Gernot Eichmann. The Analytic Structure of the Quark Propagator in the Covariant Faddeev Equation of the Nucleon

Transcription

1 Gernot Eichmann The Analytic Structure of the Quark Propagator in the Covariant Faddeev Equation of the Nucleon Diplomarbeit zur Erlangung des Magistergrades der Naturwissenschaften verfasst am Institut für Physik an der Karl-Franzens-Universität Graz Betreuer: Univ.-Prof. Dr. R. Alkofer Graz, 006

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3 Contents 1 Introduction 5 The Covariant Quark-Diquark Model 7.1 Three-quark problem and Dyson equation Nucleon bound-state equation Faddeev approximation Diquarks Relative momenta of a three-body system BSE for identical quarks The Building Blocks Quark propagator Dyson-Schwinger equations in QCD Dressed quark propagator Quark-gluon vertex Gluon propagator, ghost propagator and running coupling The analytic structure of the quark propagator Branch-cut parametrizations Diquarks Diquark amplitudes Color structure Flavor structure Dirac structure The diquark BSE in rainbow-ladder truncation Diquark propagator Solving the quark-diquark BSE Decomposition of the BS amplitudes Chebyshev expansion Solution of the BSE The Electromagnetic Current Operator The decomposition of the nucleon-photon-vertex Impulse approximation Quark-photon vertex Diquark-photon vertex Transition between axial-vector and scalar diquark

4 4 CONTENTS 4..4 Charge and flavor coefficients Exchange quark and seagull terms Coupling to the exchange quark Seagulls Charge, flavor and color coefficients Electromagnetic form factors Results General method Nucleon mass and q q correlations Form factors, magnetic moments and charge radii Technicalities Quark mass dependence Examination of the form factor contributions Summary 74 A Conventions in Euclidean Metrics 75 B Chebyshev polynomials in a nutshell 77 C Singularities and Resulting Boundaries 79

5 Chapter 1 Introduction Today it is generally accepted that Quantum Chromodynamics (QCD) is the theory describing the strong interaction, with quarks and gluons being the elementary degrees of freedom. Unfortunately, having the correct theory at hand does not mean to be able to actually solve it. A full solution of QCD is unknown, and several frameworks have emerged in the past decades, approaching the theory from different perspectives and with different scopes: perturbation theory in the high-momentum limit, quark models which incorporate QCD s basic features, but also more general techniques such as lattice QCD and Dyson-Schwinger methods. While perturbation theory in terms of a series expansion in orders of the coupling constant is well applicable over a wide momentum range in Quantum Electrodynamics (QED), in QCD such an expansion can only be performed in the high-energy regime, where the coupling constant is small and asymptotic freedom is approached. In contrast to this perturbative behavior, the low-momentum region is dominated by the hadron spectrum: quarks and gluons are bound together to form hadrons, the elementary degrees of freedom in this domain. A unified description of hadrons within a covariant field theoretical approach and with quarks and gluons as the fundamental degrees of freedom is still missing. Apart from being covariant, such a description should incorporate two major non-perturbative features: confinement and dynamical chiral symmetry breaking. Dynamical chiral symmetry breaking (DχSB) materially impacts on the strong interaction spectrum. In the limit of vanishing current quark masses (the chiral limit ), the Lagrangian of QCD should exhibit chiral symmetry (i.e., a SU(N f ) L SU(N f ) R symmetry, where N f is the number of quark flavors). Since the explicit symmetry breaking due to finite current quark masses is small, a symmetric theory should give a good description; in particular, one would expect meson parity doublets with approximately degenerate masses. These are, however, not observed: the chiral partners ρ, a 1 and ω, f 1 show substantial mass differences of 500 MeV, respectively. One concludes that chiral symmetry is spontaneously broken, i.e. the symmetry of the QCD Lagrangian is not realized in the vacuum which is therefore a non-trivial condensate including sea-quark and gluonic parts. This vacuum condensate is directly responsible for the dynamical generation of large constituent-like dressed-quark masses. On the other hand, DχSB is also accountable for the remarkably small pion mass, since pions can be identified with the would-be Goldstone bosons of the theory.

6 6 Introduction As opposed to the fact that quantum field theories generally deal with infinitely many degrees of freedom, we are thus led to believe that only the degrees of freedom of a fixed number of (constituent) quarks or observable particles like baryons and mesons are the relevant ones in the low-momentum regime. One therefore aims at a tool for treating the covariant bound-state problem in QCD. In this context, Dyson-Schwinger and Bethe-Salpeter approaches have found widespread application, for recent reviews see Refs. [1] and []. In the last 40 years, numerous baryon models have been developed with varying degrees of success as they are usually designed to describe particular aspects of the baryon s properties. Some of the frameworks are non-relativistic [3, 4, 5] and relativistic [6, 7] quark potential models, bag models [8, 9], skyrmion [10, 11] or soliton [1] models. The relativistic three-quark bound-state problem in a Faddeev framework was studied extensively within the Nambu-Jona-Lasinio (NJL) model [13, 14, 15, 16]. In addition, some complementary aspects of these models have been combined, e.g., the chiral bag model [17] and a hybrid model that implements the NJL soliton picture of baryons within a quark-diquark Bethe-Salpeter framework [18]. This thesis is concerned with the further development of a description of particularly the nucleon as a bound state of quarks and gluons in a fully covariant quantum fieldtheoretic framework based on the Dyson-Schwinger equations of QCD. Such a framework has already reached a high level of sophistication for mesons where the quarkantiquark scattering kernel is modeled as a confined, non-perturbative gluon exchange [19, 0, 1, ]. The situation is more complicated for baryons since here one additionally has to deal with genuine three-quark interactions. An efficient description in this context is the Poincaré-covariant Faddeev approach which was extended by the idea that baryons may be viewed as bound states of quarks and diquarks interacting via quark exchange [3, 4, 5]. The physical picture behind this baryon model is very natural: diquarks are allowed to decay into two quarks, one of them recombines with the third quark and forms another diquark. The hope is that, for physics relevant at small and intermediate momentum transfers, most of the complicated structure of the baryon may effectively be described by assuming strong correlations in the quark-quark channel such that the notion of diquarks to some extent parametrizes unknown nonperturbative physics within baryons. However, although our framework can provide a description of the nucleon s quark core, the effects of pseudoscalar meson cloud contributions, which are essential to an understanding of hadron observables, are not included. These effects can nevertheless be estimated by comparison with chiral perturbation theory which provides a systematic expansion of the relevant observables in terms of low-momentum and small quark-mass scales [6, 7]. While Chapt. introduces the necessary steps in order to arrive at a covariant quarkdiquark formalism when starting from the quark 6-point function of QCD, the building blocks thereof will be treated in detail in Chapt. 3. In Chapt. 4 the construction of the nucleon s electromagnetic current operator is shown, and in Chapt. 5 we will present our results for the mass of the nucleon and its electromagnetic properties such as form factors, magnetic moments and charge radii, which will be compared with predictions from chiral perturbation theory and lattice QCD. Throughout this thesis the Euclidean formulation is used, see App. A.

