The method appeared to be a very powerful calculational tool in the quantum eld theory [3, 4]. Its generalizations [5] was applied even for the consis

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1 On the gravity renormalization o shell. K. A. Kazakov, P. I. Pronin y and K. V. Stepanyantz z August 30, 997 Moscow State University, Physics Faculty, Department of Theoretical Physics. 734, Moscow, Russian Federation Abstract Using as an example the Einstein gravity with the cosmological constant, we discuss the calculation of renormalization group functions o shell. We found, that gauge dependent terms should be absorbed by the nonlinear renormalization of metric. Nevertheless, some terms can be included in the renormalization of Newton's constant. This ambiguity in the renormalization prescription is discussed.. The high energy behavior of the gravity interaction draws the attention of researchers for a very long time. It is well known, that the perturbation theory for the Einstein gravity diers much from the Yang-Mills case due to the dimensional coupling constant. Although the theory is not renormalizable and its contribution to the low energy physics is very small, a great number of new ideas and eld theory methods originate in the research of the gravity interaction. Here we should especially mention the background eld method []. In particular, it played the important role in the t'hooft and Veltman derivation of the algorithm for the one-loop divergences calculation [] (that allowed to obtain one-loop divergences for Einstein gravity). kirill@theor:phys:msu:su y petr@theor:phys:msu:su z stepan@theor:phys:msu:su

2 The method appeared to be a very powerful calculational tool in the quantum eld theory [3, 4]. Its generalizations [5] was applied even for the consistency proof of the higher covariant derivative method [6]. And nevertheless, there are some open questions associated with the application of this approach for the gravity theories. In particular, it is not quite clear why it is necessary to use motion equations for the renormalization. The application of the background eld method for Yang-Mills theory or other usual eld theory models does not require them at all. Nevertheless, in the quantum gravity the o shell result is gauge dependent. For example the special choice of gauge condition in the Einstein gravity can made the theory nite at the one-loop o shell. The main reason of motion equation using is that the on shell result was proven [7] to be gauge independent. In our opinion such renormalization is a rather special case. It will be much more natural to renormalize a theory o shell. Then the natural question is how to avoid the dependence on the nonphysical parameters? In this paper on the example of Einstein gravity with cosmological constant we formulate the prescription for the o shell renormalization so, that the renormalization of physical values is gauge independent. Gauge parameters are included in the renormalization of unphysical metric eld and Newton's constant. Using this approach we demonstrate, that the renormalization of cosmological constant is in a complete agreement with the on shell results. Our paper is organized as follows. In next section we calculate the oneloop counterterms in an arbitrary gauge for the Einstein gravity with the cosmological constant. The renormalization procedure o shell is constructed in the section 3. The nal section 4 is devoted to the discussion of the renormalization prescription ambiguity.. The action for the Einstein gravity with the cosmological constant has the following form where S =? k d 4 x p?g (R? ) +! () k = 6G; +????? ; ()

3 ! is the dimensionless coupling constant, is the cosmological constant, G is the Newton's constant and = p 3 d 4 x?gr R? 4R R + R (4) is the Euler number (topological invariant). The calculation of the one-loop counterterms can be performed in the framework of the background eld method []. In accordance with this method the dynamical eld can be rewritten as g = g + kh. The general coordinates invariance is xed by adding to the action (3) L gf = p?gg ; = p + r h? + g r h! (5) where and are an arbitrary real constants. For the quadratic in the quantum elds eective Lagrangian we have where L ef f =? h ; r +? + ( + ) ( + )! g g r (? ) + ( + ) g r r? ( + ) g r r + P h (6) P (()()) = R? g R + g R + g g (R? )? (R? ) g g (7) and ; = (g g + g g ) : (8) 3

4 The parentheses around couple of indices denote the symmetrization whereas parenthesis around four indices means the symmetrization with pairs' interchange at the same time. The ghost action obtained in the standard way is L gh = c g r? r r + R c : (9) To calculate the one-loop counterterms we use the general expressions given in [8, 5] and tensor package [9] for the analytical calculations system REDUCE. The o-shell one-loop counterterms including the contributions of both quantum and ghost elds are where? () = 6 (d? 4) d 4 x? R R? 4R R + R + (R? 4)(a R + a ) + a 3 (0R R + 5R? 60R + 0 ) (0) a = 5 (?5? 0? 5) ? 5 ; a = (? 3? 4?? ) + 4 3? 6? 9 5 ; a 3 = 60 ( ) + (?4 4? 8 3? ) + (4 4? 6? ) : () and we introduced the notation?. In particular, in the case = 0 this expression is in agreement with results [0]. The one-loop on-shell counterterms (R = g ) also coincide with the well-known result [] 53? () = 6 d 4 x : () (d? 4) 3. The above calculations show, that the eective action depends on the gauge parameters. Nevertheless, physical values must be gauge independent R R? 58 5

