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Dimensional Regularization with Non Beta-Functions

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Dimensional Regularization with Non Beta-Functions RESERACH Revista Mexicana de F´ısica 61 (2015) 149–153 MARCH-APRIL 2015 Dimensional regularization with non Beta-functions J.D. Garc´ıa-Aguilar and A. Perez-Lorenzana´ Departamento de F´ısica, Centro de Investigacion´ y de Estudios Avanzados del I.P.N. Apartado Postal 14-740, 07000, Mexico,´ D.F., Mexico,´ e-mail: jdgarcia@fis.cinvestav.mx; aplorenz@fis.cinvestav.mx Received 20 November 2013; accepted 30 January 2015 The most general method to regularize Feynman’s integrals in quantum field theory is Dimensional Regularization, in which the most common way to evaluate the associated integral involves Beta functions. We present a new method to evaluate the integral through the residue theorem. We apply our method to a toy model on universal extra dimensions and show that radiative corrections changes the shift- mass between zero and Kaluza-Klein excited modes. Keywords: Quantum field theory; renormalization; radiative corrections. Regularizacion´ Dimensional suele ser el metodo´ mas utilizado para regularizar integrales de Feynman, comunmente´ en dicho metodo,´ las integrales asociadas implican funciones Beta. Presentamos un metodo´ para evaluar tales integrales mediante el Teorema del Residuo. El metodo´ se aplica sobre un modelo de juguete con dimensiones extras universales, observando como las correcciones radiativas envuelven un corrimiento de masa entre el modo cero y los modos de Kaluza-Klein. Descriptores: Teor´ıa cuantica de campos; renormalizacion;´ correcciones radiativas. PACS: 03.70.+k; 71.38.Cn 1. Introduction In order to see how DR operates, consider for example the simplest Feynman integral Z dDp 1 It is well known that Quantum Field Theory (QFT) is plagued I (D) = ; (1) (2¼)D p2 + m2 with divergences, that appear through the integrals associated to the Feynman diagrams of beyond tree level corrections. which typically arises at on loop order corrections in scalar Dealing with those divergences, in order to obtain physical field theories. There are many ways to evaluate such an inte- (meaningful) results out of perturbation theory is the main gral. One of them is through the use of proper time represen- goal of renormalization theory [1,2], in which, regularizing tation of Feynman integrals. Other option is the introduction the divergent integrals becomes the first crucial step. of polar coordinates where one rewrites the integral as Z1 In four dimensions, for instance, it is common that the S pD¡1 I (D) = D dp ; (2) integrals associated to two and four-point correlations func- (2¼)D p2 + m2 tions diverge. However, through regularization procedures 0 they can be made finite. To achieve this a regularization pa- where the overall coefficient rameter is introduced, in away that the divergences of the in- 2¼D=2 tegrals would now appear as singularities that depend on this SD = ; (3) parameter. ¡(D=2) is the surface of a unit sphere in D dimensions. Then, the re- For example, in Dimensional regularization (DR), which sulting one-dimensional integral can be casted into the form was introduced by ’t Hooft and Veltman [3-5], the measure of an integral for the Beta function which can be expressed in of momentum integration is changed by allowing the dimen- terms of the Gamma functions, and thus, the value for (1) is sion D in the integrals to be an arbitrary complex number. given by There are several attractive features on DR. First, it preserves ¡ ¢D=2¡1 all symmetries for a non-supersymmetric theory (under this m2 scheme the gauge fields and the momentum integrals are pro- I (D) = ¡ (1 ¡ D=2) : (4) (4¼)D=2 moted to D dimensions, and the mismatch between the num- ber of degrees of freedom of gauge fields(D) and gauginos(4) One of the disadvantages of these methods arises when breaks supersymmetry [6]). Second, it allows an easy identi- two or more propagators are considered, because in the case fication of the divergences. Third, it suggests in a natural way of the proper time representation it is necessary to consider a minimal subtraction scheme (MS scheme) [7], that greatly two or more integrations, whereas, in the case of Beta func- simplifies renormalization calculation and it is capable of reg- tion it is required the use of Feynman’s parametric integral ularizing IR-divergences in massless theories. formulas. 