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 þ, 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 TrN½¼ðΠn 1 p ÞtrD½ 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 ¼ 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 þ¼ ; 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. cutoff by hand. It is rather easy to see the existence of the We assume that G and V in vacuum are made finite by inconsistency. On the other hand, it is less obvious that such some renormalization procedures or by introduction of inconsistency can be the source of the UV divergences. physical cutoffs. For theories like QCD or QED, the This paper is organized as follows. Section II is devoted 4-fermion functions are UV finite as far as we use the to further illustration of the UV problem. We consider renormalized fermion propagators and vertices. For theo- fermion pairs as the simplest example of composites and ries with contact interactions the 4-fermion functions are

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∼Λ2 ∼ Λ UV divergent and must be made finite by physically divergences. The coefficients of UV and ln UV terms motivated cutoffs [29]. In this paper we call such cutoffs are sensitive to the physics in the IR, and can differ for S “ ” S intrinsic to models physical cutoff, and distinguish it and vac. Λ from UV, a regulator which must be taken to infinity at the To see how these UV divergences arise, let us consider end of calculations. After n-point functions are made finite, for the moment the diagram such that the graphs except the Λ n p S UV shows up only when we close all the lines of -point propagator carrying momentum − is set to vac, see Fig. 2. functions to compute the zero-point energy. We consider the case where large momentum k flows into In the expression for equations of state, this 2-fermion the two-body scattering kernel, as we are interested in the propagator appears inside of the logarithm; drawing the composites at large total momenta. This contribution has corresponding closed diagrams (Fig. 1, left), we must the structure, 1= 2n attach the symmetry factor ð Þ and sum them up, Z X 1 ∼ Σ p S p − S p ring n 2vacð −Þð medð −Þ vacð −ÞÞ Φ2 ½S¼ Tr2½ðVG0Þ 2n p− n¼1 Z 1 ∼− Σ S Σ − Σ S ; 2vac vacð med vacÞ vac ð5Þ ¼ − Tr2½Lnð1 − VG0Þ p 2 − 1 −1 S ≃ S − S Σ − Σ S ¼ Tr2LnðG=G0Þ ; ð3Þ where we expand med vac vacð med vacÞ vac. 2 p (There is also the integral over þ, but it is hidden in Σ p where we wrote the energy density as a functional of S for subgraphs of the vacuum self-energy 2vacð −Þ multiplied S − S the later purpose. The expression includes bound states, to med vac.) At this point in treating the vacuum self- resonances, and the continuum in the 2-body channel. energy in the subgraph, it is necessary to mention the The form of LnðG=G0Þ reflects the fact that uncorrelated necessity to include counterterms [30,31] (Fig. 1, right), product of fermion lines in G must be subtracted to avoid the counter double counting (linked-cluster theorem). It should be also Φ2 ½S¼Tr1½ðδ2Zp= − δm2ÞS; ð6Þ clear that the term LnðG=G0Þ approaches zero at very large energy; the kinetic energy is much larger than the potential Σ δ which cancel the divergence in 2vac. We note that 2Z and energy and hence G → G0. Meanwhile, even though this δm G=G ∼ 2 are medium independent as they should, but can cancels the leading UV contributions, we expect Lnð 0Þ couple to the medium-dependent propagators S. These 1=k2 that may produce the quadratic and logarithmic counterterms renormalize the vacuum self-energy part k divergences after integrating the momentum . in the graph. Then we can focus on the medium effects We have not yet introduced a counter term or a vacuum coupled to the UV finite self-energy. We count the power of subtraction procedure for the zero point function, so we set −1 momenta as Σ2 ∼ Σ ∼ Σ ∼ p and S ∼ p .Itis Φring S vac vac med vac our reference energy density to the vacuum one, 2 ½ vac, plausible that the medium effects decouple at high S μ 0 evaluated for some propagator vac at ¼ . Then the momenta, so we expect the cancellation of the leading renormalized energy density is Σ − Σ component in med vac; then the rest behave as Σ − Σ ∼ Λ2 =p where Λ is some IR scale in Φring S Φring S − Φring S : med vac med med 2R ½ ¼ 2 ½ 2 ½ vac ð4Þ the medium. But even after this cancellation happens, the p ∼Λ2 Λ2 However, this vacuum subtraction in general is not sufficient integral over still leads to UV med. to cancel the above-mentioned quadratic and logarithmic

FIG. 2. The difference between the medium and vacuum self- energy from the 2-particle correlations. Very large momenta, p ;k→ ∞ FIG. 1. The pair fluctuation diagrams that generate quadratic þ , flows into the graph, except a fermion line with divergences. The counterterms, whose values are fixed in the momentum p− which is finite. (The counter terms to renormalize vacuum calculations, are also shown. the vacuum self-energy part is not explicitly shown.)

