Thesis for the degree of Master of Science Chiral Perturbation Theory, Weak Interactions and the Nuclear Two-body Axial Vector Current Jonathan Karlsson Department of Fundamental Physics Division of Subatomic Physics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, July 6, 2012 Chiral Perturbation Theory, Weak Interactions and the Nuclear Two-body Axial Vector Current. Jonathan Karlsson Email: karlsjona@gmail.com ©Jonathan Karlsson, 2012 Supervisor: Christian Forssén Co-supervisor: Lucas Platter Examiner: Christian Forssén Department of Fundamental Physics Chalmers University of Technology SE-412 96 Göteborg Sweden + 46 (31) 772 1000 Printed by Chalmers reproservice Göteborg, Sweden, 2012 Cover The two Feynman diagrams that give the leading contribution to the two- body axial vector current. The lines going from top to bottom represent nucleons, the dashed lines pions and the wavy lines the external field, e. g., vector bosons. Abstract In this thesis I give a practical introduction to chiral perturbation theory. This is an effective field theory of pions and nucleons. It is governed by the chiral symmetry stemming from the lightness of the up- and down quarks in quantum chromodynamics. The validity region comprises energies up to the rho meson mass. The theory is expressed as an infinite series of chiral invariant interactions, whose strengths are expressed in an infinite number of low energy constants. The interactions can be ordered and I identify the most important ones. Special care has to be taken when including nucleons in chiral perturbation theory because of the scale introduced by the nucleon mass. To facilitate straight- forward calculations I work in heavy-baryon chiral perturbation theory. In this formalism the nucleons are considered to be very heavy and the nucleon mass only appears in next-to-leading order corrections. The pions and nucleons are coupled to the charged vector bosons of the weak interactions. This interaction is determined entirely by the chiral symmetry. As an example, I compute the decay rate of charged pions. Experimental data for this observable can be used to fix one low energy constant. Finally, I compute the two-body axial vector current of nucleons in heavy- baryon chiral perturbation theory with the long wavelength approximation. This current complements the leading order one-body current operator and gives the first correction to the Gamow-Teller operator from the nuclear environment. I provide both a detailed derivation and an explicit expressions for this two-body current operator. Keywords: Chiral perturbation theory, effective field theory, meson exchange currents, Gamow-Teller operator, two-body axial vector current, nuclear currents, two-body cur- rents, Heavy baryon chiral perturbation theory Contents Acknowledgements vi Conventions vii 1 Introduction 1 1.1 Effective field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 A self-consistent framework for nuclear physics . . . . . . . . . . . . 5 1.3 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Pion-only chiral perturbation theory 7 2.1 Energy scales of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Accidental symmetries of QCD . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Chiral symmetry of pions . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Building blocks for interactions . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Leading order Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 Weak interactions of pions . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.8 Decay of charged pions . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Baryon chiral perturbation theory 23 3.1 Building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Nucleon self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Heavy baryon chiral perturbation theory 29 4.1 Heavy baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Considering several nucleons and nuclei . . . . . . . . . . . . . . . . 31 4.4 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5 Weak interactions in nuclei 34 5.1 Weak interactions of nucleons . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Current operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 Two-body axial vector current . . . . . . . . . . . . . . . . . . . . . . 38 iv 6 Conclusion 45 A Pion decay calculation 48 B Interaction Lagrangian 51 B.1 Interaction Lagrangian and vertices . . . . . . . . . . . . . . . . . . . 51 C Current operators 59 C.1 One pion exchange currents . . . . . . . . . . . . . . . . . . . . . . . . 59 C.2 Contact currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 v Acknowledgements I would like to thank my supervisor Christian Forssén for guidance and help dur- ing my work and for helping me turn this thesis into something (more) readable. Also, I thank Lucas Platter for help with theoretical questions and enlightening discussions. Further I thank Hans-Werner Hammer for help with resolving my final long-standing issues. Thanks also to everyone at the Division of Subatomic Physics for help and encouragement along the way. vi Conventions I will work with a system of units such that c =ħ= 1. This will make formulas simpler as it will not be necessary to write out these common factors. Also, energy, mass, momentum and frequency then have the same dimension while time and length have the inverse dimension. I will frequently write terms like ai ai or pµpµ with a repeated index. The meaning is that there is a sum ∑ i ai ai or ∑3 µ=0 pµpµ. Greek letters such as µ, ν, ρ, σ are Lorentz indices and run from 0 to 3; roman letters such as a, b, c . . . denote isospin indices; and i , j , k . . . are space indices – unless something else is stated. To raise and lower Lorentz indices the Minkowski metric gµν = gµν will be used and for it I will use the mostly minus convention, gµν = { 1 for µ= ν= 0 −1 for µ= ν= 1,2,3 . This means that the contraction of two four-vectors pµ and kµ is written pµkµ = pµkνgµν = p0k0 −p ·k . I will use σi to denote an operator in spin space while τa will be an operator in isospin space. These operators can be represented by the Pauli matrices, which are Hermitian and traceless complex 2×2 matrices, σ1 = τ1 = ( 0 1 1 0 ) , σ2 = τ2 = ( 0 −i i 0 ) , σ3 = τ3 = ( 1 0 0 −1 ) . Furthermore, I will use the Weyl representation of the Dirac matrices. These are 4×4 matrices γµ = ( 0 σµ σ̄µ 0 ) . where σµ = (1,σi ) and σ̄µ = (1,−σi ). Contractions between the Dirac matrices and other four-vectors are very common and to shorten expressions I will use the Feynman slash notation, /p = pµγµ. vii viii Chapter 1 Introduction An important scientific principle is that a theory should not be any more compli- cated than necessary. This idea is present in Newton’s Principia [New46] where he writes that no more causes for things than are necessary should be invented, and that this in turn leads to the necessity of assigning to many things the same cause. Put in another way, a good scientific theory should be as simple as possible while describing as many things as possible. This idea has so far been very fruitful. The explanations for almost all known phenomena are believed to have been collected into the description of just four different forces. Two of them are manifest in the macroscopic world, namely electromagnetism and gravity. The two other forces, the weak- and the strong force, can only be seen at very short length scales. This does not mean that they are unimportant. The strong force is responsible for keeping the nuclei of atoms together. It also keeps the quarks together in protons, neutrons and all other strongly interacting particles. The weak force is responsible for some decays of unstable particles, one example is beta decay. The theories of electromagnetim, the weak force and gravity have all been very successful in describing their part of the world. These theories are quite simple and describe very many different things with excellent accuracy. The theory of the strong force, on the other hand, has not been as successful in terms of Newton’s principle so far. In many cases it has been necessary to create problem specific models for each application. Many of these models are phenomenological, i. e., they are not derived from first principles but are instead based on observations of the particular subject that is studied. This leads to problems in understanding and makes it hard to make predictions. One cause for this lack of success is that there are a very large number of particles that interact strongly. In 1930 when Pauli suggested the existence of the neutrino there were only three known particles; the proton, the photon and the electron [Arn01]. Out of these only the proton is affected by the strong force. In 1932 Chadwick found the neutron [Cha32], the second known strongly interacting particle. In the thirty years that followed several hundred more particles were 1 found in cosmic radiation and accelarator experiments. The majority of these particles were strongly interacting. They are collected under the common name of hadrons. In 1964, in order to explain the large number of hadrons, Gell-Mann [GM64] and Zweig [Zwe64] independently proposed that they were not fundamental par- ticles at all. A large simplification could be made if they were composite particles made up of a new kind of particles, quarks. These quarks would need to have quite peculiar properties, for example they would need to have an electric charge that is a fraction of the electron charge. Initially there where three proposed quarks, this number has now been expanded to six. Hadrons are defined by the quarks they contain. The ones containing two quarks are called mesons and those with three quarks baryons. The quarks are be- lieved to be confined within the hadrons. Free quarks have never been observed, which gives strong support to this conjecture. A theory called quantum chromodynamics(QCD) was created to account for the force between the quarks. In this theory the quarks interact by exchanging gluons. Both the quarks and the gluons carry a type of charge called color. It is very hard to calculate properties of hadrons using QCD. At very short length scales, the quarks in a hadron are almost unaffected by the strong force – this phenomenon is called asymptotic freedom – and calculations can be made using ordinary methods. But at longer length scales the strength of the force increases enough to make it impossible to use the standard computational meth- ods. This is necessary if the quarks are to be confined within the hadrons. Because of the strength of the force it is an unsolved problem how to derive the properties of hadrons and their interactions from QCD. Based on the success of the theory at high energy it is believed to be the correct theory also at low energy. The neutrino, which Pauli suggested, was needed to explain the apparent loss of energy in beta decay. This reaction is an example of a weak interaction. Almost all particles mentioned so far, and actually all known particles, except for the gluons, interact with the weak force. This force also allows many more interactions to occur than the strong and electromagnetic interactions. On the other hand it is much weaker at low energies, which is the reason for its name. The large number of allowed interactions makes the weak interactions able to facilitate decays that the strong and electromagnetic force can not. On the other hand it makes these particles more long lived since the force is weaker. A weaker force gives a lower probablility of decay and thus a longer lifetime. Between 1960 and 1970 electroweak theory got its current form through the work of Glashow, Weinberg and Salam [Gla61, SW64, Wei67]. This theory unifies the electromagnetic and the weak interactions into one theory. Just as the electromagnetic force is mediated by a vector boson called the photon the weak force is mediated by vector bosons. The difference is that unlike the photon these vector bosons are very heavy. The range of a force is limited by the mass of the exchange particle. This makes the weak force a very short ranged force while the electromagnetic force has unlimited range since the photon is massless. 