We show that in the RN gravitation model, there is no dynamical symmetry breaking effect in the formalism of the Schwinger-Dyson equation (in flat background space-time). A general formula for the second variation of the gravitational action is obtained from the quantum corrections hμν (in arbitrary background metrics)

Kotvytskiy, A. T.

Kruchkov, D. V.

Acta Polytechnica

English

A. T. Kotvytskiy

D. V. Kruchkov

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Dynamical Symmetry Breaking in RN Quantum Gravity

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Acta Polytechnica Vol. 51 No. 4/2011Dynamical Symmetry Breaking In RN Quantum GravityA. T. Kotvytskiy, D. V. KruchkovAbstractWe show that in the RN gravitation model, there is no dynamical symmetry breaking eﬀect in the formalism of theSchwinger-Dyson equation (in ﬂat background space-time). A general formula for the second variation of the gravitationalaction is obtained from the quantum corrections hμν (in arbitrary background metrics).Keywords: quantum gravity, Schwinger-Dyson equations formalism, dynamical symmetry breaking.The study of dynamical mass generation anddynamical symmetry breaking in diﬀerent externalﬁelds [1, 2], including the gravitational ﬁeld [3–5] isan important step in studying fundamental interac-tions, which can either be considered in ﬁeld the-ory and an attempt at quantization can be madeor considered as an external interaction. This pa-per is devoted to the possibility of dynamical sym-metry breaking in such gravitation models using theSchwinger-Dyson equation formalism [6–9]. The pri-mary goal is a study of some gravity properties off(R)-gravity [10], where R is a scalar curvature. Oneof the most discussed variants of such models is RNgravity [11, 12]. In particular, we are interested in apossible eﬀect of dynamical chiral symmetry break-ing in this model under graviton and fermion inter-action, which leads to mass formation of fermions.This problem is important for several reasons. First,quantum corrections should be taken into considera-tion from diﬀerent scenarios describing the evolutionof the early universe. Second, an understanding ofthe dynamical symmetry breaking mechanism is im-portant for black hole physics. Third, RN — gravity,as a modiﬁed theory of General Relatity, is also in-teresting from the phenomenological point of view,the peculiarity of which appears in various cosmo-logical models. In particular, it is necessary to in-troduce either dark energy or a quintessence with amuch more exotic state equation to explain the ac-celerated expansion of the observed universe (beingwithin the limits of relativity theory). To sum up,it is signiﬁcant for a description of the future of theuniverse. However, if one modiﬁes gravity in a properway, it is also possible to receive interesting cosmo-logical dynamics without introducing new concepts.Here, we provide a comparative analysis of the aboveformalism both in the case of ﬂat background met-rics (Minkowski space) and in the case of an arbitrarybackground.Let us expand the four-dimensional space-timemetric as followsg˜μν  gμν + hμν , (1)where g˜μν is perturbed metric, gμν is backgroundmetric, hμν are quantum corrections. This deﬁni-tion leads to the following results. In the case ofﬂat background space-time in the RN gravity modelthere is an existence of corrections like O(hN ) or-der and higher that gives no possibility to get thenecessary equations. We can speak about the ab-sence of a symmetry breaking eﬀect (at least in theSchwinger-Dyson equation formalism). However, ifthe