoaccounttheproperty while the generator of time translations contains a nontrivial 2 interaction term � � =�, we see that this defnition is in full agreement with (25) �=�0 +� (34) 2 � �2 ⟨Ψ| � |Ψ⟩ = (⟨Ψ| �)(� |Ψ⟩) = ‖� |Ψ⟩‖ = �|Ψ⟩�� (see Table 1). It is important to note that the Hermitian (26) projection � cannot commute with this interaction and with =Υ. the total Hamiltonian �

[�, �] = [�, �] =0.̸ (35) 2.5. Interacting Representation of the PoincareGroup.´ Next we should fnd out how the quantity Υ depends on the Indeed,onlyinthiscase,thedecaylawisanontrivialfunction observer.Toobtaintheformulaforthedecaylaw(i.e.,the of time time evolution of the probability Υ), we can use either the Υ (0, 0) = ⟨Ψ| � |Ψ⟩ =1 Schrodinger¨ representation, where state vectors depend on time −(�/ℏ)�� |Ψ⟩ ∉ H�, �=0 ̸ (�/ℏ)�� −(�/ℏ)�� if (36) Υ (0, �) =(⟨Ψ| � )�(� |Ψ⟩), (27) � � � (�/ℏ)�� −(�/ℏ)�� Υ (0, � > 0) =⟨Ψ�� Ψ⟩ < 1 or, equivalently, the Heisenberg picture, where observables are time-dependent as required for any unstable particle prepared at time �=0. � � APoincare-Wigner-Dirac´ relativistic quantum descrip- � (�/ℏ)�� −(�/ℏ)�� Υ (0, �) = ⟨Ψ �(� �� )� Ψ⟩ . (28) tion of any isolated interacting system is constructed in a similar manner. In the Hilbert space H of the system one To perform these calculations, we have to specify the Hamil- defnes 10 Hermitian generators {P0, J0, K0 + Z,�0 +�}with � H tonian in the Hilbert space . In order to keep the commutators (12)–(18). Tese operators not only specify the relativistic invariance, this Hamiltonian should be consistent basic total observables of the system, but also determine with other Poincare´ generators: that is, commutation rela- how the results of observations transform from one inertial tions (12)–(18) have to be satisfed. system to another. Moreover, one can switch to the classical Using available 1-particle irreducible representations relativistic description by taking the limit ℏ→0and �(�),�(�),�(�) � � � , one can easily construct one valid represen- considering only states describable by localized quasiclassical tation of the Poincaregroupin´ H wave packets, which can be approximated by points in the phase space. In this limit, observables are replaced by 0 (�) (�) (�) �� ≡�� ⊕ (�� ⊗�� ) . (29) realfunctionsonthephasespace,quantumcommutators are represented by Poisson brackets, and time evolution is It is appropriate to call this representation “noninteracting,” approximated by trajectories in the phase space [27, 28]. because its generators {P0, J0, K0,�0} take the forms corresponding to free particles. Apparently, in this case, the 3. Results and Discussion subspace H� remains invariant with respect to all inertial transformations. In particular, noninteracting translation 3.1. Kinematical Boosts. As we mentioned at the end of generators commute with the projection � Section 2.2, Einstein’s special relativity assumes that boost transformations can be represented by exact Lorentz for- [�, �0] =0 (30) mulas (10), which are valid universally for all events and physical systems, independent on their state, composition, [�, P0] =0. (31) and involved interactions. In other words, in special relativity boosts are kinematical. According to Dirac and Weinberg [25, 26], one can In classical relativistic physics, this hypothesis is known introduce relativistic interaction by defning in H anew 0 as the condition of “invariant trajectories” or “manifest unitary representation �� =�̸ � of the Poincaregroupwith´ covariance.” Te well-known Currie-Jordan-Sudarshan the- generators {P, J, K,�}. Referring to our understanding of the orem [29] states that this condition is not compatible with 6 Advances in High Energy Physics the Hamiltonian description of dynamics presented in the must depend on interactions. Indeed, if we assume that previous section. In other words, a Poincare-invariant´ theory boosts are kinematical (Z =0), then we obtain from (17) with invariant trajectories can exist only in the absence of ��2 ��2 interactions. Tis explains the name “no-interaction the- �= [� ,� ] = [(� ) , (� ) ] =� (37) orem” ofen used for the Currie-Jordan-Sudarshan result. ℏ � � ℏ 0 � 0 � 0 Several options were tried in the literature for explaining this paradox. the absurd proposition that interaction in the Hamiltonian must vanish (�≡�−�0 =0). OneideawasthatHamiltoniandynamicsisnotsuitable �=0 ̸ Z =0̸ for describing relativistic interactions. Instead, various non- Terefore, we should have , ,which Hamiltonian theories were developed [30–35], which devi- means that we are working in the instant form of dynamics, according to Dirac’s classifcation [25]. ated from the Poincare-invariant´ Wigner-Dirac approach. So far, the predictive power of these theories remains rather limited. 