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2014 80 Alessandro Santagata Coulomb Phases: from Graphene to Quark Confinement Departamento Director/es Física Teórica Asorey Carballeira, Manuel Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Alessandro Santagata COULOMB PHASES: FROM GRAPHENE TO QUARK CONFINEMENT Director/es Física Teórica Asorey Carballeira, Manuel Tesis Doctoral Autor 2014 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Universidad de Zaragoza Facultad de Ciencias Departamento de F´ısica Te´orica COULOMB PHASES: FROM GRAPHENE TO QUARK CONFINEMENT Ph. D. Thesis Alessandro Santagata Director: Manuel Asorey Junio 2014 COULOMB PHASES: FROM GRAPHENE TO QUARK CONFINEMENT Ph. D. Thesis Alessandro Santagata ii Agradecimientos En primer lugar quer´ıa agradecer a Manolo, para haberme dado la posibilidad de realizar uno de mis sue˜nos, aportar contribuciones personales originales al campo de la f´ısica te´orica. Seguramente trabajar contigo es algo muy particular: lo que siempre me ha sorprendido ha sido tu entusiasmo, que s´olo se puede encontrar en las personas que de verdad aman lo que est´an haciendo, en el lanzarse en los problemas de la f´ısica. Lo que m´as he apreciado y que seguramente me acompa˜nar´a por mucho tiempo ha sido tu capacidad de no contentarse nunca con el primer resultado, intentando de ver siempre m´as all´a, explorando todas las posible consecuencias para sacar siempre el m´aximo partido, con gran fuerza y optimismo. Esto, seguramente, es una gu´ıa y un ejemplo que intentar´e seguir en todo lo que haga. Gracias a todos los miembros del departamento de F´ısica Te´orica, con los cuales he pasado 5 a˜nos estupendos. Siempre hab´eis estado muy disponibles y he apreciado mucho el clima casi familiar del departamento. En particular, gracias a Jos´e Maria, que ha sido muy importante para el comienzo de este trabajo, y a Matteo, Cesar, Amilcar, David, Pulak y Leo por la paciencia que hab´eis tenido en contestar a mis preguntas. Agradezco el apoyo econ´omico que recib´ı con la beca asociada al proyecto ”Grupo de Excelencia: 2009. Grupo Te´orico de Altas Energ´ıas” la beca doctoral de la DGA y el contracto de Investigador Novel de la Universidad de Zaragoza, en el proyecto ”FPA2012-35453. F´ısica del modelo est´andar iv y sus posible extensiones”, que me han permitido de realizar este trabajo. Grazie a tutti quelli che hanno condiviso in qualche modo Zaragoza. In particolare a Silvia per ”a chill pelat u sacc”, a Ola per ”mi globo”, a Daniele per il torcicollo in aula lectura, a tutti quelli del viaggio in Andalucia (Camilla, Alberto, Angelo, Adriano, Zagni, Coty, Elisabetta, Carlo), a Diana per ”ci beviamo na cosa?”, a Robertina per ridere sempre, alla Lella per non rispondere mai, a Ross per condividere la sopportazione delle ultime tre galline, al conte Ugolini per la panza piena tutte le sere, a tutti quelli del viaggio al Norte e degli aperitivi a Milano quando torno (Claudia, Andrea, Daniele, Chiara, Emilio, Marti, Leo, Vale, Michela), a Carol per avermi fatto da infermiera, a Betty Brown, Eri, Iuso, Gi`o che hanno assistito, consenzienti, a Daniela spalmarmi la cioccolata in testa, a Gennaro per i tre scudetti visti insieme, a Gabriele per la diretta gol la domenica pomeriggio, a Valeria per il cammino dell’amarezza, a Monica per le tonnellate di pasta, a Marianna e Livia per ”-todo bien? -si! pasa! pasa!”, a Margherita per aver reso infinita la mia pazienza, a Marzia per ”Caf´e y comida”. Grazie anche ad Alex, Laura e Andres per tutte le feste. Y a todos los compa˜neros de piso de estos a˜nos. Grazie a tutti i Gugnis, alle loro tifose, fidanzate, mogli e figli. Soprattutto per ”Quando sei tornato? sei arrivato a Bergamo? Quando riparti? Riparti sempre da Bergamo? quanto ti fermi? come va con lo spagnolo? quanto ti manca? qual `e la tua giornata tipo?” Grazie per avermi fatto sempre le stesse domande, tutti e 20/25, in fila, uno dopo l’altro. Ogni volta. Maledetti. E comunque no, una volta per tutte, dalla mia ricerca non si pu`o trovare la pesca senza nocciolo e non mi metto il camicie bianco in ufficio. Ma la domanda `e: ringraziandovi, ho voluto essere gentile solo per gentilezza? No, questa volta l’ho fatto davvero con gentilezza (non ne usciremo mai). In rappresentanza, per dimostrarvi che al quinto anno ho imparato queste due nuove parole, ”agradezco con humildad” Gogo per aver sfangato insieme molte giornate e aver condiviso il peso, altrimenti insostenibile, di tutte le amarezze fantacalcistiche, i due passi circolari di Alex Maro e l’Alberta per ”mi hai caricato la puntata?”. E grazie alla Vale v Cucca perch´e la lontananza non ha influito nel rendermi partecipe di un evento importante. Grazie alla mia famiglia, in particolare a mio fratello per avermi tartassato di messaggi per 5 anni. E alle mie nonne, Luisa e Carla, che solo mi hanno visto partire. 