background metric is curved, then corrections oforder appear and, particularly, speciﬁcally quadricones O(h2). They allow us to obtain the Schwinger-Dyson equations and to test them for possible dy-namical symmetry breaking. An important conclu-sion is the following: if our real Universe is describedby a curved metric, then while constructing quan-tum gravity theory, we should take into consider-ation the summands of all powers in R, as theyprovide the same degree in small quantum correc-tions h.1 Schwinger-Dyson equationsOne possible method for a dynamical symmetrybreaking study is the Schwinger-Dyson equation for-malism. Since it is impossible to write closed systemequations for all elements of the Feynman diagram,we have to use some approximations, which allow usto solve the Schwinger-Dyson equation and to ﬁndthe type of exact propagator. Exact propagators andthe vertex part are connected by the integral relation,S−1 − S−10 = iδΓ2δS(2)54Acta Polytechnica Vol. 51 No. 4/2011where S, S0 – exact and free fermion propagators,δΓ2δS– describes two particle and irreducible interac-tion diagrams, and Γ2 is a part of the eﬀective actionΓ[S] = −iSp (LogS−1 + S−10 S)+ Γ2[S]. (3)Here we conﬁne ourselves to the exact fermionpropagator, which is written down in original typeS(p) =1A(p2)pμγμ −B(p2) , (4)where A(p2), B(p2) are some unknown functions ofthe fourth momentum p, γμ are Dirac matrices. Thenthe Schwinger-Dyson equation for this propagatorcan be put down like [13–15](A− 1)pμγμ −B =∫d4q(2π)4iΓαβ(p, q − p)S′(q) ·Γ′μν(q, p− q)G′αβμν(p− q), (5)where Γ′, G′ are an exact vertex function and an ex-act graviton propagator.The inﬁnite set of SDEs determining the exactfermion and boson Green functions, as well as the fullinteraction vertex, can be solved within some trunca-tion version only. This means that the only subset ofFeynman graphs is taken into account, which natu-rally leads to the disappearance of the magical cancel-lation of gauge-dependent terms in the S-matrix ex-pansion. For this reason, the most widespread ladderapproximation gives gauge-dependent results, whenthe fermion Green function is treated to be exactonly in the case when the boson propagator and theinteraction vertex are taken to be free [14]. In thisway, we can deﬁne the functions A(p2), B(p2) for thequantum RN -gravity.The following important remark should also bemade. We have to decide what kind of gravity actionand gauge-ﬁxing term should be used. It is conve-nient to bring the following conditionSgf =−β12M2∫d4x√−g (∇λhλμ − β2∇μh) ·(gμν∇ρ∇ρ + β3∇μ∇ν)× (6)(∇σhσν − β2∇νh) ,where β1, β2, β3 are arbitrary parameters. Then,putting down the second variation of the full actionwhich we can describe as the sum of the gravitationalﬁeld action and the gauge-ﬁxing action in the formδ2 (Sg + Sgf ) =12M2∫d4xhμνHμνρσhρσ. (7)Then the gravitational ﬁeld propagator is deﬁned asan operator inverse to Hμνρσ , that isGμνρσ =M2(H−1)μνρσ. (8)Let us include the gravitational ﬁeld in our con-sideration.2 Flat background metricHere we single out the part of the quadratic actionaccording to corrections h, in order to put down theSchwinger-Dyson equations. Instead of (1) the per-turbed metric will be written down asgμν  ημν + hμν , (9)where ημν is the Minkowski metric (we choose thesignature (+−−−)).We consider the following actionSg =12M2∫d4x√−gRN . (10)Note that action (10) does not have to be a fullaction for the gravitational ﬁeld, while the Einsteinlinear gravitation on the curvature and Λ — term,and other possible variants, can be also included.However, a discussion