7 Chapter The Covariant Quark-Diquark Model In this chapter, the framework of a description of baryons in terms of quarks and diquarks will be worked out in detail. We will identify baryons with poles in the 3-quark correlation function which will lead to a bound state equation for the baryon s wave function still involving the 3-quark interaction kernel. In order to make this relativistic three-body problem tractable, we will neglect all 3-particle irreducible interactions between the quarks and assume separable correlations in the two-quark channel. While the first assumption will allow us to derive a relativistic Faddeev equation for the quark 6-point function, the latter one will introduce non-pointlike diquark correlations and reduce this Faddeev equation to a coupled set of effective quark-diquark Bethe-Salpeter equations..1 Three-quark problem and Dyson equation Our starting point is the full quark 6-point function, G(x 1, x, x 3 ; y 1, y, y 3 ) = 0 T 3 q(x i ) q(y i ) 0, (.1) where we omitted Dirac, flavor and color indices for the sake of simplicity. In the Fourier transform we extract a δ-function representing conservation of the total momentum: (π) 4 δ 4 ( i (k i p i )) G(k 1,..., p 3 ) := 3 i=1 i=1 d 4 x i d 4 y i e i(k i x i p i y i ) G(x 1,..., y 3 ). (.) Such a separation is always possible for Green functions or bound state matrix elements in momentum space due to translational invariance. Analogously, the dressed propagator of a single quark i is given by (π) 4 δ 4 (k i p i ) S(k i, p i ) := d 4 x i d 4 y i e i(k i x i p i y i ) G(x i, y i ) (.3) such that the disconnected three-quark propagator is a product of three single propagators:

8 8 The Covariant Quark-Diquark Model (π) 4 δ 4 ( i (k i p i )) G 0 (k 1,..., p 3 ) = 3 (π) 4 δ 4 (k i p i ) S(k i, p i ). (.4) The quark 6-point function satisfies Dyson s equation, which is written in momentum space in a symbolic notation: G(k, p) = G 0 (k, p) + G 0 (k, q) K(q, r) G(r, p). (.5) }{{}}{{} q r K is the 3-quark scattering kernel that contains all two- and three-particle irreducible interactions. Here, p (k, q, r respectively) stands for three different momenta p 1, p, p 3 and every integral for a four-dimensional integration d 4 p i. Aside from total momentum conservation, also the total momenta of K(q, r) and G(r, p) are conserved on their own, such that the original six integrals are reduced to four, as indicated in (.5). Moreover, if the two delta functions in G 0 are taken into account, only the two r-integrals (thus 8 integrations) remain. The sums over Dirac-, flavor- and color indices are suppressed in this notation. If we write (.5) even more symbolically, as we will from now on frequently do, we have i=1 G = G 0 + G 0 K G G 1 = G 1 0 K. (.6) This Dyson equation will be the starting point for the derivation of the quark-diquark Bethe-Salpeter equation. G K G Figure.1: The Dyson equation (.6) in pictorial form.. Nucleon bound-state equation With the nucleon in mind, we restrict our treatment of the quark 6-point function to bound states. A bound state of mass M with on-shell four-momentum P and wave function Ψ (plus appropriate discrete quantum numbers - spin, isospin,...) will appear as a pole in the 6-point function: in the vicinity of the pole (which is the only region we are interested in) G has the form of a free propagator, G(k 1,..., p 3 ) Ψ(k 1, k, k 3 ) Ψ(p 1, p, p 3 ) P + M, (.7) where P = p 1 + p + p 3 and the three-particle wave function Ψ is defined to be the transition matrix element between the vacuum and the bound state: (π) 4 δ 4 ( i p i P ) Ψ(p 1, p, p 3 ) := 3 d 4 x i e ip i x i 0 q(x 1 )q(x )q(x 3 ) P. (.8) i=1

9 .3 Faddeev approximation 9 Inserting (.7) into Dyson s equation and comparing residues yields a homogenous bound state equation at the pole, G 0 K Ψ = Ψ (G 1 0 K) Ψ = 0 G 1 Ψ = 0, (.9) which includes two momentum integrals, i.e. an 8-dimensional integration. This equation is far too complicated for a direct solution since an expression for all - and 3-particle irreducible graphs in K is not known. It is thus necessary to resort to some approximation. K Figure.: The nucleon bound state equation..3 Faddeev approximation K ~ K 1-1 ~ K -1 ~ K 3-1 Figure.3: The Faddeev approximation. lines. Dashed Lines represent amputated quark The above problem is greatly simplified if three-particle interactions are removed from the interaction kernel K so that K becomes a sum of two-particle kernels with one spectator quark (index i) each. For distinguishable quarks this reads: K = 3 K i, i=1 K i = K i S 1 i. (.10) This is the so-called Faddeev approximation. Inserting (.10) into the bound state equation (.9) yields (G 1 0 K i )Ψ = (K j + K k )Ψ. (.11) The Dyson equation for the 6-point function G i = G i S i, where only two quarks interact with each other, is given by (see Fig..4) G i = G 0 + G 0 K i G i G 1 i which, in turn inserted into (.11), leads to = G 1 0 K i (.1) Ψ = G i (K j + K k )Ψ. (.13)