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The severest UV divergence comes from the vector self- then the zero point energy is energies and is quadratic. If we ignore the modification Z of the vector self-energies as in typical mean field E p − E p ð medð Þ vacð ÞÞ model treatments, we have less divergence but it is still p ∼Λ4 Λ med ln UV. Unless we assume unreasonably stronger ∼ M2 − M2 Λ2 c2 − c2 Λ : Σ − Σ ð med vacÞ damp þð med vacÞ ln UV ð10Þ damping for med vac, we cannot avoid the UV divergences. As we will see, the divergences associated with the The first term is characterized by the damping scale. The typical model results are obtained if we entirely neglect product of IR and UV contributions will be handled by c taking into account the interplay between a composite state the terms med;vac in the UV region, but the validity of such and the constituents in it. procedures is uncertain from the physical point of view. In c general med;vac exist and are nonuniversal, leaving the logarithmic UV divergence. III. THE SINGLE PARTICLE CONTRIBUTION The situation gets even worse if we also take into account Z p In this section we look at the single particle contribution the modification of the residue function med;vacð Þ which to the equations of state. We assume that a fermion acquires appears if the vector self-energy is nonzero. The propa- S −Z p=˜ − M −1 the self-energy from the IR dynamics which is sensitive gators are med;vac ¼ med;vacð med;vacÞ . Assuming to the presence of the medium. After calculating the Z p ∼ Z d =p2 the damping med;vacð Þ univ þ med;vac , the differ- in-medium self-energy, the last step is to close the fermion ence of TrLnS between the medium and vacuum cases line and we get the single particle contribution to equations ∼ d − d Λ2 leaves the quadratic divergence ð med vacÞ UV. of state. Therefore the damping of the self-energy at high We begin with the case in which the fermion acquires momenta alone cannot be the sufficient condition for the M μ the momentum independent mass gaps, med at finite , UV finiteness of equations of state (unless extraordinary M μ 0 S and vac at ¼ . Using the propagators, med;vac ¼ damping takes place in the self-energy). In order to get − p=˜ − M −1 μ ð med;vacÞ , the contribution at is [32] physical equations of state, we need to assemble these UV Z divergent pieces with those from other origins. − S−1 −2 θ E p − μ 1 E p ; Tr1Ln med ¼ ½ ð medð Þ Þþ medð Þ p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IV. EQUATIONS OF STATE IN THE 2PI E p M2 p2; FORMALISM: SOME DEFINITIONS medð Þ¼ med þ ð7Þ A. The structure of 2PI functional where the particle and antiparticle contributions are We have seen that the quasiparticle contributions to included. After subtracting the vacuum counterpart, we get equations of state from the composites and their constitu- Z ents in general have the quadratic divergences. It is − S−1 S−1 2 θ μ − E p E p Tr1Ln med þ Tr1Ln vac ¼ ð medð ÞÞ medð Þ tempting to expect these divergences to cancel. Actually p Z these terms do not directly cancel and we must go one step − 4 E p − E p ; further to correctly handle the double counting problem. ð medð Þ vacð ÞÞ p For this purpose we use the formalism of the two particle ð8Þ irreducible (2PI) action, developed by Luttinger-Ward [33], Baym-Kadanoff [34,35], and its relativistic version by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cornwall-Jackiw-Toumbolis [36]. Its renormalizability where E ðpÞ¼ M2 þ p2. The last set of terms vanish vac vac was first discussed in [37] and since then seminal works if M ¼ M , but otherwise it leaves the quadratic med vac have followed [38]. These works gave detailed account divergence, ∼−ðM2 − M2 ÞΛ2 . med vac UV by explicitly choosing some of renormalizable theories, To improve the situation, it is tempting to consider the while we discuss the structure of the medium-induced UV momentum dependence of the gap as M → MðpÞ, and Λ divergences in more abstract fashion, so that we can demand that beyond some physical damping scale damp M p emphasize the common aspects in the renormalization the mass function ð Þ approaches to some universal programs. In particular our arguments also may include value independent of the IR physics. For instance one can the cases for non-renormalizable models. think of The 2PI action I½S; μ is a functional of a fermion S M2 p 2 ≤ Λ2 propagator , 2 med;vac ðj j dampÞ M ; ðpÞ ∼ ; med vac m2 c =p2 p 2 ≥ Λ2 I S; μ S SΣ˜ ½S;μ Φ S ; univ þ med;vac ðj j dampÞ ½ ¼Tr1Ln þ Tr1½ þ ½ Σ˜ ½S;μ S−1 − Sμ −1; ð9Þ ¼ ð treeÞ ð11Þ

036001-4 ZERO POINT ENERGY OF COMPOSITE PARTICLES: THE … PHYS. REV. D 101, 036001 (2020) which includes a tree level propagator (in Euclidean space), of some contributions; they are the origin of the UV Sμ p Sμ¼0 p˜ − p=˜ −m −1 m divergences, as we will clarify shortly. treeð Þ¼ tree ð Þ¼ ð Þ , with the fermion mass . ˜ n >0 The Tr1½SΣ term plays a role to cancel the double counted (ii) In the 2PI formalism the rernomalization of ð Þ- contributions. The Φ functional is the sum of 2PI graphs point functions in vacuum does not readily guarantee composed of dressed fermion propagators. We assume that the UV finite equations of state. The situation is Γ ϕ a set of counterterms have been introduced to renormalize different from the 1PI effective action ½ ,asa ϕ the vacuum self-energies and vertices. The number of functional of some field values, [40]. For the 1PI counterterms can be either finite or infinite depending on functional, we consider the form the renormalizability of theories [39], but this aspect is not Z important for our arguments. We also note that, because we Γ½ϕ − Γ½ϕ ¼ 0¼ ½að∂ϕÞ2 þ bϕ2 þ cϕ4 þ; write the action as a functional only for fermion propa- x gators, the graphs we consider are not necessarily 2PI with ð15Þ respect to the propagators of other possible fields. Φ S The variation of with respect to yields the graphs for where a; b; c; are some constants which must be S the fermion self-energies, and if we choose to give the renormalized. Here a; b; c; appear in the n-point extrema, it satisfies the Schwinger-Dyson equation, functions obtained from the functional derivative δnΓ=δϕn, and they get renormalized through the δI½S; μ δΦ½S ¼ Σ˜ ½S;μðpÞ − Σ½SðpÞ¼0; Σ½SðpÞ¼− : studies of these functions. With the expression (15), δSðpÞ δSðpÞ the UV finiteness of a; b; c; can be directly ð12Þ translated into the finiteness of the 1PI functional for a given distribution of ϕðxÞ. In contrast, the 2PI functional is characterized by a variable Sðx; yÞ; The solution of the Schwinger-Dyson equation is written as S Sμ p Φ S S even when the coefficients of are finite, this ð Þ. If we further differentiate ½ by , we get kernels variable by itself can generate the divergence in with the 3-, 4-, and more-external legs. The equation of x → y μ the limit of . Therefore we need the discus- state at is obtained as sions about the asymptotic behaviors of S.

μ μ 0 −PðμÞ¼I½S ; μ − I½S ¼ ; μ ¼ 0; ð13Þ B. Isolating the divergence where the pressure is set to zero at μ ¼ 0. In the following we analyze the UV divergences in the Now several comments are in order on the structure of functional. We first define the functional: μ¼0 (i) The expression of the 2PI functional differs from IR½S; μ ≡ I½S; μ − I½S ; μ ¼ 0; ð16Þ the expression of thermodynamic potential in the Gaussian pair-fluctuation theories. The latter has the which is a functional of S at finite μ. To clarify the structure, structure it is convenient to distinguish the contributions associated with the change of fermion bases and the other, since the I I Φ S ; GPF ¼ MF þ 2½ MF treatments of the UV divergences will be different for these S ;μ two contributions. We decompose I S S Σ˜ ½ MF Φ S ; MF ¼ Tr1Ln MF þ Tr1½ MF þ MF½ MF ð14Þ I S; μ I S; μ − I Sμ; μ I Sμ; μ − I Sμ¼0; μ 0 R½ ¼ð ½ ½ k Þ þ ð ½ k ½ ¼ Þ