2 Together these two theories, QCD and the unified electroweak theory, make up the Standard Model of particle physics. Everything from the binding of atoms and molecules to the highest energy accelerator experiments can be explained by these theories. 1.1 Effective field theory The Standard Model is a fundamental field theory. This implies that it aims to explain phenomena on all length scales, from the macroscopic world down to the smallest possible lengths. It is also a very simple theory utilizing only 20 parameters [Lan09]. But, this does not mean that it is trivial to use this theory for everything. In contrast to this type of theory we have effective field theories (EFTs). These are theories that do not aim to explain everything but instead focus on a specific set of problems at a certain length scale or, equivalently, at a certain energy scale. Length- and energy scales can be related. To examine a shorter length scale a shorter wavelength probe and thus a higher frequency is needed. A quantum of higher frequency will have a higher energy, and in this way energy scales and length scales can be related. The first step in creating an EFT is to determine the relevant energy scale. It is also necessary to determine a highest possible energy, the cutoff energy. Nothing above the cutoff energy will be treated explicitly in the EFT; the theory will only be valid up to this energy. At higher energies, or shorter lengths, the EFT will loose all predictive power. This is similar to how a theory about macroscopic sound waves is unable to predict how the individual molecules in the air will move. At a certain energy scale some particles will be more important than others. Those that have a mass lower than the cutoff energy of the theory will be able to be real particles, while those with a higher energy will be restricted to be virtual particles as they can not acquire enough energy to become real particles. As a consequence, the second step amounts to selecting the relevant degrees of freedom (fields or particles) that are active in the energy range of the theory. All other particles are excluded. When writing down the interactions of the remaining particles there will be more interactions than before the removal of the higher mass particles.When the removed particles where included explicitly they could show up as virtual intermediate particles allowing interactions that would not be possible when only considering the remaining low energy particles. In fact, in order to be consistent with the higher energy theory, the removal of the high mass particles must gener- ate all kinds of interactions among the remaining ones as long as the interactions are consistent with the symmetries of the underlying theory. A conjecture by Weinberg [Wei79] states that writing down the most general Lagrangian containing the selected matter content and obeying the relevant symmetries yields the most general S-matrix consistent with the fundamentals of 3 quantum field theory and the imposed symmetry. This means that among all the possible EFTs containing a set of selected particles with some symmetry there will be at least one that will give the same predictions for observable quantities as a higher energy theory for these particles. In practice, it is impossible to consider the infinite number of interactions for any given process. To be able to make predictions there must be a way to determine what interactions give the largest contributions. It turns out that the interactions can be put into a series expansion in the energy scale of the process divided by the cutoff energy of the EFT [Pic98]. The relative strength of these interactions are often not given by symmetry. The so called low energy constants (LECs) that parametrize this must be determined from the higher energy or fundamental theory. In the cases where it is not possible or known how to solve the high-energy theory the LECs can be determined from experiment. For any calculation all interactions up to the chosen order will need to be considered. Of course, going to a higher order will give a more precise result but will at the same time involve a larger number and more complex interactions. An EFT may coexist with the Standard Model even though the details differ. This is possible because of one of the basic principles of quantum mechanics. It is impossible to measure how a process takes place, it is only possible to measure the outcome. This is because a measurement of the details of a process will disturb the process enough to change it into another process. This gives us a freedom to choose any theory to describe a process as long as it gives the same outcome. The reason for using EFTs is that they provide ways to describe processes or phenomena in an easier way than in the fundamental field theory. EFTs also extend the applicability of quantum field theory. It is possible to use EFTs in cases where the fundamental theory has not been solved as in the case of QCD, or is not a quantum field theory like gravity [Don94]. An example of an EFT is Euler-Heisenberg theory, which is a low energy theory of quantum electrodynamics. The photon is massless so at low energy it can be chosen as the active degree of freedom while removing the electron. This will then generate all photon-photon interactions that are consistent with the symmetries of quantum electrodynamics. In this case new 4-photon and higher vertices are generated in an expansion in the energy or frequency of the photons divided by the electron mass [Har01]. This theory will then of course only be valid under the electron-positron pair production energy of approximately 1 MeV. In this case the LECs determining the strength of the interactions among the photons can be calculated from quantum electrodynamics. The advantage of this approach is that it is much easier to compute the self-interaction of the electromagnetic field from this EFT than it is from the full theory of quantum electrodynamics. A similar process can be applied to QCD to describe the interactions between hadrons. We take all hadrons that have a mass below a certain cutoff energy and find whatever symmetries we can impose on them from QCD and create the most general Lagrangian invariant under that symmetry. This yields a tool for 4 calculating properties and interactions of hadrons even though the underlying theory is unsolved. This last fact makes it impossible to calculate the LECs from QCD at the moment; they must be determined from experiments or by using numerical methods. A valid question is why this would be expected to be more successful than low energy QCD. The reason is that in taking the hadrons as the basic particles of our theory we have absorbed the largest part of the color force. All hadrons are color neutral and because of this the force between hadrons should be much weaker than the force between quarks. An analogy is the force between electri- cally charged particles, which is much stronger than the residual forces between neutral bound states of charged particles such as the van der Waals force. 1.2 A self-consistent framework for nuclear physics One use for the EFTs of QCD is to provide a self-consistent and general framework for nuclear physics. Historically, nuclear physics has been developed as different phenomenological models. Collectively this is called the standard nuclear physics approach (SNPA). The shortcomings of the SNPA are characterized by Krane [Kra87]: Nuclear physics lacks a coherent theoretical formulation that would permit us to analyze and interpret all phenomena in a fundamen- tal way [. . . ] As a result, we must discuss nuclear physics in a phe- nomenological way, using a different formulation to describe each different type of phenomenon, such as α decay, β decay, direct reac- tions, or fission. Another problem is that these phenomenological models do not provide a connection to the Standard Model of particle physics. This means that the Standard Model can not be tested using the SNPA and also that the SNPA is not derived from the Standard Model. With the advent of more powerful computers an alternative method has appeared. Based on brute force calculations many properties of nuclei have been calculated using realistic nucleon-nucleon potentials as the only input. This has been possible for some nuclei with up to 20 nucleons at this point [FRN11]. Many of these calculations have until recently been based on phenomenological SNPA inter-nucleon potentials. In these calculations the nuclear wave functions and the excitation energies can be found, for example, by diagonalizing the Hamiltonian in an appropriate basis. These wave functions can then be used to calculate other properties such as cross-sections, decay rates as well as static ones such as radii, electromagnetic moments and so forth. To perform the dynamical calculations the interacting parts of the nuclear wave function must be found. This is called a nuclear current. Recently it has become possible to use potentials derived from EFT to perform these ab-initio calculations. This enables the use of the same EFT to derive 5 both the nuclear wave function and the interaction nuclear currents so that calculations can be done in a self-consistent way. This is important because intermediate results are often model dependent. The infinite number of LECs in EFTs limit their predictive power. In principle these constants should be derivable from the underlying theory on which the EFT is based. For the case of low energy theories of QCD it should in principle be possible to utilize lattice field theory to compute the low energy constants. Lattice field theory is a non-perturbative framework for quantum field theory. The basic idea is that by discretizing space-time, that is, formulating the theory on a lattice, computations can be made by, for example, Monte Carlo integra- tion [Thi07]. QCD calculations using lattice field theory are collected under the term lattice QCD. Lattice QCD is only dependent on the parameters of QCD, which are the quark masses and the strong coupling constant. If the LECs could be computed using this approach the infinite number of parameters of the EFTs would be dependent only on the finite number of QCD parameters. Currently lattice QCD has not reached the level where the LECs of strong force EFTs can be computed to sufficient precision [EHM09]. Instead, experimental results must be used to fix the LECs. This limits the predictive power somewhat since it is then necessary to use a number of results as input to determine the values of the LECs. Still, once all LECs to certain order has been fixed, all other calculations have no free parameters and are fully predictive. 1.3 Purpose I will focus on the EFT part of this program for the self-consistent framework. There are two goals that I wish to achieve. The first is to give the reader a basic understanding of low energy hadron weak interactions. To achieve this an EFT is described including pions and later adding nucleons. This work will be limited to EFTs for nucleons and pions; kaons and other strange hadrons will be excluded. The second goal is to give a detailed derivation of the nuclear two-body axial vector current. This current operator is important to provide an accurate description of the weak interactions in nuclei, for example, beta decay and neu- trino nucleus interactions. The derivation of this current operator also serves as an example of an application of the EFT. The two-body axial vector current is only part of the full weak current. I will not go into detail about these other currents but focus my work on the two-body current. This work will be performed using an EFT called Chiral Perturbation Theory (ChPT). It is a low energy theory of QCD that describes the interactions of pions and nucleons. I will describe it in detail in the following chapters. 