of the eﬀects caused by theform (10) is the main goal of this paper.In the case of a ﬂat background metric, we havethe following expansion (with accuracy up to the sec-ond order approximation)√−g  1 + 12h− 14hμνhμν +18h2, (11)where raising and lowering the index is by ημν back-ground metric and h = ημνhμν .The Riemann tensor is nowRμνρσ = ∂ρΓμνσ − ∂σΓμνρ + ΓμτρΓτνσ − ΓμτσΓτνρ, (12)while the Ricci tensor is determined by the Riemanntensor convolution according to the ﬁrst and the thirdindices.Then, for the scalar curvature we getR = gμνRμν  12(ημν − hμν + hμρhνρ)× (13)(∂α∂μhαν − ∂α∂αhμν − ∂μ∂νh+ ∂α∂νhαμ +O(h2))= ∂α∂νhαν − ∂α∂αh+O(h2).This implies, in the case of RN gravitation, thedevelopment according to h has the formRN  (∂α∂νhαν − ∂α∂αh+O(h2))N ∼ O(hN ) (14)That is, the smallest order has a power N inquantum corrections. Physically this means that thegraviton propagator is missed in such a gravitationmodel, and only N -particle vertex functions exist.Therefore, it is not essential for the study of dynam-ical symmetry breaking (DSB) in this formalism. In55Acta Polytechnica Vol. 51 No. 4/2011other words, we can say that there is no dynamicalsymmetry breaking eﬀect in this approximation.Now we proceed to the quite diﬀerent situation ofnon-ﬂat background space-time.3 Curved background metricLet us choose the action for the gravitational ﬁeld inthe form ofSg =12M2∫d4x√−g˜R˜N , (15)where tilde denotes a perturbed metric, and is deter-mined by (1).Since the background metric is not the Minkowskione, the summands of all h orders appear in all rede-ﬁned constructions. So the expressions for Christoﬀelsymbols can be reduced to the form (with accuracywithin the second order on quantum corrections)Γ˜μνρ  Γμνρ +12(gμγ − hμγ) · (16)(∇ρhγν +∇νhγρ −∇γhνρ) ,where Γμνρ are the Christoﬀel symbols calculated ac-cording to the unperturbed metric gμν , and ∇ρ is thecovariant derivative relative to the same metric.The complete expression for the Riemann tensor(in this approximation) isR˜μνρσ  Rμνρσ +12(gμτ − hμτ )×(hτεRενσρ + hενRετσρ +∇ρ∇νhτσ −∇ρ∇τhνσ −∇σ∇νhτρ +∇σ∇τhνρ) +14gμγgτε[(∇σhεν +∇νhσε −∇εhνσ) ·(∇τhγρ −∇ρhγτ −∇γhτρ) +(∇σhτγ −∇τhσγ +∇γhτσ) ·(∇ρhνε +∇νhρε −∇εhνρ)].Here, the procedure for raising and lowering theindices is provided by the background metric.Hence, we ﬁnd the scalar curvatureR˜ = g˜νσR˜ρνρσ  (gνσ − hνσ + hναhσα) R˜ρνρσ. (17)Introduce the following symbolsh1 ≡ −Rνσhνσ +∇ν∇σhνσ −∇ν∇νh, (18)h2 ≡ Rνσhναhσα − hρτ (∇ρ∇σ +∇σ∇ρ)hτσ −hρτ∇ρ∇τh+ hρτ∇σ∇σhρτ +12∇σhτσ∇τh−∇σhτσ∇γhγτ − (19)14∇τh∇τh+ 12∇τh∇γhγτ +34∇σhερ∇σhρε − 12∇εhρσ∇σhρε.The expression for scalar curvature isR˜  R + h1 + h2, (20)where h1 contains the ﬁrst power of the quantum cor-rections only, and h2 contains the second powers.Then, the N -th power of the scalar curvature be-comesR˜N  RN +NRN−1 (h1 + h2) + (21)12N(N − 1)RN−2h21.In the case of arbitrary background space-time weshould take into account the expansiondet(g˜)  det(g) + hμνKμν(g) + (22)hμνhαβFμναβ(g),whereKμν(g) = εαβρσ(δμ0 δναg1βg2ρg3σ + δμ1 δνβg0αg2ρg3σ +δμ2 δνρg0αg1βg3σ + δμ3 δνσg0αg1βg2ρ).We underline the fact that if the background met-ric is the Minkowski one ημν , then function Kμν =−ημν and with accuracy within the ﬁrst order we getthe well-known formula −g˜  1 + h. The expressionfor Fμναβ(g) is cumbersome, and we do not presentit here.To consider the possibility of dynamical symme-try breaking, it is necessary to have an expression forthe second variation of the gravitational action. Letus take into consideration the following expansion√a+ x1 + x2 √a+x1 + x22√a− x218√a3, (23)where x1 + x2  a, and