1 Introduction 1.1. Quark Confinement in Quantum Chromodynamics The goal of this thesis is to develop a new approach to the more striking feature of Quantum Chromodynamics, quark confinement. Let us start with historical review of the role of Quantum Chromodynamcs in the description of strong interactions. The first theory of strong interaction is due to Yukawa and was formulated in 1934. The need of such an interaction is a consequence of the multiproton nuclei stability. The only way of compensate the electric repulsion of protons is to postulate the existence of a very strong attractive interaction. The strong interaction in the Yukawa model is mediated by the exchange of a new massive particle. The particle responsible for the interaction, the pion, was later discovered by Powell in 1947. The mass of the pion signals the short range scope of the strong interaction. The pion was one of the first of the saga of elementary particles which soon became a zoo. A very successful classification scheme, known as Eight- 6Introduction fold Way, was introduced by Gell-Mann in 1961. Gell-Mann’s way consist in organizing the families of strongly interacting particles according to irreducible representations of a new kind of internal symmetry associated to a unitary group SU(3). The success of the SU(3) model lead to Gell-Mann and Zweig to raise in 1964 the conjecture that all the hadrons are composed of more elementary constituents, the quarks. The problem generated by discovery of the Ω particle made of three identical quarks, apparently in contradiction with Pauli principle, lead to Gell-Mann to propose the existence of new quantum number, the color, associated to a new internal SU(3) symmetry. The strange property of the color quantum number is that only particles which are neutral of color are stable. This means that the quarks themselves cannot be isolated as independent particles. They can only live in combination with other quarks in a color neutral hadron. This is the first manifestation of quark confinement. However, the existence of quarks can be made manifest by scattering experiments. The study of the inner hadronic structure can be made through high energy electron scattering with large momentum transfers. The results obtained at the end of the sixties in the experiments of deep inelastic electron-proton scattering, which were reminiscent of Rutherford’s experiments, confirmed the existence of three almost-free point-like constituents inside the proton, which were called partons by Feynman and identified with the quarks. The search for a quark dynamics started right after the foundation of the quark model. The dynamics governing the quark system should have the property that the strong interaction between the quarks becomes weaker at shorter distances as observed in the deep inelastic electron-nucleon scattering experiments. The quantum field theories of relativistic interactions known at that moment had the opposite properties. For example, in Quantum Electrodynamics (QED), the quantum field theory describing the electromagnetic interaction, the strength of the interaction is increasing at short distances. The strength of an interaction are in general parameterized by coupling constant. Due to quantum corrections the value of this coupling changes 1.1. Quark Confinement in Quantum Chromodynamics 7 with the distance. In the presence of an electric charge vacuum fluctuations include virtual pair creation which means that the quantum vacuum is a polarizable medium where electric charge gets screened. The dependence of the screening with the distance to the external charge can be derived from its dependence on the momentum transfer q2carried by the mediating photon (large q2correspond to small distance). In perturbation theory QED predicts that the electromagnetic coupling constant is running as α(q2) = α(m) 1−α(m) 3πln(q2/m2) ,(1.1) valid for values of the four momentum q2m2much larger than the the electron mass m. This shows that the effective coupling α(q2) tends to increase at large q2. This screening of the effective charge associated with the vacuum polarization disappears at the Compton wavelength of the electron 1/m. Until 1973 it was believed that all the quantum field theories behaves in a similar manner, with the coupling constant increasing logarithmic with Q2. But in that year Gross, Wilczek and Politzer showed that the nonAbelian gauge field theories, introduced by Yang and Mills in 1954, satisfy the desired property for the coupling to decrease with the energy. This property was called asymptotic freedom. The quantization of Yang Mills theory was already achieved including its renormalization by ’t Hooft and Veltman in 1971. This confirmed that the strong interaction of quarks is driven by nonAbelian gauge theories. These theories differ from QED by the fact that are the internal symmetry group is non-Abelian. It is just the SU(3) symmetry group associated to the internal color quantum number. The theory of quark dynamics based on that mechanism is the celebrated quantum chromodynamics (QCD). The particle mediating the strong interaction, the gluon, plays a similar role that the photon plays in electromagnetic interactions. There are some differences photons have no electric charge but gluons carry color charge; photons do not interact between then at tree level, whereas gluons interact 8Introduction with each other even in the absence of the quarks. This property of gluons is an essential ingredient for understanding the behaviour of QCD at short distances. In particular, to understand why the quarks inside an hadron at very short distances do have light interactions, i.e. they are almost asymtotically free. As in QED the production of virtual quark and antiquark tends to screen the color charge of the quark, but now in QCD we have to consider also the auto-interaction of the gluons. The net effect of the production of virtual gluons in the vacuum is not to screen the color charge but its antiscreening. Getting closer to a quark diminishes the anti-screening effect of the surrounding virtual gluons, weakening the effective charge which decreases as we approach the quark core. Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different flavors of quark. For standard QCD with three colors and six flavors the anti-screening prevails. This is described by the running of the strong coupling constant formula, obtained in perturbation theory: αs(q2) = αs(µ2) 1 + 33 −2F 12παs(µ2) ln(q2/µ2) ,(1.2) where and Fis the number of quark flavors. Notice that if F= 6 the sign of the logarithmic term is positive and not negative as in QED (1.1) which implies that αs(q2)→0 as q2→ ∞. Notice the presence in the formula (1.2) of a new parameter µwith dimensions of mass. In QED the similar parameter was the electron mas, but in QCD quarks are not physical particles, then, there is not an apriori natural scale. The QCD scale µhas to be fixed by scattering experiments. From a theoretical viewpoint the emergence of this scale in a conformal invariant theory (in absence of quarks) is due to a conformal anomaly which not only generates a new scale, but also a induces a dimensional transmutation of some physical observables. The discovery of asymptotic freedom promoted the QCD as the theory of strong interactions and permitted to Gross, Wilczek and Politzer to win the Nobel Prize in 2004. With the birth of QCD the interaction between 1.1. Quark Confinement in Quantum Chromodynamics 9 two hadrons, for example two protons in a nucleus, which Yukawa though to be a fundamental process, must be regarded as a complicated interaction of six quarks. The high energy experiments provided the evidence that there are quarks inside the hadrons but they also confirmed the fact that no isolated quark have been ever observed. The situation is quite different respect QED: even though bound states arise also in QED, the constituent electrons and nuclei are themselves frequently observed, as processes like ionization, in which constituents are released from their bound states when a sufficiently large transfer of energy is attained. The analogs of these processes have never been seen for strongly interacting particles, even though the bound states have been exposed to collision energies that are many times the rest energy of the bound particles themselves. Quarks are absolutely confined inside baryons and mesons. Quark confinement is also a physical characteristic of strong interactions. Whether QCD can account for that behavior of the theory or not was the next question addressed by the high energy physics community in the late seventies. In opposition to asymptotic freedom, confinement has not been understood in QCD in analytic terms. Confinement is a genuine non-perturbative phenomenon. It cannot be seen at any finite order of weak-coupling perturbation theory. It requires new non-perturbative analytic tools to deal with. One technical difficulty is that the standard QED regularization techniques necessary to control the UV divergences are not gauge invariant in non-abelian gauge theories. A new kind of regularization technique, dimensional regularization, was introduced to prove the renormalizability of the theory and the emergence of asymptotic freedom in perturbation theory. However, the dimensional regularization method does not work beyond perturbation theory. The first step in the analysis of confinement was the introduction by K. Wilson of a new type of regularization method based on a space-time lattice. In lattice gauge theory the continuum space-time is replaced by a four dimensional Euclidean discretized space-time, the quarks are situated at the points of this lattice and the gauge fields on the links between the lattice points. 10 Introduction Once the theory is regularized it is possible to calculate the strength of the force between quarks. This can be done in absence of dynamical quarks by computing the expectation value of a gauge invariant observable known as Wilson loop. In the strong coupling limit gs1 one obtains by a simple analytical calculation that < W >∼ =e−σT L in the large loop approximation, where Tand Lare, respectively, the Euclidean time and space span of the the square loop. This behavior implies that the potential between a q−¯qpair of heavy static quark-antiquarks grows linearly with the distance Lbetween then, i.e. V=−σL, which implies the need of linearly increasing amount of energy to separate these quarks a large distance L. This linear behavior is radically different of the expected Coulombian behaviour V=a/L of the perturbative approach (L1). In summary, the analytic approach to QCD is successful in two opposite regimes: in the limit of weak coupling, the quarks interact with a Coulomb like potential, which leads to asymptotic freedom, while on the other hand, the limit of strong coupling shows the existence of a linear potential that confines the quarks. However this