ring counter ≡ IΔS½S; μþIΔμ: ð17Þ where Φ2½S is the sum of Φ2 ½S and Φ2 ½S I defined in Eqs. (3) and (6). The mean field part MF Φ Here the first term in the right-hand side (RHS), IΔS½S; μ, is the 2PI functional and MF includes only 2PI graphs composed of a single fermion line. Here the measures the energy gain or cost associated with the change S → S solution of the Schwinger-Dyson equation in the of bases, k (defined below), at the same chemical S ;μ μ S Σ˜ ½ MF S S IΔS 0 mean-field level, MF, was substituted and ¼ potential. Clearly at ¼ k the functional is ¼ , −δΦ =δS . We note that the structure of I has S MF jSMF MF guaranteeing the existence of for which the functional the form of the 2PI formalism. After just adding the IΔS is UV finite. On the other hand, the second term in Gaussian fluctuations Φ2½S, however, we lose such Eq. (17) compares the energies at different chemical correspondence by neglecting its impact on the potentials but with the same fermion bases; for this purpose fermion self-energy. In fact the incomplete treatment we have introduced an in-medium propagator made of the of such contributions introduces the double counting vacuum fermion bases,

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μ μ¼0 μ −1 ρvac ρvac S p ≡ S p → p˜ −p=˜ m Σ p ; where p , a are the spectral functions in vacuum kð Þ ð Þ¼½ þ þ kð Þ (4 × 4 matrices) for particle and antiparticle components, μ Sμ¼0 Σ p ≡ Σ½ p → p˜ : kð Þ ð Þ ð18Þ respectively. Or one can also write the invariant mass parametrization

That is, we changed only the variable as p0 → p0 − iμ but Z ∞ s p=˜ ρvac s ρvac s fixed all the others in the vacuum propagator. We empha- Sμ p − d V ð Þþ S ð Þ μ ð Þ¼ 2 ð20Þ μ 0 S k 0 2π p˜ − s size that, except for the ¼ case, k in general does not minimize I½S; μ, so is not the solution of the Schwinger- μ where the Lorentz invariance in vacuum ensures that μ ½S ;μ Σ ≠ Σ k −δΦ=δS μ vac Dyson equation, i.e., ¼ jS . More ρV;SðsÞ are the Lorentz scalar which depends only on k k Sμ the Lorentz scalar variable s. The vacuum spectral func- about k will be discussed in the next section. Below we will first give some concrete discussions for tions reflect that we are using the same fermion bases as in vacuum, while the modification H → H − μN (H: hamil- the in-medium propagators made of the vacuum fermion tonian, N: number of fermions) is reflected only through bases. After this preparation we discuss how IΔμ and IΔS the change of variable p0 → p0 − iμ. are made UV finite. We emphasize that the functional IΔS½S; μ is UV finite only for particular classes of S; the S B. The self-energy in-medium self-energy of must approach to the vacuum μ ½S μ counterpart sufficiently fast at large momenta. Σ k Σ Below we compare the structure of and k, then discuss both can be UV finite by the vacuum counterterms. V. IN-MEDIUM CALCULATIONS WITHIN THE These self-energies differ only when the fermion self- energy graph includes fermion loops, see Fig. 3. The key VACUUM FERMION BASES μ observation is that the shift p → p˜ in Σ affects the A. In-medium propagators made k of the vacuum fermion bases propagator connected to the external leg, but does not affect the momenta of fermion loops; the fermion propa- We first give more remarks concerning with the termi- Sμ¼0 Sμ¼0 gators in fermion loops are the vacuum one, ð¼ k Þ, nology “fermion bases.” We define them directly from the μ ½S fermion propagators, rather than the unitary transformation that does not depend on μ. On the other hand, in Σ k , all propagators, from the one connected to the external of fermion field operators as the latter need not be μ Sμ p fermion lines as well as those forming loops, are S .So manifestly treated. For the propagator kð Þ, most gen- k erally the spectral functions are the same as the vacuum the formal structure of the difference in the self-energies Sμ p can be written as one, and in this sense the propagator kð Þ is made of the μ vacuum fermion bases. In the spectral representation, ½S μ Σ k ðpÞ − Σ ðpÞ Z k Z μ ∞ w ρvac w; p ρvac w; p ½S μ μ¼0 μ d p ð Þ a ð Þ k ½S ½S S p − ; Gp k I k k − I k k ; 21 kð Þ¼ þ ð19Þ ¼ ðf gÞð ðf gÞ ðf gÞÞ ð Þ 0 2π ip0 þ μ − w ip0 þ μ þ w fkg

μ ½S μ Σ k p Σ p FIG. 3. The self-energies ð Þ and kð Þ (the 1-fermion loop case as an illustration). In the former all fermion lines are calculated Sμ Sμ¼0 with the propagator k, while in the latter all internal fermion loops are calculated with the vacuum propagator . The fermion lines μ ½S μ Σ k p Σ p connected to the external legs are common for ð Þ and kð Þ; if no-fermion loop is available, these two self-energies coincide.

