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) (2π)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 Z∞ 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- (2π)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- 2πD/2 tegrals would now appear as singularities that depend on this SD = , (3) parameter. Γ(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) = Γ (1 − D/2) . (4) (4π)D/2 moted to D dimensions, and the mismatch between the number 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 propagators (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 integral. 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 explicitly, 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 corrections for the zero mode scalar mass reduced by their interactions 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 → ∞, 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 Γ0 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) ∞ z2 + m2 Z D−1 t Γ1 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 ∞ I Z D−1 integral on such a contour is generally expressed as iDα t f (z) dz = e 2 i2α 2 R t e + m I Z 0 iα ¡ iα ¢ f (z) dz = e f e t dt Z∞ tD−1 ² − eiD(2π−α) , t2ei2(2π−α) + m2 2Zπ−α ¡ ¢ 0 + iR f Reiθ eiθdθ and thus, in the limit α → 0, we get 0 ∞ Z² Z D−1 D−2 ³ ´ ³ ´ t πim D D i(2π−α) i(2π−α) dt = − i + (−i) . (9) + e f e t dt t2 + m2 (1 − ei2πD) 0 R Z0 Therefore, ¡ ¢ + i² f ²eiθ eiθdθ µ ¶ 2πimD−2 iD + (−i)D I (D) = − ¡ ¢. (10) 2π−α i2πD D D 1 − e D 2 X 2 π Γ 2 = 2πi 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 − ei2πD . 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 µ. Let us take the limit D = 4 − ² to write ±im and k ± im. (18) 2πim2 µ²I (D) = − 1 − e−i2π² Thus, using the residue theorem, we straightforwardly find £ ¡ ¢¤−1 µ ¶ ² ² 2 2 that Γ 2 − 2 4πµ × 2 − 2 . (11) 8π m 2πi 1 I0 (D) = ¡ ¢ i2πD D D 1 − e D−1 2 The arbitrary mass parameter µ appears in a dimension- 2 π Γ 2 Ã 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 |A|2 ! µ ¶ ² D 4πµ2 2 ² 4πµ2 ¡ ¢ AiD + A∗ (−i) − = 1 + ln + O ²2 . (12) − mD−2 , (19) m2 2 m2 2k |A|2 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 Γ 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 µ 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 iπ² ¡ ¢ 2 3 ² 0 m −i2π² = − + + + O ² . (14) µ I (D) = − 1 − e 2π² 2 6 16π2 (k2 + 4m2) µ ¶ Inserting Eqs. (12-14) into (11) we ﬁnd the Laurent expan- 2 4πµ2 ² × + ln + 1 − γE + O (²) sion in : ² m2 m2 µ²I (D) = − k4 + 3k2m2 − 2m4 2km 2 − arctan 16π 16π2km (k2 + 4m2) k2 − m2 µ 2 ¶ 2 4πµ 2 2 × + 1 − γE + ln + O (²) . (15) k + 5m ² m2 + 16π2 (k2 + 4m2) 2 µ ¶ Notice that the residue of the pole is proportional to m and 2 4πµ2 independent of µ. × + ln + 1 − γE + O (²) . (20) ² k2 + m2 We should underline that the result of Eq. (15) is the same value obtained through the use of Beta functions, as it could be expected. This analysis exempliﬁes very well the advantage of the method, because we were not compelled to use the Feyn- man’s parametric integral formula to evaluate the integral 3. Two propagators (16), but rather proceed through a direct evaluation. Next, we demonstrate the method for the integration over two It is straightforward to extend the method to compute in- propagators, which is known to be convergent for D < 4. So, tegrals which involve more than two propagators, in that case we consider the integral it is necessary to consider an integral with the form Z D ∞ 0 d p 1 1 Z Yn I (D) = , (16) SD 1 D p2 + m2 (p − k)2 + m2 J (D) = pD−1dp (21) (2π) D 2 2 (2π) (p − kl) + m 0 l=1 where the external momentum k is the incoming momentum. It is easy to see that one can rewrite it as where n is the number of propagators and kl ± im are the Z∞ poles. As it is showed above, with the new method, the Feyn- S pD−1 dp I0 (D) = D . (17) man parametrizations are changed by algebra with complex (2π)D (p − k)2 + m2 p2 + m2 0 numbers.