is not a proof confinement, it is necesary to show that both behaviors arise in the same theory. This means to prove that the coupling constant flows from small value at high energy to large value at small energy without any phase transition, otherwise we should see just one of the two regimes. The formula (1.2) indicates that asincreases as the energy becomes smaller and smaller but it stop to be valid at a scale of energy beyond ΛQCD ≃200Mev as the perturbative calculus is no more valid. The QED for example also shows confinement for strong coupling, but we know that in the continuum limit electrons and photons are free. The consistency of the lattice theory with the continuum limit requires the existence a phase transition separating the strong coupling phase to the weak coupling phase. Such a transition has been clearly seen in numerical simulations when going from strong coupling to weak coupling. The proof that no phase transition occurs for intermediate value of the coupling constant in QCD would be equivalent to a proof of confinement. 1.1. Quark Confinement in Quantum Chromodynamics 11 Although the numerical simulations on the lattice do not show any trace of such a transition in the intermediate phase, the analytic proof of this fact has remained elusive for decades and still is one of the challenging theoretical questions in QCD. It must be shown that the same theory that confines on the lattice for strong coupling has a continuum limit for weak coupling that is consistent with the asymptotically free behavior at short distances of QCD. On the other hand the intermediate regime of the coupling constant is very rich of interesting phenomena, like for example the existence of a plethora of Quarkonia, that are bound states of heavy quarks. The relevant energies in a quark bound state are in most cases of the order the scale of 1 fm, the average hadron size, that correspond to the scale of energy of ΛQCD, where it is supposed to happen the crossover between the perturbative regime of asymptotic regime and non perturbative regime of confinement. So the theoretical comprehension of these phenomena is directly connected to the physics of confinement. In case of heavy quarks, like bottom and charm, the velocities of the quarks are non relativistic and the time scale associated with the binding of the gluons is smaller than the time scale associated with the quark motion: the gluon interaction between heavy quarks can be considered instantaneous. Therefore, the non relativistic quarkonium bound state can be modeled with a potential and the energy can be obtained solving the corresponding non-relativistic Schr¨odinger equation. The effective potential that can be obtained in perturbation theory from one gluon exchange, as mentioned above, has a Coulomb like form. It cannot be the final answer since it does not confine quarks and gives a spectrum incompatible with the data, but adding a linear potential, motivated by the strong coupling limit, we can form a phenomenological potential (Cornell potential) Vc(r) = −4 3 αs r+σ r, where αsand σare parameters to be fitted with the data. The Schr¨odinger equation with this potential and αs= 0.38 and σ= 0.18GeV−2gives a quite satisfactory agreement with the data. 12 Introduction 1.2. The Gribov picture of confinement To show the absence of phase transition in the intermediate regime of QCD, we need an inspiration to guess how from pure QCD dynamics a confining color string can be created out of the vacuum by the presence of a quark-antiquark pair. Some very appealing pictures have been proposed in the past, but none was completely successful. The oldest one is based on the dual superconductor scenario. In a normal superconductor the phenomenon of superconductivity is due to the condensation, below the critical temperature, of Cooper electron pairs. A related aspect is the Meissner effect: an external magnetic field permeating the superconductor is expelled from the bulk material as the temperature drops below the critical value. This implies that one magnetic monopole cannot exists in the superconductor. The amount of magnetic flux that comes out from the monopole inside the superconductor cannot be completely expelled out. Because of the Meissner effect in the presence of a monopole-antimonopole pair, the magnetic flux coming out from the monopole into the antimonopole can be expelled from a type II superconductor by forming a thin flux tube connecting both magnetic sources and inducing a potential between them growing linearly with the size of the pair. This implies that it cost more and more energy to pull the monopoles apart. This phenomenon is reminiscent of the Wilson area law, and the confinement of monopoles induced by Meissner effect suggests a possible mechanism for quarks confinement. If the QCD vacuum behaves like a dual superconductor generated by the condensation of magnetic monopoles the chromo-electric flux will be expelled from the vacuum by the dual Meissner effect. Thus, the immersion of a quark-antiquark pair in such a magnetic superconducting vacuum will generate a concentration of the chromo-electric flux along the line connecting the two particles. The flux tube behaves like a string, inducing an effective quark-antiquark potential growing linearly with the distance, i.e. we will get quark confinement. The dual superconductor picture illuminated many approaches to the confinement problem but none of then achieved a analytic formulation leading to a proof quark confinement. 