Calculation of the anomalous magnetic moment with the modified BPHZ scheme

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(1)Calculation of the Anomalous Magnetic Moment with the Modified BPHZ Scheme. Degree Project. Miguel Ángel Vargas Jara Adviser: Andrés F. Reyes Lega Department of Physics Faculty of Science Universidad de Los Andes September 2012.

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(3) Calculation of the Anomalous Magnetic Moment with the Modified BPHZ Scheme. Project degree to obtain the title of physicist. Directed by: Andrés F. Reyes Lega. Department of Physics Faculty of Science Universidad de Los Andes September 2012.

(4) Copyright c Miguel Ángel Vargas Jara.

(5) ... Amy and Ángela.

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(7) Acknowledgements I wish to thank my advisor Andrés F. Reyes Lega for all his help and dedication during the course of this project. Also I would like to thank Juan Carlos Sanabria and Rolando Roldán for all they taught me about physics and life. Last but not least I would like to thank Juan Carlos Salazar for all his guidance and support.. vii.

(8) Index Acknowledgements. vii. 1. Introduction 1.0.1. History . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Basic Formalism 2.1. Interaction Scheme . . . . . . . . . . 2.1.1. Dyson series, Wick Theorem . 2.1.2. Feynman Diagrams . . . . . . 2.1.3. S-Matrix and Cross Sections 2.1.4. QED . . . . . . . . . . . . . . 3. Regularization and Renormalization 3.1. Divergences of the Feynman integrals 3.2. Schemes of Regularization . . . . . . 3.2.1. Pauli-Villars . . . . . . . . . . 3.2.2. Dimensional Regularization . 3.3. One loop regularization of QED . . . 3.3.1. Electron self-energy . . . . . 3.3.2. Vacuum polarization . . . . . 3.3.3. Vertex correction . . . . . . . 4. BPHZ 4.1. Bogoliubov Causality . . . . . 4.2. Construction of the S-Matrix, 4.3. Zimmermann’s Forest formula 4.4. Modified BPHZ approach . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 1 2. . . . . .. 4 4 6 10 13 16. . . . . . . . .. . . . . . . . .. . . . . . . . .. 20 21 25 26 26 31 31 33 36. . . . . . . . . . . . . . . . . relation with counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. 40 40 42 47 49. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 5. Modified BPHZ 50 5.1. Electron self-energy . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2. Vertex correction in QED . . . . . . . . . . . . . . . . . . . . 53. viii.

(9) Index. ix. 6. Conclusions. 57. A. Feynman Parameters. 59. B. Wick Rotation. 61. C. Electron’s Gyromagnetic Moment. 63. D. Alternative Vertex Calculation. 64. E. γ matrix algebra 70 E.1. γ matrix algebra: identities and traces . . . . . . . . . . . . . 70 E.2. γ matrix algebra for an arbitrary spacetime dimension d . . . 71 Bibliography. 72.

(10) List of figures 2.1. Vacuum expectation value of equation (2.30) 2.2. For each propagator . . . . . . . . . . . . . . 2.3. For each external point . . . . . . . . . . . . . 2.4. For each vertex . . . . . . . . . . . . . . . . . 2.5. For each propagator . . . . . . . . . . . . . . 2.6. For each external point . . . . . . . . . . . . . 2.7. For each vertex . . . . . . . . . . . . . . . . . 2.8. Representation of the Scattering process . . . 2.9. Dirac Propagator . . . . . . . . . . . . . . . . 2.10. Photon Propagator . . . . . . . . . . . . . . . 2.11. QED vertex . . . . . . . . . . . . . . . . . . . 2.12. External fermions . . . . . . . . . . . . . . . . 2.13. External antifermions . . . . . . . . . . . . . 2.14. External photons . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 11 12 12 12 13 13 13 15 18 18 18 18 18 18. 3.1. Radiative corrections for the Feynman diagram due to the Coulomb scattering at the third order in the electric charge and at the first order in the external field . . . . . . . . . . . 3.2. Electron self-energy . . . . . . . . . . . . . . . . . . . . . . . 3.3. Vacuum polarization . . . . . . . . . . . . . . . . . . . . . . . 3.4. Vertex correction . . . . . . . . . . . . . . . . . . . . . . . . .. 21 31 34 37. 4.1. Loop in φ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Forest operator Rγ . . . . . . . . . . . . . . . . . . . . . . . .. 48 49. B.1. Wick’s Contour. 61. . . . . . . . . . . . . . . . . . . . . . . . . .. x.

(11) Chapter 1. Introduction As it is known, the attempt to apply a formal perturbation theory to a relativistic quantum field theory leads in general to a set of ill-defined expressions for the elements of the S-matrix, i.e., inconsistent integrals which are supposed to describe different probability amplitudes for scattering processes. To overcome this difficulty there are two conceptually rather different lines of attack in QFT: one consists of employ a set of empirical regularization schemes which are designed and introduced into the ill-defined expressions to replace the divergent integrals in Feynman diagrams by convergent ones. All of these procedures are justified by their usefulness in practical applications to physical processes. The other line of attack follows from the approach developed by Epstein and Glaser, which is a method based on causality and locality in coordinate space, mathematically rigorous within perturbation theory, but not so useful in practice to make calculations. Among the regularization procedures of the first group the classical method of Bogoliubov, Parasiuk, Hepp and Zimmermann (BPHZ), by its conceptual clarity, is the most rigorous. In this method divergent integrals are regularized by means of appropriate Taylor subtractions of the integrand. Although this approach is clear by means of the Zimmermann’s forest formula, when it comes down to its implementation in higher orders of the perturbation theory, its applicability is not justified. For this reason, from a practical point of view, other methods of regularization are still better tools in calculations [1]. In this work it will be study the modified BPHZ procedure, which in principle combines the practical usefulness of dimensional regularization, an 1.

(12) 2 empirical regularization scheme, with the simplicity of the classical BPHZ renormalization.. 1.0.1.. History. In 1900, Max Planck introduced the idea that energy is quantized in order to derive a formula that could predict the observed frequency dependence of the energy emitted by a black body. In 1905, Einstein explained the photoelectric effect by postulating that light energy comes in quanta called photons. In 1913, Bohr explained the spectral lines of the hydrogen atom, again by using quantization. In 1924, Louis de Broglie put forward his theory of matter waves. These theories, though successful, were strictly phenomenological: they provided no rigorous justification for the quantization they employed. They are collectively known as the old quantum theory. The phrase "quantum physics"was first used in Johnston’s Planck’s Universe in Light of Modern Physics. Modern quantum mechanics was born in 1925 when Werner Heisenberg developed matrix mechanics and Erwin Schrödinger invented wave mechanics and the Schrödinger equation. Schrödinger subsequently showed that these two approaches were equivalent. Heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation took shape at about the same time. Starting around 1927, Paul Dirac unified quantum mechanics with special relativity. He also pioneered the use of operator theory, including the influential bra-ket notation, as described in his famous 1930 textbook. During the same period, John von Neumann formulated the rigorous mathematical basis for quantum mechanics as the theory of linear operators on Hilbert spaces, as described in his likewise famous 1932 textbook. These, like many other works from the founding period, still stand and remain widely used. Beginning in 1927, attempts were made to apply quantum mechanics to fields rather than merely to single particles, resulting in what are known as quantum field theories. Early workers in this area included Dirac, Wolfgang Pauli, Weisskopf, and Jordan. This area of research culminated in the formulation of quantum electrodynamics by Feynman, Freeman Dyson, Julian Schwinger, and Sin-Itiro Tomonaga during the 1940s. Quantum electrodynamics is a quantum theory of electrons, positrons, and the electromagnetic field, and serves as a role model for subsequent quantum field theories..

(13) 3 QED was the first quantum field theory in which the difficulties of building a consistent, fully quantum description of fields and of the creation and annihilation of quantum particles were satisfactorily resolved. Tomonaga, Schwinger and Feynman received the 1965 Nobel Prize in Physics for its development, their contributions involving a covariant and gauge invariant prescription for the calculation of observable quantities. Feynman’s mathematical technique, based on his diagrams, initially seemed very different from the field-theoretic, operator-based approach of Schwinger and Tomonaga, but was later shown by Freeman Dyson to be its equivalent. The renormalization procedure for making sense of some of the infinite predictions of quantum field theory also found its first successful implementation in quantum electrodynamics (though Feynman would later refer to renormalization as a "shell game.and "hocus pocus") [2]. As such, Quantum field theory is in principle the set of ideas and tools that provides the theoretical framework to describe and analyze a large extension of theories in modern particle physics, in which highlights the physics of elementary particles. Its formalism, being a generalization of quantum mechanics in the case of an infinite number of degrees of freedom in which the number of particles is not preserved, together with principles of relativity and the notion of fields, allows a description of processes of creation, annihilation, decay and scattering of particles through a set of well defined rules. It is itself a theory whose impact involves contributions in areas such as condensed matter, nuclear physics, cosmology, quantum gravity, high energy physics and pure mathematics “It is literally the language in which the laws of Nature are written. ” [3]..