6 Chapter 2 Pion-only chiral perturbation theory The goal of any EFT is to descibe a certain set of processes. To create the theory the energy scale of these processes must first be identified. I want to describe the weak interactions of the lightest hadrons and nuclei. The lightest hadrons are the pions with a mass of approximately 140 MeV. The nucleons are much heavier with a mass of about 1 GeV. As a consequence, in a system with a center-of-momentum energy much below the nucleon mass and only incoming pions there will not be any resulting nucleons. In this chapter I will consider only such systems. How to include nucleons will be covered in the next two chapters. An important symmetry of pions and nucleons is the chiral symmetry. Chiral symmetry has its roots in QCD and this symmetry is central to the low energy interactions of hadrons. The symmetry will be described in detail in this chapter. The most general theory of pions with the chiral symmetry is called chiral per- turbation theory (ChPT). It is governed by a chiral invariant Lagrangian with an infinite number of terms representing an infinte series of interactions. I will show how this Lagrangian can be ordered in a series in p/Λwhere p is the external pion four-momentum andΛ is the high energy cutoff of this theory. There is an additional expansion in the symmetry breaking parameters, the quark masses mq . Because of the quark masses’ role in creating the pion masses each quark mass factor counts as p2. The cutoff is given by the next massive particle that can be created by the strong interactions. The kaons can be ignored since they must be produced in pairs due to conservation of strangeness, which leaves the rho-meson as the next lowest mass state. The result is a series expansion in p/Mρ . The low energy constants of the theory need to be determined from experiment or by computational methods, e. g., lattice QCD, since low energy QCD has not been solved. Using this Lagrangian I will show how to couple the pions to the vector bosons of the weak interaction. As an example of how this is performed I will show a 7 detailed calculation of the decay rate of charged pions. This will demonstrate how to get to the explicit interaction Lagrangian and also how an experimental value can be used to determine one of the low energy constants of an EFT. 2.1 Energy scales of QCD When choosing an energy scale and a cutoff for an EFT the matter content of the underlying theory plays a major part. In general, it is not fruitful to arbitrarily choose an energy scale and cutoff. Instead, in order to create a successful EFT, the mass spectrum must be studied in detail. In the case of QCD there is a very large number of hadrons. It would be helpful if we could study only a subset of all of these particles. Luckily, as we can see in table 2.1, the pions are by far the lightest mesons. We will choose them as the matter content of our EFT. This also fixes the energy scale of our theory to that of the pion mass, about 140 MeV. Table 2.1: Selected meson masses [PDG10] Meson Mass π0 135 MeV π± 140 MeV K ± 493 MeV K 0, K̄ 0 498 MeV η 547 MeV ρ 775 MeV There are three different pions, which differ by their electric charge. The two charged pions are each other’s antiparticles and thus have the same properties except for their charge. The neutral pion is slightly lighter, and it is its own an- tiparticle. None of the pions are stable even though they are the lightest strongly interacting particles. The charged pions decay weakly, mostly to muons and muon neutrinos; this gives them a comparatively long half life. The neutral pion mostly decays electromagnetically to two photons, which makes its half life much shorter. All pions are pseudoscalar particles, i.e., they are represented by scalar fields that pick up an extra minus sign under parity transformations. After choosing the energy scale and matter content we need to determine the cutoff energy. This simply corresponds to the mass of the lightest excluded particle. The kaons must be produced in pairs so the lightest single particle that can be produced is the rho meson. Thus, the theory will be valid up to the rho meson mass but to assure reasonable convergence we should remain well below that energy; I will go into more detail about the series expansion in section 2.5. 8 2.2 Accidental symmetries of QCD The next step is to find the relevant symmetries for the matter content of the theory. Since the underlying theory is QCD we look and see if there are any symmetries that our theory can inherit. Specifically, since we are interested in the dynamics of pions, the focus will be on the quark sector of QCD since the defining property of a hadron is its quark content. Quarks are the matter particles of QCD. They have never been observed by themselves and are thought to be confined within the hadrons. There are 6 flavors of quarks which can be seen in table 2.2. Furthermore, each quark also has a color, which can be red, green or blue. This is of course not a real color but is just a name for the charge of the strong force known as the color charge. Table 2.2: Quark flavors [PDG10] Flavor Charge Mass up +2/3 2.5 MeV down −1/3 5 MeV strange +2/3 101 MeV charm −1/3 1.27 GeV top +2/3 172 GeV bottom −1/3 4.2 GeV The color charge arises from an SU(3) gauge symmetry. This local symmetry gives rise to the force particles of QCD, that is, the gluons. These are massless vector bosons, in a way similar to the photon, but with the fundamental differ- ence that the gluons themselves carry a color charge. There are 8 different color combinations of gluons corresponding to the 8 generators of SU(3). The dynamics are governed by the QCD Lagrangian [Lan09] LQCD =−1 4 ∑ i F i µνF iµν+∑ r q̄rαiγµi D α µ βqβr −∑ r q̄ α r qrαmr . (2.1) In the first term, F i µν is the field strength of the gluon fields. The index i runs over the 8 gluon fields. This term contains both the kinetic energy terms for the gluons and the gluon-gluon interaction. It does not contain any quark fields so we will ignore it. In the second term, qβr are the quark fields. The index r runs over the quark flavors and the index β runs over the colors. D α µ β is the color covariant derivative. It contains the kinetic term for the quarks and also the coupling to the gluon fields. The gluon field coupling is there to ensure SU(3) gauge invariance of the term. We note that the covariant derivative D α µ β does not contain a flavor index r so its action is the same on all 6 quark flavors. The last term, which gives rise to the quark masses is not really part of the QCD Lagrangian but comes from the coupling to the Higgs field and the spontaneous 9 symmetry breaking in the weak sector of the Standard Model. In our case we can write these terms like this since we are not concerned with the details of gauge symmetry or spontaneous symmetry breaking. As can be seen in table 2.2, the three lighter quarks; the up-, down- and strange quark; are much lighter than most hadrons and than the three heavier quarks. Of these three quarks the up- and the down quark are especially light. Compared to the cutoff energy scale at the rho meson mass of ∼ 770MeV these two lightest quarks are almost massless. So for the next part we will ignore the heavier quarks and consider these two lightest quarks to be massless. Then, ignoring the gluon part, we have the Lagrangian L ′ QCD = ∑ r∈{u,d} q̄rαiγµi D α µ βqβr = q̄αiγµi D α µ βqβ. (2.2) We have put the quark fields in a vector qα = (uα,dα). If we make a unitary transformation between the two quark flavors we get qα→U qα (2.3) L ′ QCD → q̄αU †iγµi D α µ βU qβ. (2.4) Since the covariant derivative acts the same on all the different flavors we can just commute the unitary transformation through it; q̄αU †iγµi D α µ βU qβ = q̄αU †Uiγµi D α µ βqβ = q̄αiγµi D α µ βqβ =L ′ QCD. (2.5) This transformation leaves the Lagrangian unchanged which means that there is an approximate U(2) symmetry of the light quarks in QCD. But it is possible to find a larger symmetry than this. If we take a closer look at how the Dirac matrices connect the spinor components we see that the Lagrangian can be divided into two independent parts. We write each Dirac spinor as q = (qL , qR ) where qL , qR are two-component Weyl spinors. This decomposition can be seen explicitly by inserting the Weyl spinors into the Lagrangian, q̄γµDµq = q †γ0γµDµq = ( q L q R )† ( 0 1 1 0 )( 0 σµ −σ̄µ 0 ) Dµ ( q L q R ) = q † Lσ µDµq L −q † R σ̄ µDµq R . (2.6) Since we have ignored the quark mass terms there is no part of the Lagrangian connecting qL to qR . Now, by the same argument as above each of these two terms can be independently flavor rotated. So we have two independent U(2) 10 symmetries, i.e., a U(2)×U(2) symmetry. We will call this group U(2)L ×U(2)R where the group U(2)L acts on the left handed spinors and U(2)R on the right handed spinors. This decomposition is manifest because of the choice of the Weyl representation of the Dirac algebra. Naturally, it is possible to achieve this decomposition in any representation. However, this symmetry only holds on the classical level. During quantization parts of the symmetry are destroyed by an anomaly and only a SU(2)L ×SU(2)R × U(1)V symmetry is left [PS95, p. 672]. The subscript V on U(1)V denotes that it acts on the whole Dirac spinor at the same time. SU(2) is the group of all unitary 2×2 matrices with determinant 1. The gen- erators are three anti-Hermitian traceless matrices, iτi with τi being the Pauli matrices. This means that any SU(2) matrix r can be written as r = exp iΘiτi . The direct product group SU(2)×SU(2) is defined as the set of all pairs of elements in SU(2) with the product defined as (a,b)(c,d) ≡ (ac,bd). (2.7) However, we will use a different parametrization of the group than just in- dependently rotating the left handed and the right handed fields. We will write it as SU(2)V ×SU(2)A . The vector subgroup, SU(2)V , corresponds to both fields being rotated the same way with one element written as (v, v), v ∈ SU(2). The axial vector subgroup in turn corresponds to the fields being rotated the opposite way (a, a−1), a ∈ SU(2). An element in SU(2)V ×SU(2)A is written ( va, va−1 ) and any element (l ,r ) in SU(2)L ×SU(2)R can be written this way. We can see this for a given element (l ,r ) ∈ SU(2)L ×SU(2)R by choosing a ∈ SU(2)A such that l = r a2 ⇔ l a−1 = r a. (2.8) If we then choose v ∈ SU(2)V such that l = va we have v−1l a−1 = 1, (2.9) but by equation 2.8 we have that v−1r a = v−1l a−1 = 1. (2.10) Extracting r from this equation gives, r = va−1, (2.11) just like we wanted to show. If we go back to table 2.2 we see that the strange quark mass is also com- paratively low, both compared to the other quarks and to the typical hadron mass of about 1 GeV. Including the strange quark would yield an approximate SU(3)×SU(3)×U(1) symmetry. This is also a usable theory since the strange quark mass is low compared to the hadron mass scale. Most of the results presented here can be carried over to that theory with some minor modifications. 