1.2. The Gribov picture of confinement 13 A complementary picture came from the AdS/CFT correspondence. The main point of this correspondence is that strongly coupled gauge theories are dual to perturbative string theories. This connection created a bridge between two fields of physics. Problems that are unaccessible in gauge theory can be in principle translated in the language of string theory and viceversa. This connection inspired the search of a string theory dual to QCD in order to make progress in the analytic comprehension of non perturbative effects of strong interactions. In particular one promising duality was represented by IIB string theory in a background of five-dimensional anti-de Sitter space times a five-sphere and the large N limit of N= 4 supersymmetric Yang-Mills theory in four dimensions. However, the resulting dual gauge theory is conformally invariant and is not confining. Both pictures can help in the understand how the mechanism should work, but they are not connected with real QCD. There is an alternative picture, proposed by Gribov, that was inspired by real QCD, which is based on Coulomb phase instability. Gribov was motivated by the instability of the relativistic hydrogenoid atoms, pointed out by complex values of the energies E=mc2"1 + Zα n−(j+1 2) + q(j+1 2)2−(Zα)2!2#−1 2 (1.3) when Zα > 1 for j=1 2, for example. This means that there is a critical value for the atomic number Zc= 137, beyond which the vacuum becomes unstable and generates electron-positron pairs: the electrons, with negative energy, fall into the heavy nucleus reducing its charge Z and the positrons are left free. If the new atom is still unstable the process repeats until the charge of the nucleus becomes sufficiently small. This phenomenon is possible only due to the existence of light fermion. Based in this analogy, Gribov suggested that a similar phenomenon could be re root of the confinement of quarks in QCD. The existence of very light quarks, can drastically change the vacuum structure. He claimed that understanding the physics of light quarks was directly connected with the solution of the confinement 14 Introduction problem. He pointed out that in QCD already the color charge of a single quark could be supercritical, in contrast to QED where a very large charge Zc= 137 is required. In QCD the role of the supercritical charge is in fact played by the strong coupling constant which becomes large at low momentum scales because of the anti-screening phenomena (1.2). The confinement of heavy quarks in Gribov’s scenario is very similar to the supercritical binding in QED. In the presence of a single quark, due to its supercritical effective charge, the vacuum becomes unstable and decays into quark-antiquark pairs. In this sense the quark exists only as a resonance and cannot be observed as a free particle. To provide a quantitative support to the picture, Gribov analyzed the Green function of light quarks, to get a full dynamical physical picture of the supercritical charge regime. He derived the Green function from the corresponding Dyson-Schwinger equation and he found a critical coupling αc=2πN N2−11−r2 3, for any SU(N) gauge theory. In particular, αc≃0.43 for real QCD (N= 3). For couplings bigger that αcthe Green function changes its behavior pointing out a chiral symmetry breaking. Although the light quark Green function did not correspond to confined quarks, Gribov argued that taking into account the effects of pions which arise as Goldstone bosons of the symmetry broken, the modified effective Green function would display an the analytic structure of confined quarks. Unfortunately, in 1997 Gribov died right after claiming to have a full proof of confinement, but before concluding the papers ([38], [39]). 2 Coulomb impurities in graphene The main motivation of Gribov picture of confinement was based on the instability of Coulomb phase in QED. However, this instability requires Z > 137 and is very difficult to obtain nuclei with such a large electric charge. Although they can be created by heavy nuclei collisions and an associated positron abundance has been observed, the nuclear instabilities do now allow us to get a clean picture of the physical phenomena. However, very recently with the discovery of graphene a similar phenomenon occurs in the presence of charged impurities, but with a much lower critical charge Zc= 1. In this case the instability yields to screening of the charge impurity which in pure QED cannot occur. The phenomenon has been very recently experimentally observed. Motivated by this new physical phenomenon the analysis of the instability has been deeply analyzed shedding some new light in the Gribov approach. The novel viewpoint shows that unitarity is never lost by the introduction of charge impurities. In fact the formal transmutation of the bounded spectrum of the Hydrogen-like atom analysis into complex values does not means the lost of hermiticity of the corresponding 22 Coulomb impurities in graphene effective Hamiltonian. It