(14) Chapter 2. Basic Formalism It will be assume from the reader a basic knowledge of classical fields, notions of canonical quantization and a previous treatment of free fields equations of motion given by linear partial differential equations; Concretely, fields without interactions. Here it will be discussed various aspects and principles of classical fields, quantum mechanics and special relativity, given that one wishes to understand processes that occur at very small (quantummechanical) scales and very large (relativistic) energies, together with its implications, defined the minimum and necessary theoretical framework before approaching to an interacting QTF.. 2.1.. Interaction Scheme. In order to obtain an accurate description of the real world, one in which particles interact with each other and the results of these interactions can be measured and verified experimentally, it is necessary to include nonlinear terms in the Hamiltonian, or Lagrangian, that will couple different Fourier modes, and the particles that occupy them, to one another. Thus, the terms describing the interaction will be of the form: Z Z Hint = d3 xHint [φ(x)] = − d3 xLint [φ(x)]. For now, only theories in which Hint is a polynomial function of the fields will be considered. To illustrate the kind of (small) perturbations that can be added to the theory one may consider consider, as an easy example, the Lagrangian for a real scalar field:. 4.

(15) 2.1. Interaction Scheme. 5. X λn 1 1 L = ∂µ φ∂ µ φ − m2 φ2 − φn . 2 2 n!. (2.1). n≥3. Here the coefficients λn are scalar coupling constants and the restrictions on them, to ensure that the additional terms are small perturbations, can be identified by dimensional analysis given the convention described in the introduction [3]. For now, by demanding that the theory be renormalizable, i.e. to contain coupling constant with possitive mass dimensions, and by excluding the case of strongly coupled field theories, the allowed interaction terms can be listed: For theories involving only scalars1 „ the allowed interaction terms will have the form µφ3 and λφ4 , with [µ] = 1 and [λ] = 0 . In the same way, spinor fields can be added such that the Yukawa term g ψ̄ψφ arises; vector fields and can be added also . For example, one of the most familiar term of interaction in QED would be the vector-spinor eψ̄γ µ ψAµ , less important terms can be constructed out of Weyl and Majorama spinors. These simple elements can be used to exhaust the list of possible terms of the Lagrangian involving scalar, spinor and vector particles. It is important to highlight that the currently accepted models of strong, weak and electromagnetic interactions include all of the types of interactions listed. Now, given a set of particles and couplings, it is necessary to work out a description that leads to a complete solution of the dynamics of the interacting field theory, i.e. find the unitary time evolution of the system, as it can be done in the free theories. A useful viewpoint in quantum mechanics to describe situations where one works with small perturbations of a well-understood Hamiltonian, consists in returning to the familiar ground of quantum mechanics with a finite number of degrees of freedom, where in the Schrödinger picture, the states evolve as d|ψiS = H|ψiS . (2.2) dt Observables (operators) in this picture while the operator OS are independent of time. In contrast, in the Heisenberg picture the states are fixed and the operators change in time according to i. OH (t) = eiHt OS e−iHt. (2.3). As an alternative to the Schrödinger and the Heisenberg pictures, one can use the interaction picture, a ’hybridóf the two,and split the Hamiltonian as 1. In order to be consistent with this convention of dimensions, it is important to clarify that in the units c = h=1. So in the system [lenght] = [time] = [energy]−1 = [mass]−1.

(16) 2.1. Interaction Scheme. 6. H = H0 + Hint .. (2.4). Here it is assumed that the complete Hamiltonian has two parts, the Hamiltonian H0 for a free field theory, which is responsable of the time dependence of operators, and the interactive Hamiltonian Hint , that governs the time dependence of states. The states and operators in the interaction picture will be denoted by a subscript I and are given by |ψ(t)iI = eiH0 t |ψ(t)iS ,. (2.5). OI (t) = eiH0 t OS e−iH0 t .. (2.6). The last equation applies also to Hint , which is time dependent. The interaction Hamiltonian in the interaction picture would be HI ≡ (Hint )I = eiH0 t (Hint )S e−iH0 t. (2.7). Then the Schrödinger equation for states in the interaction picture can be derived starting from the Schrödinger picture. i. d|ψiS d = HS |ψiS ⇒ i (e−iH0 t |ψiI ) = (H0 + Hint )S e−iH0 t |ψiI dt dt d|ψiI ⇒i = eiH0 t (Hint )S e−iH0 t |ψiI (2.8) dt. By using (2.7) it can be identify that (2.8) can be written as d|ψiI = HI |ψiI (2.9) dt This is the way in which the states of the system evolve. Also, the time derivative of the operators in this representation would be i. i∂t OI = [OI (t), H0 S ] Which implies that the field operator retains the properties of the free fields (a complete derivation and disscution about the different pictures and the way to change the description from one representation to another can be found in [4]).. 2.1.1.. Dyson series, Wick Theorem. Since the interaction effects are contained in |ψiI , then (2.9) must be the starting point of a perturbation treatment. In order to solve it, one can.

(17) 2.1. Interaction Scheme. 7. define the time-evolution operator, called Dyson operator, U(t, t0 ). This is a unitary operator that defines a connection between the state vectors in t0 and t as |ψi(t)I = U(t, t0 )|ψi(t0 )I. (2.10). satisfying   U(t, t) = 1, U(t, t1 ) · U(t1 , t0 ) = U(t, t0 ),   U(t, t0 ) = U −1 (t0 , t).. (2.11). Now, adopting the Dyson operator, the evolution of the interaction picture given by (2.9) requires that d (U(t, t0 )|ψi(t0 )I ) = HI (U(t, t0 )|ψi(t0 )I ) dt d ⇒ i U(t, t0 ) = HI U(t, t0 ) dt i. (2.12). It is advantageous to transform the differential equation (2.12) into an equivalent integral equation within the boundary conditions defined by (2.11): t. Z U(t, t0 ) = 1 + (−i). dt0 HI (t0 )U(t0 , t0 ).. (2.13). t0. Equation (2.13) is an integral equation of Volterra type, i.e., the independent variable enters as an integration limit. Equations of this type can be solved by iteration under some general and defined conditions. The process of successive re-insertion of the left side of (2.13) leads to the Neumann series, which consist of multiple integral involving products of the interaction Hamiltonian H(ti ) taken at different times, i.e.: Z. t. U(t, t0 ) = 1 + (−i). dt1 HI (t1 ) Z t Z t1 2 + (−i) dt1 dt2 HI (t1 )HI (t2 ) t0. t0. t0. + ... n. Z. t. + (−i). tn−1. dt1 ... t0. + .... Z. dtn HI (t1 )HI (t2 )...HI (tn ) t0. (2.14). It can be noticed that the time arguments are sorted in descending order, so the mutually dependent upper boundaries makes these integrals really.

(18) 2.1. Interaction Scheme. 8. diffucult to evaluate. To deal with this problem one can follow the idea put forward by Dyson2 , such that the integrations can be rewritten in a way that the all cover the full time interval [t0 , t]. In order to achive this, it is neccesary to use the time-ordered product ( Dyson product) as a tool. The operators in the product are put in the order of descending time argument:. T (HI (t1 )HI (t2 )...HI (tn )) = HI (ti1 )HI (ti2 )...HI (tin ) where ti1 ≥ ti2 ... ≥ tin The easiest way to exploit this idea, illustrated in the simple case of two operator products, consist in using the unit step function of Heaviside:. T (HI (t1 )HI (t2 )) = Θ(t1 − t2 )HI (t1 )HI (t2 ) + Θ(t2 − t1 )HI (t1 )HI (t2 ). (2.15). Remark: Notice that the T − operator in eq (2.15) is only defined for t2 > t1 or t2 < t1 , but not for t1 = t2 (cf.4.2). Equation (2.15) would imply that one can write the intregals as Z. t. Z. t1. dt1 t0. t0. 1 dt2 HI (t1 )HI (t2 ) = 2. Z. t. Z. t. dt2 T (HI (t1 )HI (t2 )). dt1 t0. (2.16). t0. By using (2.15) and (2.16), one can generalize the previous result by mathematical induction: Z. t. Z. tn−1. dt1 ... t0. t0. 1 dtn HI (t1 )...HI (tn ) = n!. Z. t. Z. t. dtn T (HI (t1 )...HI (tn )).. dt1 ... t0. t0. (2.17) With (2.17) one finds that the perturbation seriesof the time-evolution operator is Z t Z t Z t ∞ X 1 n U(t, t0 ) = (−i) dt1 dt2 ... dtn T (HI (t1 )...HI (tn )) n! t0 t0 t0. (2.18). n=0. The main use of the time-evolution operator lies in its applications to scattering processes. Also, as an alternative and more compact way to write 2. F.Dyson, Phys. Rev. 74, 486 and 1736 (1949).