11 2.3 Chiral symmetry of pions In the previous section we saw that QCD has an SU(2)×SU(2)×U(1) symmetry. Our theory of pions should have this symmetry as well. In the following, we will see how to implement the transformation of this group on the pions and use that to create a chiral-invariant Lagrangian. There is, however, one more thing we need to take into account. Nature does not seem to fulfill the full chiral symmetry. Only the SU(2)V ×U(1)V part of it seems to be present when looking at particles found in nature. Thus, if we still want to believe in QCD, the symmetries must be hidden in some way. This can be realized in the framework of spontaneous symmetry breaking where the ground state, i.e., the vacuum, breaks the symmetry. This is believed to be the case in low energy QCD [GL84]. As we will see, this can also give an explanation for the very low masses of the light mesons which gives further support to this idea. Spontaneous symmetry breaking is a mechanism where the dynamics of a theory destroys the invariance of the vacuum under one or more of its symme- tries. For example, this may occur by some field taking a non-vanishing vacuum expectation value. When applying a symmetry transformation this expectation value is changed. Therefore, the symmetry has been broken. According to Goldstone’s theorem there will be a massless, scalar particle for each broken continuous symmetry [GSW62]. These particles are called Goldstone bosons. If we break SU(2)V ×SU(2)A to SU(2)V we have broken the SU(2)A sub- group. This subgroup contains three independent symmetries so there should be three massless bosons. The three pions are unnaturally light and have the correct quantum numbers; so they are good candidates to be the Goldstone bosons. The quark masses are only small, they do not vanish completely. The con- sequence of this is that the chiral symmetry is only approximate. This gives an explanation for the non-vanishing mass of the pions. Since the broken symmetry is only approximate the pions are only approximate Goldstone bosons. The question is now how these bosons will transform under SU(2)V ×SU(2)A . We will use the conventions of Scherer [Sch03] regarding the transformations of the pions. The ground state must be invariant only under the unbroken SU(2)V part of the chiral group. We introduce three real scalar fields φa which are linear combinations of the pion fields π−,π+ and π0. To each field φa we pair a generator of SU(2), i. e., a Pauli matrix τa , φ= τaφa = ( φ3 φ1 − iφ2 φ1 + iφ2 −φ3 ) = ( π0 p 2π+ p 2π− −π0 ) . (2.12) We exponentiate this matrix to get an SU(2) matrix with the fields φa as transfor- mation parameters, U = exp ( i φ fπ ) = exp ( i φaτa fπ ) . (2.13) 12 The new constant fπ is the pion decay constant; it is one of the LECs of this theory of pions. It is possible to identify this group element with a representative of a left coset in SU(2)L ×SU(2)R /SU(2)V . We write the group SU(2)L ×SU(2)R as {(l ,r ) : l ∈ SU(2),r ∈ SU(2)} and the SU(2)V -subgroup as {(v, v) : v ∈ SU(2)}. We can then write an arbitrary coset as (l̃ , r̃ )SU(2)V = (1, r̃ l̃−1)(l̃ , l̃ )SU(2)V = (1, r̃ l̃−1)SU(2)V . (2.14) Thus the coset can be represented by the element r̃ l̃−1. We will write a matrix representation with the same capital letter, for example, L =D(l ). The represen- tative of the coset then has the matrix representation R̃L̃†. Since L̃, R̃ represent arbitrary elements of SU(2) also the product R̃L̃† represents an arbitrary element and because of this we can identify it with the matrix U . To get the transformation behaviour of U we look at the left multiplication of a group element, (l ,r ) ∈ SU(2)L ×SU(2)R , and a left coset represented by u, (l ,r )(1,u)SU(2)V = (l ,r u)SU(2)V = (1,r ul−1)(l , l )SU(2)V = (1,r ul−1)SU(2)V . (2.15) Thus we can let U transform as U → RU L†. (2.16) If the transformation is part of the unbroken group SU(2)V the ground state should be invariant. The ground state is the same as the vacuum state, i.e. all fields φi = 0 and U0 = 1. We let this transform under SU(2)V , U0 →V U0V † =V 1V † =V V † = 1 =U0. (2.17) We see that our construction of U0 is invariant is this case. On the other hand for a transformation in the broken subgroup SU(2)A we get U0 → A†U0 A† = A† A†. (2.18) The vacuum is not invariant which is consistent with spontaneous symmetry breaking. It can also be shown [Sch03] that the pion fields φa transform linearly under the unbroken subgroup SU(2)V . This is not true for the broken subgroup SU(2)A . 2.4 Building blocks for interactions The next point to address is how to use the pion matrix U to create chiral invariant Lagrangian terms. The Lagrangian is a Lorentz scalar, and as such all of its terms must be Lorentz scalars. Since the QCD Lagrangian is invariant under parity transformations, the ChPT Lagrangian should also have this symmetry. 13 We need some way to turn the pion matrix U into a scalar while also ensuring that it is invariant under the required symmetry transformations. One way to turn a matrix into a scalar is to take its trace. The trace of only one pion matrix, TrU is a scalar but it is not chiral invariant. But, if we take a pair of matrices A,B that transform as A,B → R AL†,RBL† we can form a chiral invariant, Tr ( AB † ) → Tr ( R AL†LB †R† ) = Tr ( R AB †R† ) = Tr ( R†R AB † ) = Tr ( AB † ) . (2.19) Here we have exploited the cyclic property of the trace. The trace of the product UU † is trivial since U is unitary. One allowed term is then Tr ( ∂µU∂µU † ) . But to create an accurate low energy theory of QCD the global SU(2)L ×SU(2)R symmetry must be upgraded to a local (or gauge) symmetry [Leu94]. This means that the transformations L,R are made to be dependent on space- time. The transformation of U is the same as in the global case, U (x) → R(x)U (x)L(x)†. (2.20) But for terms containing derivatives of U we also get derivatives of the transfor- mation matrices; ∂µU (x) →∂µ ( R(x)U (x)L†(x) ) = ∂µR(x)U (x)L†(x) +R(x)∂µU (x)L†(x) +R(x)U (x)∂µL†(x). (2.21) This destroys the invariance of Tr ( ∂µU (x)∂µU †(x) ) . In the following, I will not write the spacetime dependence explicitly. To repair the invariance we introduce the covariant derivative Dµ, which transforms in a way as to cancel the terms that destroy the invariance. Following Scherer [Sch03] we introduce the gauge fields Rµ and Lµ. Their transformation properties are given by Rµ→ RRµR† + i R∂µR†, (2.22) Lµ→ LLµL† + i L∂µL†. (2.23) The covariant derivative acting on U is defined as DµU ≡ ∂µU − i RµU + iU Lµ. (2.24) We look at how this transforms, DµU →∂µ(RU L†)− i (RRµR† + i R∂µR†)RU L† + i RU L†(LLµL† + i L∂µL†) = ∂µRU L† +R∂µU L† +RU∂µL† − i RRµU L† +R∂µR†RU L† + i RU LµL† −RU∂µL†. (2.25) 14 The unwanted terms containing the derivative on L† are immediately cancelled. To see the cancellation of the terms with the derivative on R we need to use that R is unitary, R†R = 1, (2.26) 0 = ∂µ ( R†R ) = ∂µR†R +R†∂µR; ⇒ ∂µR†R =−R†∂µR. (2.27) Then we can rewrite the term R∂µR†RU L† = −∂µRU L†, which cancels the re- maining unwanted terms. So we are left with R∂µU L† − i RRµU L† + i RU LµL† = RDµU L†. (2.28) This is the same transformation property as for the pion matrix U . As a conse- quence DµU is a building block that can be used in the same way to create a chiral-invariant Lagrangian. There is an added benefit of introducing the gauge fields Lµ,Rµ; they can be used to facilitate the coupling to external fields or particles that are not part of ChPT. For the gauge fields we can write down the field strength tensors f R µν ≡ ∂µRν−∂νRµ− i [Rµ,Rν], (2.29) f L µν ≡ ∂µLν−∂νLµ− i [Lµ,Lν]. (2.30) These transform as f R µν → R f R µνR† and f R µν → L f L µνL†. Following Scherer [Sch03, p.102] who is following the convention of Gasser and Leutwyler we introduce the linear combination χ≡ 2B0(s + i p), (2.31) where s and p are scalar and pseudoscalar external fields respectively. These fields are matrices; we let them transform as s, p → RsL†,RpL†. Concerning the quark masses we here employ a trick and let them be in s even though they will not have the correct chiral transformation. In fact the quark masses will contribute symmetry breaking terms. We can now write down a few more building blocks transforming as A → R AL† [Sch03]: χ, U f L µν, f R µνU . (2.32) Out of these only the combinations χU † and Uχ† have non-vanishing trace. In order to get a parity-invariant Lagrangian only the combination χU † +Uχ† can be accepted [Sch03, pp.103-104]. 15 2.5 Ordering When creating the most general Lagrangian we will end up with an infinite num- ber of terms containing products of U , the derivative of U and external fields including the quark masses. In front of each of these independently invariant terms there will be an unknown proportionality constant, an LEC. To be able to use this theory to make any kind of prediction we must have a way of ordering the terms by their relative contribution. We start by dividing the Lagrangian into parts corresponding to the power of the quark masses and derivatives. Each derivative will generate a pion four- momentum p for the corresponding vertex in the Feynman rules and each quark mass is equivalent to two powers of the pion four-momentum [Sch03]. Because the dimension of each Lagrangian term must be fixed, each pion momentum or quark mass must be divided by some other mass or energy. The only other energy scale available is the cutoffΛ= Mρ and as such the expansion will be in p/Mρ . The contribution to the order of a diagram by a vertex of type i is [Wei79] νi = di +ei −2, (2.33) where di is the number of derivatives and ei the number of external fields. We will denote the Lagrangian terms by the contribution to the order of the resulting vertices. When determining the order of a diagram we need to look both at the order of the vertices included and the number of loops in the diagram. With Vi the number of vertices of type i and L the number of independent loops the order is given by [Wei79] ν= 2+2L+∑ i Viνi (2.34) The order ν is the exponent in the expansion, that is, a diagram of order ν is proportional to (p/Λ)ν. For example if we want all diagrams of order ν= 2 we can have any number of vertices from L π 0 and no loops. At order ν= 4 we can have any number of vertices from L π 0 and one vertex from L π 2 or one loop. When calculating any quantity to a given order all diagrams of lower orders need to be included. The reason for counting the order of the Lagrangian terms the way that we do is to make it consistent with ChPT including multiple nucleons. In this case it becomes more clear how a certain Lagrangian term will contribute to a process no matter if it contains two, four or zero nucleons. Some authors(e.g. ref. [Sch03]) denote the order of Lagrangian terms by the number of derivatives di plus the number external fields ei . In this case the order of each Lagrangian term is increased by 2. 16 2.6 Leading order Lagrangian Of course, all terms in the Lagrangian must be Lorentz invariant as well as chiral invariant. The only available four-vector is the covariant derivative, Dµ. There are only two ways of turning four-vectors into scalars, either through contracting them with the metric gµν or the Levi-Civita tensor εµνρσ. This means that we can only have an even number of covariant derivatives in a Lagrangian term. If we also recall that the each quark mass counts as two powers of momentum, we can see that our expansion will only contain even order terms. Then the Lagrangian can be written as L π =L π 0 +L π 2 +·· · . (2.35) So to create L π 0 we want to find all terms of order p2/M 2 ρ . This means that we can have either two derivatives or one quark mass. There is one allowed term with two derivatives. As we saw in section 2.4, we need to take the trace of the resulting chiral matrix to get a chiral invariant term, Tr ( DµU ( DµU )† ) . (2.36) There is also a term including the quark masses. The quark masses are con- tained in χ, and we saw in section 2.4 that the only allowed expression is χU † +Uχ†, (2.37) which, since χ contains one quark mass is of order p2/M 2 ρ . Combining these expressions we get the leading order Chiral perturbation theory Lagrangian, L π 0 = f 2 π 4 ( Tr ( DµU ( DµU )† ) +Tr ( χU † +Uχ† )) . (2.38) This Lagrangian contains two LECs, the pion decay constant, fπ; and the parameter B0, which is contained in χ. Both of these are free parameters that need to be fixed either by fitting them to experimental data or by deriving their values from QCD with, for example, lattice QCD. In the last section of this chapter we will see one possible way to determine the value of fπ. To be able to make calculations we need the interaction Lagrangian. By this we mean the Lagrangian expressed in its constituent fields, for example, π,Lµ, and not the more abstract building blocks of the Lagrangian like U . When we have done this the Feynman rules can be easily read off. 2.7 Weak interactions of pions In order to understand how to implement the interactions between the pions in ChPT with the vector bosons W ±, we will look at how the interaction is realized in the underlying theory, i. e, the Standard Model. By comparing the symmetries 17 q L π W Chiral SU(2) Gauge Coupling Chiral SU(2) Gauge Coupling Standard Model Chiral Perturbation theory Figure 2.1: The coupling of pions and W ± bosons follows from the coupling of quarks and W ± bosons. of the Standard Model with that of ChPT we can transfer the interactions of the quarks to the pions. In the Standard model, the electroweak interaction is realized by postulating a local SU(2)×U(1) symmetry of the lepton and quark fields, giving rise to four vector boson fields. Three of the vector boson fields are given masses by the Higgs-induced spontaneous symmetry