Acta Polytechnica Vol. 51 No. 4/2011 Dynamical Symmetry Breaking In RN Quantum Gravity A. T. Kotvytskiy, D. V. Kruchkov Abstract We show that in the RN gravitation model, there is no dynamical symmetry breaking effect in the formalism of the Schwinger-Dysonequation (inflatbackgroundspace-time). Ageneral formula for the secondvariation of thegravitational action 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 and dynamical symmetry breaking in different external fields [1, 2], including the gravitational field [3–5] is an important step in studying fundamental interac- tions, which can either be considered in field the- ory and an attempt at quantization can be made or considered as an external interaction. This pa- per is devoted to the possibility of dynamical sym- metry breaking in such gravitation models using the Schwinger-Dyson equation formalism [6–9]. The pri- mary goal is a study of some gravity properties of f (R)-gravity [10], where R is a scalar curvature. One of the most discussed variants of such models is RN gravity [11, 12]. In particular, we are interested in a possible 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 evolution of the early universe. Second, an understanding of the 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 a much more exotic state equation to explain the ac- celerated expansion of the observed universe (being within the limits of relativity theory). To sum up, it is significant for a description of the future of the universe. However, if one modifies gravity in a proper way, it is also possible to receive interesting cosmo- logical dynamics without introducing new concepts. Here, we provide a comparative analysis of the above formalism both in the case of flat background met- rics (Minkowski space) and in the case of an arbitrary background. Let us expand the four-dimensional space-time metric as follows g̃μν � gμν + hμν , (1) where g̃μν is perturbed metric, gμν is background metric, hμν are quantum corrections. This defini- tion leads to the following results. In the case of flat background space-time in the RN gravity model there is an existence of corrections like O(hN ) or- der and higher that gives no possibility to get the necessary equations. We can speak about the ab- sence of a symmetry breaking effect (at least in the Schwinger-Dyson equation formalism). However, if the background metric is curved, then corrections of order appear and, particularly, specifically quadric ones 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 described by a curved metric, then while constructing quan- tum gravity theory, we should take into consider- ation the summands of all powers in R, as they provide the same degree in small quantum correc- tions h. 1 Schwinger-Dyson equations One possible method for a dynamical symmetry breaking study is the Schwinger-Dyson equation for- malism. Since it is impossible to write closed system equations for all elements of the Feynman diagram, we have to use some approximations, which allow us to solve the Schwinger-Dyson equation and to find the type of exact propagator. Exact propagators and the vertex part are connected by the integral relation, S−1 − S−10 = i δΓ2 δS (2) 54 Acta Polytechnica Vol. 51 No. 4/2011 where S, S0 – exact and free fermion propagators, δΓ2 δS – describes two particle and irreducible interac- tion diagrams, and Γ2 is a part of the effective action Γ[S] = −iSp ( Log S−1 + S−10 S ) + Γ2[S]. (3) Here we confine ourselves to the exact fermion propagator, which is written down in original type S(p) = 1 A(p2)pμγμ − B(p2) , (4) where A(p2), B(p2) are some unknown functions of the fourth momentum p, γμ are Dirac matrices. Then the Schwinger-Dyson equation for this propagator can be put down like [13–15] (A − 1)pμγμ − B = ∫ d4q (2π)4i Γαβ (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 exact fermion and boson Green functions, as well as the full interaction vertex, can be solved within some trunca- tion