(19) 2.1. Interaction Scheme. 9. this result, one can formally sum up the series of U(t, t0 ), arriving at the time-ordered éxponential’function Z t n o U(t, t0 ) = T exp[−i dt0 HI(t0 ) ] . (2.19) t0. Since in any local field theory the Hamiltonian can be expressed as an integral over a Hamiltonian density H, which consist of products of fields operators, the result would take the final form U(t, t0 ) = T. n. Z. t. exp[−i. o d3 x0 HI(x0 ) ]. (2.20). t0. From Dyson’s formula, now, one can compute quantities like hf |T {HI (x1 )...HI (xn )}|ii, where |ii and |f i are assumed to be eigenstates of the free theory. In this expresion the ordering of the operators is fixed by the time-ordering. However, since the HI ’s contain creation and annihilation operators, it is convenient therefore to find a way to simplify the extensive calculations once one makes the distribution of the terms into the product. This can be done by using normal ordering, which consist in moving all annihilation operators to the right so they can start killing things in |ii. In principle, Wick’s theorem tells how to go from time ordered products to normal ordered products. Wick’s prescription can be easily evaluate and generalize by starting from the case of a real scalar field. To begin one descomposes the field φI (x) into positive and negative frecuency parts: − φI (x) = φ+ I (x) + φI (x). (2.21). where. φ+ I (x) φ− I (x). Z = Z =. d3 p 1 p ap e−ip·x 3 (2π) 2Ep. (2.22). d3 p 1 p a†p e+ip·x 3 (2π) 2Ep. (2.23). This descomposition can be done for any free field and can be exploited because − φ+ I (x)|0i = 0 and h0|φI (x) = 0. Choosing x0 > y 0 , the time-ordered product of two fields is.

(20) 2.1. Interaction Scheme. 10. + − − + − − + T {φI (x)φI (y)} = φ+ I (x)φI (y) + φI (x)φI (y) + φI (x)φI (y) + φI (x)φI (y) + − + − + − − = φ+ I (x)φI (x) + φI (y)φI (x) + φI (x)φI (y) + φI (x)φI (y) − + [φ+ I (x), φI (y)]. (2.24). In every term of (2.24), except for the commutator, all the ap ’s are to the right of all the a†p . Such terms are said to be in normal order and the reason to use this convention is because it has a vanishing vacuum expectation value, which simplifies a lot the calculations. The notation for terms in normal order will be : φ1 φ2 : . Now, if instead of x0 > y 0 one considers x0 < y 0 , the result is the same − as the one in (2.24), but this time the commutator would be [φ+ I (y), φI (x)]. With this in mind one can define one more quantity, the contraction of two fields, as ( − 0 0 z }| { [φ+ I (x), φI (y)] for x > y (2.25) φ(x)φ(y) ≡ − 0 0 [φ+ I (y), φI (x)] for x < y . This quantity is the Feynman propagator: Z z }| { φ(x)φ(y) = DF (x − y) =. d4 k ieik·(x−y) (2π)4 k 2 − m2 + i. (2.26). As it was mentioned, Wick established how to go from time ordered products to normal ordered products according to z }| { h0|T {φ(x)φ(y)}|0i = φ(x)φ(y), z }| { T {φ(x)φ(y)} = : {φ(x)φ(y) + φ(x)φ(y)} :. (2.27). The generalization to arbitrarily many fields, made by induction and proved in [5], which is known as Wick’s theorem, can be write down as. T {φ(x1 )φ(x2 )...φ(xm )} = : {φ(x)φ(y) + All possible contractions} : (2.28) Where the phrase all possible contractions refers to each possible way of contracting the m fields in pairs.. 2.1.2.. Feynman Diagrams. Wick’s theorem allows to turn any vacuum expectation value of timeordered products, with the form h0|T {HI (x1 )...HI (xn )}|0i, into a sum of.

(21) 2.1. Interaction Scheme. 11. products of Feynman propagators. Now, although this theorem simplified a lot the calculations, its implementation can be even more useful by developing a diagrammatic interpretation. A practical way to understand the implementation of Wick’s theorem is to consider the case of m = 4 in (2.27), where by notation it will be use φ(x1 ) as φ1 z }| { z }| { z }| { T {φ1 φ2 φ3 φ4 } = : φ1 φ2 φ3 φ4 : + φ1 φ2 : φ3 φ4 : + φ1 φ3 : φ2 φ4 : + φ1 φ4 : φ2 φ3 : z }| { z }| { z }| { z }| { z }| { + φ2 φ3 : φ1 φ4 : + φ2 φ4 : φ1 φ3 : + φ3 φ4 : φ1 φ2 : + φ1 φ2 φ3 φ4 z }| { z }| { z }| { z }| { + φ1 φ3 φ2 φ4 + φ1 φ4 φ2 φ3 (2.29) When one takes the vacuum expectation value of (2.29), any term in which remain uncontracted operators, gives zero, as claimed. The only ones which survive are the full contracted terms, i.e.. h0|T {φ1 φ2 φ3 φ4 }|0i = DF (x1 − x2 )DF (x3 − x4 ) + DF (x1 − x3 )DF (x2 − x4 ) + DF (x1 − x4 )DF (x2 − x3 ). (2.30). Now, for a diagrammatic interpretation of this example, if one represents each of the points xm , for m = 4, by a dot (vertex) and each propagator DF (x − y) by a line that joins x and y, then (2.30) can be represented as the sum of three diagramas, called Feynman diagrams. Figura 2.1: Vacuum expectation value of equation (2.30) Although these diagrams are not exactly measurable quantities, they are actually correlation functions, they suggest an important interpretation; a description of how particles are created at two spacetime points, then each propagates to one of the other points, and how then they are annihilated. As Figure 2.1 indicates, the description can happen in three different ways corresponding to the three ways of contracting the four fields in pairs. As result, the total amplitude of the process would be the sum of the three diagrams. The diagrams actually interpret the analytic formula as a process of particule propagation, creation and annihilation which take place in spacetime..

(22) 2.1. Interaction Scheme. 12. The rules for associating these analytic expressions with pieces of diagrams are called then the Feynman rules. For the simple case of a scalar theory like the φ4 theory, the rules would be:. Figura 2.2: For each propagator. Figura 2.3: For each external point. Figura 2.4: For each vertex. Finally, divide by a symmetry factor (number of different contractions that give the same expressions). According to this description, lines in these diagrams will be referred as propagators (because of their association with the propagation amplitude DF ). The internal points where lines meet will be referred as vertices. The vertex factor (−iλ) can be thought as the for emission or absorR amplitude 4 tion of a particle at a vertex. The term d x is a direct consequence of the superposition principle of quantum mechanics, instructing to integrate over all the points where this process can occur (In general, a process can happen in different ways, so the amplitudes for each one of that possible was must be added.). These rules are called position-space Feynman rules, since they are written in terms of spacetime points. By introducing the Fourier expansion of each propagator, one can express them in terms of momenta as : Z DF (x − y) =. d4 p i e−ip·(x−y) 4 2 (2π) p − m2 + i. (2.31). This new representation assigns a 4-momentum p (with an arrow that indicates a direction) to each propagator. Then, as consequence, when several lines meet at a vertex, one gets a Dirac delta ’functionín terms of the sum.

(23) 2.1. Interaction Scheme. 13. of all momenta flowing into the vertex. In other words, this establishes that momentum is conserved at each vertex. By using the momentum integrals one has the following momentum-space Feynman rules:. Figura 2.5: For each propagator. Figura 2.6: For each external point. Figura 2.7: For each vertex. Impose momentum conservation at each vertex R Integrate over each undetermined momentum d4 p/(2π)4 Divide by a symmetry factor Finally, it must be indicated that the amplitude or contribution resulting from the different diagrammatic expressions will be denoted by M(Feynman amplitude). This amplitud is given by M=. ∞ X. M(n) ,. (2.32). n=1. where M(n) comes from the nth order or perturbation terms S (n) . To summarize, the Feynman amplitude M(n) is obtained by drawing all the topologically different and connected Feynman diagrams in momentum space which contains n vertices and a set of correct external lines.. 2.1.3.. S-Matrix and Cross Sections. The experiments that probe the behavior of elementary particles, especially in the relativistic regime, are scattering experiments. In general, one collides two beams of particles with well-defined momenta in order to observes what comes out. The probability of a particular final state can be.