036001-6 ZERO POINT ENERGY OF COMPOSITE PARTICLES: THE … PHYS. REV. D 101, 036001 (2020) where Gp is the fermion line connected to the external the μ-dependence appears. Next we pick up poles from fermion legs with momentum p, and the function I is the each propagator by deforming the contour C in the standard diagram attached to Gp. They are connected by lines carrying way. Here we consider the poles of the ith propagator in the k k ;k ; fqg a set of momenta, f g¼ð 1 2 Þ, which will be loop Fl . For this calculation the most general expression integrated. The difference in the self-energies is summarized can be obtained by using the spectral representation, in I through the difference of fermion propagators in the fermion loops. From the definition it should be clear that the μ¼0 S ð−il0 þ qi0; l þ qiÞ Z two self-energies coincide if there are no fermion loops. ∞ vac vac dwi ρ ðwi; l þ qiÞ ρ ðwi; l þ qiÞ ¼ − p þ a ; 0 2π l0 − wi þ iqi0 l0 þ wi þ iqi0 1. Fermion loops μ μ 0 ð24Þ ½S ½S ¼ To see how I k ðfkgÞ and I k ðfkgÞ are calculated, we focus on the subdiagrams in which there is one fermion where qi0 is left Euclidean. After picking up the residues loop and external lines with momenta q1;q2; … are l0 ¼wi − iqi0, the statistical factors in Eq. (28) for attached, see Fig. 4. Let l be the loop momenta. Then particles become, in the zero temperature limit, S μ all fermion propagators k in the loop has the -dependence ˜ 1 only through the combination of l0 ¼ l0 − iμ. More explic- → θ μ − w ; wi−μ−iqi0 ð iÞ ð25Þ itly such a loop contains the structure (reminder: e T þ 1 μ μ¼0 S ðpÞ¼S ðp → p˜ Þ) k and for antiparticles fqg ˜ μ¼0 ˜ μ¼0 ˜ Fl ðl0Þ¼−tr½S ðl þ q1ÞV1;2S ðl þ q2Þ 1 −w −μ− q → 1: ð26Þ μ¼0 ˜ i i i0 × V2;3S ðl þ q3Þ; ð22Þ e T þ 1 q in which several external lines with momenta i¼1;2; are The antiparticle contributions are totally made of the attached to the fermion loop with the vertices V. For the vacuum quantities; after all the two loop graphs are moment we introduce an infinitesimal temperature T and expressed as the sum of the vacuum pieces and the medium ˜ use the Matsubara formalism with l0 → ωn − iμ where dependent part, ω 2n 1 πT μ l˜ Z Z Z n ¼ð þ Þ [32]. Here is hidden in the variable 0, X ∞ l fqg ˜ fqg so the integral over 0 leads to Fl ðl0Þ¼ Fl ðl0Þ − dwi l l i 0 Z X Z Z q q Ff g l˜ → T Ff g ω − μ ˜ fq;ig l ð 0Þ l ð n i Þ × θðμ − wiÞFl ð−iwiÞ; ð27Þ l n l l Z Z fqg dl0 F ð−il0Þ l where (lw ¼ð−iwi; lÞ and qi;j ¼ qi − qj) ¼ − l −μ ; ð23Þ 2π 0 l C i e T þ 1 ˜ fq;ig μ¼0 vac F ð−iwiÞ¼−tr½S ðlw þ qi−1;iÞVi−1;iðρ ðwi; lÞÞ where C is the contour surrounding the poles of l p μ 0 l0−μ −1 V S ¼ l q : ðe T þ 1Þ , and now this factor is the only place where × i;iþ1 ð w þ iþ1;iÞ ð28Þ

FIG. 4. The 1-fermion loop graph with many line insertions, see the expression Eq. (22). The loop momentum is l. Using the standard technique of the analytic continuation one can isolate the vacuum piece and the medium dependent piece, as shown in Eq. (28). The box in the last diagram specifies where we pick up the pole.

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l ρvac p → p˜ After the integration of 0, the vacuum spectral density p former is the vacuum self-energy with replacement μ i ½S remains as contributions from the th propagator. Note that and hence is UV finite; thus Σ k ðpÞ is also UV finite. we have shifted variables l þ qi in the spectral density so The case with more fermion loops does not introduce any that the jth propagator depends on fqg through the essential modifications. Essentially such graphs can be combination qj;i. regarded the vacuum n-point functions contracted to the Now we can see the domain of the spatial momenta l is medium θ-functions. The vacuum functions are assumed to bound and the integral over l is UV finite. To confirm this be UV finite, and the contraction only leads to integrals we use the fact the energy of the vacuum spectral function pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi whose domains are limited, so the results of the integration behaves as w ¼ m2 þ l2 for a state with the invariant are all finite. mass m2. Then we notice that there is the energy bound We note that the present argument does not ask whether θðμ − wÞ which in turn limits the domain of l. The result is the theory is renormalizable or not. If all vacuum functions entirely expressed by the vacuum spectral density and μ,as are made UV finite for some fermion bases, and if we keep μ μ 0 we used the propagators S and S ¼ . using the same bases from the vacuum to a medium, then k k the in-medium self-energies are UV finite. The key fact was that the UV contributions exactly cancel in Eq. (27).Ifwe 2. Integrating lines connected to fermion loops used different bases for the vacuum and medium, the We have just verified the fermion loops in the subgraphs cancellation was not exact; in such cases we must consider are divided into the vacuum pieces plus UV finite pieces. the power counting for asymptotic momenta, as we will The last step for the evaluation of the self-energies is to discuss in the next section. integrate out momenta connecting the fermion loops to the external fermion lines. C. The UV finiteness of IΔμ As an example a one-fermion loop graphs for the self- μ Finally we consider IΔμ. The naive power counting of the ½S μ energy difference, Σ k ðpÞ − Σ ðpÞ, are shown in Fig. 5. ∼μ2Λ2 k dimensionality indicates the presence of terms like UV ∼μ4 Λ The vacuum part in the fermion loop was already canceled or ln UV. The vacuum subtraction, however, com- by taking the difference. Note that one of fermion lines in pletely cancels these UV divergences as far as we use the the loop is replaced with the vacuum spectral density with same fermion bases for the vacuum and medium. This is θ μ − E the restriction ð lþqi Þ attached, as in Eq. (28). The what happens in perturbation theory and has been discussed graph in Fig. 5 can be evaluated by using the vacuum in the standard textbook [32]. For the later purpose we 4-point functions which are renormalized and UV finite. briefly review this cancellation in somewhat abstract The only question is whether closing two legs with fashion, looking at general graphs which may include momenta l generates the UV divergence, but its integration infinite number of loops. domain is restricted, so no new UV divergences arise. The discussion goes in the very similar way as one given μ ½S μ Σ k p − Σ p for the self-energy, see Fig. 6 for the zero-point energy. So we have proved that ð Þ kð Þ is UV finite. μ There may be several fermion loops. But as we have μ ½S Σ p Σ k p Moreover kð Þ and ð Þ are separately UV finite; the already discussed, we can decompose each loop into the vacuum term and medium part with θ-function inserted. Thus by considering all fermion loops we find the product of the vacuum term and the medium terms. The potentially worst UV divergence would appear if we pick up the vacuum contributions from all fermion loops, but it is the vacuum quantity and can be canceled by the vacuum subtraction. The next-to-worst divergence appears if we pick up only one medium piece but choose the vacuum pieces for all the rest of fermion loops. But such diagrams can be regarded as vacuum fermion 2-point functions contracted with the medium piece. The vacuum 2-point function is UV finite by our assumption on the vacuum renormalization, and its contraction of the external fermion legs with the medium FIG. 5. The next-to-worst UV contribution with 1-medium μ insertion (the 1-fermion loop case as an illustration). The dependent piece, bound by , does not yield any addi- vacuum-medium piece is UV finite, as we can regard the graph tional UV divergence. Thus the 2PI graphs with one originated from the contraction between θ-function and the medium piece are UV finite. The graphs with more vacuum 4-point function which can be renormalized by the medium pieces can be discussed in the same way. With vacuum counterterms. this we conclude IΔμ is UV finite.