breaking; one is left massless. Another feature of the Standard Model is that it contains no regular fermion mass terms. Instead, all fermion masses are generated by Yukawa couplings to the Higgs field. With no mass terms, the natural basic building blocks for fermions are not the four-component Dirac spinors but the two-component Weyl spinors like we saw in section 2.2. The W ± bosons only interact with the left handed Weyl spinors of the quark fields. To get the interaction of the up- and down quarks we create an SU(2) doublet of the left handed fields QL = ( uL dL ) , (2.39) then for the physical up and down quarks the interaction term is [PS95, pp. 704, 714], L W 0 = Q̄L L̃µγ µQL , (2.40) where, L̃µ = Vud gp 2 ( W + µ T ++W − µ T − ) . (2.41) 18 Here L̃µ is part of the gauge current of the SU(2) subgroup in the SU(2)×U(1) group. W ± are the W-boson fields and the matrices T + and T − give the coupling to the quarks fields, T −† ≡ T + ≡ ( 0 1 0 0 ) , (2.42) while g is the weak interaction coupling strength. Even though the weak inter- action has a universal strength due to gauge invariance there is an additional coupling constant Vud . The reason is that the quark mass eigenstates are not the same as those that are created and destroyed by the interaction with the W -bosons; Vud parametrizes this overlap. This is nothing new, the SU(2) group acts on exactly the same quark doublet as the SU(2)L subgroup of the chiral group. To implement the weak interactions of the pions we only need to identify the gauge fields in the Standard Model with the corresponding ones in ChPT as illustrated in figure 2.1. In ChPT we have already introduced the gauge field Lµ for the gauged SU(2)L subgroup. To implement the weak interaction we simply identify Lµ ≡ L̃µ. (2.43) To get the vertices at leading order we set Rµ = 0 in the effective Lagran- gian (2.38) and expand the covariant derivative Tr ( DµU ( DµU )† ) = Tr (( ∂µU + iU Lµ )( ∂µU † − i LµU † )) = Tr ( ∂µU∂µU † + iU Lµ∂ µU † − i∂µU LµU † +U LµLµU † ) . (2.44) The first term only contains the pion matrix U so it will not give rise to any interactions with the weak sector and the last term does not contribute at leading order since it is not linear in Lµ. The pion matrix U is by definition unitary since it is an element of SU(2). Thus we can use the result of equation (2.27), ∂µU †U =−U †∂µU . We use this to simplify equation (2.44). Together with the cyclic property of the trace we see that the second- and third term are equal. Thus, the relevant part of the Lagrangian can be written L W 0 = f 2 π 2 Tr ( i Lµ∂µU †U ) . (2.45) To get the interaction in terms of the pion fields we make an expansion of U in the matrix φ U = exp ( iφ fπ ) = 1+ iφ fπ +O ( φ2) . (2.46) We insert this expression into the Lagrangian and keep terms linear in φ and Lµ. The matrix φ is Hermitian since it is a real linear combination of the Hermitian 19 Pauli matrices. We can then write the leading order interaction Lagrangian as L πW 0 = fπ 2 Tr ( Lµ∂µφ † ) = gVud fπ 2 p 2 Tr ( W + µ T +∂µφ+W − µ T −∂µφ ) . (2.47) We insert the pion fields as in equation (2.12) and look at the first term of the trace, Tr ( W + µ T +∂µφ ) = Tr ( W + µ ( 0 1 0 0 ) ∂µ ( π0 p 2π+ p 2π− −π0 )) = Tr ( W + µ ∂ µ (p 2π− −π0 0 0 )) = p 2W + µ ∂ µπ−. (2.48) The second term works out in the same way and we get the leading order interac- tion Lagrangian, L πW 0 = Vud g fπ 2 ( W + µ ∂ µπ−+W − µ ∂ µπ+ ) . (2.49) This gives rise to two vertices in the Feynman rules coupling W bosons to pions. Each derivative of the pion fields gives a factor of −i pµ for a field with momentum pµ. We get the vertices π+, pµ W + = Vud g fπ 2 pµ (2.50) π−, pµ W − = Vud g fπ 2 pµ. (2.51) 2.8 Decay of charged pions W +π+ µ+ νµ Figure 2.2: Tree level diagram giving the leading order contribution to the process π+ →µ++νµ The overwhelming majority of positive pion decays go to a positive muon and a muon neutrino [PDG10]. By comparing the observed decay rate with one 20 calculated from chiral perturbation theory we get a constraint on one of the LECs of ChPT. At order p2, i.e., at tree level with only vertices from L π 0 , only the diagram in figure 2.2 contributes to the amplitude. The W +-boson propagator [PS95, p.743] can be approximated because of the very high mass of this particle compared to the mass of the pion. For a vector boson momentum k2 ≈ m2 π << m2 W we have −i k2 −m2 W ( gµν− kµkν m2 W ) ≈ i gµν m2 W . (2.52) Further the vertex coupling the leptons to the W + is given by [PS95, p.705] W + µ+ νµ =−i g 2 p 2 γµ(1−γ5). (2.53) Both the pion and the leptons couple to the W + with the strength g and with the approximated W-boson propagator it is convenient to use the Fermi constant GF = p 2 8 g 2 m2 W . (2.54) Using these components we can write down the amplitude for the diagram in figure 2.2. With the incoming pion momentum p, muon momentum k ′ and neutrino momentum k we get iM = Vud g fπ 2 pµ i gµν m2 W ūs(k) ( − i g 2 p 2 ) γν ( 1−γ5)v r (k ′) =GF Vud fπūs(k) /p ( 1−γ5)v r (k ′). (2.55) This can be simplified by using standard methods, for further details look in appendix A. To compute the decay rate we need the square of the amplitude |M |2, which we will compute in the rest frame of the pion. With Eν the neutrino energy and Eµ the muon energy we get |M |2 = 8G2 F V 2 ud f 2 πm2 π ( EνEµ+k ·k ′) . (2.56) Combining the two-body phase space [PS95, p.107] with the square of the matrix element we get the decay rate,∫ dΓ= 1 2mπ ∫ dΠ2 |M |2 = 1 2mπ ∫ dΩ 1 16π2 |k | mπ 8G2 F V 2 ud f 2 πm2 π ( EνEµ−k2) . (2.57) 21 After performing the integration and simplifying the resulting expression we get the pion decay rate Γ= G2 F V 2 ud f 2 π 4π mπm2 µ ( 1− m2 µ m2 π )2 . (2.58) All parameters in this expression except for fπ are known from other experi- ments. By measuring the decay rate of charged pions we can now determine the value of the pion decay constant fπ. When we know the value we can reuse it in any other calculation dependent on fπ. We could also turn this argument around. By determining fπ from another experiment, perhaps pion-pion scattering, we can predict the pion decay rate. If we find a way to determine B0 from an experiment we have fixed all the LECs of leading order ChPT. This means that we could then predict the result of any experiment to leading order. So, by using one set of observables to determine the LECs we can then predict the results of all other experiments. The result will always be approximative since there are an infinite number of higher order corrections which are ignored. 22 Chapter 3 Baryon chiral perturbation theory Building on the previous chapter I will review how to add nucleons to chiral perturbation theory. Because of the different structure of the Lagrangian terms when including fermions it will be necessary to develop new transformation rules for the nucleons. There are issues that make it hard to create a consistent power-counting scheme in the nucleon-pion sector. The reason for this is that a new mass scale is introduced when including nucleons, namely the nucleon mass. I will explore an example of this problem in the end of the chapter with the calculation of the nucleon self-energy. This chapter is minimal and I will not venture into nearly as much detail as the previous one. The reason being that I will not be making the major calculations in this formalism, instead it will only be used for further theoretical developments. 3.1 Building blocks We will consider the nucleons to be point-like, spin-1/2 particles. As such we introduce Dirac spinor fields p and n for the proton and the neutron. Just as in the pion-only sector all the internal structure of the nucleons will be encoded in the LECs. The nucleons are then inserted into a chiral vector, the nucleon doublet, Ψ= ( p n ) . (3.1) The nucleons are low energy QCD particles so they will transform under the chiral group. We will let them transform as Ψ→ K (L,R,U )Ψ. (3.2) 23 Here K is a function of the SU(2) transformations L,R and the pion matrix U , K (L,R,U ) = √ LU †R†R p U . (3.3) This is also an SU(2) transformation [BKM95]. Since the pion matrix U is a local matrix the transformation of the nucleon field will be local. So, in order to get the correct transformation for the derivative of the nucleon field ∂µΨwe need to use a covariant derivative [GSS88], DµΨ= ∂µΨ+ΓµΨ Γµ = 1 2 [ u†,∂µu ] − i 2 u† ( Vµ+ Aµ ) u − i 2 u ( Vµ− Aµ ) u†. (3.4) The pion fields are contained in the matrix u, which is defined by u2 =U . (3.5) The gauge fields Vµ = Rµ + Lµ and Aµ = Rµ − Lµ are exactly the same gauge fields as the ones in chapter 2 and have the same transformation behavior. The transformation of the covariant derivative of the nucleon field is the expected DµΨ→ K DµΨ. (3.6) We now have two covariant derivatives, which to use will be determined by the context. Both will be written as Dµ. There is one more building block that we need for the nucleon ChPT Lagran- gian. It is an axial vector object similar to the connection. It is called the chiral vielbein and is given by [GSS88] ∆µ = 1 2 { u†,∂µu } − i 2 u† ( Vµ+ Aµ ) u + i 2 u ( Vµ− Aµ ) u†. (3.7) Under a chiral transformation the vielbein transforms covariantly ∆µ→ K∆µK †. (3.8) 3.2 Lagrangian Any number of operators transforming as B → K BK † can be sandwiched between the nucleon fields, or covariant derivatives of the nucleons fields, to form a chiral invariant term. We want to have the minimal number of derivatives in order to get the lowest order Lagrangian. There is one term with no derivatives, Ψ̄Ψ. (3.9) With one covariant derivative we get the term Ψ̄γµDµΨ. (3.10) 24 The vielbein ∆µ also contains one derivative. In order to create a parity invariant term we must contract it with another axial vector, Ψ̄γµγ5∆µΨ. (3.11) Each of these three terms will have a preceding LEC and an arbitrary phase factor to follow conventions and ensure the reality of coupling constants. One of the LECs can be removed by redefining the nucleon fieldΨ. With this in mind, the most general chiral-invariant Lagrangian with the smallest number of derivatives is [GSS88] L N = Ψ̄( i /D −mN + i g Aγ µγ5∆µ ) Ψ, (3.12) with mN being the nucleon mass. A new LEC, the axial coupling constant g A , has been introduced. It gives the coupling strength of the nucleon to a single pion as we will see in the next section. This Lagrangian only describes the nucleons and their interactions with the pions. The dynamics and interactions of the pions are still described by the same Lagrangian of equation (2.38). 