10 10 The Covariant Quark-Diquark Model ~ G i ~ K i ~ G i ~ T i i Figure.4: DSE for G i. We now define the Faddeev components of the baryon s wave function Ψ by Ψ i := G 0 K i Ψ, 3 i Ψ i = Ψ, and secondly the two-quark scattering matrix T i = T i Si 1 by amputating all quark legs from the connected part of G i such that (second part of Fig..4) G i = G 0 + G 0 T i G 0. (.14) Then we have Ψ = G i (K j + K k ) Ψ (.14) = (1 + G 0 T i ) G 0 (K K i ) Ψ }{{} Ψ Ψ i, (.15) from which we obtain the Faddeev equations Ψ i = G 0 T i (Ψ j + Ψ k ) = S j S k Ti (Ψ j + Ψ k ) (.16) which relate the Faddeev components Ψ i among each other using the full two-quark correlation function T i instead of the three-quark kernel K. Due to total momentum conservation in T i we are left with one integral: the Faddeev equations constitute a set of coupled four-dimensional integral equations, which is a considerable simplification of the original 8-dimensional integral-equation problem defined in (.9). k j ~ ~ T i T i Ψ i Ψ j Ψ k i Figure.5: The Faddeev equations. In terms of the Faddeev amplitudes Φ i, defined as amputated wave functions by Ψ i = G 0 Φ i, the Faddeev equations read Φ i = T i G 0 (Φ j + Φ k ) = T i S j S k (Φ j + Φ k ). (.17).4 Diquarks In order to solve the Faddeev equations (.16), we would first have to determine the two-quark correlation function T i from its own Dyson equation (obtained by comparing (.1) with (.14)), T 1 i = K 1 i S j S k, (.18)

11 .4 Diquarks 11 which still includes the unknown two-quark kernel Ki. Therefore, it is desirable to employ a further simplification. This is done by approximating T i as a sum over separable correlations: T i (k j, k k, p j, p k ) = a χ (a) i (k j, k k )D (a) i (k j + k k ) χ (a) i (p j, p k ). (.19) These separable correlations are called diquarks. They consist of the diquark propagator D (a) i and the diquark amplitudes χ (a) i, with conjugates χ (a) i, and will be treated with greater detail in Sec. 3.. The index (a) refers to different representations in Dirac and flavor space, for example, scalar and axial-vector diquark structures, which we will later discuss to be the presumably most important correlations: T αβ,γδ (k j, k k, p j, p k ) = χ 5 αβ (k j, k k ) D(k j + k k ) χ 5 γδ (p j, p k ) + + χ µ αβ (k j, k k ) D µν (k j + k k ) χ ν γδ (p j, p k ). (.0) β ~ χ i T i S a (a) D i α (a) - χ (a) i δ γ Figure.6: The separable matrix T i We choose an analogous ansatz for the Faddeev components Ψ i (Fig..7): Ψ i (p i, p j, p k ) = a = a G 0 χ (a) i (p j, p k ) D (a) i (p j + p k ) φ (a) i (p i, p j + p k ) S j S k χ (a) i (p j, p k ) ψ (a) i (p i, p j + p k ), (.1) where we introduced the Bethe-Salpeter (quark-diquark) wave functions ψ (a) i and the Bethe-Salpeter amplitudes φ (a) i (the latter being amputated wave functions, or: effective vertex functions of the baryon with quark and diquark). They only depend on the relative momentum between spectator quark (with momentum p i ) and diquark (momentum p j + p k ) thus we can write just as well φ (a) i (p i, P ), with the total momentum P = p i + p j + p k. Inserting this into the Faddeev equations (.16) yields coupled integral equations for the BS amplitudes of the following structure: (φ i ) a α = { } χ b jt S T k χ a i T S j φ c j α Dbc j + { } χ b k ST j χ a i S k φ c k α Dbc k (.) These are the Bethe-Salpeter equations for distinguishable quarks. The integration is carried out over one relative momentum between quark and diquark each, and the Lorentz indices a, b, c are already written in a form which is convenient for later use (cf. (.0)).

12 1 The Covariant Quark-Diquark Model k j Ψ i χ i f i i Figure.7: The ansatz for Ψ i.5 Relative momenta of a three-body system In order to obtain a more concise shape for Eq. (.) it is advantageous to rewrite the momenta {p i, p j, p k } therein in terms of relative momenta. For a three-body system these are given by the Jacobi momenta P = p i + p j + p k (.3) p rel i := (1 η)p i η(p j + p k ) = p i ηp (.4) q rel i := σp j (1 σ)p k (.5) with even permutations of (i, j, k). Applied to our problem, P is the total momentum of the quark-diquark system, p rel i is the relative momentum between quark i and the diquark consisting of quarks j and k, and q rel i is the relative momentum within the diquark. We have introduced two momentum partitioning parameters η, σ [0, 1] here which distribute relative momenta between quark and diquark and within the diquark, respectively; this is possible since in our covariant framework there is no unique definition of the relative momentum. Physical observables, like the mass of the bound states, form factors etc., should of course not depend on these parameters, so we could assign fixed values to them. Since σ is of no practical importance, we will set σ = 1/ in all our calculations, in particular because this is of numerical advantage [5]. Nevertheless we keep η as a variable due to the technical benefit of broadening several quantities calculation ranges which may be restricted by the occurrence of singularities in propagators and amplitudes (see also App. C). In terms of these relative momenta, we can write the Bethe-Salpeter equation (.) as follows: φ a i (p rel i, P ) = χ b T j (q rel j, p D j ) Sk T (p k) χ a i T (q rel i, p D i ) S j (p j ) Dj bc (p D j ) φ c j(p rel j, P ) + }{{}}{{} p rel j KBS(p ba rel i, p rel j, P ) (G q dq 0 ) bc j (p rel j, P ) + { } j k, χ χ T, (.6) where p i = p rel i + ηp, p D i := p j + p k = (1 η)p p rel i (the expressions for j and k are analogous via cyclic permutation - note that all appearances of j and k are interchanged in the second term). (G q dq 0 ) j denotes the disconnected quark(-j)-diquark propagator. The quark-diquark interaction kernel K BS describes a quark exchange between quark and diquark: a quark with momentum p k (in the second term: p j ) defects from diquark to quark, thus swapping their roles. To arrive at an equation for a nucleon, we have to