only shows the existence of non-trivial spectral densities in the continuum spectrum which correspond to the existence of resonances in the scattering approach. To clarify the possible application to the Gribov picture we analyze in this chapter the transition between from subcritical regime to a supercritical in the graphene impurities problem. In particular we address a rigorous definition of the Dirac Hamiltonian in the presence o charged impurities potential as self adjoint operator. The problem requires a renormalization the singularity due the Coulomb potential which is more severe for relativistic interactions. Finally the problem is reduced to setting the right boundary conditions near the singularity. This will show us how to proceed in the analysis of the confinement problem from a Coulomb regime in QCD. A similar discussion for conformal potential in Quantum Mechanics is provided in Appendix A. 2.1. 2D Dirac Hamiltonian in a Coulomb background The electron states in the graphene in presence of a charge impurity are described by two dimensional Dirac Hamiltonian with Coulomb potential: H=−i(σ1∂x+σ2∂y) + mσ3−α r,(2.1) where σi, i=1,2,3, are the Pauli matrices, αthe charge of the impurity, m the mass and r=px2+y2. We assume that the speed of light is normalized to 1. The singularity at the origin requires some renormalization mechanism. First we introduce a space a cut-off r0>0 and then a boundary condition at the cut-off which defines a self adjoint Hamiltonian H. Then, we introduce a renormalization prescription and remove the cut-off. The final result will lead to a selfadjoint Hamiltonian defined over R2\{0}. In the cut-off domain r≥r0shielding the impurity, the domain of the self adjoint extension of the hamiltonian can be defined by imposing the following boundary conditions (1 + ˆ n/)ψ(r0) = U(r0)σ3(1 −ˆ n/)ψ(r0),(2.2) 2.1. 2D Dirac Hamiltonian in a Coulomb background 23 where U(r) is an unitary operator defined on the square integrable spinors defined over the circle of radius r0. and ˆnis its normal unit vector, i.e. ˆn=r/r. Using polar coordinates rand θa general spinor ψcan be expanded, as ψ(r, θ) = ∞ X l=−∞ (Fl(r)Φ+ l(θ) + Gl(r)Φ− l(θ)), with l∈Zin terms of the orthogonal eigenfunctions of the total angular momentum Jz=Lz+Sz=−i∂ ∂θ +1 2σ3, with semi-integer eigenvalues j=l+1 2 Φ+ l=ei l θ 0and Φ− l=0 i ei(l+1) θ. The space of spinors can be then decomposed as orthogonal sum of subspaces with fixed total angular momentum j,H=⊕Hj,i.e. general spinor can be written as ψ=Pjψj, where ψjwill denote the spinor component in the subspace of total angular momentum j, ψj(r, θ) = Fj(r)ei(j−1 2)θ iGj(r)ei(j+1 2)θ,!.(2.3) If the boundary condition defined by Upreserves rotational symmetry, U reduces, on each subspace of fixed angular momentum j, to a U(1) phase, i.e. the boundary condition (2.2) becomes (1 + ˆ n/)ψj(r0) = ei βj 0σ3(1 −ˆ n/)ψj(r0).(2.4) Explicitly, Fand Gsatisfy on the boundary: ei βj 0=Fj(r0) + iGj(r0) Fj(r0)−iGj(r0),(2.5) where F(r0)/G(r0) are real functions. 24 Coulomb impurities in graphene Renormalization of the UV divergences and boundary conditions Let us now introduce now a fundamental ingredient in our discussion, asymptotic zero modes near the origin. They will play a fundamental role in the renomalization of the singularity introduced by the singularity. We can by means of these zero-modes define the self adjoint extension of the Hamiltonian as a limit when the cut-off goes to zero in terms of appropriate the boundary conditions. The form of the zero modes is strongly dependent on the value of the charge. To find the zero modes near the singularity of the Hamiltonian (2.1) we have to keep only the leading terms of of the asymptotic expansion around the impurity. Using the expansion (2.3) is easy to show that they have to satisfy the following coupled equations dFj 0 dr −j−1 2 rFj 0+α rGj 0= 0,(2.6) dGj 0 dr +j+1 2 rGj 0−α rFj 0= 0.(2.7) We search a solution of (2.6) and (2.7) of the form Fj 0(r) = rs Gj 0(r) = C rs This requires that s=−1 2+ν, C+=j−ν α, or s=−1 2−ν, C−=j+ν α, where ν=pα2−j2. These solutions are only valid for α26=j2. For α2=j2the two solutions are instead are given by Fj 0(r) = r−1 2 Gj 0(r) = C r−1 2 2.1. 2D Dirac Hamiltonian in a Coulomb background 25 and Fj 0(r) = r−1 2log(Λ r)) Gj 0(r) = j |j|r−1 2(log(Λ r)) −1 j), where is an arbitrary parameter with dimension [L]−1. Notice that for α2< j2the parameter νis real, while for α2> j2it is purely imaginary. Taking also into account the local L2normalizability at the origin, there are four different types of regimes depending on the strength of the charge impurity. i) For α2≤j2−1 4the solution with s=−1 2−νis not locally normalizable at the origin, so it must be excluded. The general zero mode is in this case of the form Fj 0(r) = r−1 2+ν Gj 0(r) = C+r−1 2+ν.