5 So, ambiguous terms should be absorbed by the renormalization of unmeasurable values, for example metric eld. For this purpose we will use the following nonlinear renormalization [, 0] g! g B = g +! B = + c 4 6 (d? 4) k ; 6 (d? 4)c Rg + c g + c 3 R ; G! G B = G + c 5 6 (d? 4) G : (3) Then the bare Lagrangian takes the form L(g) B = L(g ) + 6 (d? 4) p?g h g (R? )? R (c Rg + c g + c 3 R )? c 4? c 5 (R? ) i + O(R 3 ): (4) L(g B ) + L should be nite. It leads to the following equations for the coecients c : : : c 5 : They can be rewritten as?c 3 + 0a 3 = 0; c + c 3 + a + 5a 3 = 0;?4c + c? c 3? c 5? 4a + a? 60a 3 = 0;?4c? c 4 + c 5? 58 5? 4a + 0a 3 = 0: (5) c =?a? 0a 3 ; c = c 5? a + 30a 3 ; c 3 = 0a 3 ; c 4 =? 9 5? c 5: (6) 5

6 4. (6) means, there is an ambiguity in the renormalization: gauge dependent terms can be absorbed in the renormalization either of metric tensor or of Newton's constant. Is it necessary to nd a "true" prescription of renormalization? We believe, that it is not. Really, the ambiguity does not aect physical values. The metric eld is not measurable, because motion of a classical particle is completely dened by connection. As for the Newton's constant, in the considered model it is a pure multiplicative factor in the Lagrangian, or by the other words an unessential constant []. Moreover, we are able to avoid the ambiguity by introducing so that the Lagrangian will be = k ; G = k g ; (7) L = p?g (R(G)? ) +!: (8) (Here will already be an essential constant.) The renormalization of G and does not include an arbitrary constant as above, G! G + 6 (d? 4) (?a? 0a 3 )Rg + (?a + 30a 3 )c g +0a 3 R ;!? (d? 4) : (9) and coincides with the on shell result. So, we see, that the ambiguity comes from the fact, that in this particular model Newton's constant is only multiplicative factor and is not contained in the motion equations. If matter elds are added to the Lagrangian, the generalization of (7) will made them dimensionless, for example! = k. Therefore, this substitution allows to avoid the specication of the mass scale and renormalize only physical dimensionless values. 6

7 References [] B.DeWitt Dynamical Theory Groups and Fields (Gordon and Breach, New York, 965). [] t'hooft G. and Veltman M., Ann. Inst. Henri Poincare 0, 69, (974). [3] C.Lee and C.Rim, Nucl.Phys. B 55, 439, (985). [4] S.Ichinose and M.Omote, Nucl.Phys. B 03,, (98). [5] P.Pronin and K.Stepanyantz, Nucl. Phys. B 485, 57, (997). [6] P.Pronin and K.Stepanyantz, hep-th/ [7] R.Kallosh and I.Tuitin, Sov.Journal of Nucl.Phys., 7, 98, (973). [8] P.Pronin and K.Stepanyantz, in: "Gravity, Particles and Space-time", ed.: P.Pronin and G.Sardanashvili, World Scientic, Singapure, (996), (hep-th/ ). [9] P.Pronin and K.Stepanyantz, in: "New Computing Technick in Physics Research. IV.", ed.: B.Denby and D.Perred-Gallix, World Scientic, Singapure, (995). [0] R.Kallosh, O.Tarasov and I.Tyutin, Nucl. Phys. B 37, 45, (978). [] S.Christensen and M.Du, Nucl. Phys. B 70, 480, (980). [] S.Weinberg, in: "General relativity", ed.: S.Hawking and W.Israel, Cambridge University Press, Cambridge, (979). 7

Gravity and running coupling constants

Gravity and running coupling constants 1) Motivation and history 2) Brief review of running couplings 3) Gravity as an effective field theory 4) Running couplings in effective field theory 5) Summary 6)