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FIG. 6. An example of the Φ-functional with 2-fermion loops. In each loop we decomposed the vacuum and medium dependent parts, and then considered the product. The vacuum-vacuum product has the UV divergence that can be subtracted off. The 1-medium insertion graphs are UV finite, as we can regard the graph originated from the contraction between θ-function and the renormalized vacuum 2-point self-energy. The same logic is applied for higher orders of medium insertions.

VI. UV CONTRIBUTIONS INDUCED BY modulo possible logarithmic factors in p. Here we consider CHANGES OF FERMION BASES the power of p˜ , not p, because the counting based on the latter involves the expansion of μ=p which is not helpful. A. The UV finiteness of IΔ and constraints S We characterize S by this damping. The asymptotic on the fermion bases S − S γ behaviors of k are classified by the value of : We have already seen that the 2PI functional in medium (a) The γ ¼ 0 case will be called “canonical” and we regard is finite if we keep using the same fermion bases as in the it as our baseline; powers are reduced by 2 compared to vacuum. But the medium effects often deform the fermion S − S ∼ p˜ −3 the single fermion propagators, k ; bases, typically in a nonperturbative manner. The purpose (b) The γ ¼ −1 case happens when we omit the vector here is to examine that to what extent we can change the self-energies and include only mass self-energies, fermion bases without encountering the UV problems. If −4 leading to S − S ∼ p˜ ; we met UV divergences, we would be forced to keep the k (c) The γ ¼ 1 case happens if we include only mass bases same as the vacuum one (for which IΔS is guaranteed self-energies as in (b) but assume that the medium- to be zero). It is important to emphasize that, unless induced constant shift of the self-energies, e.g., correctly handling the UV contributions, we might artifi- ΔΣ∼ΔM ¼ const, survives to large momenta, leading cially exclude physical solutions. to S − S ∼ p˜ −2; For these reasons we analyze in detail the UV structure k (d) The γ ¼ 2 case happens if the medium-induced shift in of the functional IΔS (defined in Eq. (17)) and the condition required for S. We classify the strength of the UV the residue function survives to large momenta and couples to the p=-component, e.g., ΔΣ ∼ ΔZp=, with contribution by choosing the expansion parameter as −1 S;μ −1 μ −1 ΔZ ¼ const, leading to S − S ∼ p˜ ; (reminder: Σ˜ ½ S − S which becomes the k ¼ ð treeÞ γ > 2 physical self-energy when we choose S to be the solution (e) The case is regarded unlikely, so this case will of the Schwinger-Dyson equation) be excluded from our considerations, unless otherwise stated. −1 −1 S;μ μ S;μ In the following, models in our mind are those of Yukawa S ðpÞ − S ðpÞ¼Σ˜ ½ ðpÞ − Σ ðpÞ ≡ ΔΣ½ ðpÞ: ð29Þ k k or gauge theory types in which fermion couples to bosons whose propagators behave as ∼p−2 in the UV domain. To This measures the difference between S and S . We count k simplify the presentation we assume the absence of tadpole- Σ˜ ½S;μ ∼ Σμ ∼ p p S k at large . For the physical solution of , type graphs. If there are tadpoles, the self-energies acquire a we expect that at large p the IR effects decouple; if this is constant independent of the external momenta, as in the the case ΔΣ → 0 as p → ∞. The question is how fast this cases (c) and (d) which will require extra discussions of damping is. To characterize the damping, we write shifting the (bosonic) field, ϕ → ϕ0 þ ϕ. Such procedures will not be performed in this paper. ΔΣ ∼ p˜ −1þγ: ð30Þ B. Preparation: Power counting for the subgraphs with which The discussion of the UV behaviors get involved because of loops in subgraphs. To disentangle various UV diver- S p − S p ∼−S p ΔΣS p ∼ p˜ −3þγ; ð Þ kð Þ kð Þ kð Þ ð31Þ gences shared by different loops, it is useful to specify the

036001-9 TORU KOJO PHYS. REV. D 101, 036001 (2020) soft and hard parts in the graphs. We assign powers for For the graph (c), all lines are hard, so fermion propagators as 8 −1 −4 −2 2 p × ðp p × p Þ ∼ p : ð38Þ p−1 ðhardÞ S ∼ ð32Þ ∼p2 M−1 ðsoftÞ The graphs (b) and (c) are the order of and apparently have the quadratic divergence whose origin is the hard where M is a constant with the mass dimension one. fermion loop. The leading quadratic divergence is cancelled Accordingly the fermion self-energy is by the vacuum subtraction, but generally that procedure still leaves the logarithmic divergence coupled to quantities p ðhardÞ dependent on the fermion bases. To eliminate such con- Σ ∼ ð33Þ tributions we need special cares by summing a proper set of M ðsoftÞ graphs. We will come back to this point when we discuss For boson propagators we assign the boson self-energy in Sec. VI D. p−2 ðhardÞ C. The self-energies for fermions D ∼ ð34Þ M−2 ðsoftÞ Before looking at the zero-point energy, we compare the self-energies for different bases and examine how the Finally we also assign powers for loop integration, difference can be made finite. More explicitly we will Z assume γ ∼ 0 for the counting and then check that the self- p4 hard ∼ ð Þ energy graphs are finite under this assumption; if not we 4 ð35Þ l M ðsoftÞ would run into inconsistencies. Here it is sufficient to discuss the μ ¼ 0 case for the divergences associated with where the domain of integration is restricted to the IR the change of bases; as we have already seen, the in- region for the soft part. medium and vacuum self-energies for the same bases differ As an illustration we consider a 2-loop graph for the only by the UV finite value. fermion self-energies where one-fermion loop is connected The structure of fermion self-energies looks relatively to external lines by 2-boson lines (Fig. 7). We use a box to simple because from the dimensional ground they appa- indicate that inside the box the lines participating in the rently have only the logarithmic divergences. This is indeed loops are hard. the case if there are no fermion loops inside of the self- For the graph (a), one fermion line in the loop is kept soft energy graphs. Extra cares are necessary for graphs with but all the other lines are set hard. Then the counting is internal fermion loops in the subgraphs. To see it we consider the difference in the self-energies for propagators 4 4 −1 −4 −1 −1 3 −1 μ¼0 μ¼0 M p × ðp p p M Þ ∼ M p ; ð36Þ S and S ,