3.3 Nucleon self-energy The purpose of this section is both to give an example and to show a problem with naive baryon ChPT. We will see that in the chiral limit, i. e., with zero pion mass, the nucleon mass is renormalized. This indicates that it will be hard to count the order of the contribution of a given diagram, more about this in the end of the section. The Lagrangian (3.12) contains a free Dirac Lagrangian for each nucleon. With this as the free theory each nucleon will have the propagator, i SF (p) = i /p −m̊N + iε . (3.13) This expression is modified by the self-energy Σ(p) in the usual way by including all irreducible diagrams in a geometric series (for more details see ref. [PS95, p. 220]), i SF (p) = i /p −m̊N −Σ(p)+ iε . (3.14) For a real nucleon the four-momentum must satisfy p2 = m2. For a small pertur- bation the change in the nucleon mass should be small so that to zeroth order m = m̊N . We will therefore evaluate Σ(p) at p2 = m̊2 N . Interaction Lagrangian and Feynman rules To find the self-energy we will need an explicit expression of the interaction between nucleons and pions; we can derive it from the Lagrangian in (3.12). By 25 deriving the interaction Lagrangian and the Feynman rules it will be possible to find and calculate the leading order self-energy diagrams. There is an N Nπ vertex in the leading order Lagrangian which comes from the term L N 1 ′ = Ψ̄g Aγ µγ5i∆µΨ. (3.15) We set all external fields to zero and expand the pion matrix u in the vielbein i∆µ =− 1 2 fπ ∂µφ aτa +O (π2). (3.16) Inserting this into the Lagrangian term gives the leading order N Nπ Lagrangian L N Nπ 1 =−Ψ̄ g A 2 fπ γµγ5∂µφ aτaΨ (3.17) This gives rise to the vertex k, a = − g A 2 fπ γ5 /kτa . (3.18) We will also need the pion propagator, which is the Klein Gordon propagator in momentum space [PS95], k = i k2 + iε . (3.19) Here the pion mass is set to zero since we do this calculation in the chiral limit. Feynman diagrams and their evaluation Using the N Nπ-vertex, the leading order contribution to the self-energy is given by the diagram in figure 3.1. p −k k p p Figure 3.1: Nucleon self-energy diagram When evaluating this diagram we get a factor 3 from summing over the Pauli matrices, iΣ= 3g 2 A 4 f 2 π ∫ d 4k (2π)4 /kγ5 ( /p − /k +m̊N ) /kγ5 (k2 + iε) (( p −k )2 −m̊2 N + iε ) . (3.20) 26 We will use dimensional regularization to control the divergent integrals. To avoid problems when including γ5 we will use only the following identities to simplify the expression [Sch03], {γµ,γν} = 2gµν, gµµ = n, {γµ,γ5} = 0, γ2 5 = 1. (3.21) Starting with the numerator, /kγ5 ( /p − /k +m̊N ) /kγ5, (3.22) we can put it in a form that will cancel the denominator in a nice way. For the first term kµγ µγ5pνγ νkργ ργ5 = kµpνkργ µγνγρ = (3.23) kµpνkρ (−γµγργν+2γµgρν )=−k2 /p +2/kpνkν The second term, /kγ5 /k /kγ5 = /kk2. (3.24) And finally the third, /kγ5m̊N /kγ5 =−k2m̊N . (3.25) We assemble the whole numerator and put it in a more useful form. − ( /p +m̊N ) k2 + (p2 −m̊2 N )/k − (( p −k )2 −m̊2 N ) /k (3.26) The first and the last terms cancel parts of the denominator and the middle term is zero since we demand that p2 = m̊2 N . We can now assemble the expression for the self-energy, iΣ= 3g 2 A 4 f 2 π ∫ d 4k (2π)4 ( −/p −m̊N (p −k)2 −m̊2 N + iε − /k k2 + iε ) . (3.27) The last term is odd in /k and thus integrates to zero. We are now left with a single term with a quadratic polynomial in /k in the denominator. If we would just Wick-rotate and change to spherical coordinates we get a quadratic divergence of the integral. To control this divergence we use dimensional regularization. We Wick-rotate and change to Euclidean spherical coordinates. Then we take the number of dimensions to be integrated over as a complex parameter. For non-integer dimension the integrals will then converge, and we can take the limit when going to four dimensions in a controlled way. 27 Looking at just the integral we then have∫ d nk (2π)n 1 (p −k)2 −m̊2 N + iε = ∫ d nk (2π)n 1 k2 −m̊2 N + iε =−i ∫ d nkE (2π)n 1 k2 E +m̊2 N + iε =−i 1 (4π)n/2 Γ(1−n/2) Γ(1) ( 1 m̊2 N )1−n/2 . (3.28) The last integral comes from ref. [PS95, p.251]. Now by letting n = 4−ε we can find the behaviour near the n = 4 pole, − i ( 1 16π2 + ε ln4π 32π2 +O (ε2) )( 2 ε −γ+O (ε) )( m̊2 N + εm̊2 N lnm̊−2 N 2 +O (ε2) ) . (3.29) We keep all terms up to order ε0: − i m̊2 N 16π2 ( 2 ε −γ+ ln 4π m̊2 N ) +O (ε). (3.30) The whole expression for the self-energy then becomes Σ= 3g 2 A 4 f 2 π ( /p +m̊N ) m̊2 N 16π2 ( 2 ε −γ+ ln 4π m̊2 N +O (ε) ) . (3.31) The self-energy clearly diverges when we let the number of dimensions ap- proach 4 and ε approach 0. To absorb this infinity the nucleon mass must be renormalized [Sch03]. Also the coupling constant g A is renormalized by pion loop diagrams [Sch03]. This means that loop diagrams contribute at order p which precludes the possibility to create a simple ordering scheme like in the pion-only sector. In that sector the loop would add two powers of momenta to the diagram which in this case would produce a diagram of order p3. The reason for this break-down is the nucleon mass which does not vanish in the limit of zero quark masses [BKM95]. Derivatives of the nucleon field will yield a nucleon four momentum that is of the same order as the nucleon mass, which in turn is of the same order as the cutoff energy of our theory. In essence, the nucleon momentum p is not a small momentum. This does not mean that it is impossible to use baryon ChPT for calculations. As per usual it is necessary to include all terms up to the given order. For a one- loop calculation in baryon ChPT, there will be the three Lagrangian terms of order p, loop contributions of order p2 and p3, and finally counter terms of order p2 and p3 [BKM95]. In the next chapter we will see how the power counting can be restored and this picture simplified by considering the nucleons to be very heavy. 28 Chapter 4 Heavy baryon chiral perturbation theory To make calculations in a practical manner we must have a way of ordering the contributions by powers of momenta as in pion-only chiral perturbation theory. As we saw in the previous chapter this is not possible with the direct inclusion of the nucleons. This problem is solved by considering the nucleons to be very heavy, an ap- proach pioneered by Jenkins and Manohar [JM91]. This removes the dependence on the nucleon mass from the leading order Lagrangian at the cost of remov- ing manifest Lorentz covariance. Making this approximation the result is heavy baryon chiral perturbation theory. In this chapter I will describe how to construct this theory from the covariant formalism of the previous chapter. Ultimiately I wish to describe properties of nuclei. These are bound states of nucleons and as such there are some complications that need to be considered. The presence of shallow bound states in itself indicates a breakdown of perturba- tion theory [Wei90]. I will go into some detail on how this can be controlled by only considering irreducible graphs in time-ordered perturbation theory. I will also look at the ordering that arises in the heavy baryon formalism and present the leading order and next to leading order Lagrangian. Together this will set the scene for computing the two-body axial vector current which is the topic of the next chapter. 4.1 Heavy baryons The building blocks of the Lagrangian in the heavy baryon formalism are similar to those of the covariant baryon ChPT Lagrangian. The pion fields are treated in the same way as in the pion-only sector. The difference lies in the treatment of the nucleon fields. By factoring out an on-mass-shell field already at the Lagrangian level the dependence on the nucleon mass can be eliminated in the leading order Lagran- 29 gian. By doing this the correspondence between the momentum expansion and the loop expansion is made as simple as in the pion-only case. Since the nucleon is very heavy compared to the typical energy scale of mπ, it will leave an interaction with almost the same momentum that it entered with. Also, all external particles need to be on shell so we can write the nucleon momentum as pµ = mN vµ+kµ (4.1) where vµvµ = 1. The two four-vectors are vµ, the four-velocity of the nucleon, and kµ, the small residual momentum of the nucleon, which is of the order of mπ. The space dependent part of the positive energy solution to the Dirac equa- tion with momentum p = mN v is just a plane wave exp(i mN vµxµ). By introduc- ing the velocity projection operators P± v , P± v ≡ 1± vµγµ 2 , (4.2) we can decompose each nucleon field into two parts, N = e i mN v x P+ v Ψ, H = e i mN v x P− v Ψ. (4.3) For the special case of vµ = (1,0) these parts correspond exactly to the light and heavy components of the spinor field [BKM95]. We have also factored out the dominant part of the solution, the plane wave with four-momentum mN v . Derivatives on the new field N yields −i kµ [JM91] instead of the full nucleon momentum −i pµ from the full nucleon fieldΨ, ∂µN |p〉 = ∂µ ( e i mN v x P+ v Ψ ) |p〉 = (i mN vµ− i pµ)e i mN v x P+ v Ψ|p〉 =−i kµP+ v e i mN v xΨ|p〉 =−i kµN |p〉. (4.4) The heavy component field H can be completely eliminated from the Lagr- angian at the cost of introducing an infinite series of corrections of increasing order in k/mN . This process also completely removes the nucleon mass mN from the leading order Lagrangian. For details see for example the lecture notes by Scherer [Sch03] or the review by Bernard et al. [BKM95]. For the fundamentals on heavy baryon fields see the short review by Georgi [Geo90]. 4.2 Ordering The motivation for the heavy baryon approach was to restore the power counting of the pion sector. This has been achieved but only for single nucleons. As we will see in the next section bound states introduce complications that also need to be accounted for. The order of Lagrangian terms is characterised by three quantities. The small four-momentum p, which corresponds to pions, external fields or the residual 30 four-momentum of the nucleons; the cutoff scale Mρ ; and the nucleon mass mN . The latter two are of the same order of approximately Λ= 1GeV. There are two simultaneous expansions, one is the ordinary chiral expansion in small momenta over the cutoff p/Mρ , and the other is a relativistic expansion in the small mo- menta divided by the nucleon mass p/mN . Combined we get an expansion in p/Λ both for the terms of the Lagrangian and for the diagrams. The point of using the heavy baryon approximation for the nucleon field is that it leads to a sensible ordering. Loops can be absorbed order for order as in the pion-only sector; for example, diagrams with vertices from the first order Lagrangian and one loop will only require the renormalization of terms in the third order Lagrangian. The order of a vertex in equation (2.33) must be changed to account for the nucleon line [Wei92] resulting in νi = di +ei + ni 2 −2, (4.5) where di is the number of derivatives and pion masses, ei the number of external fields and ni the number of nucleon fields. We will continue to assign an index to the Lagrangian terms by the order of the resulting vertices. Each diagram is then characterised by its order ν which is defined as the power of the factor (p/Λ)ν in that diagram. This order is given by [Wei92] ν= 4− A−2C +2L+∑ i Viνi (4.6) Here Vi is the number of vertices of type i , L the number of loops, A the number of nucleons. C is the number of separately connected parts in a diagram. In a process with A nucleons, a diagram can have up to A separately connected parts. 4.3 Considering several nucleons and nuclei There are very loosely bound states of nucleons. An example is the deuteron that has a binding energy of only 2.2 MeV, which is very low compared to the typical momentum scale of mπ. These states will naturally play a very large role when considering more than one nucleon. Indeed, their existence implies a breakdown of perturbation theory in the form of infrared divergences [Wei90]. This means that the naive ordering of section 4.2 can not be correct for all Feynman diagrams. It is possible to separate out the part that leads to the divergence. By aban- doning the manifestly covariant formalism of Feynman diagrams, and instead considering time-ordered graphs, we can find the source of the divergence. In this framework it is meaningful to talk about intermediate states. The divergence of the perturbation series comes from intermediate states consisting only of nucleons [Wei90]. The solution is then simply to cut all graphs apart at each time where there is a nucleon-only intermediate state, an example can be seen in figure 4.1. We are then left with irreducible graphs, that is, graphs that have no 31 purely nucleonic intermediate state. These graphs will be of the order indicated by equation (4.6). (a) Reducible graph (b) Irreducible graphs Figure 4.1: A reducible and corresponding irreducible graphs The sum of irreducible graphs can then be used as an effective potential from which the S-matrix can be computed by the use of the Lippmann-Schwinger equation [Wei90]. A general graph consists of a number of irreducible graphs glued together by purely nucleonic intermediate states. Although we require that the full graph is connected, each component irreducible graph does not need to be fully con- nected [Wei92]. The result is that for a process involving A nucleons we must in principle consider one-, two- . . . A-body graphs. From equation (4.6) we can see that a diagram is enhanced by −2C powers of momenta where C is the number of separately connected parts. The leading contribution comes from graphs where all nucleons are disconnected from each other. In this case C = A. The next con- tribution in terms of connected graphs is when two nucleons are connected and the rest are spectators, which gives C = A−1 separately connected parts. We call the leading contribution a one-body graph and the next order graph a two-body graph. For a higher N -body graph the number of disconnected parts is decreased as more nucleons are connected by pion lines. The number of disconnected parts in the general case is C = A−N +1. When considering interactions of nuclei we must take care to avoid double counting. In many cases we will have a nuclear wavefunction that is the result of solving the Schrödinger equation with an effective potential. This effective potential should then not be included in the interaction graphs since it has already been accounted for. This is very similar to the ordinary amputation of the self energy from external lines in Feynman diagrams. This is also not part of the interaction but rather it gives the asymptotic particles their physical properties. For a bound state the binding energy is the self-energy of the bound state minus self-energy of the constituent particles if they were free. The requirement that external particles need to be on the mass shell is valid for the complete composite particle. For the constituent particles this condition is modified so that they should have the 32 correct bound state wavefunction. 