13 .6 BSE for identical quarks 13 project this configuration onto the nucleon quantum numbers. As it will turn out, the quark exchange will generate the attractive interaction that binds quarks and diquarks to form a nucleon. p j +p k q i rel p j +p k q i rel p i rel -χ i p i rel -χ i p i p k p j p i p j p k χ j rel p j χ k rel p k q j rel q k rel p i +p k p i +p j Figure.8: Momentum dependence of the diquark amplitudes.6 BSE for identical quarks For the treatment of identical quarks antisymmetrization is required. In this case we can simply drop the single particle indices i on the quark propagators S i, diquark propagators D i and diquark amplitudes χ i since their functional form does not depend on them. We rename the momenta as follows ( k refers to incoming, p to outgoing, and the numbers 1, in the brackets to the corresponding diagrams): p rel j p rel i p (1), p rel k () k p k (1), p j () q = (1 η)p p k p i p q = p + ηp p j (1), p k () k q = k + ηp p D i p d = (1 η)p p p D j (1), p D k () k d = (1 η)p k q rel i (1) p r = k + (1 σ)p + (ησ (1 σ)(1 η)) P () p r σ (1 σ) q rel i q rel k () k r = p + (1 σ)k + (ησ (1 σ)(1 η)) P (1) k r σ (1 σ) q rel j

14 14 The Covariant Quark-Diquark Model The BSE then takes the final form: φ a (p, P ) = b,c d 4 k (π) 4 χb (k r, k d ) S T (q) χ a T (p r, p d ) }{{} K ba BS(p, k, P ) S(k q ) D bc (k d ) }{{} (G q dq 0 ) bc (k, P ) φ c (k, P ) (.7) p f a P p d k r q χ- a χ b p r k q k d k c f P p q K BS Figure.9: Quark-diquark Bethe-Salpeter equation The diquark amplitudes have to satisfy χ(p 1, p )! = χ T ( p 1, p ) σ (1 σ), (.8) where the additional ( ) comes from the antisymmetrization. Due to the color and flavor structure, the kernel picks up an additional factor 1 ( ) 1 3, (.9) 3 1 where the first row (column) represents the scalar part of the kernel and the second row (column) the axial-vector part. While 1 comes from the color trace, the rest is obtained by taking the trace over the flavor matrices. This will be explained in detail in Secs. 3.. and 3..3.

15 Chapter 3 The Building Blocks In this chapter we will specify the building blocks that enter the quark-diquark Bethe- Salpeter equation such as quark propagator, diquark propagator and diquark amplitudes. We will collect them from several previous calculations such that they become as realistic and consistent with each other as available at the moment. The simplest form to start with is using free quark and diquark propagators and simple parametrizations for the diquark amplitudes. This has been studied in [8]. Another possibility is to employ parametrizations with entire functions (e.g., in [9]) which have the advantage of mimicking confinement in the sense that they are pole-free at the timelike axis. These, however, suffer from severe shortcomings such as essential singularities at infinite timelike momenta but are nevertheless useable when dealing only with spacelike and small timelike momenta, which is the region sampled by the quarkdiquark BSE. Recently, a combined solution of quark, gluon and ghost Dyson-Schwinger equations beyond rainbow truncation has been obtained [30, 31] which also agrees very well with lattice data [3, 33]. It is the goal of this work to implement these results in the application to three-quark problems and test their effects on experimentally accessible observables. In Sec. 3.1 we will concentrate on the quark Dyson-Schwinger equation and elucidate the steps towards its solution. In Sec. 3. we will have a closer look at the diquark approximation carried out earlier and show how the ingredients thereof can be obtained by use of the ansatz (.19) only. However, there are still inconsistencies: while the quark propagator was obtained by leaving rainbow truncation behind, the diquark BSE can at the moment only be solved in rainbow-ladder truncation, an approach that has, on the other hand, been most successful in describing light pseudoscalar mesons and their electromagnetic properties. When applied to diquarks, there are two caveats in our approach: firstly, the inconsistency of working effectively with two different quark propagators; secondly, the diquark mass poles in the -quark scattering matrix obtained in this trunctation are, in contrast to the respective meson masses, rather sensitive to the effective coupling employed and moreover disappear when involving higher-order corrections (which, however, agrees with the observation that diquarks should be confined).

16 16 The Building Blocks 3.1 Quark propagator Dyson-Schwinger equations in QCD Generally, the fully dressed quark propagator is obtained from its Dyson-Schwinger equation (also called the QCD gap equation ) [1]: where S 1 (p, s) = Z (s, Λ ) S0 1 (p, Λ ) g (s) Z 1F (s, Λ ) 4 3 Λ q d 4 q (π) 4 γ µs(q, s) i Γ ν (p, q, s) D µν (k, s), (3.1) s = µ is the renormalization point and Λ the regularization parameter in the momentum integral, which is removed at the end of all calculations by taking the limit Λ, Z S0 1 with S0 1 (p, Λ ) = i /p m 0 (Λ ) is the inverse tree level propagator (with renormalization constant Z ) i Z 1F γ µ is the tree level quark-gluon vertex, in contrast to the fully dressed quark-gluon vertex Γ ν, D µν is the dressed gluon propagator with gluon momentum k = q p, and, for an arbitrary number N C of colors, the color factor 4/3 stemming from the color trace of the loop would read C F = N C 1 N C. All the renormalization constants appear in the renormalized QCD Lagrangian, cf. Eq. (81) in [1]. In pictorial form the quark DSE is written as: -1 = -1 Figure 3.1: Quark DSE involving quark propagator, dressed gluon propagator and dressed quark-gluon vertex. Dashed lines represent amputated quark legs. The quark DSE involves both dressed gluon propagator and dressed quark-gluon vertex. A priori, both are unknown functions and have to be determined from their own Dyson- Schwinger equations which in turn include higher Green functions (satisfying their own DSEs), such that we end up with an infinite system of coupled integral equations. For example, the pictorial form of the DSE for the gluon propagator given in Fig. 3. involves 3- and 4-gluon vertices, the quark-gluon vertex, but also the ghost propagator and the ghost-gluon vertex. A self-consistent solution of the whole array of DSEs is impossible; only within a suitable truncation scheme and/or via the implementation of ansätze for certain Green functions one can obtain solutions for a subset of the tower of DSEs.