(2.8) For α2> j2−1 4both the solutions are normalizable so the most general zero mode is a linear combination of the two behaviors. The only constraint is the ratio F/G have to be a real functions. The structure strongly depend on the value of the charge. We will parametrize the different linear combinations (up to a normalization constant N) in terms of an angle θ∈[0, π) and the scale parameter Λ. ii)j2−1 4< α2< j2: Fj 0(r) = N r−1 2cos θ(Λ r))ν−sin θ(Λ r))−ν Gj 0(r) = N r−1 2cos θ C+(Λ r))ν−sin θ C−(Λ r))−ν,(2.9) iii)α2=j2: Fj 0(r) = N r−1 2cos θ+ sin θlog(Λ r)) Gj 0(r) = N C r−1 2cos θ+ sin θlog(Λ r)) −1 j,(2.10) 26 Coulomb impurities in graphene iv)α2> j2: Fj 0(r) = N r−1 2exp(−i θ) (Λr)ν+ exp(i θ) (Λr)−ν Gj 0(r) = N r−1 2exp(−i θ)C+(Λr)ν+ exp(i θ)C−(Λr)−ν,(2.11) In the UV limit the cut-off r0→0 and any eigenfunction of Hsatisfying the boundary condition (2.5) must also become an asymptotic zero mode. Thus, we can associate to each boundary condition in the four regimes an asymptotic zero mode and reciprocally each zero mode defines one class of boundary conditions: Fj 0(r0) + iGj 0(r0) Fj 0(r0)−iGj 0(r0):= ei βj 0.(2.12) In other terms, once the cut-off is fixed r0, we can associate to each boundary condition parametrized by βj 0, a particular asymptotic zero mode (Fj 0, Gj 0) satisfying (2.12). Again, the relation depends on the regime of the coupling constant. i) For α2≤j2−1 4since we have only one asymptotic zero mode means that there is only a selfadjoint extension. In fact the constraint (2.12) completely fixes βj 0, independently from r0: βj 0= arccos pj2−α2. This means that we have only one self adjoint extension on H. For α2> j2−1 4there is a family of asymptotic zero modes which can be parametrized by the phase βj 0at the boundary cut-off r0. For simplicity we define ζ= tan θand y= tan(βj 0/2). ii) For j2−1 4< α2< j2, for each βj 0∈[0,2π) we can associate a zero mode by the relation: ζ=(j−ν)−α y (j+ν)−α y(Λ r0)2ν.(2.13) 2.1. 2D Dirac Hamiltonian in a Coulomb background 27 iii) For α2=j2, fixed r0, to each βj 0∈[0,2π) we can assign a zero mode by ζ=j−|j|y 1−(j−|j|y) log(Λ r0).(2.14) iv) For α2> j2, if we define ξ=−tan (|ν|log(Λ r0)), we can associate to each βj 0∈[0,2π) a zero mode by the relation: ζ=j−ξ|ν|−α y j ξ +|ν|−α ξ y.(2.15) In summary, we have defined the boundary condition on the cut off r0by means of an unitary operator, which for each angular momentum is given by a phase which is in one-to one correspondence with a locally normalizable asymptotic zero mode. The next step is to remove the cut-off, sending r0to zero. For any other cut-off r< r0the boundary condition can be determined by a running phase βj that can be defined in terms of the initial one βj 0which converges to a well defined boundary condition when the cut-off is removed, which gives a well defined Hamiltonian for the dynamics of electrons in presence of the impurity in the graphene sheet. The running phase βj at different ris renormalized according to ei βj (r)=Fj 0(r) + iGj 0(r) Fj 0(r)−iGj 0(r),(2.16) in terms of the zero mode (Fj 0, Gj 0) associated to the initial phase βj 0at r0by the relations (2.13), (2.14) or (2.15), That is, the initial phase βj 0 defines a zero mode (Fj 0, Gj 0), and the boundary phase βj runs with the cut-off while keeping fixed the zero mode. Once more the explicit form of the renormalization function for βjdepends by the value and regime of the charge. ii)Case j2−1 4< α2< j2 tan βj  2=−j+ν α1 + 2 tan θ (Λ r)2ν−tan θ, 28 Coulomb impurities in graphene iii)Case α2=j2 tan βj 2=j−tan θ+jtan θlog(Λ r) |j|(1 + tan θlog(Λ r)) , iv)Case α2> j2 tan βj  2=1 αj+|ν|tan θ+ tan (|ν|log(Λ r)) tan θtan (|ν|log(Λ r)) + 1; Finally we can define the selfadjoint extensions of the Hamiltonian on R2\{0}in the limit r→0. Once we impose the boundary condition (2.5) on the cut off r0, we impose that ei βj (r)=Fj(r) + iGj(r) Fj(r)−iGj(r), with β(r) running according (2.16). Explicitly, if we remove the cut-off r→0 the boundary condition in terms of the zero mode is given by lim r→0Fj(r)Gj 0(r)−Gj(r)Fj 0(r)= 0.(2.17) Any selfadjoint extension of Hon R2\{0}is then defined by boundary conditions in terms of a zero mode, and the family of self adjoint extensions for α2> j2−1 4is parameterized by the zero-mode phase θ. The explicit forms of (2.17) depends again on the regime of the charge α: i)Case α2≤j2−1 4 lim r→0(−j+ν)Fj(r) + αGj(r)= 0; (2.18) 2.2. Bound states spectral problem 29 ii)Case j2−1 4< α2< j2 lim r→0(Λr)2ν(−j+ν)Fj(r) + αGj(r)+tan θ(j+ν)Fj(r)−αGj(r)= 0; (2.19) iii)Case α2=j2 lim r→0j G(r)1 + log(Λr) tan θ−F(r)|j|+−1 + jlog(Λr)tan θ= 0. (2.20) iv)Case α2> j2 lim r→0(Λr)2ν(−j+ν)Fj(r) + αGj(r)−exp(2iθ)(j+ν)Fj(r)−αGj(r)= 0. (2.21) 2.2. Bound states spectral problem Once the boundary conditions give rise to a self adjoint Hamiltonian, we can find the bound states in the spectrum, Hψ =Eψ. (2.22) Using the ansatz (2.3), in each subspace of fixed j, the eigenvalue problem reduce to the following coupled differential equation: dFj dr −j−1 2 rFj+ (E+m+α r)Gj= 0,(2.23) dGj dr +j+1 2 rGj−(E−m+α r)Fj= 0.(2.24) Replacing Fj(r) = √m+E 2ra(r)−b(r), Gj(r) = √m−E 2ra(r) + b(r), 30 Coulomb impurities in graphene x= 2 r,  =pm2−E2,(2.25) gives the following coupled equations for a(x) and b(x): xd dxa(x) + x 2−1 2−αE a(x) + αm +jb(x) = 0,(2.26) xd dxb(x)−x 2+1 2−αE b(x) + −αm +ja(x)=0.