4 M μ 0 μ¼0 δΣαβ where the first is the factor from the phase space; is the ½S ¼ ½S μ¼0 μ¼0 4 ðΣ −Σ Þαβ ¼ ðS −S Þγδ þ ð39Þ phase space for the soft fermion line and p for the other δSγδ Sμ¼0 loop momenta. The factor in the parenthesis is from propagators. Therefore this case (a) is ∼p−1 and hence where the first term in the RHS represents the sum of p → ∞ μ¼0 μ¼0 UV finite for . graphs in which one fermion line is set to S − S and For the graph (b), the fermions in the loop are hard but μ¼0 all the others are S . the others are all soft. The power counting is As an example, we consider a 2-loop graph shown in Figs. 8 and 9. The above expansion produces two types of M4p4 M−1M−4 p−2 ∼ p2: × ð × Þ ð37Þ graphs. In the first type (the graph in Fig. 8), a subtracted

FIG. 7. An example of 2-loop graph. In a box, lines participating in a loop carry hard momenta.

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to consider the divergence from the fermion loop. The loop include a subtracted fermion propagator of ∼l−3, but this reduction of powers is not sufficient to make the loop convergent; there would remain the logarithmic divergence. No vacuum counterterm is available to eliminate this divergence. Hence this divergence must be cancelled by assembling the divergences from other graphs. This divergence in the fermion loop also propagate to the fermion self-energies. μ 0 FIG. 8. A graph generated from the expansion of Σ½S ¼ − Before discussing how to handle this divergence, it is Sμ¼0 μ 0 μ 0 ½ ¼ ¼ more convenient to rephrase it in terms of boson self- Σ (with 2-boson lines) to the linear order of S − S . The S divergence from the vacuum fermion loop is cancelled by the energies. Let Π½ a correlator of quark bilinear currents. counter term in the second graph, to leave powers ∼l2.Itis This is the fermion contributions to the boson self-energies. combined with the boson propagators and a subtracted fermion We note an algebraic relation, propagator, leading to the UV finite result. μ 0 ½Sμ¼0 ½S ¼ μ¼0 μ¼0 μ¼0 Π ¼ Π − 2Tr½S VðS − S ÞV μ 0 μ 0 μ 0 μ¼0 ¼ μ¼0 ¼ μ¼0 ¼ − Tr½ðS − S ÞVðS − S ÞV; ð40Þ propagator S − S belongs to the line shared with the external legs. In the second type (the graph in Fig. 9), one μ¼0 μ¼0 line in the fermion loop is set to S − S . where the second term in the RHS is what we are In the first type, a momentum l enters the fermion loop discussing. The first term in the RHS is the vacuum made of the vacuum propagators. As we saw in the last functions and can be made finite by the vacuum section, the UV divergence would happen only in the case counterterm, while the integral of the third term is −2 where hard momentum enters the fermion loop. The loop convergent and is the order of l . Therefore proving is made of the vacuum bases, so with our assumption on the UV finiteness of the second term in the RHS is the vacuum renormalization, it is made UV finite by the equivalent to proving the UV finiteness of the left-hand counter term (or some cutoff). Then, from the dimension- side (LHS). μ 0 ality the loop leaves powers ∼l2 (modulo logarithm) with Actually the UV finiteness of Π½S ¼ is necessary but UV finite coefficients. Now this piece is contracted with not sufficient condition to make the fermion self-energy μ 0 two gluon propagators attached to the fermion line, to yield finite. From the dimensional ground Π½S ¼ ∼ l2,butwe 2 −2 2 −3 −5 l l l ∼ l Sμ¼0 Sμ¼0 0 factors × ð Þ × (factors from two boson Π½ − Π½ ∼ l μ 0 need , i.e., the leading contributions Sμ¼0 − S ¼ 2 propagators; the internal fermion loop; and ). of l to cancel. Otherwise the self-energy difference l 2 −2 Thus the integral over leaves UV finite functions of of ∼l couple to two boson propagators to yield l . fermion external momenta. Then the fermion self-energy may have the logarithmic Next we consider the graph in Fig. 9, where one line in divergence. μ¼0 μ¼0 the fermion loop is set to S − S . Again it is sufficient From these arguments, now our problem about the fermion self-energies is now translated to the problem of proving the boson self-energies to be UV finite and ∼l0. Before proceeding to the boson self-energies, we mention what happens if there are more boson lines attached to the fermion loop and to the fermion line connected to the external legs. An example is shown in Fig. 10. In this case the analyses are actually simpler than the case shown in Fig. 9; the fermion loop more than two boson line insertions is by itself UV convergent, so that there is no danger to get the UV divergent factor from this subgraph. Also, the 2-loop subgraph with 3-boson lines can be μ 0 FIG. 9. A graph generated from the expansion of Σ½S ¼ − regarded as the vacuum 4-point fermion functions is Sμ¼0 μ 0 μ¼0 convergent by itself. So if one of the loop is soft, the Σ½ (with 2-boson lines) to the linear order of S ¼ − S . result is convergent. Finally, when all loops are hard, The bubble graph contains a single subtracted propagator μ 0 μ 0 −2 μ¼0 ¼ −3 μ¼0 ¼ the 4-point function is ∼l and S − S ∼ l ,so S − S . Naive counting leads to ∼ ln ΛUV in the fermion loop. To eliminate it other graphs not shown here (with more the integration over l leaves UV finite quantities. Therefore boson line insertions) must be added. The way of the cancellation subgraphs, in which the fermion loop with only two boson depends on models. line insertions, is exceptional and requires special cares.