(24) 2.1. Interaction Scheme. 14. expressed in terms of the cross section, a quantity that is intrinsic to the colliding particles and therefore allows comparison of differents experiments with similar setups. As it was mentioned, when in Dyson’s formula one wants to compute quantities like hf |T {HI (x1 )...HI (xn )}|ii, |ii and |f i are assumed to be eigenstates of the free theory, this means that one is conceiving that asymptotically the states |ii at t → −∞ and |f i at t → ∞ are eigenstates of the free Hamiltonian H0 . This assumption can make sense at some level, since one can argue that at t → −∞, or at t → ∞, the particles in a scattering process are far separated and don’t ’feel’the effects of each other. Furthermore, one should expect these states to be eigenstates of an individual number operators N , which commute with H0 , but not with Hint . Then, as the particles approach each other they interact briefly, before departing again, each going on its own way (Although from the physics point of view this is something reasonable, from the mathematician point of view this it not obvious (Figure 2.8) [3].) The amplitude to go from |ii to |f i is lı́m hf |U(t, t0 )|ii ≡ hf |S|ii. t→∞ t0 →−∞. (2.33). where the unitary operator S is known as the S-matrix. Then the Smatrix is just the limit of the time-evolution operator in the interaction picture: S = lı́m U(t, t0 ) t→∞ t0 →−∞. (2.34). Then, according to (2.34) and to (2.18), one can express the S-matrix explicitly as. S=. Z ∞ Z ∞ Z ∞ ∞ X 1 (−i)n dt1 dt2 ... dtn T (HI (t1 )...HI (tn )) n! −∞ −∞ −∞. (2.35). n=0. Typically, in experiments related with particle physics one uses well prepared beams of particles with definite momentum, polarization, etc, in order to mesure these same quantities for the final states. Now, one can consider a general scattering process with a set of initial particles with four momenta of the kind pi = (Ei , pi ) which collide and produce a set of final particles with four momenta pf = (Ef , pf ). Each external photon line contributes with (1/2V E)1/2 , whereas each external fermionic line with (m/V E)1/2 ..

(25) 2.1. Interaction Scheme. 15. Figura 2.8: Representation of the Scattering process The conservation of the total four momentum gives a term of X X (2π)4 δ 4 Pi − Pf , i. f. and, If one excludes the uninteresting case f = i, by working in a momentum representation, one can express the result for a generic matrix element of the S-matrix in the following form. 4 4. hf |S|ii = Sf i = (2π) δ. X. Pi −. i. X f. Y r 1 Pf 2EV ext.bos. Y ext.f erm. r. m M VE (2.36). where it is important to indicate tha the Feynman amplitude M is obtained by drawing at a given perturbative order all the connected and topologically inequivalent Feynman’s diagrams according to the rules defined. If one considers the typical case of a two particle collision giving rise to N particles in the final state, the probability per unit time of the transition is given by w = |Sf i |2 with w = V (2π)4 δ 4 (Pf − Pi ). Y i. 1 Y m |M|2 2Ei V V Ef. (2.37). f. then w gives the probability per unit time of the transition of a state well defined. Since one is interesed in knowing the momenta between p~f and p~f + d~ pf , according to the normalization of wave function, one can express easily that the cross-section for the final states with momenta defined between p~f and p~f + d~ pf is given by. dσ = w. V Y V d 3 pf vrel (2π)3 f. (2.38).

(26) 2.1. Interaction Scheme. 16. here vrel is related with the relative velocity of the flux of incoming particles. One can obtaing finally an expression for the cross-section. dσ = (2π)4 δ 4 (Pf − Pi ). 1 4E1 E2 vrel. Y. (2m). f ermions. Y f. d3 pf |M|2(2.39) (2π)3 2Ef. which is in principle a sign of the total interaction probability, for the prepared particles, or ratio of the area of the section of the solid to the total targeted area.. 2.1.4.. QED. The first constructed theory that fulfilled the principles demanded by QFT was quantum electrodynamics (QED), a theory of light interacting with charged matter. Since it describes mathematically all phenomena involving electrically charged particles interacting by means of exchange of photons3 , it is one of the most importants theories in physics. The study of interacting electron-positron and electromagnetic fields are describe by the Lagrangian density4 L = L0 + LI with the free-field Lagrangian density L0 = : [ψ̄(x)(i∂/ − m)ψ(x) − 1/2(∂ν Aµ )(∂νAµ )] :. (2.40). and the interaction Lagrangian density [6] / LI = : [−S µ (x)Aµ (x)] : = e : {ψ̄(x)A(x)ψ(x)} :. (2.41). The QED interaction Hamiltonian density is. / HI (x) = −LI (x) = −e : {ψ̄(x)A(x)ψ(x)} : + − /+ / − )(ψ + + ψ − ) = −e : (ψ̄ + ψ̄ )(A + A. x. : (2.42). Since Lorentz invariance requires the interaction Hamiltonian HI be a product of an even number of spinor fields, there are no many difficulties to make a generalization of a theorie containing fermions. To start with the 3. It represents the quantum counterpart of classical electrodynamics giving a complete account of matter and light interaction 4 The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field is given by the real part of L = ψ̄(i∂/ − m)ψ − 1/4(Fµν )2 − eψ̄γ µ ψAµ.

(27) 2.1. Interaction Scheme. 17. treatment, in order to apply Wick’s theorem, it is necessary to generalize the definitions of normarl-ordering and time-ordering to include fermions. The time-ordering operator T acting on two spinor fields is conveniently defined with and additional minus sign ( ψ(x)ψ̄(y) for x0 > y 0 T {ψ(x)ψ̄(y)} ≡ (2.43) −ψ̄(y)ψ(x) for x0 < y 0 With this definition, the Feynman propagator for the Dirac field is Z SF (x − y) =. i(p d4 p / + m) −ip·(x−y) e = h0|T {ψ(x)ψ̄(y)}|0i (2.44) 4 2 (2π) p − m2 + i. When one has to deal with products of more that two spinor fields, the timer-ordered product gains a minus sign for each interchange of operators in order to put the fields in time order, e.g. T {ψ1 ψ2 ψ3 ψ4 } = (−1)3 ψ3 ψ1 ψ4 ψ2 if x03 > x01 > x04 > x02 Likewise, for the normal-ordered product of spinor fields one needs to put an extra minus sign for each fermion interchange. There are many conventions to write the normal-ordered product due to the anticommutation properties, here it will be use : {ap aq a†r } : = (−1)2 a†r ap aq The generalization of Wick’s theorem is straightforward by using these modifications. In analogy with (2.27), the time-oreder operator will define the contraction of two fields by z }| { T [ψ(x)ψ̄(y)] = : ψ(x)ψ̄(y) : + ψ(x)ψ̄(y) Where the contraction for the Dirac field is ( z }| { {ψ + (x), ψ̄ − (y)} for x0 > y 0 ψ(x)ψ̄(y) = SF (x − y) ≡ −{ψ̄ + (y)ψ − (x)} for x0 < y 0 z }| { z }| { ψ(x)ψ(y) = ψ̄(x)ψ̄(y) = 0. (2.45). (2.46). (2.47). With these conventions, Wick’s theorem takes the same for as before in (2.28) :. T [ψ1 ψ̄2 ...ψm ..] = : [ψ1 ψ̄2 ...ψm .. + All possible contractions] :. (2.48). By denoting scalar particles by dashed lines and fermions by solid lines, one obteines the following momentum-space Feynman rules:.

(28) 2.1. Interaction Scheme. 18. Figura 2.9: Dirac Propagator. Figura 2.10: Photon Propagator. Figura 2.11: QED vertex. Figura 2.12: External fermions. Figura 2.13: External antifermions. Figura 2.14: External photons.

(29) 2.1. Interaction Scheme Impose momentum conservation at each vertex R Integrate over each undetermined momentum d4 p/(2π)4 Figure out the overral sign of the diagram. 19.

(30) Chapter 3. Regularization and Renormalization Once established the Feynman rules and their corresponding role in order to obtain the matrix elements Sf i for any kind of collision process in QED, one can make fairly simple calculations and decent predictions in the lowest order of the perturbation theory developed. As one goes to higher orders in perturbation, the Feynman diagrams become topologically complicated and many of them may contain internal propagators whose momenta are not uniquely determined in terms of the external momenta. Instead, some of the internal propagators may involve loop momenta that has to be integrated over all possible values. The inconvenient on doing such calculations is that one encounters divergent integrals. As result, one needs to find a way of extracting all kind of meaningful results from such a quantum field theory. The process to overcome these difficulties, according to this aproach, involves three steps. Firstly, one regularizes the theory, which consist in modifies it such that it remains well-defined in all orders of perturbation, this is known as regularization. In the second step one needs to realize that there are non-interacting particles, leptons and photons, from which the perturbation theory starts and which are not real physical particles that interact (bare particles). Since the interaction modifies in different ways the properties of the particles, one must recognize a way to express the theory in terms of the properties of the actual interacting particles (physical particles). The second step is called renormalization, and is the process of redefining the infinities that are encounter in perturbation theory. Broadly, one needs to introduce a regularization procedure which gives a meaning to the divergent Feynman integrals by isolating the divergent parts of the diagram, relating in that way the properties of the physical particles to those of the bare particles by 20.