17 3.1 Quark propagator = Figure 3.: Gluon DSE: the filled blobs mark three- and four-gluon vertices, ghost-gluon vertex and quark-gluon vertex Dressed quark propagator The quark DSE s solution is the dressed quark propagator which takes the form: S(p) = i /p σ v (p ) σ s (p 1 ) = i /p A(p ) + B(p ) = A 1 (p ) i /p + M(p ) = A 1 (p ) i/p M(p ) p + M (p ) (3.) The dressing functions σ v, σ s and A, B (M is the quark mass function and A 1 Z f the quark wave function renormalization ) are related to each other via σ v = σ s = A p A + B = 1 1 A p + M (3.3) B p A + B = 1 M A p + M (3.4) M = B A = σ s σ v (3.5) Except for the renormalization-point independent quark mass function M(p ), they all (including S itself) additionally depend on the renormalization point s = µ. Asymptotic freedom implies that for large momenta perturbation theory should be recovered, i.e. the propagator reduces to the tree level form. On the other hand, the DSE solution accounts for dynamical chiral symmetry breaking in the infrared domain (p GeV ) in terms of a dynamically generated quark mass. This is a result of the gluon dressing and is represented by M(p ): since the quark propagator contains a singularity at p 0 + M(p 0 ) = 0, M(p 0 ) can be viewed as a constituent-quark mass which is usually much higher than the typical current-quark masses (cf. Fig. 3.4). The renormalization constant Z relates the renormalized dressing functions A, B with the bare ones A 0, B 0 : A(p, s) = Z (s, Λ ) A 0 (p, Λ ) (3.6) B(p, s) = Z (s, Λ ) B 0 (p, Λ ) (3.7) It is advantageous that in Landau gauge the loop corrections to the vector self-energy are finite: 0 Z (s, Λ ) 1. On the other hand, the one-loop perturbative behaviour of Z in Landau gauge is Z (s, Λ ) = 1 Λ. Since for large momenta p perturbation theory entails A(p, s) Z (s, Λ ), (3.8)

18 18 The Building Blocks Figure 3.3: A finite renormalization from a perturbative point s to a nonperturbative point t for the vector self-energy A (taken from [30]). we have A(, ) = 1. This is schematically depicted in Fig The quark mass function is independent of the renormalization point and, for perturbative momenta p = s, approaches the renormalized current-quark mass m s, M(p ) = B(p, s) A(p, s) p s m s, (3.9) which is connected to the bare current quark mass via the mass renormalization constant Z m, m s = m 0(Λ ) Z m (s, Λ ). (3.10) With the renormalization condition A(s, s) = 1 s one thus gets the tree-level form for the perturbative quark propagator (if s is in the perturbative range): S 1 (p, s) = i/p A(p, s) B(p, s) S 1 (s, s) = i/p m s, Quark-gluon vertex S(s, s) = i/p m s p + m. (3.11) s In order to solve the quark DSE (3.1) one has to specify the dressed gluon propagator and the quark-gluon vertex. The Dirac structure of the latter involves in general twelve Lorentz invariant functions V i (p, q) [34]: i Γ µ (p, q) = γ µ V 1 (p, q) + (p + q) µ V (p, q) + (/p + q/)(p + q) µ V 3 (p, q) +... (3.1) The fully dressed vertex which also accounts for the color structure is given by i Γ i µ = λi i Γ µ (3.13) where λ i, i = are the Gell-Mann matrices representing the gluon color octet. The quark-gluon vertex is, in principle, determined by its own DSE whose solution is, however, rather difficult and can be circumvented by making an ansatz for Γ µ.

19 3.1 Quark propagator 19 Before discussing the quark-gluon vertex in QCD, let s recall the quite analogous situation in QED: there the fermion-photon vertex Γ µ (p, q) has to satisfy a Ward-Takahashi identity (WTI) (p q) µ Γ µ (p, q) = S 1 (p) S 1 (q) (3.14) in order to ensure QED s gauge invariance. This links the longitudinal part (with respect to the photon momentum k = p q) of Γ µ to the structure of the fermion propagator S: for a tree-level propagator (3.11) we have to choose a bare vertex ( rainbow approximation ) i Γ µ = γ µ (3.15) for (3.14) to be satisfied, whereas the dressed propagator (3.) requires a more sophisticated structure of Γ µ. The simplest possible form that respects the WTI is the (purely longitudinal) Ball-Chiu vertex [35]: i Γ BC µ (p, q) = A(p, s) + A(q, s) γ µ i B(p, s) B(q, s) p q (p + q) µ A(p, s) A(q, s) p q (/p + q/)(p + q) µ. (3.16) One easily confirms that (3.14) is satisfied. One can also add a suitable transverse part since this does not affect the WTI; a popular form is the Curtis-Pennington (CP) vertex [36]: i Γ CP µ (p, q) = i Γ BC µ (p, q) + A(p ) A(q ) p + q (p q ) + (M (p ) + M (q )) ((p q )γ ν (/p q/)(p + q) µ ). (3.17) In QCD, the WTIs are replaced by the non-abelian Slavnov-Taylor identities (STIs) which now also involve the ghost dressing function G(k ) and the ghost-quark scattering kernel H(q, p), with gluon momentum k = p q: G 1 (k )(p q)γ µ (p, q) = S 1 (p) H(q, p) H(q, p) S 1 (q). (3.18) Because of the factor G 1 in the STI, G must also appear in the quark-gluon vertex Γ µ. Since the non-perturbative structure of H(q, p) is unknown, Γ µ cannot be determined exactly; one thus again has to use an ansatz. This is carried out by using an Abelian part from above, multiplied with a non-abelian factor containing the ghost dependence, Γ µ (p, q, s) = Γ Abel µ (p, q, s) G (k, s) Z 3 (s, Λ ), (3.19) }{{} ΓqAbel (k,s) where Z 3 is the renormalization constant for the ghost propagator. In Landau gauge, the Slavnov-Taylor identity for the quark-gluon vertex-renormalization factor Z 1F implies Z 1F = Z Z 3. (3.0) The non-abelian part in (3.19) has been chosen such that the quark mass function M(p ) is renormalization-point independent (shown in [30]) and that the anomalous dimension γ m of the mass function known from perturbation theory is recovered in the ultraviolet (cf. Sec ). If one employs a bare vertex (3.15) in the Abelian part, an extra factor Z has to be attached to satisfy the first condition.