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For gauge theories, the calculations are technically involved, but the required principle is clear-cut: the gauge invariance. The situation is schematically shown in Fig. 11. The gauge invariance protects the polarization function from the quadratic divergences, and for this we need to include a proper set of 2PI-graphs which keeps the (truncated) 2PI functional gauge invariant. In particular V → V S the vertex must be improved as tree ½ , when we take into account the change of fermion bases [41,42]. Once this is done for all S, the leading divergence of Π½S always starts with the logarithmic one, and after single subtraction Sμ¼0 Sμ¼0 S S0 FIG. 10. A graph contributing to Σ½ − Σ½ with 3-boson ðΠ½ − Π½ Þ becomes UV finite. μ¼0 μ¼0 lines. The linear order of S − S appear in the fermion loop. For Yukawa type theories, it is not easy to express the This subgraph is UV finite as each boson line insertion reduces recipe in simple terms. Typically we have no symmetry the power by one. The other subgraphs made of 4-point fermion restriction on the appearance of bosonic mass terms functions are also UV finite. nor the quadratic divergence. After taking the difference between two self-energies, the quadratic divergence cancels, while the logarithmic divergence in general couples D. The self-energies for bosons to quantities that depend on the fermion bases. Such The discussion for the bosonic self-energies should be logarithmic divergences must be combined with those much more involved than the fermion cases, since the naive appearing in the 4-point boson functions with fermion power counting indicates its leading divergence to be the loops. (Unlike the gauge theory cases, the 2-point boson quadratic order. The leading divergence can be cancelled in functions start with quadratic divergence so that 2-more the difference of the self-energies, while the next-to-leading boson insertions can reduce the divergence only to the order UV divergence (logarithmic divergence) may couple logarithmic one.) In short, the renormalization program of to the quantities dependent on the fermion bases. Our 2-point boson functions requires resummed 4-point boson task is to identify the recipe to remove the logarithmic functions in the subgraphs. In the end the problem is divergence. reduced to the renormalization of the Bethe-Salpeter Such recipe requires the detailed discussions which amplitudes appearing in the subgraphs [37]. After this depend on models. There have been detailed analyses in is done for all S, the single subtraction makes the boson the literatures which will not be repeated here; we give only self-energies UV finite. the outline of the necessary discussion. Two examples are In what follows, we will assume that we prepare the 2PI considered: gauge and Yukawa theories. functional which includes the proper set of graphs to make In both cases it is not sufficient to look at only a single the boson self-energies UV finite. graph; the UV divergence is eliminated only by summing Now we discuss the power counting of the difference up a proper set of graphs, or using the improved vertices of boson energies, Π½S − Π½S0, for different bases S and S0. which depend on our choice of bases for S. The choice of Before the subtraction the coefficients of l2 contain the graphs depend on the graphs used to calculate the fermion logarithmic divergence together with l-dependent functions self-energies. dependent of the fermion bases,

FIG. 11. A gauge theory example for the fermion loop contributions to the boson self-energies. The fermion bases are Sð≠StreeÞ. The 1-loop result with a unimproved vertex results in the violation of the gauge invariance and quadratic divergence. They are associated S → S with the change of bases tree . Improving the vertex to recover the gauge invariance eliminates the quadratic divergence.

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2 2þγ S 2 Λ 0 is the order of Λ . This piece can be canceled by terms Π½ ðlÞ ∼ c l ln UV þ Oðl Þ: ð41Þ UV univ l2 þ f½S from the Φ-functional,

0 μ where f½S is basis-dependent quantity of ∼l , and c μ ½S μ univ Φ½S − Φ½S ¼−Tr1½Σ k ðS − S Þ is the universal coefficient of the logarithmic divergence. k k X 1 δnΦ At l much larger than the nonperturbative scale (l ≫ f½S), μ n þ Trn n ðS − S Þ ; ð45Þ S0 n! δS μ k subtracting the self-energy for yields n 2 S S ¼ ¼ k Π½S l − Π½S0 l ∼ c f S − f S0 ∼ O l0 : ð Þ ð Þ univð ½ ½ Þ ð Þ ð42Þ Φ S Sμ where we expand ½ around k. The meaning of the As we have seen, this is necessary for the UV finiteness of second trace is that we replace the n-propagators in the the fermion self-energies. In the next section we will also Φ S − Sμ -functional by k, and then set the rest of propagators see that the same is required for the zero-point energy. Sμ to k. As shown in Fig. 12, this is purely an algebraic S S − Sμ Sμ operation: we can write ¼ð kÞþ k and draw all E. The zero-point energy S − Sμ Sμ possible diagrams made of k and k. n n Finally we discuss the zero-point energy assembling the δ Φ=δS μ We note that the vertex functions jS¼S are UV discussions given in the previous sections. k Let us begin our analyses with the single particle finite. Its evaluation can be done as performed in Sec. VB; μ n contribution TrLnS. Its expansion starts from the linear we can isolate -independent pieces of -point vertex order in ΔΣ, functions through the analytic continuation as in Eq. (28),

μ μ n n Tr1LnS − Tr1LnS −Tr1Ln 1 S ΔΣ δ Φ δ Φ μ 0 k ¼ ð þ k Þ f μ; S ¼ ; n ¼ n þ finite½ ð46Þ n δS μ δS μ¼0 μ¼0 X −1 S¼S S¼S ¼S ð Þ μ n k k ¼ Tr1½ðS ΔΣÞ : ð43Þ n k n 1 ¼ f μ; Sμ¼0 μ Sμ¼0 where finite½ is a functional of and . Here are n 1 ∼Λ2þγ δnΦ=δSn In our counting, the ¼ term is the order of UV that two remarks concerning with . First, its UV n n μ¼0 corresponds to the terms producing the quadratic diver- finiteness follows from the fact that δ Φ=δS at S is n 2 ∼Λ2γ n gences in Sec. III. The ¼ term is the order of UV the vacuum -point vertices which are UV finite. Second, −2 3γ S (∼ ln Λ for γ ¼ 0), and the n ¼ 3 terms are ∼Λ þ , and taking derivative with respect to , at each time the UV UV S so on. (Clearly this expansion is useless for γ ≥ 2.) integration over loop momentum disappears while the ˜ in the denominator adds one power of momentum, so the Meanwhile the Tr1½SΣ terms in the 2PI action yield powers of δnΦ=δSn are ∼p4−3n. S μ S μ μ S SΣ˜ ½ − S Σ˜ ½ k S ΔΣ S − S Σ˜ ½ ; Assembling the above-mentioned three pieces of con- Tr1½ Tr1½ k ¼Tr1½ k þTr1½ð kÞ tributions to IΔS, we find ð44Þ X −1 n S;μ −1 μ −1 S −1 ð Þ μ n Σ˜ ½ S − S Σ˜ ½ k S − IΔS½S; μ¼ Tr1½ðS ΔΣÞ ¼ ð treeÞ where we used ¼ k n k Sμ −1 Σ S − Sμ ≃−Sμ ΔΣ Sμ ∼ n¼2 ð treeÞ ¼ k. In our counting k kð Þ k μ μ ˜ ½S;μ ½S p−3þγ þ Tr1½ðS − S ÞðΣ − Σ k Þ . Here we observe that the first term in the RHS k can be used to cancel the n ¼ 1 term in the Tr1LnS terms in X 1 δnΦ μ n þ Trn n ðS − S Þ : ð47Þ Eq. (43), i.e., the strongest UV term in the single particle n! δS μ k n 2 S¼S contribution. On the other hand the second term in the RHS ¼ k

Φ S S Sμ FIG. 12. An example for the expansion of ½ around ¼ k. Shown is the Fock term.