(31) 3.1. Divergences of the Feynman integrals. 21. expresing the predictions in terms of the known masses and charges of the physical particles. The last step consist in going from the regularized theory to QED. In principle, one must remove the regularization dependence of the theory and as result, the original ill-defined quantities will apear in releations between physical and bare particles. Since these relations are unobservable, one needs to recognize that the observable predictions of the theory may be express in terms of the actual measured carges and masses of the particles [6].. 3.1.. Divergences of the Feynman integrals. When one considers the scattering of electrons or positrons by an éxternalélectromagnetic field Aαe (x), such as the Coulomb field of a heavy nucleus, the S˘matrix expansion in QED given by (2.34) to evaluate S (n) and by (2.42), as the QED interaction Hamiltonian density, leads to expand up to the third order of n to find and treat physical processes whose contribution is not zero or unphysical. The relevant Feynman diagrams up to this order are. Figura 3.1: Radiative corrections for the Feynman diagram due to the Coulomb scattering at the third order in the electric charge and at the first order in the external field The Feynman amplitudes of the interesting diagrams in the Fig. 3.1 result from the Feynman rules and are given as follows:.

(32) 3.1. Divergences of the Feynman integrals. 22. For the self-energy contribution to one of the external photons one has the two cases Ma = ū(p0 )(−ieγ µ )Aext p0 − p~) µ (~ Mb = ū(p0 )[ie2 Σ(p0 )]. i [ieΣ(p)]u(p) p / − m + i. i µ ext 0 p − p~)u(p) 0 − m + i (−ieγ )Aµ (~ p /. (3.1) (3.2). where −ig µν i d4 k µ (−ieγ ) (−ieγν ) (2π)4 p / − k/ − m + i k 2 + i Z d4 k 1 1 γµ = −e2 γµ 2 / (2π)4 p k + i − k − m + i / Z. 2. ie Σ(p) =. so Z Σ(p) = i. p d4 k 1 / − k/ + m γµ γµ 2 4 2 2 (2π) (p − k) − m + i k + i. (3.3). (3.4). Now one can express the amplitudes for the vacuum polarization contribution as Mc = ū(p0 )(−ieγ µ ). −igµν 2 νρ ie Π (q)Aext p0 − p~)u(p) ρ (~ q 2 + i. (3.5). with q = p0 − p, where. 2. Z. µν. ie Π (q) = (−1). h i d4 k i i µ ν T r (−ieγ ) (−ieγ ) (2π)4 k/ + p k/ − m + i / − m + i (3.6). and therefore Πµν (q) = i. Z. i h d4 k i i µ ν T r γ γ (2π)4 k/ + p / − m k/ − m + i. (3.7). The last contribution amplitude due to the Vertex correction is Md = ū(p0 )(−ie)e2 Λµ (p0 , p)u(p)Aext p0 − p~) µ (~. (3.8). where ρσ d4 k i i ρ µ σ −ig eγ γ eγ 0 (2π)4 k 2 + iε p / − k/ − m + iε p / − k/ − m + iε Z 4 d k ρ 1 1 1 γµ γρ 2 γ 0 (3.9) = −i 4 (2π) p / − k/ − m + iε p / − k/ − m + iε k + iε. e2 Λµ (p0 , p) = i2. Z.

(33) 3.1. Divergences of the Feynman integrals. 23. Since the three integrals have divergences, one needs to give some sense to the theory so one can define properly the integrals. This is what is known as the regularization procedure. The purpose is to work out a prescription to make the integrals finite. In order to analyze the divergences of the different Feynman integrals one can begin by studing the degree of divergence of a generalize graph. To start one may consider a diagram of order n, i.e. with n vertices, E external lines, I internal lines and L loops in which the space-time has d dimensions. One ask then for the superficial degree of divergence D of this diagram5 , given by D = dL − 2I It is convenient to express D in terms of E and n in order to eleminate I and L. Since there are I internal momenta; there is a momentum conservation at each one of the n vertexes, but there is also overall momentum conservation such that there are n − 1 relations between the momenta, so the number of independent momenta is I − n + 1. These leads to express L as L=I −n+1 Now, in doing a leg count for each vertex for a general φr theory, rn = E + 2I. such that. D =d−. d. hr i − 1 E + n (d − 2) − d 2 2. (3.10). which, when d = 4, gives. D = 4 − E + n(r − 4). (3.11). For a φ4 theory, for example, (3.11) reduces to D = 4−E, which indicates that diagrams with more external legs that 4 will all converge6 . Now, in the same way in which the scalar theory was treated, for QED, a general formula for D, according to [7], can be written as:. D = dL − 2Pi − Ei 5. (3.12). This quantity is of interest because according to Weinberg’s theorem, a Feynman diagram converges if its degree of divergence D, together with the possible degrees of divergences of all its subgraphs, is negative. 6 This seems to indicated that in a φ4 theory in four space-time dimensions, D does not depends on the order in perturbation theory, so one would expect that the effects of a small number of divergent graphs on it can be eliminated by making a renormalisation of various physical quantities..

(34) 3.1. Divergences of the Feynman integrals. 24. where L = number of loops Pi = number of internal photon lines Ei = number of internal electron lines d = dimension of the space-time In addition one can define. n = number of vertices Pe = number of external photon lines Ee = number of external electron lines As before, L is the number of independent momenta for integration and can be expressed like L = Ei + Pi − n + 1 In doing the same counting for a φ4 theory, when d = 4, one can get the result:. D =4−. 3Ee − Pe 2. (3.13). For instance, one can explicitly calculate the superficial degree of divergence for the different divergent elements from the integral amplitudes:.

(35) 3.2. Schemes of Regularization. 25. Tabla 3.1: Superficial degree of divergence D of the one-loop diagrams in QED Diagram. Ee. Pe. D. 2. 0. 1. 0. 2. 2. 2. 1. 0. Therefore, there are only three divergent one-loop diagrams and all the divergences can be brought back to the three functions Σ(p), Πµν and Λµ (p0 , p). This doesn’t mean that an arbitrary diagram is not divergent, but it can be made finite if the previous functions are such. In such a case one has only to show that eliminating these three divergences (primitive divergences) the theory is automatically finite. In particular, it can be show that the divergent part of Σ(p) can be absorbed into the definition of the mass of the electron and a redefinition of the electron field (wave function renormalization). The divergence in Πµν , the photon self-energy, can be absorbed in the wave function renormalization of the photon (the mass of the photon is not renormalized due to the gauge invariance). And finally the divergence of Λµ (p0 , p) goes into the definition of the parameter e [7].. 3.2.. Schemes of Regularization. In order to distinguish between convergent and divergent integrals, It will be use the method of power-counting based on a cut-off of the domain of integration. In terms of physics, this cut-off corresponds to the introduction of an upper bound for the admissible energies (resp. momenta). For the regularization of divergent integrals, it will be discuss different methods used by physicists in renormalization theory: Pauli-Villars, the method of.

(36) 3.2. Schemes of Regularization. 26. subtractions and Dimensional Regularization.. 3.2.1.. Pauli-Villars. “Regularize divergent integrals by introducing additional ghost particles of large masses.” One of the earliest regularization techniques, developed by Wolfgang Pauli and Felix Villas in 1949, which consist in introduce massive auxiliary fields called regulators in order to eliminate singularities from propagators and other ill-definend field functions. This technique gives in principle a prescrition to use these regulators in QED such that the theory remains gauge invariant at each order of perturbation theory. By means of this technique one needs to regard these regulators as a purely formalism cutoff procedure. So one counteracted divergences with fictitious or auxiliary particles which modify the structures of the integrals. e.g.7 : 1 1 1 −→ − 2 2 2 (k − p) + i (k − p) + i (k − p) − Λ2 + i. 3.2.2.. Dimensional Regularization. Dimensional regularization8 consist in calculate the Feynman diagrams as analytic functions with spacetime dimensionality d, so that they become finite. For d small enough, any loop-momentum integral will converge and the final expression for any observable quantity will have a well-defined limit9 as d → 4 . In first place, one generalizes from a four-dimensional to a d -dimensional space with a positive integer d. In a d -dimensional space, the metric tensor g µν = gµν of this space is defined by g 00 = −g ii = 1, i = 1, 2, ..., d − 1 g µν = 0, µ 6= ν. (3.14). Correspondingly, a four-vector lµ is remplaced by a vector with d -components 7. Pag 640 Ashok Das - Lectures on Quantum Field Theory Developed by G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44 (1972) 189 9 Dimensional regularization was introduced by Gerardus â Hooft and Martinus Veltman in 1972.19 They showed that, in contrast to the 1938 Fermi model, the electroweak Standard Model in particle physics is renormalizable. For this, Hooft and Veltman were awarded the Nobel prize in physics in 1999. 8.

