CCosmological constant as a finite temperatureeffect
I. Y. Park † Department of Applied Mathematics, Philander Smith CollegeLittle Rock, AR 72223, [email protected]
Abstract
The cosmological constant problem is examined by taking an Einstein–scalar with a Higgs-type potential and scrutinizing the infrared struc-ture induced by finite temperature effects. A variant optimal per-turbation theory is implemented in the recently proposed quantum-gravitational framework. The optimized renormalized mass, i.e., therenormalized mass determined by the variant optimal perturbationtheory, of the scalar field turns out to be on the order of the tempera-ture. This shifts the cosmological constant problem to compatibility ofthe consequent perturbative analysis. The compatibility is guaranteedessentially by renormalization group invariance of physical quantities.We point out the resummation behind the invariance. a r X i v : . [ h e p - ph ] J a n Introduction
The cosmological constant (CC) problem [1] (see, e.g., [2–5] for reviews )arises from the loop effects of Standard Model (SM) particles, such as theHiggs particle. Since the electroweak scale is much higher than, say, thetemperature of the cosmic microwave background (CMB), it is a standardpractice to apply the zero-temperature setup to formulate and tackle theproblem. However, since the CC as a vacuum energy is an infrared effectand thus governed by the low energy sector, the infrared structure of thetheory must be important and its meticulous description is desirable. Thefact that the CC is a vacuum energy also implies that quantization andrenormalization of gravity must be involved in its systematic treatment. Inparticular, the solution of the problem would require renormalization of thevacuum energy. In this work we show that, when properly taken into accountin the quantum gravitational setup, the finite temperature effects allow oneto avoid the CC fine-tuning problem.Are there indications that the finite temperature effects may a priori beimportant for CC analysis, despite the fact that the electroweak scale ismuch higher than the CMB temperature? First of all, it should definitely bepossible, and is natural, to obtain zero-temperature results as a vanishing-temperature limit of the corresponding finite-temperature results. A hintof an indication comes from zero-temperature loop analysis, which typicallyyields logarithmic factors such as ln mµ , where m is the mass of the field and µ the renormalization scale. For the benefit of convergence, it is necessaryto choose µ ∼ m . By the same token it will be necessary to take µ ∼ m ∼ T once the temperature enters. In the present work this scaling is achieved inthe course of improving the perturbative analysis by optimal perturbationtheory (OPT) as well as standard thermal resummation: we show that thereis an OPT procedure that enforces the scaling.In the body we reformulate the CC problem as a zero-temperature limitof the finite temperature counterpart. A potential obstruction to any finite- The inspiring review by J. Sola [5] is largely based on the previous works by him andhis collaborators, [6–8] (and some others), that have been further developed in [9–11]. Oneof the main themes of these works is the so-called running vacuum. These works will befurther commented on in the conclusion and [35]. OPT is based on the variational principle. Other thermal physics techniques based onthe variational principle include screened perturbation theory [12] [13] and 2PI formalism[14]. thus removing the root of theCC fine-tuning problem.With the renormalized mass determined, the following task still remains:the zero-temperature theories, such as the zero-temperature Standard Model,have been quite successful. There, the renormalized masses turn out to beclose to the pole masses, usually within a few percent. In the case of theSM Higgs field, for instance, the renormalized mass is close to 125 GeV,the pole mass value. If one now wants to take the renormalized mass to bearound the CMB temperature, which is much smaller than the pole mass,one must yet maintain compatibility with the zero-temperature analysis: theresulting perturbation theory should preserve the success of the original zero-temperature theory. In the body, after examining vast freedom in choosingrenormalization conditions, we invoke renormalization group invariance ofphysical quantities to affirm this. We note that a certain resummation isbehind the invariance.With the renormalized mass being of the order of the temperature, the CCproblem is avoided: the CMB temperature in terms of eV is 6 . × − eV.Approximating this as 10 − GeV, the vacuum energy contribution associatedwith the thermal mass of a Higgs-type field is ∼ − GeV . This is roughlyof the same order as the observed CC value ∼ − GeV . It has been revealed in the recent works of [25, 26] that quantum corrections canqualitatively change the