150 J.D. GARCIA-AGUILAR´ AND A. PEREZ-LORENZANA´ In this work we present a new method that make use of Residue theorem which allows to evaluate the one- dimensional integral without considering Beta functions. In- stead, it is only necessary the use of one specific contour for the integration. As we will discuss, this provide an easier way for the evaluation of the integrals with two or more propaga- tors (similar techniques are shown in Ref. 8). The outline of our paper is as follows: Sec. 2 presents the contour and their use to evaluate the one-dimensional in- tegral. Sec. 3 shows a detailed example where the method is used to evaluate the radiative corrections for the scalar field mass due to a Yukawa interaction. Here as we will show ex- plicitly, the use of extra integrations becomes unnecessary. Next, in Sec. 4 we apply the method to another example of physical interest. There, we consider a toy model on an extra- dimensional space-time [9] to compute the radiative correc- tions for the zero mode scalar mass reduced by their interac- tions with the excited modes. Sec. 5 lists the conclusions. 2. Evaluation of Integral FIGURE 1. Contour used to evaluate the integral (5). Definite integrals appear repeatedly in problems of mathe- matical physics. Three moderately techniques are used in Next, by Jordan’s lemma when R ! 1, we have Z evaluating definite integrals: (1) Numerical quadrature which zD¡1 is not commonly used on QFT; (2) conversion to Gamma or dz = 0; z = Reiθ; (7) z2 + m2 Beta functions which, as we mentioned in the introduction, is ¡0 rather very used on the literature, and (c) contour integration which is rarely used in the problem at hand. Demonstrating whereas, when ² ! 0 Z the advantages of the use of the last technique to evaluate the zD¡1 usual integral dz = 0; z = ²eiθ: (8) 1 z2 + m2 Z D¡1 t ¡1 A = dt; (5) t2 + m2 0 It is important to remark the necessity to use analytic contin- is the purpose of this section. uation to solve the integral (5), because the Eqs. (7) and (8) To proceed, we choose the contour shown in Fig. 1 to are valid only for values D < 2. avoid the pole in the origin (because in it arises a branch) and From this the integral is reduced to the possible poles over the real positive axis. Then, a given 1 I Z D¡1 integral on such a contour is generally expressed as iD® t f (z) dz = e 2 i2® 2 R t e + m I Z 0 i® ¡ i® ¢ f (z) dz = e f e t dt Z1 tD¡1 ² ¡ eiD(2¼¡®) ; t2ei2(2¼¡®) + m2 2Z¼¡® ¡ ¢ 0 + iR f Reiθ eiθdθ and thus, in the limit ® ! 0, we get 0 1 Z² Z D¡1 D¡2 ³ ´ ³ ´ t ¼im D D i(2¼¡®) i(2¼¡®) dt = ¡ i + (¡i) : (9) + e f e t dt t2 + m2 (1 ¡ ei2¼D) 0 R Z0 Therefore, ¡ ¢ + i² f ²eiθ eiθdθ µ ¶ 2¼imD¡2 iD + (¡i)D I (D) = ¡ ¡ ¢: (10) 2¼¡® i2¼D D D 1 ¡ e D 2 X 2 ¼ ¡ 2 = 2¼i residues (within the contour); (6) This Feynman integral is UV-divergent in even dimen- ¡ ¢¡1 where we will, of course, take f(z) = zD¡1=(z2 + m2). sions, which is reflected by the poles in 1 ¡ ei2¼D . The Rev. Mex. Fis. 61 (2015) 149–153 DIMENSIONAL REGULARIZATION WITH NON BETA-FUNCTIONS 151 poles can be subtracted in various ways, parametrized by an The poles of the integral on the Eq. (17) are located at arbitrary mass parameter ¹. Let us take the limit D = 4 ¡ ² to write §im and k § im: (18) 2¼im2 ¹²I (D) = ¡ 1 ¡ e¡i2¼² Thus, using the residue theorem, we straightforwardly find £ ¡ ¢¤¡1 µ ¶ ² ² 2 2 that ¡ 2 ¡ 2 4¼¹ £ 2 ¡ 2 : (11) 8¼ m 2¼i 1 I0 (D) = ¡ ¢ i2¼D D D 1 ¡ e D¡1 2 The arbitrary mass parameter ¹ appears in a dimension- 2 ¼ ¡ 2 à less ratio with the mass. It is this kind of terms which contains D¡1 2 A (B¤) ¡ A¤BD¡1 IR-divergences in the limit m ! 0. They are expanded in £ i powers of ²: 2km jAj2 ! µ ¶ ² D 4¼¹2 2 ² 4¼¹2 ¡ ¢ AiD + A¤ (¡i) ¡ = 1 + ln + O ²2 : (12) ¡ mD¡2 ; (19) m2 2 m2 2k jAj2 The ²-expansion over the Gamma function in (11) reads where A = k + 2im and B = k + im. 1 ² ¡ ¢ ¡ ¢ = 1 + (1 ¡ γ ) + O ²2 (13) Therefore, the divergence for D = 4 is contained in the ² E 2 ¡ 2 ¡ 2 denominator, since the overall coefficient possesses poles at D = 4; 6;:::. The remaining parameter integral is finite for where γE is the Euler-Mascheroni constant. 2 Next step is to consider the ²-series expansion of the func- any D as long as m 6= 0. In terms of ² and ¹ parameters in- tion associated with the exponential, for which we take its troduced above, the expresion for the integral (16) reads (see Laurent expansion (see Appendix A) Appendix B) 1 i 1 i¼² ¡ ¢ 2 3 ² 0 m ¡i2¼² = ¡ + + + O ² : (14) ¹ I (D) = ¡ 1 ¡ e 2¼² 2 6 16¼2 (k2 + 4m2) µ ¶ Inserting Eqs. (12-14) into (11) we find the Laurent expan- 2 4¼¹2 ² £ + ln + 1 ¡ γE + O (²) sion in : ² m2 m2 ¹²I (D) = ¡ k4 + 3k2m2 ¡ 2m4 2km 2 ¡ arctan 16¼ 16¼2km (k2 + 4m2) k2 ¡ m2 µ 2 ¶ 2 4¼¹ 2 2 £ + 1 ¡ γE + ln + O (²) : (15) k + 5m ² m2 + 16¼2 (k2 + 4m2) 2 µ ¶ Notice that the residue of the pole is proportional to m and 2 4¼¹2 independent of ¹.
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