20 0 The Building Blocks Gluon propagator, ghost propagator and running coupling Up to now the gluon propagator has not yet been determined. Moreover, also knowledge of the ghost propagator is necessary, since the ghost dressing appears in the quark-gluon vertex (3.19). The dressed gluon propagator can be written (in Landau gauge) as a transversal projector times a dressing function: D µν (k, s) = D(k, s) k ( δ µν k µk ν k ) = D(k, s) D free µν (k ) (3.1) with the gluon dressing function D(k, s) being the solution of the corresponding gluon DSE (Fig. 3.). The ghost propagator is that of a massless scalar particle but for an opposite sign which reflects the ghost anticommutivity: D G (k, s) = G(k, s) k. (3.) The whole input from the Yang-Mills sector (i.e., both ghost and gluon dressing functions) can be absorbed into the expression for the renormalization-point independent running coupling in Landau gauge: α(k ) := g (s) 4π D(k, s) G (k, s). (3.3) Inserting (3.19), (3.0), (3.1) and (3.3) into the quark DSE (3.1) yields: S 1 (p, s) = Z (s, Λ ) S0 1 (p, Λ ) Z (s, Λ ) 3π 3 Λ q d 4 q α(k ) D free µν (k) γ µ S(q, s) i Γ Abel ν (p, q, s). (3.4) One way to solve this equation is to employ an ansatz for α(k ). A parametrization which has commonly been used in the context of meson Bethe-Salpeter studies, together with a rainbow approximation for the quark-gluon vertex, was given in [37], α(k ) = π d ω 6 k4 e k ω + π γ m (1 e k /m t ) ( ), (3.5) 1 ln e 1 + (1 + k /Λ QCD ) where γ m = 1/(11N C N f ) is the one-loop anomalous dimension of the quark mass function (here for N f = 4), and with the parameters ω = 0.3 GeV, d = GeV, m t = 1.0 GeV and Λ QCD = 0.34 GeV. For large momenta, the above fit for α(k ) reproduces the form known from perturbation theory (i.e., in one-loop order): α(k ) π γ ( m ). (3.6) ln k /Λ QCD Recently more fundamental Dyson-Schwinger results for the gluon propagator have been obtained [31, 30]. The truncation used therein neglects four-gluon interactions and applies ansätze for the three-gluon and ghost-gluon vertices which ensure that

21 3.1 Quark propagator 1 the running coupling α(k ) is renormalization-point independent and the anomalous dimensions of the ghost and gluon propagators are reproduced at the one-loop level for large momenta. The corresponding results for the gluon dressing function show that D(k ) vanishes for k 0 as 1 ( ) k D(k κ 0, s) (3.7) s with 0.5 κ 0.7, depending on the details of the truncation. For intermediate momenta ( 1 GeV) a maximum shows up, followed by relatively flat momentum dependence above this scale. This is in excellent agreement with lattice data [3]. On the contrary, the ghost propagator is infrared singular, with ( ) k G(k κ 0, s). (3.8) s The resulting running coupling can be fitted by α(k ) = 1 { ( x α 0 + π γ m x + 1 ln x x )}, x = k x 1 Λ QCD (3.9) where Λ QCD = 0.71 GeV is obtained by requiring α(m Z ) = α[( GeV) ] = This fit not only reproduces the correct perturbative form (3.6), but also the infrared behavior inferred from the DSE: α(0) = 8.915/N C. (However, the DSE results for the quark propagator were not obtained by using (3.9) but rather by taking α(k ) directly from the ghost and gluon solutions.) The analytic structure of the quark propagator The results for the quark mass function M(p ) and the inverse vector self-energy 1/A(p ) obtained from (3.4) by using a Curtis-Pennington quark-gluon vertex (3.17) are shown in Fig The solution for the mass function exhibits a substantial increase in the infrared domain which is a typical feature of non-perturbative QCD and expresses the quark mass generation by the gluon coupling. This mass enhancement can be interpreted as the quark s constituent mass. The results show two quantitatively different mass scales: the value M(0) GeV (for current masses m s = MeV), but also the value at timelike psing, defined by p sing + M (p sing ) = 0, where (if a solution exists and A(p sing ) is finite, cf. (3.3,3.4)) the quark propagator has a singularity. The latter is usually higher, M(p sing ) GeV. The results are in very good agreement with lattice data [33] and depend clearly on the structure of the quark-gluon vertex [31]: carrying out the same calculation with a bare vertex gives not enough mass generation, i.e. the mass function will be of a sizeable amount smaller. Since the DSE results for the quark propagator obtained in [31] are so far only available for (a finite number of) real spacelike momenta (p > 0), an analytic continuation for complex momenta is not unique and allows for different analytic structures of the quark 1 In fact, the behavior for k 0 can be inferred analytically from the respective untruncated gluon and ghost DSEs.

22 The Building Blocks Figure 3.4: Quark dressing functions M(p ) and A 1 (p ) as DSE solutions (with CP vertex) for different current quark masses [30]. dressing functions. However, due to asymptotic freedom, the dressed quark propagator must reduce to a free fermion propagator at large momenta, i.e. σ v,s p 0 (3.30) in all directions of the complex p plane [38]. Since a function which is analytic over the whole complex plane (even at infinity) is constant, and σ s, σ v are not constant (otherwise they would be zero everywhere), they cannot be analytic in the whole complex plane, i.e. they must contain singularities. The use of entire functions for σ v, σ s with an essential singularity at infinity (which have frequently been applied, for example in [9] or [39]) is thus clearly ruled out in general, but nevertheless justified in the framework of Bethe-Salpeter equations, since the domain of the quark propagator which is sampled in such calculations lies inside a parabola with a negative real (timelike) point p as the apex (cf. Appendix C), where discrepancies in the analytic structure in the negative half-plane are not so important. Two simple examples for the analytic structure of a propagator in quantum field theory are a real pole and a pair of complex conjugate poles. The propagator of a real, massive, scalar particle has a single pole on the real timelike (p < 0) momentum axis (a masslike pole), and its propagator is given by σ(p ) = 1 p + m. (3.31) For a pair of complex conjugate poles the masses become complex, m = a ± ib; such a propagator could describe a short lived excitation: σ(p ) = 1 p + (a + ib) + 1 p + (a ib). (3.3) A practical tool for examining the analytic structure of propagators is the Schwinger function, which for a generic propagator function (x y) is given by (t) := d 3 z (z) = 1 π dp 4 e i p 4 t (p 4) = 1 π 0 dp 4 cos(p 4 t) (p 4) (3.33)