classical solution. In the present work we see a similar novelty: thefinite temperature non-perturbative effects dictate, small temperatures notwithstanding,the renormalized mass. a ) ( b )Figure 1: Diagrams for one-loop self-energy: (a) one-loop correction for prop-agator (b) its counter-term In section 3, we put forth an OPT that leads to the renormalized mass ofthe order of the temperature. This raises a question on consistency of theresulting perturbative analysis, since the new renormalized mass is muchsmaller than the actual physical mass. Although in nature the modificationamounts to finite renormalization and thus must not affect the physics, itwill be useful to take a close look at how the perturbative analysis modifiesin the new scheme. As we now demonstrate by considering two-loop renor-malization of the propagator of a real scalar theory with a quartic potential,the key to preserving the success of the zero-temperature lies in vast freedomin choosing subtraction schemes.Let us consider the following scalar system in a flat background S = − (cid:90) d x (cid:104) ∂ µ ζ∂ µ ζ + 12 m ζ (cid:105) − (cid:90) d x λ ζ . (1)The two-point proper vertex is defined asΓ (2) ≡ k + m − Σ( k ) (2)where Σ denotes self-energy. At one-loop, Σ can be computed by consideringthe diagrams in Fig. 1: the one-loop two-point divergence introduces thecounter-term: : − m λ (4 π ) (cid:16) ε + c m (cid:17) ζ (3)where ε ≡ − D and D denotes the spacetime dimension; c m is a constant to bedetermined by one’s subtraction scheme. For instance, the modified minimal4igure 2: Diagrams for two-loop self-energysubtraction (MS) corresponds to setting c m = 0. Fixing the renormalizedmass according to the OPT principle of minimal sensitivity (as we will insection 3) is in contrast to the usual practice in zero-temperature: there, onefixes c m by one’s subtraction scheme, and then the renormalized mass by thepole mass condition, k + m − Σ( k ) (cid:12)(cid:12)(cid:12) k = − m P = 0 (4)where m P denotes the physical pole mass. In the new scheme, it is thecoefficient c m that is determined by the pole mass condition, while the renor-malized mass is fixed by the OPT.As stated in the introduction, the advantage of the new scheme is obvious:it realizes the scaling mentioned in the introduction and thereby allows oneto avoid the CC fine-tuning problem. To see the disadvantage, let us notethat the pole mass condition in the new scheme yields c m m ∼ m P − m (5)and thus implies a larger value of c m , compared with the standard approachwhere the pole mass condition typically leads to m (cid:39) m P . In general, formass-related quantities it will not be possible at tree-level to achieve suitableagreement with experimental values, since the renormalized mass is pre-fixed:it will be necessary to go to one-loop where one has the freedom of adjustingthe finite parts. The two-loop-relevant diagrams are given in Fig. 2. Thecircle in the last diagram in Fig. 2 represents the counter-term for the one-loop four-point diagram (not explicitly shown). Adding all up, the totaltwo-loop self-energy Σ (2) isΣ (2) = m λ µ ε (4 π ) (cid:104) ε + 14 ε ( − c m + 3 c λ ) − p m ε + · · · (cid:105) (6)5here c λ denotes the finite part of the counter-term of the one-loop four-pointamplitude and p the momentum entering through one end of the diagrams.The ε -pole terms will have to be removed by two-loop counter-terms. Letus focus on the finite parts. The constant c m will also appear in the finiteparts represented by the ellipses. By imposing the pole mass condition andsolving it, say, interactively for c m , it will be possible to determine its two-loop correction. Since Σ (2) has an additional λ and (cid:126) compared with one-loop,the large value of c m cannot disturb the series in any significant way. After all,the new perturbation can be viewed as finite renormalization. We will pointout later that a certain resummation is behind this finite renormalization. The crux of the variant optimal perturbation theory (OPT) can be capturedby considering a scalar system in a flat spacetime. Since the UV divergencesoriginate locally from a short distance, they are insensitive to global geome-try. For this reason, the zero-temperature UV regularization can be employedin a finite-temperature theory. As for quantities depending on the infraredstructure, the prime example of which is vacuum energy, one must considerin principle the actual background. The difference between using the curvedbackground and the flat one lies in the finite parts. (However, the finite partsare adjusted by the renormalization conditions anyway; we refer to [27] forfurther discussion.)In thermal field theory, convergence of perturbative analysis is improvedby resummation. The convergence can be further enhanced with a touch ofnon-perturbative techniques, such as OPT. The OPT implemented in thiswork is a relatively minor, but nonetheless