4.4 Lagrangian At zeroth order we have the Lagrangian [BKM95], L HB 0 = N̄ ( i vµDµ+2i g ASµ∆µ ) N + (contact terms), (4.7) where N is the two component heavy baryon field and Sµ is the spin matrix, Sµ = i 2 γ5σ µνvν. (4.8) The other building blocks are carried over from the previous chapter. Note that the nucleon mass is not explicitly included in the leading order Lagrangian. The contact terms are not relevant for the calculations in this thesis and will be ignored. We take the first order Lagrangian from ref. [BKM97] for the two-nucleon-field part and from ref. [PMS+03] for the four-nucleon-field part. The singlet gauge field has been excluded so we have the Lagrangian L HB 1 = N̄ { vµvν− gµν 2mN DµDν+ g A mN { SµDµ, vν∆ν }+ c1 Tr ( χ+ ) +4 ( c2 − g 2 A 8mN )( vµi∆µ )2 +4c3i∆µi∆µ+ ( 4c4 + 1 mN )[ Sµ,Sν ] i∆µi∆ν + c5 ( χ+− 1 2 Tr ( χ+ ))− i (1+κv ) 4mN [ Sµ,Sν ] f + µν } N −4i d1N̄ Sµ∆µN N̄ N +2i d2ε abcεµνλδvµ∆νa N̄ Sλt b N N̄ Sδt c N . (4.9) Here we have, apart from the usual p/Mρ terms, also terms that are part of the relativistic expansion, which are suppressed by powers of p/mN . These terms are the result of the heavy baryon approximation and the elimination of the heavy component fields from the Lagrangian. The singlet gauge field is needed to write the full electromagnetic current. Including it would give one more term and modify the covariant derivative. We are primarily interested in the charged weak currents so we avoid this unnecessary complication. In this chapter we have seen how to overcome two problems involving di- vergences. First by identifying the nucleon as heavy we could remove the mass from the leading order Lagrangian; the result being a sensible power counting in the one nucleon sector. The extension to more than one nucleon introduced the problem of bound states. This was overcome by only considering the effective potential of irreducible time-ordered graphs. For the case of interactions we saw that only irreducible graphs will contribute to the amplitude. In the next chapter we will see how to apply this Lagrangian to model weak interactions in nuclei. Then we will apply the tools developed in this chapter. 33 Chapter 5 Weak interactions in nuclei In this chapter I will examine the weak interactions of systems of nucleons; of which the most familiar, and also the most important special case, is atomic nuclei. Just as for pions, the weak interactions of nucleons are determined by the chiral symmetry. This means that there is no need to introduce additional LECs to parametrize the coupling strength. The existing LECs together with the Standard Model weak interaction coupling constants are sufficient. I will also compute the two-body axial vector current operator, which is the leading order two-body contribution to the weak currents. This current operator is important both for weak interaction phenomenology but also for determining LECs in HBChPT in order to create realistic three-nucleon potentials. All computations will be performed in the HBChPT formalism, which was introduced in the previous chapter. 5.1 Weak interactions of nucleons Perhaps the most important consequence of nuclear weak interactions is the beta decay of nuclei. This is a collective name for charge changing decays that do not change the mass number of the nuclei. There are three basic processes to be considered: Beta decay N (Z , A) → N (Z +1, A)+e−+ ν̄e Electron capture N (Z , A)+e− → N (Z −1, A)+νe Beta-plus decay N (Z , A) → N (Z −1, A)+e++νe 34 The neutrinos in these processes are there to conserve the electronic lepton number. Another process is the scattering of nuclei and neutrinos. The neutrino only interacts weakly so to understand it we must understand nuclear weak inter- actions. Many neutrino experiments use the interactions between nuclei and neutrinos to detect neutrinos. Just like in the case of pion-only weak interactions, which was covered in section 2.7, the nucleon weak interactions are completely determined by the chiral symmetry. This means that the HBChPT Lagrangian already contains the appropriate ingredients to facilitate the description of weak interactions. Once again it is the left-handed gauge field Lµ =Vµ− Aµ that can be identified with the current of charged vector bosons W ±. 5.2 Current operators By four-current we mean a four-vector field satisfying some conservation princi- ple. This means that we can associate a conserved charge to the field. A general four-current can be considered to consist of two components, Jµ(x) = (J 0(x), J (x)), (5.1) where J 0(x) is the charge density and J (x) is the three-vector current. The role of currents is to describe the source of force fields. The most familiar and archetypal current is the electric four-current Jµem. The time component is the electric charge density, ρ, and the space components are the regular electric current j i . This four-current is included in the electrodynamic Lagrangian through the interaction term: Jµem Aµ. In quantum electrodynamics the four-current is formed from the electron field, Jµem = i eψ̄γµψ; e is the electric charge and ψ the electron-positron spinor field. The nuclear four-currents we will be looking at are very similar. By a weak four-current we mean a four-vector field that interacts with one of the weak fields. Just as the electromagnetic field is mediated by the photon, the weak field is mediated by the W ±- and Z bosons. Because of the very high masses of these bosons the field falls off in a very short distance from a weakly interacting particle. The word operator in current operator refers to an operator in the quantum mechanical sense. This means that we must express the current as a linear opera- tor on the Hilbert space of states. This operator must, in general, satisfy the same symmetries as the classical current, although in some cases quantization may destroy classical symmetries. An important point to remember is that there is often no manifest classical current and corresponding force field to the quantum mechanical current opera- tors. In the case of electromagnetism the force carrier is massless and thus the field has infinite range. This makes it easy to observe the field in the macroscopic world. In many other cases the force particles are massive and as a consequence 35 the range of the field is often microscopic. This means that the classical descrip- tion of the current and force field has very few applications. This also becomes apparent in how we derive the currents, not starting from classical physics, but instead from quantum mechanics. We will look at the current operators that represent the weak currents of hadrons. To see how such an operator may come about we can examine a weak interaction of nucleons and leptons. Using the heavy vector meson approxi- mation from chapter 2, the weak force can be described as a current-current interaction and the leading order matrix element can be written M =−i GFp 2 〈N ′, l |JµH Jlµ|N〉. (5.2) Here N describes the nucleon state and l the lepton state. If the interaction between the leptons and the nucleons can be considered to be very weak then the current JH does not affect the leptons and vice versa. In this case the matrix element can be decomposed into a product of a nucleon matrix element and a lepton matrix element M =−i GFp 2 〈N ′|JµH |N〉〈l |Jlµ|0〉. (5.3) The lepton matrix element is easy to evaluate in the Standard Model. The nucleon matrix element is significantly harder. The reason is that the nucleon states hidden under the labels N , N ′ are much more complicated. Let the nucleon states N and N ′ consist of A nucleons. The current operator JH must operate on the whole space of A nucleons. But, recalling that the leading order contribution comes from minimally connected graphs, we can assume that the leading order contribution will come from the operators that affect as few nucleons as possible. Forgetting about the Lorentz index for a while we write the current operator as a sum JH = J 1B H + J 2B H +·· ·+ J AB H . (5.4) Each operator on the right hand side is a sum of all operators affecting one nucleon, a pair of nucleons, and so on. The first two operators are written J 1B H = A∑ i Ji , J 2B H = A,A∑ i> j Ji j . (5.5) Here the first sum is over all nucleons and the second sum is over all pairs of nu- cleons. The operators Ji are one-body operators and Ji j are two-body operators. The nucleon matrix element has been decomposed into different parts de- pending on the number of nucleons participating in each reaction. To simplify 36 the calculations we insert a complete set of A-nucleon momentum states in the expression for the nucleon matrix element. 〈N ′|JµH |N〉 = ∫ d p1 · · ·d p Ad p ′ 1 · · ·d p ′ A× 〈N ′|p1 · · ·p A〉〈p1 · · ·p A|JµH |p ′ 1 · · ·p ′ A〉〈p ′ 1 · · ·p ′ A|N〉 (5.6) Given that we can write down the nucleon wavefunctions 〈p1 · · ·p A|N〉 all that remains is to calculate the matrix element 〈p1 · · ·p A|JµH |p ′ 1 · · ·p ′ A〉. These wave- functions must of course be totally antisymmetric under interchange of the identical nucleons. Remember that JH is composed of a sum of N -body operators. The one-body operators will only affect one nucleon, it will thus be a function of only one nucleon momentum. For the rest of the momenta it will only give a momentum conserving delta function. Let us take a closer look at the two-body matrix element, 〈p1 · · · p A|J 2B H |p ′ 1 · · ·p ′ A〉 = A−1,A∑ i< j 〈p1 · · · p A|Ji j |p ′ 1 · · ·p ′ A〉. (5.7) If we look at just one term of the sum on the right hand side we have 〈p1 · · · p A|J12|p ′ 1 · · ·p ′ A〉 = 〈p1p2|J12|p ′ 1p ′ 2〉〈p3 · · ·p A|p ′ 3 · · ·p ′ A〉. (5.8) All terms will be identical to this, only with different nucleon labels. The second factor will only contribute a product of momentum-conserving delta-functions. If we can compute the matrix element 〈p1p2|J12|p ′ 1p ′ 2〉 as a function of the mo- menta p1, p2, p ′ 1, p ′ 2 it will only be a matter of summing the contributions from the different combinations of nucleons in order to get the matrix element from the full current operator J 2B H . Now that we have decomposed the general A-nucleon matrix element into one-body-, two-body- and many-body matrix elements we will have a closer look at the two-body matrix elements. First, to better understand the full picture, we will take a quick look at the leading order one-body operators and sketch how they can be realized in HBChPT. If we go back to the specific goal of calculating weak currents we remember that it is only the left handed current that interacts weakly. In chapter 2 we saw that a left handed current can be written as Lµ = V µ − Aµ. When computing nuclear matrix elements it is more convenient to work in the latter basis. In phenomenological models of the weak interactions the leading contribu- tions come from the one-body Fermi and Gamov-Teller operators [Suh07]. The Fermi operator is also called the charge operator and it is the time component of the charge changing, four-vector nuclear current V µ±. The Gamov-Teller operator is the charge changing, axial vector current A± of Aµ± and changes the spin state of the nucleus. These are nuclear operators and should not be confused with external vector- and axial vector fields. 