3.3. Decays Caused by Boosts. Our conclusion about the Another idea is to abandon particles and replace them by dynamical character of boosts disagrees with the usual (quantum) felds [36–39], because “there are no particles, there special-relativistic “geometrical” view on boosts. In particu- are only felds” [40]. Tis approach goes as far as claiming lar, we can no longer claim that that there is no point in discussing observables (positions and Any event that is “seen” in one inertial system is momenta) of interacting particles, their wave functions, and “seen” in all others. For example if observer in one also their time evolutions in the interacting regime. system “sees” an explosion on a rocket then so do Te foregoing discussion suggests that the the- all other observers. Polishchuk [21] ory will not consider the time dependence of Returning to our example of unstable particle, we can say particle interaction processes. It will show that that when the observer at rest sees the pure unstable particle in these processes there are no characteristics �, moving observers may see also its decay products �+�with precisely defnable (even within the usual limi- some probability. We can prove an even stronger statement: tationsofquantummechanics);thedescription if all (both resting and moving with diferent rapidities �) of such a process as occurring in the course of observers see the particle � at �=0with 100% probability time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. Υ (�, 0) =1 (38) Te only observable quantities are the properties (momenta, polarizations) of free particles: the then this particle is stable with respect to time translations as initial particles which come into interaction, and well. the fnal particles which result from the process (L. Suppose that (38) is true; that is, for any |Ψ⟩ ∈ H�, D. Landau and R. E. Peierls, 1930). Berestetskii et � −(��/ℏ)� � (��/ℏ)� �� al. [41] � � � � Υ (�, 0) ≡ ⟨Ψ �� Ψ⟩ =1. (39) Te more one thinks about this situation, the H more one is led to the conclusion that one should Tis means that all boosts leave the subspace � invariant notinsistonadetaileddescriptionofthesystem (��/ℏ)� � � � |Ψ⟩ ∈ H ,∀� in time. From the physical point of view, this � (40) isnotsosurprising,becauseincontrasttonon- and that the interacting boost operator �� commutes with the relativistic quantum mechanics, the time behavior � of a relativistic system with creation and annihi- projection . Ten commutators (17) and (31) and the Jacobi lation of particles is unobservable. Essentially only identity imply scattering experiments are possible, therefore we ��2 retreat to scattering theory. One learns modesty in [�, �] =− [�, [� ,� ]] ℏ � 0� feld theory. Scharf [42] (41) ��2 ��2 Wecannotacceptthispointofview,becauseithas = [� ,[� ,�]]+ [� ,[�,� ]] = 0 ℏ � 0� ℏ 0� � nothing to say about such interacting time-dependent system as the unstable particle. which contradicts the fundamental property (35) of unstable particles. To resolve this contradiction, we have to admit that 3.2. Dynamical Boosts. Our preferred way to resolve the (��/ℏ)� � the boosted state � � |Ψ⟩ does not correspond to the Currie-Jordan-Sudarshan controversy is to abandon the � hypothesis of “invariant trajectories” and admit that boost particle with 100% probability. Tis state must contain an admixture of decay products even at the initial time �=0 transformations are dynamical. Actually, even in the original Dirac’s paper [25], it was mentioned that, in a theory with (��/ℏ)� � � � |Ψ⟩ ∉ H , �=0. ̸ kinematical space translations (32) and rotations (33), boosts � if (42) Advances in High Energy Physics 7

Tismeansthat,fromthepointofviewofthemoving 4. Conclusions observer,thestatevector’sprojectiononthesubspaceofdecay products is nonzero We applied Poincare-Wigner-Dirac´ theory of relativistic interactions to unstable particles. In particular, we were � (��/ℏ)� � �2 interested in how the same particle is seen by diferent moving 0<�(1−�) � � |Ψ⟩� � � observers. We proved that the decay probability cannot be � −(��/ℏ)� � (��/ℏ)� �� invariant with respect to boosts. Diferent moving observers =⟨Ψ�� (1−�)(1−�) � � � Ψ⟩ � � may see diferent internal compositions of the same particle. (43) � −(��/ℏ)� � (��/ℏ)� �� In spite of being very small, this efect is fundamentally =⟨Ψ�� (1−�) � � � Ψ⟩ � � importantasitsetsthelimitofapplicabilityforspecial � −(��/ℏ)� � (��/ℏ)� �� relativity. � � � � =1−⟨Ψ�� Ψ⟩ and that the nondecay probability is less than unity Conflicts of Interest

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