crucial, variation of the widely-studied one. In the widely-used OPT, an artificial mass term is subtractedout after adding. This is one way of ensuring artificial-mass independence ofthe full closed results. In our case, the renormalized mass itself serves as theOPT parameter to be fixed by the OPT principle of minimal sensitivity. Since the issue under consideration belongs to the mass, one may well set c λ = 0in the spirit of the MS. Similarly, the two-loop analogues of c m , c λ (and the finite partassociated with the wavefunction renormalization) can be set to zero. In other words, toomuch freedom in choosing the finite parts can be a burden: the freedom remaining afterdetermining c m can be fixed just as in a convenient subtraction scheme, such as the MS,to facilitate next steps, e.g., solving the renormalization group equations. S = 1 κ (cid:90) d x √− g R − (cid:90) d x √− g (cid:16) g µν ∂ µ ζ∂ ν ζ + V ( ζ ) (cid:17) (7)where κ = 16 πG with G being Newton’s constant. The potential V ( ζ ) is V ( ζ ) = λ (cid:16) ζ + 6 λ ν (cid:17) . (8)Note the notation change as compared with section 2: the mass has been de-noted by ν . One conceptual hurdle is the justification of the complete-squareform of the potential instead of the usual V = ν ζ + λζ . The value ofCC depends, of course, on whether one uses the complete-square form or theform without the constant piece. A shift of potential by a constant is imma-terial in flat spacetime quantum field theory. The same is not true, however,in the quantum gravitational context. Whether one should use the complete-square form or the more usual form is not part of the CC problem. A closelyrelated question, whose answer is not currently known [28], is why the mini-mum value of the classical Higgs potential should be taken to be zero. It isan independent problem that must ultimately be answered experimentally.Our goal here is to show that in the setup dictated largely by convergenceof thermal perturbation theory, the fine tuning-problem is not present; thisgoal can be achieved more conveniently with the complete-square form.To set the stage for the refined BFM [27, 29–31], we shift the fields as g µν → h µν + ˜ g µν , ζ → ˆ ζ + ˜ ζ (9)with ˜ g µν ≡ g c µν + ϕ µν , ˜ ζ ≡ ζ c + ξ (10)where g c µν , ζ c denote the classical solutions, ϕ µν , ξ the background fields,and h µν , ˆ ζ the fluctuation fields. The loop analysis is based on the followingtwo-point functions (see [31] for the conventions). For the metric, < h µν ( x ) h ρσ ( x ) > = ˜ P µνρσ ˜∆( x − x ) (11)where the tensor ˜ P µνρσ is given by˜ P µνρσ ≡ ¯ κ (cid:16) ˜ g µρ ˜ g νσ + ˜ g µσ ˜ g νρ −
12 ˜ g µν ˜ g ρσ (cid:17) ; (12)7here ¯ κ ≡ κ and satisfies˜ P µνκ κ ˜ P κ κ ρσ = ˜ P µνρσ . (13)˜∆( x − x ) is Green’s function for a (massless) scalar theory in the backgroundmetric ˜ g µν : < ˆ ζ ( x ) ˆ ζ ( x ) > = ˜∆( x − x ) . (14)The explicit form of ˜∆ for a massive scalar theory will be given and utilizedin section 3.2 where the curved space analysis of the matter-involving sectoris conducted. As stated in the beginning, the crux of our OPT is captured by considering ascalar system in a flat spacetime. We employ the MS subtraction scheme inthe present section. We will come back to the deviation from the MS schemein section 3.3.The starting point of the OPT-improved thermal resummation can betaken as the following renormalized action S ( ζ ) = − (cid:90) d x ∂ µ ζ∂ µ ζ − (cid:90) d x (cid:16) M ζ + λ ζ (cid:17) − (cid:90) d x ν λ (15)with M ( T ) ≡ ν + λ T (16)Shift the field ζ → ˆ ζ + ˜ ζ (17)where ˆ ζ, ˜ ζ denote the fluctuation field and background field, respectively.Since we are interested in the potential as opposed to the action, the back-ground field ˜ ζ can be treated as a constant. Then the potential can beeffectively computed by considering the field-dependent mass term M ( T ) → ˜ M ( T, ˜ ζ ) = ν + λ T + λ ζ (18)8nd integrating out the fluctuation field ˆ ζ . A remark is in order beforeproceeding to determination of the OPT-induced renormalized mass. Westated earlier that although we consider a CMB-order temperature, the high-energy expansion can be utilized. The temperature being high or low isrelative to the mass and we will show below that our OPT implies ˜ M ∼ (cid:126) / T ( (cid:126) will be kept implicit). One can now tell why the high-energy expansion isjustified: since the auxiliary mass ˜ M satisfies ˜ M /T ∼ (cid:126) / , the intermediateanalysis corresponds to that of high temperature: ˜ MT << M - and ˜ M - dependence isimportant. Let us focus on the one-loop potential; after carefully followingthese terms, one gets V opt( ˜ ζ ) = 3 ν λ − π T − ˜ M π ln ¯ µe γ E πT + 124 ˜ M T + 12 (cid:16) ν + λ T (cid:17) ˜ ζ − π ˜ M T + 14! λ ˜ ζ + O (cid:16) ˜ M T (cid:17) (19)where ¯ µ ≡ µ (cid:16) πe γE (cid:17) / is a scaling parameter of dimensional regularization(with the MS scheme). The field equation associated with (22), ∂∂ ˜ ζ V opt = 0,yields λ ζ + M − λ (4 π ) ˜ M ln ¯ µe γ E πT + 124 λT − π ˜ M T = 0 (20)up to terms of two-loop order. The solution is˜ ζ ( M ) (cid:39) − M L + 14 π (cid:18) − π T + 3 √ πT √− M − M ln ¯ µe γ E πT (cid:19) . (21)With this, one gets the following onshell potential: V opt = 3 ν λ − π T − ˜ M π ln ¯ µe γ E πT + 124 ˜ M T + 12 M ˜ ζ ( M ) − π ˜ M T + 14! λ ˜ ζ ( M ) + O (cid:16) ˜ M T (cid:17) . (22)By solving the following PMS condition for ν ∂V opt ∂ν = 0 (23)9ne gets ν = − λT . (24)which leads to ζ = − T . (25)Once the potential (22) is evaluated with this value, one gets V opt = − π T ∼ T (26)which then allows one to avoid the fine-tuning problem.The complete two-loop offshell form of the potential will be presentedin [35]. There one encounters a novelty: the potential develops an imaginarypart, signaling instability of the vacuum. More on this in the conclusion.
Whereas what we referred to as the second-layer perturbation in [27] is nec-essary for the pure gravity sector computation, there exists, for the mattersector, a powerful shortcut based on the first-layer perturbation, the “one-stroke” method. In the present work the first-layer perturbation will be usedexclusively. From the results obtained, it becomes evident that the qualita-tive conclusion of the flat space analysis remains unchanged.
Graviton sector
Let us recall the zero-temperature case first. In [30] and [27], we conductedthe computation in a brute-force manner by employing the second-layer per-turbation and viewing the classical CC as the graviton mass term. As shown, A similar result of ν ∼ λT was obtained in [36] (see also [37] and [38]) by consideringrenormalization group and choosing appropriate renormalization conditions. More on thisin section 3.3. Strictly speaking, the potential itself remains real even at two-loop. However, this isbecause the source of the imaginary parts, ˜ M , does not contribute even at two-loop. Dueto the expected contribution of ˜ M term (and the terms with higher odd-integer powersof ˜ M that should appear in higher-loop computations), it is expected that the complexityof the potential will become manifest at three-loop and on. ∼ (cid:90) (cid:112) − ˜ g. (27)Strictly speaking, for a flat background, the one-loop results vanish in di-mensional regularization in the absence of the classical CC treated as thegraviton mass term. For consistency with the observed value of the CC, theclassical (i.e., renormalized) CC will have to be set to ∼ T in the renormal-ization program described below, and thus will not qualitatively affect theproposed resolution of the CC problem.The results of n -loop with n ≥ P µνρσ and (13). An arbitrary n -loop graph with n ≥ < h α α h α α h α α · · · h β β h β β h β β · · · > (28)where the upper and lower indices are fully contracted. Contractions of thefields lead to ˜ P µ ν ρ σ ˜ P µ ν ρ σ · · · (29)where again, all the indices are contracted one way or another with ˜ g µν ’s.Whenever a pair of ˜ P ’s have a pair of the indices contracted, the explicitexpression for ˜ P given in (12) can be used. One can also use (13) to reducethe total number of ˜ P ’s until only one ˜ P remains at the end. Since all ofthe indices must be contracted, the final expression must be ∼ ˜ P µν µν , andtherefore vanishes. One important implication of this analysis is that it isnot necessary to conduct resummation or OPT in the gravity sector. Matter-involving sector
The matter-sector diagrams can be subdivided, depending on whether or notthey involve a graviton loop. The matter-involving vertices do not, unlike thepure graviton vertices, come with κ . Since the graviton propagator comeswith κ , the diagrams leading in κ are those with a matter loop, which areour focus.There exists a highly effective “one-stroke” method of computing the matter-involving part of the effective action, in which the flat spacetime analysis can11 a) (b) Figure 3: Resummation behind the new scheme: (a) a connected diagram inMS scheme (b) resummation required in the new schemebe entirely carried over. For this, note the explicit form of ˜∆( x − x ) canbe written as˜∆( x − x ) = (cid:90) d k (2 π ) (cid:112) − ˜ g ( x ) e ik · ( x − x ) ik µ k ν ˜ g µν ( x ) . (30)Defining “flattened” momentum and coordinates as K α ≡ ˜ e µα k µ , X β ≡ ˜ e βν x ν , ˜ e µα ˜ e νβ ˜ g µν = η αβ (31)where the underlined indices are flattened, one gets the flattened propagator:˜∆( X − X ) = (cid:90) d K (2 π ) e iK γ ( X − X ) γ iK α K β η αβ . (32)When computing a diagram, one can pull out all of the background fields andcontract the fluctuation fields. The propagators can then be transformed tothe above. Afterward, the steps become parallel to those corresponding tothe flat cases. The matter part of the effective action can thus be computedexactly in the same manner in which it is computed in the flat case. With the renormalized mass around the CMB temperature, one