version only. This means that the only subset of Feynman 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 ladder approximation gives gauge-dependent results, when the fermion Green function is treated to be exact only in the case when the boson propagator and the interaction vertex are taken to be free [14]. In this way, we can define the functions A(p2), B(p2) for the quantum RN -gravity. The following important remark should also be made. We have to decide what kind of gravity action and gauge-fixing term should be used. It is conve- nient to bring the following condition Sgf = −β1 2M2 ∫ d4x √ −g ( ∇λhλμ − β2∇μh ) · (gμν∇ρ∇ρ + β3∇μ∇ν ) × (6) (∇σhσν − β2∇ν h) , where β1, β2, β3 are arbitrary parameters. Then, putting down the second variation of the full action which we can describe as the sum of the gravitational field action and the gauge-fixing action in the form δ2 (Sg + Sgf ) = 1 2M2 ∫ d4xhμν Hμνρσ h ρσ. (7) Then the gravitational field propagator is defined as an operator inverse to Hμνρσ , that is Gμνρσ = M2 ( H−1 )μνρσ . (8) Let us include the gravitational field in our con- sideration. 2 Flat background metric Here we single out the part of the quadratic action according to corrections h, in order to put down the Schwinger-Dyson equations. Instead of (1) the per- turbed metric will be written down as gμν � ημν + hμν , (9) where ημν is the Minkowski metric (we choose the signature (+ − −−)). We consider the following action Sg = 1 2M2 ∫ d4x √ −gRN . (10) Note that action (10) does not have to be a full action for the gravitational field, while the Einstein linear gravitation on the curvature and Λ — term, and other possible variants, can be also included. However, a discussion of the effects caused by the form (10) is the main goal of this paper. In the case of a flat background metric, we have the following expansion (with accuracy up to the sec- ond order approximation) √ −g � 1 + 1 2 h − 1 4 hμν h μν + 1 8 h2, (11) where raising and lowering the index is by ημν back- ground metric and h = ημν hμν . The Riemann tensor is now Rμνρσ = ∂ρΓ μ νσ − ∂σ Γ μ νρ + Γ μ τ ρΓ τ νσ − Γ μ τ σΓ τ νρ, (12) while the Ricci tensor is determined by the Riemann tensor convolution according to the first and the third indices. Then, for the scalar curvature we get R = gμν Rμν � 1 2 ( ημν − hμν + hμρhνρ ) × (13)( ∂α∂μh α ν − ∂α∂ αhμν − ∂μ∂ν h + ∂α∂ν hαμ + O(h2) ) = ∂α∂ ν hαν − ∂α∂ αh + O(h2). This implies, in the case of RN gravitation, the development according to h has the form RN � ( ∂α∂ ν hαν − ∂α∂ αh + O(h2) )N ∼ O(hN ) (14) That is, the smallest order has a power N in quantum corrections. Physically this means that the graviton propagator is missed in such a gravitation model, and only N -particle vertex functions exist. Therefore, it is not essential for the study of dynam- ical symmetry breaking (DSB) in this formalism. In 55 Acta Polytechnica Vol. 51 No. 4/2011 other words, we can say that there is no dynamical symmetry breaking effect in this approximation. Now we proceed to the quite different situation of non-flat background space-time. 3 Curved background metric Let us choose the action for the gravitational field in the form of Sg = 1 2M2 ∫ d4x √ −g̃R̃N , (15) where tilde denotes a perturbed metric, and is deter- mined by (1). Since the background metric is not the Minkowski one, the summands of all h orders appear in all rede- fined constructions. So the expressions for Christoffel symbols can be reduced to the form (with accuracy within the second order on quantum corrections) Γ̃μνρ � Γ μ νρ + 1 2 (gμγ − hμγ ) · (16) (∇ρhγν + ∇ν hγρ − ∇γ hνρ) , where Γμνρ are the Christoffel symbols calculated ac- cording to the unperturbed metric gμν , and ∇ρ is the covariant derivative relative to the same metric. The complete expression for the Riemann tensor (in this approximation) is R̃μνρσ � R μ νρσ + 1 2 (gμτ − hμτ ) ×( hτ εR ε νσρ + hεν R ε τ σρ + ∇ρ∇ν hτ σ − ∇ρ∇τ hνσ − ∇σ∇ν hτ ρ + ∇σ∇τ hνρ) + 1 4 gμγ gτ ε[(∇σhεν + ∇ν hσε − ∇εhνσ ) · (∇τ hγρ − ∇ρhγτ − ∇γ hτ ρ) + (∇σhτ γ − ∇τ hσγ + ∇γ hτ σ) · (∇ρhνε + ∇ν hρε − ∇εhνρ)]. Here, the procedure for raising and lowering the indices is provided by the background metric. Hence, we find the scalar curvature R̃ = g̃νσR̃ρνρσ � (g νσ − hνσ + hναhσα) R̃ ρ νρσ. (17) Introduce the following symbols h1 ≡ −Rνσhνσ + ∇ν∇σhνσ − ∇ν∇ν h, (18) h2 ≡ Rνσhναhσα − h ρτ (∇ρ∇σ + ∇σ∇ρ) hτ σ − hρτ ∇ρ∇τ h + hρτ ∇σ∇σhρτ + 1 2 ∇σhτ σ∇τ h − ∇σhτ σ∇γ hγτ − (19) 1 4 ∇τ h∇τ h + 1 2 ∇τ h∇γ hγτ + 3 4 ∇σhερ∇σhρε − 1 2 ∇εhρσ∇ σhρε. The expression for scalar curvature is R̃ � 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- comes R̃N � RN + N RN−1 (h1 + h2) + (21) 1 2 N (N − 1)RN−2h21. In the case of arbitrary background space-time we should take into account the expansion det(g̃) � det(g) + hμν Kμν (g) + (22) hμν hαβ F μναβ (g), where Kμν (g) = εαβρσ ( δ μ 0 δ ν αg1βg2ρg3σ + δ μ 1 δ ν βg0αg2ρg3σ + δ μ 2 δ ν ρ g0αg1β g3σ + δ μ 3 δ ν σg0αg1βg2ρ ) . We underline the fact that if the background met- ric is the Minkowski one ημν , then function K μν = −ημν and with accuracy within the first order we get the well-known formula −g̃ � 1 + h. The expression for F μναβ (g) is cumbersome, and we do not present it here. To consider the possibility of dynamical symme- try breaking, it is necessary to have an expression for the second variation of the gravitational action. Let us take into consideration the following expansion √ a + x1 + x2 � √ a + x1 + x2 2 √ a − x21 8 √ a3 , (23) where x1 + x2 � a, and also (x1) 2 ∼ x2. Then,√ −g̃ � √ −g − hμν Kμν − hμν hαβ F μναβ � √ −g − hμν K μν + hμν hαβ F μναβ 2 √ −g − (24) (hμν K μν ) 2 8 √ −g3 . Thus, we obtain the final form of the second vari- ation δ(2)Sg = 1 2M2 ∫ d4x ( √ −g · ( N RN−1h2 + 1 2 N (N − 1)RN−2h21 ) − N 2 √ −g hμν K μν RN−1h1 − (25) hμν hαβ F μναβ 2 √ −g RN − (hμν K μν )2 8 √ −g3 RN ) . 56 Acta Polytechnica Vol. 51 No. 4/2011 Note, that in the case of the quadric gravitation (N = 2) and a flat background (R = 0), instead of (25) we obtain δ(2)Sg = 1 2M2 ∫ 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) and the Schwinger-Dyson equations (5). This indicates that the effect of dynamical symmetry breaking is possible in RN -gravity. 4 Conclusions Some quantum properties of model RN -gravity have been considered in the paper. A comparative analysis for two cases: a) a flat background space-time, and b) an arbitrary curved background has been carried out. Expanding this model of gravity with quantum corrections hμν , we found that in the first case the smallest order of quantum corrections is N . This means that in the quantum theory of the RN -gravity graviton propagator (for N > 2) does not exist, and there is a vertex function of graviton-graviton inter- action that is not used in this formalism. Thus, there is no Schwinger-Dyson equation (5) and, therefore, there is no effect to discuss. In case b, a general formula (25) for the second variation of gravitational action on the quantum cor- rections hμν is obtained, which in the limit R → 0 coincides with the previously known results. It is determined that in this formulation of the problem under the effects of dynamical symmetry breaking research 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 same order in quantum fluctuations as the Einstein action (N = 1). Therefore, if we represent a full gravitational ac- tion in the form of L = α1R 1 + α2R 2 + α3R 3 + . . ., where αi are some factors of a necessary dimension, then in the case of quantum gravity, each term will contribute to the propagator of a graviton (∼ h2). And we cannot neglect any term. In fact, this means that there exist Schwinger-Dyson equations for any N , and, hence, the effect of dynamical symmetry breaking is possible. 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Kotvytskiy D. V. Kruchkov E-mail: kotw@mail.ru Department of Theoretical Physics V. N. Karzin Kharkov National University Svobody Sq. 4, 61077 Kharkov, Ukraine 58