Rev. Mex. Fis. 61 (2015) 149–153 152 J.D. GARCIA-AGUILAR´ AND A. PEREZ-LORENZANA´

4. λφ4 on Extra Dimension Now we make use of the definition of Riemann’s zeta function [12] An interesting application of the proposed method, is found µ ¶ when dealing with multiple field theories, where such mul- X∞ X∞ 1 −2 n2 = = ζ (−2) = 0, (29) tiplicity amounts to more complicated integrals. Such is the n case of models with extra dimensions, whose effective 4D n=1 n=1 theory contains infinite towers of excited states, numbered which implies that the zero mode mass correction is given by by a Kaluza-Klein (KK) index [10]. λ0 X∞ To see how the method works on such cases, we next con- δm2 = n2 ln n. (30) sider a scalar field toy model on an extra space-time dimen- 0 2 8 (πR) n=1 sions, whose Lagrangian is given by We should notice that this result is still divergent, due to † M λ 4 L = ∂M φ ∂ φ − |φ| . (22) infinite number of KK modes from the tower. For each level, 4 however, poles have been removed (by renormalization pro- To be specific we assume the compactification is realized cedures). To attain a finite value we need to explicitly intro- over a circle of radius R, and thus, the KK wave functions go duce a cut-off on the KK levels. An example that introduce iny/R as φn ∝ e . We are interested in computing the radiative the cut-off is given in Ref. 13 in which the authors present corrections for the mass of the zero mode, which essentially other mechanism to regularize extra-dimensional models. are given by the effective interactions λ0 X 4.1. Splitting mass between zero mode and the KK L = − |φ |2 |φ |2 , (23) modes int 4 0 n n=0 As we saw in the last section, there is a change for the zero where λ0 = λ/2πR, is the effective 4D coupling constant. mode mass due to radiative corrections. That would be also The corrections we shall consider are given by two inte- the case for the excited KK modes. However for them, the grals first integral on the right hand of Eq. (24) is changed into Z Z dDp 1 X dDp 1 2 Z D δm0 ∝ 4 + ¡ ¢2 . (24) d p 1 (2π)D p2 (2π)D p2 + n 4 ¡ ¢ . (31) n=1 R (2π)D 2 n 2 p + R First integral does not contribute, because its pole is not present within the contour. This is actually expected as a re- Therefore after regularizing, the change on the KK excited sult of Veltman’s theorem [11]. masses is given by

Next we make the use of the new variable x = pR and 0 2 µ 2 2 ¶ 2 λ n 2 4πµ R the sum δmn = − + 1 − γE + ln + O (²) (4πR)2 ² n2 X 1 X∞ 1 1 = − 0 X∞ x2 + n2 x2 + n2 x2 λ 2 n=1 n=0 + m ln m. (32) 8 (πR)2 π 1 m=1 = coth (πx) − , (25) 2x 2x2 That allows to compute the total shift mass between zero which allows to write the second integral on Eq. (24) as and excited KK modes taking into account the radiative cor- Z · ¸ rections under MS scheme 1 dDx π 3 coth (πx) − . (26) 2 µ 0 2 2 ¶ D−2 D 2 n λ 4πµ R 2R (2π) x x m2 − m2 = 1 − ln . (33) n 0 R2 16π2 n2 After performing the integration with the use of the method, we get 5. Conclusions − D 2 X∞ D D 2 2πi π 2 R (in) + (−in) δm0 ∝ − ¡ ¢ , (27) Loop corrections on Quantum Field theories can play an im- 1 − e2πD (2R)D Γ D n2 n=1 2 portant role in the phenomenology, often these corrections and in order to regularize it we take the limit D → 4−², such lead to non physical divergences over parameters of the the- that, above expression becomes ories, one way to remove these divergences is through renormalization. Any procedure to achieving this requires the use X∞ 2 1 2 of a regularization method that isolates the divergences from δm0 ∝ − 2 n (4πR) n=1 ﬁnite physical contributions. µ ¶ In this paper we present a method to perform Dimen- 2 4πµ2R2 × + 1 − γE + ln + O (²) . (28) sional Regularization without the use of Beta functions nor ² n2 Feynman’s Integral parametrization. We believe this method

Rev. Mex. Fis. 61 (2015) 149–153 DIMENSIONAL REGULARIZATION WITH NON BETA-FUNCTIONS 153 has the advantage of a direct evaluation of the loop integrals, B. Complex Logarithm by using an appropriate integration contour in complex space. To exemplify this we have applied our method to usual loop For the propose of present discussion it is important to re- integrals with one and two propagators. We have also pre- mind the reader the properties of the Logarithm over complex sented the use of this method to regularize models on extra numbers, for which space-time dimensions, as an illustrative example. ln (x + iy) = ln (r) + iθ, (B.1) Appendix where p y A. Laurent Series r = x2 + y2; θ = arctan , (B.2) x When it is not possible to express a function in a Taylor se- and we need to properly choice the branch not to overvalue rie, we can expand that function by its complex extension and the angle. This has been used on Sec. 3, where have make express it by its Laurent serie. use of the algebra When x is zero, the function 1/(1 − ex) is not deﬁned, however we can express it by a Laurent serie, such that µ µ (p + im) Ã !−1 ln = ln 1 X xn 1 p − im p2 + m2 x = − = − p 1 − e n! x (1 + u) µ p2 + m2 m n=1 = ln + i arctan . X xn 1 ¡ ¢ p2 + m2 p u = = − 1 − u + u2 − ... (n + 1)! x n=1 1 1 x Acknowledgments = − + − + O(x2). (A.1) x 2 12 Note the existence of the term which includes a negative de- This work was supported in part by Conacyt, Mexico,´ grant gree. no. 132061.

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Rev. Mex. Fis. 61 (2015) 149–153