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μ S S − S 1 μ The replacement of with accompanies the reduc- μ μ μ ½S μ k IΔS½S; μ ≃− Tr1½ðS − S ÞðΣ − Σ Þ tion of the powers from p−1 to ∼p−3þγ, so the terms in the 2 k k μ μ 4−nð2−γÞ μ μ ½S ½S ∼Λ γ < 2 1 S − S Σ − Σ k last sum of the RHS is UV and for the þ Tr ½ð kÞð Þ 2γ divergence is at most ∼Λ . 1 δΣ UV μ μ n 2þγ − Tr2 ðS − S Þ : ð49Þ We note that the worst divergences, ∼Λ , have been 2 δS μ k UV S¼S cancelled. Indeed the powers of ΔΣ start from the quadratic k order, ∼ðΔΣÞ2, in which the powers are reduced by p4−2γ. Finally we expand In the context of the quadratic divergences discussed in Sec. II and III (which were discussed with the assumption μ μ δΣ S ½S μ μ of γ ¼ 0), we first add them, and then subtract the double Σ½ Σ k S − S ¼ þ ð kÞþ ð50Þ δS S Sμ counted contributions; this results in the cancellation of the ¼ k worst divergences. For this cancellation it is essential to keep track all the effects associated with the change of with which we end up with fermion bases; some appears from the single particle μ contributions and the other from composite particles or μ 1 μ μ ½S μ 3 IΔS½S ; μ¼− Tr1½ðS − S ÞðΣ k − Σ Þ þ OðS − S Þ : backgrounds. Now the leading divergence is weaker than 2 k k k ∼Λ2γ naive expectations, and is the order of UV which ð51Þ ∼ Λ γ 0 becomes ln UV at ¼ . We can proceed further. From now on we focus on our −2þ3γ The last term yields ∼Λ and will be neglected. We canonical case, γ ≃ 0, and examine the logarithmic diver- UV could not find a way to further simplify the expression of gence. We pick up the UV divergent pieces in Eq. (49) as Eq. (51), so at this point we stop transforming it, and then look into more details. μ 1 ½S μ μ ˜ ½S;μ μ Σ k − Σ IΔS S; μ ≃− Tr1 S − S Σ − Σ First we recall that the structure of is given by ½ 2 ½ð kÞð kÞ k μ Eq. (21) or Fig. 3. Therefore Eq. (51) yields graphs shown μ S;μ ½S S − S Σ˜ ½ − Σ k þ Tr1½ð kÞð Þ in Fig. 13 where we pick up graphs with 1-fermion loop as θ 1 δΣ examples. If we decompose the graph into the -function − S − Sμ 2 : Tr2 ð kÞ ð48Þ part and the others, the latter can be regarded as the self- 2 δS S Sμ ¼ k energy graph in which single subtraction is done for the fermion loop. We have analyzed the UV structure of these μ μ μ μ graphs in Sec. VI C looking at graphs in Figs. 9 and 10.We where we used S ðΔΣÞS ≃−ðS − S Þ, ΔΣ½S ¼ Σ˜ ½S − Σ , k k k k concluded these graphs to be finite (after the vertex 2 2 and δ Φ=δS ¼ −δΣ=δS. improvement). Contracting these UV finite self-energies So far we have not used specific properties of S except its with the θ-function leaves UV finite results. asymptotic behaviors. Now we substitute the solution of the I μ With this we conclude that the ΔS is UV finite, as far as Schwinger-Dyson equation, S, in place of S. Then, by the propagator S leads to the UV finite self-energies for Sμ μ −1 μ −1 Sμ Σ˜ ½ S − S Σ½ definition ¼ð Þ ð treeÞ ¼ , and we obtain fermions and bosons.

μ ½S μ Σ k − Σ FIG. 13. A 1-fermion loop example of Eq. (51). The self-energy difference k is indicated by a box with blue lines. It is Sμ − Sμ contracted with the subtracted propagator k. The box with green lines indicates the self-energy graph with one subtraction. They have the same UV structure as graphs shown in Figs. 9 and 10 and are UV finite. As a result these graphs are UV finite after contracting with the θ-function.

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VII. SUMMARY us such computation and cancels the apparent UV divergences which are expected from naive quasiparticle pic- We have investigated the UV divergences in the zero- tures for the constituents and composites. point energy of composite particles that are associated with We emphasize again that our attention in this paper is not the change of fermion bases. We use the 2PI functional to necessarily restricted to renormalizable theories. Rather our keep track all the impacts due to the change of bases, and intention is to understand the UV structure in general show that the UV divergences all cancel, provided that the context so that we will be able to construct models or invent in-medium fermion propagator at high energy approaches efficient cutoff schemes for effective theory calculations sufficiently fast to the vacuum counterpart. While we started with the examples of 2-particle corre- with physically motivated UV cutoff. After the removal lation or composite particles made of 2-particles, in the of the artificial UV divergences, only the dependence on discussion of 2PI functional we did not refer to the specific the physical cutoff remains so that our task is reduced to form of the Φ-functional, so the discussion can accom- quantifying such cutoff scale from the microscopic calcu- modate also the 3-, 4-, and infinite particle correlations. lations. The practical implementation of composite par- In phenomenology, the present discussions may be ticles will be presented elsewhere. important for the QCD equation of state at high baryon density where baryons merge exhibiting quark d.o.f. ACKNOWLEDGMENTS The relevant effective d.o.f. change, and there must be a framework which simultaneously treat baryons and quarks This work was supported by NSFC Grants while avoid the double counting. The 2PI formalism allows No. 11650110435 and No. 11875144.

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