23 3.1 Quark propagator 3 with t = z 4 (thus being the propagator in momentum space, with p = 0). The condition of positivity of the Schwinger function, (t) 0, is the special case of the Osterwalder- Schrader axiom of reflection positivity for a two-point correlation function (x y) in a Euclidean quantum field theory [31]: the existence of asymptotic states ( free particles) requires (t) to be positive definite, which corresponds to the existence of timelike poles in the propagator. Thus, a sufficient criterion for confinement is the violation of positivity: if negative-norm contributions in the Schwinger function for a given propagator type are encountered for certain t ranges, then the corresponding particle is confined. The Schwinger functions for the two simple propagator functions given above are (t) e m t, (t) e a t cos(b t + δ). We see that complex conjugate poles lead to an oscillatory behavior in the Schwinger function such that the corresponding particle is confined. On the other hand, the masslike pole propagator shows no zero crossings, which, however, does not necessarily mean that the associated particle is free: only the reverse is true. Thus, propagators with real poles in momentum space can still describe confined particles. This will become important in the case of the quark propagator, where (p ) is one of the scalar propagator functions σ v (p ) and σ s (p ). The quark DSE solution in rainbow approximation produces complex conjugate poles in σ v, σ s on the timelike half of the complex momentum plane with the first pair being dominant (and, as far as one can guess by investigating this domain, probably entire functions for A, B). This can be parametrized by σ v (p ) = σ s (p ) = N i=1 { λi p + m i N { λi m i i=1 p + m i λ } i + p + m i } + λ i m i p + m i (3.34) (3.35) In [1], where the rainbow approximation is carried out together with the ansatz (3.5) for the running coupling α(k ), the dominant pole pair appears at m 1 = ± 0.303i GeV. For comparison, if instead one employs the DSE fit (3.9), one gets m 1 = 0.09 ± 0.101i GeV [31]. As before, complex conjugate poles lead to zero crossings in the Schwinger function and can thus be viewed as an indicator for confinement. However, such negative norm contributions are not observed in the Schwinger function of the quark DSE solution employing a Curtis-Pennington vertex, which we better trust since it reproduces the lattice data much better than rainbow approximation. Thus, no statement about confinement is possible from the structure of the quark propagator. What can we infer for the analytic structure itself? Since masslike poles show no negative norm contributions, complex conjugate poles are likely to be artifacts of the rainbow truncation. Actually the fits which reproduce both the DSE solutions with It is noteworthy that the Schwinger function of the gluon propagator (obtained from the DSE solution) shows in fact zero crossings which is in accordance with gluons being confined.

24 4 The Building Blocks CP vertex and the corresponding Schwinger function best have turned out to entail a dominant real pole at m sing 0.50 GeV, a pair of complex conjugate poles very close to the real momentum axis, or branch point singularities. It has turned out that the crucial term in the quark-gluon vertex which is responsible for the qualitatively different behavior of the Schwinger function is the scalar coupling B µ, i.e. the second term in (3.16): actually, a mere reduction by about 0% of this term is enough to generate positivity violations. Furthermore, when inserting a scalar coupling, also the complex conjugate poles disappear in favor of a singularity on the real timelike momentum axis. On the other hand, the question of positivity violation does not depend on the details of the running coupling 3 α(k ). Im p Im p -m Re p Re p Figure 3.5: Possible singularity structures of the dressing functions σ v,s. Left panel: parametrization with singularities and branch cut as indicated by the DSE solution with CP vertex (3.40, 3.41), right panel: complex conjugate poles obtained with the rainbow approximation Branch-cut parametrizations Actually, branch cut singularities in the quark DSE arise in a quite natural way. To sketch this, let s consider the quark DSE (3.4): by taking the Dirac trace once with and once without multiplying the equation with /p we can project out the mass function M(p ) and the vector self-energy A(p ). For the simpler case of a bare vertex, i Γ Abel ν = Z γ ν, we arrive at M(p ) A(p ) = Z m 0 + Z π 3 A(p ) = Z + 1 p Z 3π 3 d 4 q α(k ) k σ s (q ) (3.36) ( ) d 4 q α(k ) k σ v (q (q k)(p k) ) p q + k (3.37) with k = p q (the renormalization point and cutoff dependence has been omitted). In general, the integral on the right hand side can be written as I(p ) = d 4 q α(k ) k σ(q ) K(p, q, p q). (3.38) 3 It is nevertheless possible to tune the parameter ω in (3.5) such that the complex conjugate poles come very close towards the real axis; this can be achieved by choosing ω 0.5 GeV [40]. We will make use of this property in the course of the diquark BSE solution.

25 3.1 Quark propagator 5 Im q Im q p Re q Re q p Figure 3.6: Possible integrations paths in in the complex q plane for the radial integral in I(p ) that do not cross the circular branch cut (dotted lines), see Appendix B of [31]. Assuming that σ(q ) has a singularity at q = m sing and α(0) 0 (with the kernel K having no further singularities), the only singularities in the integrand are located at k = (p q) = 0 and at q = m sing. With the choice p = (0, 0, 0, p ), p C and p q = p q cos ψ, the integral reads: I(p ) = 4π 0 π 0 = Λ 0 Λ dq q σ(q ) dψ sin ψ α(p + q p q cos ψ) p + q p q cos ψ K(p, q, p q cos ψ) dq σ(q ) K(p, q ). (3.39) Stemming from the angular integral, the kernel K now has a circular branch cut at q = p e iφ, 0 < φ < π (the point q = p itself is not included). In order not to cross the branch cut, the integration path of the radial integral from 0 to Λ has to be analytically continued into the complex q plane such that it passes q = p. If there is an additional singularity at q = m sing, one can find two integration paths that cannot be deformed into each other without picking up residual contributions from m sing (Fig. 3.6). Therefore, I(p ) becomes a multi-valued function with a branch point singularity at p = m sing. The natural choice for the branch cut is along the negative real axis between (, m sing ).

