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Zero Point Energy of Composite Particles: the Medium Effects PHYSICAL REVIEW D 101, 036001 (2020) Zero point energy of composite particles: The medium effects Toru Kojo Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China (Received 18 November 2018; accepted 19 January 2020; published 4 February 2020) We analyze the zero point energy of composite particles (or resonances) which are dynamically created from relativistic fermions. We compare the zero point energies in medium to the vacuum one, taking into account the medium modification of the constituent particles. Treating composite particles as if quasiparticles, their zero point energies contain the quadratic and logarithmic UV divergences even after the vacuum subtraction. The coefficients of these divergences come from the difference between the vacuum and in-medium fermion propagators. We argue that such apparent divergences can be cancelled by consistently using fermion propagators to compute the quasi-particle contributions as well as their interplay, provided that the self-energies of the constituents at large momenta approach to the vacuum ones sufficiently fast. In the case of quantum chromodynamics, mesons and baryons, which may be induced or destroyed by medium effects, yield the in-medium divergences in the zero point energies, but the divergences are assembled to cancel with those from the quark zero-point energy. This is particularly important for unified descriptions of hadronic and quark matter which may be smoothly connected by the quark-hadron continuity. DOI: 10.1103/PhysRevD.101.036001 I. INTRODUCTION One might think that the composite particle dissociates at some momentum which provides a cutoff on the Strongly correlated systems of fermions typically develop integral. But in general there are also states whose composite objects or collective modes [1–3].Theexamples energy spectra survive to high energy and behave as include fermion pairs in superfluidity [4], composite fermions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E k ∼ m2 v2k2 m in quantum hall systems [5], hadrons in quantum chromo- compð Þ comp þ ( comp: the energy in the rest dynamics (QCD), and many others. Those states often play frame; k: spatial momentum; v: velocity approaching dominant roles in transport phenomena or in thermal equa- to the light velocity in the UV limit). The number of tions of state, especially when fermions as constituents have 3 their states growsR as ∼jkj ; in this case the zero the excitation energies larger than the composites. point energy is ∼ d3kE ðkÞ¼aΛ4 þ bm2 Λ2 þ In principle the composites can also contribute to the comp UV comp UV cm4 ln Λ þÁÁÁ, where Λ is some UV scale much zero point energy, i.e., the vacuum energy of a system or comp UV UV equations of state at zero temperature through their quan- bigger than the natural scale of the theory. The first term is tum fluctuations. The consideration for these contributions saturated by the physics at high energy and hence is may be potentially important for, e.g., QCD equations of universal, so can be eliminated by the vacuum subtraction. state at high baryon density where the relevant degrees of This is not the case for the second and the third terms which freedom (d.o.f.) change from hadrons to quarks [6–14]. arise from the coupling between universal hard part and This paper will address problems of the in-medium zero soft dynamics; the soft part can be easily affected by the point energy of composite particles. The UV divergences environment and is not universal. These arguments clearly appear if the interplay between the composites and their require closer inspections based on more microscopic d.o.f. Another problem is that, going from zero to high fermion constituents is not properly taken into account. The UV divergences to be discussed are associated with density, composite states may not keep the one-to-one the summation of states for composite objects. Composite correspondence to their vacuum counterparts, e.g., some particles have the total momenta which must be integrated. composite states in vacuum might disappear in medium. This mismatch in the d.o.f. likely leaves the mismatch in the UV contributions which in turn appear as the UV divergences in equations of state. A more general question Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. is how excited states, including the resonances and con- Further distribution of this work must maintain attribution to tinuum, should be taken into account. To discuss these the author(s) and the published article’s title, journal citation, issues, some integral representations should be used to and DOI. Funded by SCOAP3. include all possible states in a given channel [15–19]. 