(37) 3.2. Schemes of Regularization. 27. lµ = (l0 , l1 , l2 , l3 ) → (l0 , l1 , ..., ld−1 ) ≡ (l0 , l) and 2. 0 2. 2. 0 2. l = (l ) − l = (l ) −. d−1 X. (li ). 2. (3.15). (3.16). i=1. Further, any loop integral will become integral in d dimension with volume element d d4 l 4−d d l (3.17) → µ (2π)4 (2π)d where µ is an energy scale parameter introduced to keep the physical dimension. In order to show how the technique works, one can consider spacetime to have one time dimension and (d -1) space dimensions. Omitting for now the normalzation given by (3.17), let be Z Id =. dd lE. 1 ≡ d (l2 + ∆)n (2π) E. Z. Z. dΩd. ∞. dlE. (2π)d. 0. lE d−1 2 + ∆)n (lE. (3.18). where it has been defined d -dimensional solid angle. Now, with d = 4, this term converges if n > 2. In is log-divergent if n = 2, but if one defines d < 4 the integral is made to converge , i.e., one needs to take at the final d → 4 to go back to QED. To compute the solid angle of a unit sphere in d -dimensions, in (3.18), one can use the following trick: √. d. ( π) =. Z Z. −x2. dx · e Z. d. Z =. d X 2 d x · exp − xi d. i=1. ∞ d−1. −x2. dΩd dx · x ·e 0 Z Z ∞ 2 = dΩd · 1/2 d(x2 ) · (x2 )d/2−1 · e−x 0 Z Γ(d/2) √ d dΩd ( π) = 2 =. (3.19). in (3.19) it was used the definition of the gamma function Γ(z) = R ∞ Wherez−1 · ex . Then the area of a d -dimensional unit sphere is 0 dx · x Z dΩd = returning to (3.18). 2π d/2 Γ(d/2). (3.20).

(38) 3.2. Schemes of Regularization. Z. dΩd. Id =. 28. Z. ∞. dlE. lE d−1 2 + ∆)n (lE. (2π)d 0 Z Z ∞ dΩd 1 lE d/2−1 2 = d(l ) E 2 + ∆)n (lE (2π)d 2 0. by defining x ≡. ∆ 2 +∆ lE. Z. (3.21). and by using the definition of the beta function:. 1. dx xα−1 (1 − x)β−1 ≡. 0. Γ(α)Γ(β) Γ(α + β). one can obtain an expression for Id and then one can relate this result with (3.18). Id =. π d/2 Γ(n − d/2) 1 n−d/2 Γ(n) ∆ (2π)d. (3.22). then Z ⇒. dd lE. Γ(n − d/2) 1 n−d/2 1 1 = 2 + ∆)n Γ(n) ∆ (4π)d/2 (2π)d (lE. (3.23). Now Γ(0) = ∞; but Γ(z > 0) = finite, then Id = finite for d = 4 as long as n > 2 according to (3.23). The manifestation of differences in I4 for d = 4 is expressed as follows I4 ∼ Γ(n − d/2) ∼ Γ(d − d/2) → Γ(0) as d → 4 Since one wants to recover QED in the limit, it is necessary to take ≡ 4 − d → 0+ . Now, since zΓ(z) = Γ(z + 1), then Γ(2 − d/2) = Γ(/2) =. /2 Γ(/2) Γ(/2 + 1) 2 = −−→ →0 /2 /2. So the divergence is manifested as a pole in . One can find also the finite parts, since that is where physics lies Γ(/2) = Since Γ0 (1) = constant), one gets. R∞ 0. Γ(/2 + 1) 1 + /2Γ0 (1) + O(2 ) = /2 /2. e−x ln x dx ≡ −γ ' −0,577216(Euler-Mascheroni. Γ(/2) = Similarly, one has. (3.24). 2 − γ + O() . (3.25).

(39) 3.2. Schemes of Regularization. 29. π d/2 1 1 = ' (1 + ln 4π) 2 d d/2 16π 2 (2π) (4π). (3.26). The intregral (3.18) is then Z. dd lE. 1 1 2 − log∆ − γ + log(4π) + O() − − − → 2 + ∆)n d→4 (4π)2 (2π)d (lE (3.27) From (3.27) one confirms that the 1/ pole in dimensional regularization corresponds to a logarithmic divergence in the momentum integral. Also, the scale of the logarithm is hidden in the 1/ term, since ∆ is a dimensionful quantity. The actual right expression has to be dimensionless for the logarithm, so, by taking into account the normalization given by (3.17) one gets I4 =. Ie2 →. 1 2 ∆ − log − γ + log(4π) + O() (4π)2 µ2. (3.28). Now, to generalize even more a prescription to deal with the calculations, one can take the result (3.20) and use it to write Id , in (3.21), as π d/2. 1 Id = d Γ(d/2) (2π). Z. ∞. xd/2−1 F (x)dx. (3.29). 0. 2 and F (x) = (l2 + a2 )−n and a2 = ∆. The integrals that will with x = lE E appear later are of the type. IdM. dd l. Z =. d. (2π). (l2. 1 − ∆ + i)n. (3.30). with l as a vector in a d -dimensional Minkowski space. It can be perform an anticlockwise rotation of π/2, Wick’s rotation, in the complex plane l0 without hitting any singularity. The one can Pdo a2 change of variables from l0 → il0 , and use the definition l2 → −l2 = N i=1 li to obtain IdM. Z =. dd l. 1 = i(−1)n Id 2 − a2 )n (−l (2π) d. (3.31). 2 + a2 )−n , and at the Where Id is the integral discussed with F (x) = (lE same time the result (3.22). Then, again, this result leeds to a general form of (3.23 ).

(40) 3.2. Schemes of Regularization. Id =. 1 π d/2 Γ(n − d/2) d n−d/2 2 Γ(n) (2π) (a ). 30. (3.32). This shows that Id has simple poles located at d = 2(n, n + 1, ...). Therefore the integral will be well-defined at all d such that d 6= 2(n, n + 1, ...)( although at the end of the procedure, one has to take the limit d → 4). Then, the original integral in Minkowski space is Z. d/2 Γ(n − d/2) dd l 1 1 n π = i(−1) 2 2 n d d n−d/2 2 Γ(n) (2π) (l − a ) (2π) (a ). (3.33). This formula will be useful to derive another formula. Putting l = l0 + k and b2 = k 2 − a2 in (3.33) d/2 Γ(n − d/2) dd l0 1 1 n π = i(−1) 2 d d n−d/2 2 0 0 2 2 n Γ(n) (2π) (l + 2l · k + k − a ) (2π) (a ) (3.34) from which, by changing the name of the dummy variable. Z. d/2 Γ(n − d/2) 1 dd l 1 n π = i(−1) 2 2 n d d 2 Γ(n) (2π) (l + 2l · k + b ) (2π) (k − b2 )n−d/2 (3.35) Differentianting with respect to kµ , one gets various useful relations like. Z. d/2 Γ(n − d/2) −kµ lµ dd l n π = i(−1) 2 Γ(n) (2π)d (l2 + 2l · k + b2 )n (2π)d (k − b2 )n−d/2 (3.36) and. Z. Z. lµ lν dd l π d/2 (−1)n =i × Γ(n − d/2)lµ lν (2π)d (l2 + 2l · k + b2 )n (2π)d Γ(n)(k 2 − b2 )n−d/2 1 2 2 − gµν (k − b )Γ(n − 1 − d/2) (3.37) 2. Finally, in order to consistently track of the terms O(), one needs to re-examine all identities that might change with dimension. To begin with, in d -dimension g µν obeys gµν g µν = T r[g 2 ] = T r[I dxd ] = d ⇒ gµν g µν = d which implies that the numerator of a symmetric integrand conteining should be remplace with. lµ lν ,.

(41) 3.3. One loop regularization of QED. 31. 1 2 µν l g (3.38) d In QED, the Dirac matrices can be manipulated as a set of d matrices satisfying lµ lν →. {γ µ , γ ν } = 2g µν · 14×4 ,. T r[1] = 4. (3.39). Since the actual number of γ µ ’s i now d 6= 4, the contraction identities are modified in d = 4 − to γ µ γ ν γµ = −(2 − )γ µ γ µ γ ν γ ρ γµ = 4g νρ − γ ν γ ρ γ µ γ ν γ ρ γ σ γµ = −γ σ γ ρ γ ν + γ ν γ ρ γ σ. 3.3.. (3.40). One loop regularization of QED. In this section it will be show how to regularize the relevant quantities in QED, Σ(p), Πµν (q) and eΛµ (p0 , p) by using dimensional regularization. In order to identify the counterterms, one needs to analyze the expressions which become divergent in the limit when d → 4.. 3.3.1.. Electron self-energy. Figura 3.2: Electron self-energy The corresponding self-energy contribution displayed in Figure 3.2 is due to one virtual photon and can be described by applying the Feynman rules, as it was indicated in (3.4): Z Σ(p) = i. p 1 d4 k / − k/ + m γµ γµ 2 . 4 2 2 (2π) (p − k) − m + i k + i. By using the prescription to generalize Σ(p) into a dimension d, one has Σ(p) = iµ. 4−d. Z. p dd k 1 / − k/ + m γ γµ µ (2π)d (p − k)2 − m2 + i k 2 + i. (3.41).