also (x1)2 ∼ x2.Then,√−g˜ √−g − hμνKμν − hμνhαβFμναβ √−g − hμνKμν + hμνhαβFμναβ2√−g − (24)(hμνKμν)28√−g3 .Thus, we obtain the ﬁnal form of the second vari-ationδ(2)Sg =12M2∫d4x(√−g ·(NRN−1h2 +12N(N − 1)RN−2h21)−N2√−ghμνKμνRN−1h1 − (25)hμνhαβFμναβ2√−g RN − (hμνKμν)28√−g3 RN).56Acta Polytechnica Vol. 51 No. 4/2011Note, that in the case of the quadric gravitation(N = 2) and a ﬂat background (R = 0), instead of(25) we obtainδ(2)Sg =12M2∫d4x√−gh21, (26)which coincides with [13].So, it has been shown that in the case of any arbi-trary background metric for any N there is an equa-tion (25). Hence, we obtain the propagator (8) andthe Schwinger-Dyson equations (5). This indicatesthat the eﬀect of dynamical symmetry breaking ispossible in RN -gravity.4 ConclusionsSome quantum properties of model RN -gravity havebeen considered in the paper. A comparative analysisfor two cases: a) a ﬂat background space-time, andb) an arbitrary curved background has been carriedout.Expanding this model of gravity with quantumcorrections hμν , we found that in the ﬁrst case thesmallest order of quantum corrections is N . Thismeans that in the quantum theory of the RN -gravitygraviton propagator (for N > 2) does not exist, andthere is a vertex function of graviton-graviton inter-action that is not used in this formalism. Thus, thereis no Schwinger-Dyson equation (5) and, therefore,there is no eﬀect to discuss.In case b, a general formula (25) for the secondvariation of gravitational action on the quantum cor-rections hμν is obtained, which in the limit R → 0coincides with the previously known results. It isdetermined that in this formulation of the problemunder the eﬀects of dynamical symmetry breakingresearch the terms of all powers from the scalar cur-vature should be considered in action for the grav-itational ﬁeld, because they give exactly the sameorder in quantum ﬂuctuations as the Einstein action(N = 1).Therefore, if we represent a full gravitational ac-tion in the form of L = α1R1 + α2R2 + α3R3 + . . .,where αi are some factors of a necessary dimension,then in the case of quantum gravity, each term willcontribute to the propagator of a graviton (∼ h2).And we cannot neglect any term. In fact, this meansthat there exist Schwinger-Dyson equations for anyN , and, hence, the eﬀect of dynamical symmetrybreaking is possible.We would like to note the analogy with the Fierz-Pauli model, in which all undesirable degrees of free-dom in the ﬂat metric background are cancelled,whereas in a curved background one undesirable de-gree of freedom (the Boulware-Deser mode) appearsagain in the spectrum [16].References[1] Sitenko, Yu. A.: Polarization of the Mass-less Fermionic Vacuum in the Background ofa Singular Magnetic Vortex in 2 + 1 Dimen-sional Space-Time, Ukr. J. Phys. 43 (1998),1 513–1525.[2] Sitenko, Yu. A.: Induced vacuum condensatesin the background of a singular magnetic vortexin 2+1 dimensional space-time, Phys.Rev. D60(1999), 125 017.[3] Miransky, V. A.: Dynamical symmetry breakingin quantum ﬁeld theories, Singapore : World Sci-entiﬁc, 1993.[4] Inagaki, T., Odintsov, S. D., Shil’nov, Yu. 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G.: Infrared-modiﬁed gravities and massive gravitons, Usp.Fiz. Nauk 51 (2008), 759–792.A. T. KotvytskiyD. V. KruchkovE-mail: kotw@mail.ruDepartment of Theoretical PhysicsV. N. Karzin Kharkov National UniversitySvobody Sq. 4, 61077 Kharkov, Ukraine58

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Dynamical Symmetry Breaking in RN Quantum Gravity

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