2470-0010=2020=101(3)=036001(16) 036001-1 Published by the American Physical Society TORU KOJO PHYS. REV. D 101, 036001 (2020) In this paper we analyze the structure of UV divergences discuss their contributions to the equations of state. In which come from the composite particles and their con- Sec. III we discuss the “single particle” contribution with stituents. We argue that, to handle the UV divergences the self-energy and its UV divergent contribution to the associated with the composite particles, it is most essential equation of state. In Sec. IV we briefly overview the to take into account the interplay between the composites structure of the 2PI-functional and present the general and their constituents. In short, they transfer the UV strategy to handle the UV divergences. We isolate the divergences one another when the fermion bases (or medium contributions induced by the medium modifica- propagators) are deformed in a nonperturbative way. As tions of fermion propagators. In Sec. V we first discuss the a consequence, graph by graph cancellations by the vacuum medium contributions without modifications of the vacuum subtraction do not take place; only the sum cancels. To keep fermion bases. In Sec. VI we examine the impacts of track the impact of changes in fermion bases, we apply the the change of fermion bases and discuss how to handle the two-particle irreducible (2PI)-formalism. UV divergences associated with it. The interplay between In order to make the zero point energy UV finite, we single particle’s and composite’s contributions is discussed demand: (i) all vacuum n-point functions are made finite in within the 2PI-formalism. Section VII is devoted to some way; (ii) the fermion self-energies approach the summary. R R R R R R 4 d4p d3p universal limit sufficiently fast at high energy. With these We use the notation ¼ d x, ¼ 4, ¼ 3, R x p ð2πÞ p ð2πÞ conditions the quadratic divergences in the zero-point energy N and TrN½ÁÁÁ¼ðΠn 1 p ÞtrD½ÁÁÁ where trD are trace over cancel, as we can see from the general structure of the 2PI ¼ n the Dirac indices. The momenta p˜ μ ¼ðp0 − iμ;pjÞ will be functional. Meanwhile the removal of the logarithmic divergences depend on models and requires more sophisti- also used. cated discussions, but after all the origin of the problems can be traced back to the in-medium self-energies. II. THE PAIR FLUCTUATIONS The discussions were originally motivated for the appli- As the simplest example, we consider the fermion pair- cations to QCD equations of state at finite baryon density – – fluctuation contributions to the equations of state. The [8 14,20 23]. But we consider the problems in more resummation of 2-body graphs generates bound states, general context, aiming at not only renormalizable theories resonance poles, and continuum. The popular approach is but also nonrenormalizable ones. The latter is often used for the Gaussian pair-fluctuation theory in which the contri- practical calculations of equations of state, but usually butions from the 2-particle correlated contributions are requires introduction of a UV cutoff which always leaves simply added to the mean field equations of state. In this questions concerning with impacts of the cutoff effects or section we examine how the UV divergence appears in artifacts. In particular whether equations of state depend on equations of state in theories of relativistic fermions. the model cutoff quadratically or logarithmically makes We start with a free streaming 2-quasiparticle propagator, important difference. It is important to identify the origins; the strong cutoff dependences may be intrinsic to the k G0 p; k S p S p ;pp ; structure of models whose origin can be traced back to αα0;ββ0 ð Þ¼ αα0 ð þÞ ββ0 ð −Þ Æ ¼ Æ 2 ð1Þ the microphysics, but some may be due to inconsistent approximations and are artifacts to be eliminated. In this where k is the total momentum of the two particle which paper we try to structure our arguments in such a way that is conserved, while p is the relative momentum. The self- we will be able to disentangle the intrinsic and artificial energy corrections are included into the definition of S. cutoff dependence. We write the 2-body interaction as V, then the resummed To make our arguments concrete, we will often refer to 2-fermion propagator is given in a symbolic form as the Gaussian pair-fluctuation theory in which the fermion pair fluctuations are added on top of mean field results. G0 G ¼ G0 þ G0VG0 þÁÁÁ¼ ; ð2Þ For nonrelativistic fermions, this theory has been successful 1 − VG0 descriptions in the context of the BEC-BCS crossover [24–26]. On the other hand, these theories have incon- where the summation over internal momenta and other sistency in the treatments of the single-particle and of quantum numbers is implicit. The poles are found by composite particles, and, in the case of relativistic fermions searching for the values of kμ ¼ðEðkÞ; kÞ such that [27,28], it leads to the UV divergences which must be 1 − VG0 ¼ 0.
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