26 6 The Building Blocks m s [GeV] m sing [GeV] C DCSB C CQM C 4 C Table 3.1: Parameters for the branch cut fits (3.40, 3.41) for 3 different current quark masses m s [31] (not to be confused with the strange quark mass!). In my calculations, I have used parametrizations for σ v and σ s which fit the numerical Dyson-Schwinger solutions with a CP vertex and α(k ) from eq.(3.9) and were given in Ref. [31]. They have a singularity at p = m from which a branch cut extends to p =, and a second singularity at p = m Λ QCD. Away from the real axis, there is no further singularity.: with σ v (p ) = σ s (p ) = 1 ( p + m 1 α(p + m ) π ) + C 5 B chiral (p + m ) 1 ( p + m C CQM α(p + m ) γ m + B chiral (p + m ) C DCSB ) (3.40) (3.41) B chiral (p ) = p + Λ α(p ) 1 γm + QCD (p + Λ. (3.4) QCD ) The two parameters C DCSB and C CQM are obtained by comparison with the perturbative behaviour (p s) of M(p ) in the chiral limit, M(p ) C DCSB p α(p ) 1 γ m! = π γ m N C C 4 qq p ( 1 ln p /Λ QCD ) 1 γm, (3.43) where qq is the renormalization-point independent vacuum quark condensate with the phenomenological value qq = [0.75(75) GeV] 3, and for non-vanishing current quark masses, M(p ) C CQM α(p ) γ m! = m s, (3.44) with the renormalized current quark mass m s. The remaining parameters C 4 and C 5 were fitted to the numerical DSE solution. Finally, Λ QCD was taken to be Λ QCD = 0.70 GeV to improve the fit, and the renormalization point was chosen at s = (10 GeV). The values of these parameters are given in table (3.1) for three selected current quark masses m s.

27 3. Diquarks 7 3. Diquarks In Sec. (3.), diquarks or diquark correlations have been introduced in the course of the separable approximation for the quark T matrix as quark-quark bound states. Such states are colored and are therefore expected to be confined. However, nothing prevents us from assuming that diquark correlations can play a role inside baryons: two quarks correlated in a color antitriplet configuration can couple to a quark to form a color-singlet baryon. The existence of such strong quark-quark correlations within the baryon was also suggested by lattice calculations [41]. The basic procedure is similar to that of Sec..: we start from the Dyson equation for the quark 4-point function G (cf. (.1), now without particle indices and without the tilde to denote pure two-quark quantities), G = G 0 + G 0 K G G 1 = G 1 0 K, (3.45) where G 0 is the disconnected -quark propagator and K the -quark interaction kernel which contains all irreducible two-particle interactions. The corresponding 4-point correlation function T, defined by satisfies the Dyson equation G = G 0 + G 0 T G 0, (3.46) T = K + K G 0 T T 1 = K 1 G 0. (3.47). Assuming that T (resp. G) contains a pole at a certain particle s mass m (that particle would be the diquark), we employ the ansatz (.0) for the 4-point function correlation function: T αβγδ (p, q, P ) = λ χ λ αβ (p, P ) Dλ (P ) χ λ δγ (q, P ), (3.48) where χ are the diquark amplitudes (i.e., diquark wave functions with truncated quark legs 4 ) and χ their charge conjugates, P is the total diquark momentum, the greek subscripts are quark indices and λ is the Lorentz index of the diquark. The pole approximation entails that the diquark propagator D(P ) has to satisfy D(P ) P m 1 P + m (3.49) on the diquark s mass shell in order to allow an interpretation for χ as on-shell diquark amplitude. Of course we cannot simply stick to studying the on-shell behavior since our 4-point function (3.48) ultimately enters the baryon s 6-point function (.1). Furthermore, since diquarks do not exist as asymptotic states, the assumption of a real mass pole is no necessary condition (as opposed to (.7)) but only an ansatz, however not in contradiction with confinement (see the discussion in Sec ). There may well be complex poles instead of masslike ones, or an even more difficult analytic structure. 4 The notion diquark amplitudes is actually only justified on the mass shell P = m strictly speaking, we should call them quark-quark-diquark vertices.

28 8 The Building Blocks How much information on diquark propagator and amplitudes can we infer already from the above equations? Inserting (3.48) into (3.47) yields the (symbolically written) equation D 1 = χ ( K 1 G 0 ) χ (3.50) which reduces on-shell (D 1 0) to a homogenous Bethe-Salpeter equation for the diquark amplitudes, χ = K G 0 χ. (3.51) The full index structure of (3.51) and (3.50) is written down in Secs and 3..6, respectively. As it will turn out (and was already announced in the introduction of this chapter), this diquark BSE yields bound states at real diquark masses in rainbow-ladder truncation, but not so if higher-order terms in the kernel K are taken into account [34]. The on-shell diquark amplitudes are completely determined by (3.51) and an additional normalization condition obtained by taking the derivative of (3.50) on the mass shell. The full diquark propagator from (3.50) includes the general form of the diquark amplitudes for off-shell momenta P which is unknown. In section 3..4 we will calculate D 1 by the simplifying assumption that off- and on-shell amplitudes can be identified, i.e. the dependence on P is weak Diquark amplitudes Before dealing with the diquark BSE, we will explore the general structure of the diquark amplitudes which have the form χ λ αβ (p, P )k ab D AB = g λ V (p, P ) λ αβ t k ab }{{}}{{} Lorentz/Dirac Flavor TAB D }{{} Color (3.5) where the upper indices are diquark indices and the lower ones correspond to the two quarks; the coupling constants g λ determine the strength of the effective quark-quarkdiquark coupling. x x 4 p p P _ p -p x 1 x 3 Figure 3.7: Diquark amplitudes in the pole approximation q -p The diquark wave function ξ, i.e. the on-shell diquark amplitude with external quark legs attached, is the transition matrix element between a bound state with momentum P and the vacuum (cf. Fig. 3.7 for the labeling of the indices): ξ αβ (x 1, x, P ) = 0 q α (x 1 ) q β (x ) P. (3.53)