(42) 3.3. One loop regularization of QED. 32. In order to use the equations developed, one has to combine together the denominators of this expression into a single one. By using the prescription of the Feynman Parameters, appendix (A) 1 = ab. Z. Z. 1. 1. 0. dz [az + b(1 − z)]2. then one gets Σ(p) = iµ4−d. Z dz. 0. p dd k / − k/ + m µ γ γ . µ D2 (2π)d. (3.42). Where one takes initially, D = (p − k)2 z − m2 z + k 2 − 2p · kz. This denominator can be written in the following way: D = p2 z − m2 z + k 2 − 2p · kz k0. (3.43). The term p · k can be eliminated through the change of variables k = + pz. One finds D = (p2 − m2 )z + (k 0 + pz)2 − 2p · (k 0 + pz)z = (k 0 )2 − m2 z + p2 z(1 − z).. (3.44). If one puts k 0 = k, 1. p dd k /(1 − z) − k/ + m γ γµ, (3.45) µ 2 − m2 z + p2 z(1 − z)]2 d [k (2π) 0 R but the linear terms don’t contribute to the dd k integral since they are odd with respect to k. Then, by neglecting linear terms in k, so Σ(p) = iµ4−d. Σ(p) = iµ. 4−d. Z. Z. dz. 1. Z. Z dz. 0. p dd k /(1 − z) + m γµ 2 γµ d [k − m2 z + p2 z(1 − z)]2 (2π). (3.46). Integrating over k, by the prescription and the formulae (3.35) with n = 2,. Σ(p) = iµ. 4−d. 1. Z. dz 0. p iπ d/2 Γ(2 − d/2) /(1 − z) + m γµ 2 γ µ (3.47) d Γ(2) (2π) [m z − p2 z(1 − z)]2−d/2. Now, by taking = 4 − d one gets. Σ(p) = −µ. . Z. 1. dz 0. p π (4−d)/2 /(1 − z) + m Γ(/2)γ γµ µ (2π)4− [m2 z − p2 z(1 − z)]/2. (3.48).

(43) 3.3. One loop regularization of QED. 33. by contracting the γ matrices, using (3.40). Σ(p) = −. Γ(/2) 16π 2. 1. Z. dz (4πµ2 )/2. 0. ( − 2)p /(1 − z) + (4 − )m [m2 z − p2 z(1 − z)]/2. ,. (3.49). one obtains Z Γ(/2) 1 dz [2p Σ(p) = /(1 − z) − 4m − (p /(1 − z) − m)] 16π 2 0 h m2 z − p2 z(1 − z) i−/2 × . (4πµ2 ). (3.50). For simplicity, by defining A = 2p /(1 − z) − 4m B = −p /(1 − z) + m m2 z − p2 z(1 − z) C= (4πµ2 ). (3.51). and expanding for → 0, i.e. using (3.25),. Σ(p) = = = +. Z 1 h 2A ih i 1 dz + 2B − γ A 1 − log C 16π 2 0 2 Z 1 h 2A i 1 dz − A log C + 2B − γ A 16π 2 0 1 1 (p (p / − 4m) + /(1 + γ) − 2m(1 + 2γ)) 2 8π Z 16π 2 1 4πµ2 1 dz [2 p (1 − z) − 4m] log . (3.52) / 16π 2 0 m2 z − p2 z(1 − z). This is the final result, and from it, one is able to identify the finite and infinite parts as Σ(p) =. 3.3.2.. 1 (p / − 4m) + finite terms. 8π 2 . (3.53). Vacuum polarization. The corresponding vacuum polarization contribution, displayed in Figure 3.3, can be described as it was indicated in (3.7) by: Πµν (q) = i. Z. h i d4 k i i µ ν T r γ γ . (2π)4 k/ + p / − m k/ − m + i.

(44) 3.3. One loop regularization of QED. 34. Figura 3.3: Vacuum polarization Again, using the prescription to generalize Πµν (q) into a dimension d, one has h i dd k i i µ ν T r γ γ (2π)d k/ + p / − m k/ − m + i Z d d k T r[γµ (k/ + m)γν (k/ + /q + m)] = iµ4−d (2π)d (k 2 − m2 )((k + q)2 − m2 ). µν. Π (q) = iµ. 4−d. Z. (3.54). Using the Feynman parameters trick for the denominator, µν. Π (q) = iµ. 4−d. Z. 1. Z dz. 0. dd k Nb , (2π)d D. (3.55). where it has been taken Nb = T r[γµ (k/ + m)γν (k/ + /q + m)] and initially, D = [(k 2 − m2 )(1 − z) + ((k + q)2 − m2 )z]2 . The denominator can be written in the following way: D = k 2 + q 2 z − m2 + 2k · qz. (3.56). through the change of variables k = k 0 − qz one can cancel the mixed term to obtain D = (k 0 − qz)2 + 2(k 0 − qz) · qz + q 2 z − m2 2. = k 0 + q 2 z(1 − z) − m2. (3.57). by setting k 0 = k 1. dd k T r[γµ (k/ − /qz + m)γν (k/ + /q(1 − z) + m)] [k 2 + q 2 z(1 − z) − m2 ]2 (2π)d 0 (3.58) Since the integral of the odd terms in k is zero, it is only neccesary to calculate the contribution of the even term to the trace µν. Π (q) = iµ. 4−d. Z. Z. dz.

(45) 3.3. One loop regularization of QED. 35. T r[..]even = T r[γµ (k/ + m)γν (k/ + /q] + m2 T r[γµ γν ] = T r[γµ k/γν k/] − T r[γµ /qγν /q]z(1 − z) + m2 T r[γµ γν ]. (3.59) If one defines the γ-matrices of dimension 2d/2 × 2d/2 one can repeat the standard calculation by obtaining a factor 2d/2 instead of 4. Therefore T r[..]even = 2d/2 [2kµ kν − gµν k 2 − (qµ qν − gµν q 2 )z(1 − z) + m2 gµν ] = 2d/2 [2kµ kν − 2z(1 − z)(qµ qν − gµν q 2 ) − gµν (k 2 − m2 + q 2 z(1 − z))]. (3.60). One finds then, by replacing this result, that 1. 2kµ kν dd k Π (q) = iµ 2 dz d [k 2 + q 2 z(1 − z) − m2 ]2 (2π) 0 2z(1 − z)(qµ qν − gµν q 2 ) gµν − 2 − 2 (3.61) [k + q 2 z(1 − z) − m2 ]2 [k + q 2 z(1 − z) − m2 ] µν. 4−d d/2. Z. Z. From the result (3.33), one can evaluate the third term of (3.61), such that with n = 1 Z. dd l π d/2 Γ(1 − d/2) 1 = −i (2π)d (l2 − a2 ) (2π)d (a2 )1−d/2. . Similarly, one can use the relation (3.37), with n = 2, to evaluate the first term in order to get Z. dd l 1 π d/2 Γ(1 − d/2) = −ig µν (2π)d (l2 − a2 )2 (2π)d (a2 )1−d/2. Therefore the first and the third contributions to the vacuum polarization cancel out and one is left with Πµν (q) = −iµ4−d 2d/2 (qµ qν − gµν q 2 ) Z 1 dd k 1 × 2z(1 − z)dz . 2 2 d (2π) [k + q z(1 − z) − m2 ]2 0. (3.62). It is important to notice that the original integral was quadratically divergent, but due to the cancellation the divergence is only logarithmic. The reason is the gauge invariance. In fact, it is possible to show that this implies q µ Πµν (q) = 0. Now, by using the relation (3.33), setting n = 2, one can perform the momentum integration suth that.

(46) 3.3. One loop regularization of QED. Πµν (q) = −iµ4−d 2d/2 (qµ qν − gµν q 2 ) Z 1 Γ(2 − d/2) iπ d/2 .. × 2z(1 − z)dz d 2 (2π) [m − q 2 z(1 − z)]2−d/2 0. 36. (3.63). By putting again = 4 − d and expanding for → 0, Z 1 z(1 − z)dz Πµν (q) = 2µ 22−/2 (qµ qν − gµν q 2 ) 0 Z iπ 2−/2 Γ(/2) × 4− 2 (2π) [m − q 2 z(1 − z)]/2 Z 1 h m2 − q 2 z(1 − z) i−/2 2 2−/2 2 z(1 − z)dzΓ(/2) = 2 (q q − g q ) µ ν µν 16π 2 4πµ2 0 Z 1 2 2 2 = (4 − 2 log 2)(qµ qν − gµν q ) z(1 − z)dz −γ 16π 2 0 × [1 − log C] (3.64) 2 where as before, C is defined by (3.51). One gets then Z 1 8 2 2 (q q − g q ) z(1 − z)dz − 4γ − 4 log 2 [1 − log C] µ ν µν 2 8π 2 0 Z 1 1 1 γ = 2 (qµ qν − gµν q 2 ) − + z(1 − z)dz 2π 3 6 0 4πµ2 × log 2 . (3.65) m − q 2 z(1 − z). Πµν (q) =. Again, this is the final result and from it, one is able to identify the finite and infinite parts as. Πµν (q) =. 3.3.3.. 1 1 (qµ qν − gµν q 2 ) + finite terms. 2 6π . (3.66). Vertex correction. The corresponding divergent integral, displayed in Figure 3.4,is related with the vertex correction needed for this Feynman diagram, and can be describe in a general way, as it was indicated in (3.9) as:. e2 Λµ (p0 , p) = −i. Z. 1 1 1 d4 k ρ γµ γρ 2 γ 0 . 4 (2π) p / − k/ − m + iε p / − k/ − m + iε k + iε.