should makesure that that framework preserves the success of the zero-temperature theorysuch as that seen in zero-temperature SM. The new perturbation has been12llustrated in section 2 by taking a simple scalar theory. In the case of the SM,each sector in the SM, i.e., the Higgs, gauge, and fermion, the renormalizationprocess should be modified from that of the standard renormalization scheme,e.g., MS. More specifically, in the modified scheme, the pole mass value mustbe realized by adjusting the finite parts of the divergent integrals that arechosen differently from those in the standard renormalization scheme: inanalysis with MS, the pole mass condition is part of the MS scheme, and therenormalized mass is not determined prior to the pole mass condition. It isthe pole mass condition that determines the renormalized mass. In contrast,in the new scheme the renormalized mass is determined, as demonstrated insection 3.1, by the OPT. The finite parts are to be determined by the physicalpole mass condition. What it implies is that it is necessary to go to the one-loop level to achieve the accuracy of amplitudes obtained in MS. In orderto match the values of, e.g., the tree amplitudes computed in MS, one mustinsert one-particle-irreducible diagrams to the internal lines. (We anticipatedthat one-loop insertion will normally be sufficient.) The insertions amountto a certain resummation. The situation is generally illustrated in Fig. 3. Inlight of this, it is also worth noting that the ν ∼ T scaling was previouslyobtained in [36] by choosing appropriate renormalization conditions. Since some of the Standard Model particles, such as the Higgs, are massive,the matter contributions to the CC are naively expected to be larger thanthat of the graviton. The variant OPT reveals, however, that the ultimatedetermining factor of the CC is the temperature. We believe that this iden-tifies the cosmological constant problem at its root. This leads to necessityof employing a new renormalization scheme. In the new scheme, one needsto go a few orders higher and perform a certain resummation to achieve thesame level of proximity to the values of the physical observables as in thezero-temperature standard schemes. Given what it brings, this seems to arelatively small price to pay.The finding in the present work that a non-traditional renormalizationscheme is needed to tackle the cosmological constant problem is in line withthe observation, e.g., in [7]. The overarching umbrella for this and [6, 8–11]is the so-called running vacuum with time-varying and dynamical energy, aview that we find very natural. In these works, the cosmological constant13roblem was coherently tackled from different angles and various cosmolog-ical implications of such a vacuum were discussed. (Even the relevance oftemperature to the cosmological constant problem was discussed to someextent.) Several different regularization methods and their pertinent renor-malization schemes were analyzed. We believe that the present work lendssupport, by providing independent motivation to consider a non-standardrenormalization scheme, to those approaches. A more in-depth account ofthose works will be presented in [35].We end with three ramifications of our results and future directions. Theresults obtained in this work suggest that, earlier in the thermal history ofthe Universe, the value of the CC should have been larger. In other wordsthe CC becomes time-dependent through the temperature, and the presentsmall value must be due to the age of the Universe [28]. More quantitatively,the following is the basis for this anticipation. In [30] it was shown that thereexists a time-dependent solution that approaches the minimal value of thepotential. By using the one-parameter family of the potentials labeled by thetemperature, one can repeat the analysis with the present finite temperaturepotential. One can then analyze the resulting quantum-level action, and itshould be possible, at the time-dependent solution level, to establish a CCthat decreases to a small value. One may introduce a renormalized CC,Λ ren . To be consistent with the fact that the observed value of the CCis small, one will have to take Λ ren to be small. It will be of interest topursue this line of study. Another ramification, not unrelated to the first,is that the large value of the CC becomes natural when the temperature ison the order of the EW scale. (This may be something profound, and haveimplications for the hierarchy problem in the SM.) Since the Universe was athigher temperatures in the previous eras, it will be a meaningful endeavor toexplore whether one could come up with a streamlined description coveringthe entire temperature range, say, from the near-Planck era to the present.With the recent progress in the OPT literature, the present results indicatetoward an affirmative answer. 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