37 These operators also give the leading order contribution in HBChPT. Let us take a closer look at the order of different diagrams to understand how this is. For a given process we will always have A nucleons. We take A = 2 since it is the minimal number to have two-body currents. We could just as well have taken any higher number, it does not have any impact on the following derivation. By inserting this into equation (4.6) we see that the order of the operators is given by ν= 2−2C +2L+∑ i Viνi . (5.9) It is evident that the only way to lower the order of a diagram is to increase the number of separately connected parts, C . For two nucleons the maximum number of separately connected parts is C = 2. Then, with no loops and only vertices with νi = 0, we have the order ν=−2, (5.10) which we will also denote leading order (LO). The one-body Fermi operator is derived from the first LO HBChPT Lagrangian term in equation (4.7), L HB 0 ′ = N̄ i vµDµN , while the Gamov-Teller operator comes from the second term, L HB 0 ′′ = N̄ 2i g ASµ∆µN . Since loops only increase the order of a diagram the one-body Fermi and Gamov- Teller operators have the minimal order ν=−2. There is a relativistic correction to the Gamov-Teller operator at order ν= 0, or N2LO [PMS+03]. 5.3 Two-body axial vector current The next correction to the Gamow-Teller operator comes from the two-body axial vector current  a 12. The index a is a chiral vector, or isospin, index. The subscript numbers are the nucleon labels. The isospin index needs to be contracted with an isospin index on the external field. The charged vector bosons consist of the combinations W + = W 1 + iW 2 and W − = W 1 − iW 2. The corresponding combinations of the axial vector current,  + 12 =  1 12 + i  2 12 and  − 12 =  1 12 − i  2 12, give the two charge changing Gamow-Teller operators. A two-body operator connects the two nucleons together, thus lowering the number of separately connected diagrams to C = 1. Then, with no loops, and only vertices with νi = 0, we have the order ν= 0 or N2LO. Note that changing the number of nucleons A will change the number of separately connected diagrams; in the general case C = A − 1 for a two-body operator. This also changes the absolute order of the operator ν but the order relative to LO is unchanged. Thus, 38 a two-body operator will always have its first possible contribution at N2LO regardless of the total number of nucleons considered. We will only consider the case where the momentum carried by the external field goes to zero. As we will see, then the first non-vanishing contribution to the two-body, axial vector current requires a vertex from the next to leading order Lagrangian L HB 1 . As a result the order is increased by one, giving ν= 1 or N3LO. To get an expression for the two-body axial vector current we will evaluate the matrix element 〈p1p2|Âa 12|p ′ 1p ′ 2〉 in the formalism of HBChPT. In practice this is done by considering all irreducible graphs that couple to A, the external axial vector field. This field is already included in the HBChPT Lagrangian as the space components of the axial gauge field Aµ. The calculation will be performed as the momentum qµ that is carried by the external field goes to zero. The asymptotic states will be nuclei so there will be no need to consider reducible graphs since all purely hadronic subgraphs can be absorbed into the binding of the nuclei. To find all irreducible graphs we need to know which vertices exist. In the following we will see how to derive the vertices and the interaction Lagrangian. 5.3.1 Interaction Lagrangian There is a large number of possible vertices that comes from the LO and NLO Lagrangians. But when the momentum of the external field goes to zero it turns out that all non-vanishing graphs can be built out of only three vertices. Each vertex will be denoted by the external legs it has. For example, a vertex with two nucleon legs, one pion leg and one coupling to the external axial current will be called a N NπA-vertex. With this notation in place we can list the candidate vertices: πA, ππA, N Nπ, N Nππ, N NπA, 4N A and 4Nπ. These are the vertices that can be included in fully connected diagrams with four nucleon lines, one external field coupling and no loops. We will go through these one by one; deriving the interaction Lagrangian, structure and strength of the vertices. In chapter 2 we saw how to get a πLµ-vertex from L π 0 when computing the pion decay rate. The derivation of the πA vertex is very similar, and it is not possible to remove the momentum dependence of the vertex. This means that any diagram containing this vertex will be proportional to the momentum transfer q and will vanish when q → 0. Because the πA-vertex vanishes when q → 0 we can ignore the N Nππ and 4Nπ-vertices because they would only contribute in conjunction with the πA- vertex. Turning to the next pion vertex ππA we look in the LO pion Lagrangian L π 0 of equation (2.38). The possible candidate term is L π 0 ′ = f 2 π 4 Tr ( DµU (DµU )† ) . (5.11) 39 The other term in the LO pion Lagrangian does not contain any coupling to the external axial vector field. This vertex vanishes, for more details see appendix B.1. We have seen that there are no vertices coupling pions to the external field that gives a contribution at vanishing momentum transfer. Now we continue with the vertices coupling the external field to the nucleon fields instead. Too see these derivations in more detail see appendix B.1. First is the N Nπ-vertex for which the derivation can be seen in appendix B.1.2. It comes from the second term of the LO Lagrangian L HB 0 ′ and the resulting vertex is k, a =− g A 2 fπ σ ·kτa . (5.12) In all vertices and diagrams the baryon number and nucleon momenta flow from the bottom of the diagram(the in-state) to the top of the diagram(the out-state). In HBChPT we do not allow for the creation or annihilation of nucleons so this defines all fermion line directions completely. There is no N NπA-vertex in the LO Lagrangian (4.7). In HBChPT this vertex is kinematically suppressed, i. e., the Lorentz vector Aµ is contracted with the field velocity vµ. Consequently there will be no contribution including A; there will only be a vertex proportional to A0. We look instead in the two nucleon part of the NLO Lagrangian (4.9) for this vertex. Out of the eight terms in the Lagrangian only three contribute to this vertex. The two terms c1 Tr ( χ+ ) , c5 ( χ+− 1 2 Tr ( χ+ )) , (5.13) can be ignored since they do not include the external field A. The two terms proportional to vµ∆µ, g A mN { SµDµ, vν∆ν } , 4 ( c2 − g 2 A 8mN )( vµi∆µ )2 , (5.14) can be ignored since, with vµ = (1,0), they only contain A0, not A. Finally, the term i (1+κv ) 4mN [ Sµ,Sν ] f + µν (5.15) does not contribute when q → 0 since the resulting vertex is proportional to q . What is left are the three terms that do contribute: L HB 1 ′ = N̄ { vµvν− gµν 2mN DµDν︸ ︷︷ ︸ A +4c3i∆µi∆µ︸ ︷︷ ︸ B + ( 4c4 + 1 mN )[ Sµ,Sν ] i∆µi∆ν︸ ︷︷ ︸ C } N (5.16) 40 When expanding this Lagrangian to find the explicit interaction Lagrangian we want to find any terms with one pion field, one external field and two nucleon fields. The details of the derivation of the interaction Lagrangian and the resulting vertex can be found in appendix B.1.3. The resulting interaction Lagrangian is L N NπA 1 = i 4 fπmN { Aa ·∇φbεabc N̄τc N +2Aa i φ bεabc N̄τc∂i N } + 2c3 fπ ∇φa · Aa N̄ N − 1 fπ ( c4 + 1 4mn ) Aa i ∂kφ bεi j kεabc N̄σ jτc N . (5.17) Quantizing this Lagrangian yields the following vertex p p ′ Aa k ,b = i 2 fπmN p+p ′ 2 εabcτc − 2c3 fπ kδab + 1 fπ ( c4 + 1 4mN ) (σ×k)εabcτc . (5.18) The LECs of this vertex are included in the nucleon-nucleon potential and one- body operators and are well-known. There is one type of vertex left that we have not discussed, the 4N A vertex. It comes from the contact terms of the NLO Lagrangian (4.9), L HB 1 ′′ =−4i d1N̄ Sµ∆µN N̄ N︸ ︷︷ ︸ CT1 +2i d2ε abcεµνλδvµ∆νa N̄ Sλt b N N̄ Sδt c N︸ ︷︷ ︸ CT2 . (5.19) From this Lagrangian we derive the interaction Lagrangian with exactly one external axial vector field and no pion fields. For details see appendix B.1.4. L 4N A 1 = d1 Ai ,a N̄σiτa N N̄ N + 1 2 d2ε abcεi j k Ai ,a N̄σ jτb N N̄σkτc N (5.20) This gives rise to the vertex Aa = i d1 ( σ1τ a 1 +σ2τ a 2 ) +i d2 (σ1 ×σ2) (τ1 ×τ2)a . (5.21) The coupling constant of this vertex is new for the two-body operators and also shows up in the three-nucleon potential. 41 Figure 5.1: N2LO terms that are canceled by recoil contributions 5.3.2 Irreducible graphs and Feynman diagrams Now that we have all the contributing vertices we can compute the matrix ele- ments of the current operator by writing down all diagrams and evaluating them. In doing this we will consider the two nucleons to be distinct; the crossed contri- butions must be handled in the anti-symmetrization of the initial- and final state wavefunction. The two nucleons will be labeled 1 and 2. Figure 5.2: Recoil contributions In figure 5.1 there are two graphs which show up at N2LO. To evaluate these graphs we need to use time-ordered perturbation theory since they only form part of a Feynman diagram. However, these graphs are canceled by the recoil contributions in figure 5.2, which can be found in the set of reducible graphs at N2LO. For details on how this cancelation takes place see Pastore et. al. [PSG08]. Instead, the first contribution comes at N3LO. The reason that the order is increased by one is that there is no term generating a N NπA vertex in the LO Lagrangian; as we saw in the previous section this vertex is generated by three terms in the NLO Lagrangian. In total there will be two types of graphs that will give a contribution to the matrix element. The first are the one pion exchange graphs, which can be seen in figure 5.3. Here the different time-orderings can be summed into one Feynman diagram. 42 Aa p1 p2 p ′ 1 p ′ 2 + Aa p1 p2 p ′ 1 p ′ 2 = Aa p1 p2 p ′ 1 p ′ 2 Figure 5.3: Meson exchange graphs and corresponding Feynman diagram The blob represents the N NπA vertex in equation (5.18). Remember that in all diagrams fermion lines point from the bottom of the diagram to the top. We will also include the diagram where the external field is attached to the other nucleon. This amounts to exchanging the labels of the nucleons in the final expression. p1 p2 p ′ 1 p ′ 2 Aa Figure 5.4: The 4N contact diagram contributing to the two-body axial vector current The other diagram is the contact diagram in figure 5.4. This diagram repre- sents the exchange of heavier, excluded particles between the nucleons. In this case only one time-ordering appears and the Feynman diagram is equal to the time-ordered perturbation theory graph. The 4N A-vertex is the vertex in equa- tion (5.21). In this case the exchange of the nucleon labels is included already in the vertex and we need not take any further care to include all contributions. We evaluate these diagrams. After having derived the vertices this is straight- forward, we only need to take care to get the correct signs due to the directions of the momenta along the particle lines. We use general external nucleons and calculate the matrix element depending on the nucleon spin- and isospin states. The details of this calculation can be found in appendix C. The resulting two-body 43 axial vector current of nucleons is,  a 12 = g A 2mN f 2 π σ2 ·k k2 −m2 π (( i p1 +p ′ 1 2 (τ1 ×τ2)a ) + ( 2ĉ3kτa 2 ) + (( ĉ4 + 1 4 ) (τ1 ×τ2)aσ1 ×k )) + (1 ↔ 2) + g A mN f 2 π ( d̂1 ( τa 1σ1 +τa 2σ2 )+ d̂2 (τ1 ×τ2)aσ1 ×σ2 ) . (5.22) This is an operator in the two nucleon momentum space. It is derived through quantum field theory methods but the matrix elements can be carried over into a quantum mechanical operator. Also, since we are working in the formalism of HBChPT the nucleons are already non-relativistic and there is no need to make any further non-relativistic reduction. This exhausts the possibilities for the two-body axial vector current at zero momentum transfer. Together with one-body contributions up to N3LO this gives the axial vector current and the Gamov-Te