(47) 3.3. One loop regularization of QED. 37. Figura 3.4: Vertex correction Using the prescription to generalize Λµ (p0 , p) into a dimension d, one has 0. µ. Λ (p , p) = −iµ. 4−d. Z. p dd k ρ p 1 / − k/ + m /0 − k/ + m γµ γρ 2(3.67) . γ 0 2 2 2 2 d (p − k) − m (p − k) − m k (2π). The general formula to reduce n denominators into a single one, according to what is discussed in the appendix A, leads to get µ. 0. 4−d. Z. Λ (p , p) = −2iµ Z 1−x × dy 0. dd k (2π)d. Z. 1. dx 0. γ ρ (p /0 − k/ + m)γ µ (p / − k/ + m)γρ . [k 2 (1 − x − y) + (p − k)2 x − m2 x + (p0 − k)2 y − m2 y]3 (3.68). by expanding the terms in the denominator one has 2. [...] = k 2 − m2 (x + y) + p2 x + p0 y − 2k · (px + p0 y). (3.69). Now, changing variables such that k = k 0 + px + p /0 y 2. [...] = (k 0 + px + py)2 − m2 (x + y) + p2 x + p0 y − 2(k 0 + px + py) · (px + p0 y) 2. 2. = k 0 − m2 (x + y) + p2 x(1 − x) + p0 y(1 − y) − 2p · p0 xy. (3.70). setting k 0 → k and returning to (3.68) Z 1 Z 1−x dd k dx dy Λ (p , p) = −2iµ (2π)d 0 0 γ ρ (p /0 (1 − y) − p /x − k/ + m)γ µ (p /(1 − x) − p /0 y − k/ + m)γρ . [k 2 − m2 (x + y) + p2 x(1 − x) + p0 2 y(1 − y) − 2p · p0 xy]3 (3.71) µ. 0. 4−d. Z.

(48) 3.3. One loop regularization of QED. 38. The odd terms in k are zero after integration and the terms in k 2 are logarithmically divergent, whereas there is a remaining part that is convergent. (1) (2) Separating the divergent piece, Λµ , from the convergent one, Λµ : (2) Λµ = Λ(1) µ + Λµ. (3.72). Where one gets the first term as 1. 1−x. dd k (2π)d 0 0 γ ρ γλ γ µ γσ γρ · k λ k σ [k 2 − m2 (x + y) + p2 x(1 − x) + p0 2 y(1 − y) − 2p · p0 xy]3 Z 1 Z 1−x iπ d/2 (−1)3 1 dx dy = −2iµ4−d ˘ Γ(2 − d/2) Γ(3) 2 0 0 ρ µ λ γ γλ γ γ γρ 2 2 [m (x + y) − p x(1 − x) − p0 2 y(1 − y) + 2p · p0 xy]2−d/2 Z 1−x Z 1 1 4−d 1 d/2 dx dy = µ Γ(2 − d/2) 2 4π 0 0 γ ρ γλ γ µ γ λ γρ . [m2 (x + y) − p2 x(1 − x) − p0 2 y(1 − y) + 2p · p0 xy]2−d/2 (3.73) R d Here, (3.37) was used to evaluate d k. Now, using (3.40) one can replace ρ γ γλ γ µ γ λ γρ = (2 − d)2 γ µ . If one changes to = 4 − d, 0 Λ(1) µ (p , p). 0 Λ(1) µ (p , p). = −2iµ. 4−d. Z. Z. dx. Z. dy. Z 1 Z 1−x 1 1 2−/2 2 = µ Γ(/2)( − 2) γµ dx dy 2 4π 0 0 1 2 2 [m (x + y) − p x(1 − x) − p0 2 y(1 − y) + 2p · p0 xy]/2 Z 1−x Z 1 i2−/2 1 h2 dy dx = −γ [4 − 4]γµ 32π 2 0 0 2 m (x + y) − p2 x(1 − x) − p0 2 y(1 − y) + 2p · p0 xy −/2 (3.74) 4πµ2. Now, one can use a trick of notation and make an expansion to first order in the following way: a /2. a = e−/2 ln(a/b) ' 1 − + O(2 ) b 2 b By using this trick to rewrite the last term in (3.74) and in that way integrate the first terms that don’t involve dependence in the variables x and y. One finds.

(49) 3.3. One loop regularization of QED. 0 Λ(1) µ (p , p). 39. Z 1 Z 1−x γµ 1 γ 1 = 2 γµ − γµ − 2 γµ + 2 dx dy 8π 16π 2 8π 8π 0 0 4πµ2 . ln m2 (x + y) − p2 x(1 − x) − p0 2 y(1 − y) + 2p · p0 xy (3.75). This is the partial result from which a splitting into finite and infinite parts, like 0 Λ(1) µ (p , p) =. 1 γµ + finite terms 8π 2 . (3.76). In the convergent part we can put directly d = 4 such that Z 1 Z 1−x i iπ 2 (−1)3 dx dy 8π 4 0 Γ(3) 0 ρ 0 µ γ (p / (1 − y) − p /x + m)γ (p /(1 − x) − p /0 y + m)γρ. 0 Λ(2) µ (p , p) = −. m2 (x + y) − p2 x(1 − x) − p0 2 y(1 − y) + 2p · p0 xy Z 1 Z 1−x 1 =− dx dy 16π 2 0 0 γ ρ (p /(1 − x) − p /0 y + m)γρ /0 (1 − y) − p /x + m)γ µ (p m2 (x + y) − p2 x(1 − x) − p0 2 y(1 − y) + 2p · p0 xy. (3.77).

(50) Chapter 4. BPHZ The Bogoliubov, Parasiuk, Hepp, Zimmermann (abbreviated BPHZ) renormalization scheme is a mathematically consistent method of rendering Feynman amplitudes finite while maintaining the fundamental postulates of relativistic quantum field theory (Lorentz invariance, unitarity, causality). Technically it is based on the systematic subtraction of momentum space integrals.. 4.1.. Bogoliubov Causality. A brilliant example of the creation and application of new mathematical methods was the development of an axiomatic approach to quantum field theory undertaken by Bogoliubov in the 1950’s, following previous works by Stuckelberg. At that time, the ultraviolet divergence was an important problem in quantum field theory when using the Hamiltonian formalism (canonical quantization). When studying a stabilization problem of condensate in nonideal systems, Bogoliubov developed the method of quasi-mean. This turned out to be a universal tool for the investigation of systems whose main state is unstable under small perturbations. A little earlier, however, it must be mentioned the development of the formalism of generalized functions, distributions, by Sobolev and Schwartz. Its place for dealing with linear problems in physics managed to stand out in multiple applications. Nevertheless, there were no nonlinear problems applications since, like Schwartz demonstrated, the simplest nonlinear operation is the multiplication and it is practically impossible to properly define the multiplication of distributions in a reasonable manner. However, due to the important role that plays the multiplication of generalized functions in quantum field theory (namely for the construction of the S operator of Scattering by means of perturvatives methods), the problem 40.

(51) 4.1. Bogoliubov Causality. 41. of developing a theory of the multiplication of the special class of so-called çausal functionsïs a central problem of mathematical physics. Physicists have attempted to solve this problem by making use of the fact that the Fourier transform of a causal function (= distribution) can be represented through a succession of normal functions, from which they have attempted to define a "Fourier transform product. " As an example in one dimension one can defined the Fourier transform as: Z 1 Ff (k) ≡ fˆ(k) = √ f (x)e−ikx dx 2 the the convolution would be: Z f ∗ g(x) = f (x − y)g(y)dy with the relationship between them defined as F(fg) = F(g) ∗ F(g) Then if f and g are distributions whose product does not make sense as a distribution, one can try to define its Fourier transform by the previous identity. Lets consider the following example: By taking f = δ y g = Θ (step function, seen as distribution), the one has: Z 1 1 δ̂(k) = √ δ(x)e−ikx−x dx = √ 2 2π by taking the limit when − > 0: Z 1 −i Θ̂(k) = √ Θ(x)e−ikx−x dx = lı́m √ −>0 2 2π(k − i) by taking the product: Z −i dk 0 Θ̂ ∗ δ̂(k) = 2π k − k 0 − i ˆ ˆ one has On the other hand Θ̂ ∗ δ̂(k) = ”(Θδ)”. In order to obtain ”Θδ” to calculate the Fouriers transformation of Θ̂ ∗ δ̂, i.e.: Z Z −i e−ikx dk 0 F (Θ̂ ∗ δ̂)(x) = 3/2 dk k − k 0 − i 2π this integral contains a logarithmic divergence. For this reason appeared many attempts to make finite sense to these divergent quantities. The efforts led to the development of a special technique of subtraction. −1.