(2.27) It is now possible to decouple a(x) and b(x). From (2.26) one can get b(x) = (2αE +− x)a(x)−2 x a0(x)) 2(αm +j),(2.28) and plugging it in (2.27), gives rise to an equation for a(x) d2a(x) d2x+−1 4+ 1 2+αE  x+ 1 4−j2+α2 x2a(x)=0,(2.29) which can be solved in terms of Whittaker functions W and M a(x) = A W(1 2+αE , ν, x) + B M(1 2+αE , ν, x),(2.30) where A and B are constants. Substituting (2.30) in (2.28), we have b(x)=(j−αm )A W(−1 2+αE , ν, x) + αm −j αE +νB M(−1 2+αE , ν, x). (2.31) Inserting these expressions into (2.25) we have that the general solution of 2.3. Bound states and the spectral flow 37 for any integer k∈Z. Another interesting property of the spectral flow has a monotonic character, i.e. EII (θ)< EII(θ0) if θ < θ0. In particular this implies the standard sandwich inequalities between the Hydrogenoid and meta-Hydrogenoid energy levels −m<Eh 0< EH 0< Eh 1< EH 1< ... < Eh n−1< EH n−1< Eh n< EH n< ... < m (2.48) For negative angular momentum j < 0, we have the same behaviour as in the positive case j > 0. The only difference is the absence of zero levels (n=0) for the Hydrogenoid and meta-Hydrogenoid energy levels. Thus the sandwich inequalities become −m<Eh 1< EH 1< ... < Eh n−1< EH n−1< Eh n< EH n< ... < m. (2.49) Another interesting fact in the subcritical regime is that from (2.36) and (2.42) it follows that for n > 0 the spectra EH nand Eh nwith j > 0 and j < 0 are degenerate. The boundary condition (2.19) for θ6=kπ and θ6=2k+1 2breaks this degeneracy and creates a gap between the energies corresponding to j > 0 and j < 0. The situation is described in figure 2.2 for α= 1.45. For θ= 0 we have the energy corresponding to n= 1 of the Hydrogen spectrum EH 1that is degenerate for j=±3 2. As we increase the parameter θa gap appears between the states j=±3 2. The energy of the state j=−3 2becomes lower than the one corresponding to j=3 2. The gap becomes disappears again for θ=π 2, where we have the degenerate energy level corresponding to n= 2 of the meta-Hydrogenoid spectrum Eh 2. Increasing more the parameter θ, a gap appears again. This time with the energy corresponding to j=3 2lower than the one corresponding to j=−3 2. Finally, for θ=πthe two energy levels become again degenerate at the level n= 2 of EH 2. •For α2=j2, as we anticipated before, the Hydrogenoid and metaHydrogenoid spectra coincide EIII n(0) = EIII n(π 2) and are given by (2.36) with the only difference that for α2=j2and j > 0, E0= 0. Once more we can understand this in terms of the identification of boundary conditions. Analyzing how the spectrum EIII n(θ) changes with θ, we find that the 38 Coulomb impurities in graphene 0 Π 4 Π 2 3Π 4 Π Θ 0.70 0.75 0.80 0.85 Em Figure 2.2: Gap between the energies corresponding to j= 3/2 (blue) and j=−3/2 (red) for α= 1.45, m= 10 and Λ = 1 in the subcritical regime. correspondence in this case is, for j > 0, lim θ→0EIII n(θ) = EH n,(2.50) lim θ→πEIII n(θ) = Eh n+1 (2.51) (2.52) The spectrum is also periodic in this case, i.e. it is the same at θand θ+π, but the flow shifts the energy levels by one unit each time that we increase θby π. In general, for fixed angular momentum and charge we have EIII n(θ+kπ) = EIII n+k(θ),(2.53) for any integer k∈Z. Also in this case the spectral flow has monotonic character, i.e. EIII(θ)< EIII(θ0) if θ < θ0. However, in this case inequalities between the Hydrogenoid and meta-Hydrogenoid energy levels (2.49) become the trivial inequality of the Hydrogenoid levels EH n< EH n+1. •Let us now analyze the supercritical regime of charge α2> j2. As anticipated before, in this regime, the spectra EH nand Eh nare not defined, 2.3. Bound states and the spectral flow 39 so we cannot establish any correspondence between the impurity spectrum and the Hydrogenoid or meta-Hydrogenoid spectrum for any particular value of θ. In fact for any value of θ EIV n(θ) contains an infinity number of bound states that accumulate near E=m. In figure (2.3) we plot the flow of some eigenvalues of the spectrum EIV n(θ) when parameter θflows from 0 to π. In that figure we see how one eigenvalues pops up from the continuum E < −mat a particular value of the parameter θ. Π 4 Π 2 3Π 4 Π Θ -0.5 0.0 0.5 1.0 Em Figure 2.3: Spectral flow for α= 1.55, Λ = m/10 and j= 3/2 in the supercritical regime. The lowest bound state energy level emerges from the continuum spectrum for θclose to θ=π. This is the only signal of the instabilities pointed out by the fact that the analytic expressions of Hydrogenoid or meta-Hydrogenoid energy levels become formally complex. In order to analyze the transition from the subcritical regime to the supercritical regime we choose an appropriate parameter θfor EII n(θ), EIII n(θ) and EIV n(θ). By changing αwe can move each energy level of the subcritical region into one of the critical region by a continuous path. Notice that for each θ6= and θ6= there is a bound state in the subcritical regime that merge into the continuum at a particular value of α < j2and in the supercritical regime a bound state emerges form the continuum for any θ6=π 40 Coulomb impurities in graphene For all other levels we the following relation: lim α→|j|− EII n(θ) = EIII n(0) = lim α→|j|+ EIV n(θ0); (2.54) whenever θ6=π 4and θ06=π 2. This means that, fixed any values of θ6=π 4and θ06=π 2,EII n(θ) and EIV n(θ0), converge to EIII n(0) as α→α=|j|, pointing out the continuity of the flow of energy levels in the transition from the subcritical regime to the critical one (See figure 2.4). In the exceptional 1.49 1.50 1.51 1.52 Α 0.0 0.2 0.4 0.6 0.8 Em Figure 2.4: Energy level flow in charge impurity αfor boundary conditions θ=3π 4, and θ0= 0, for j= 3/2 (Λ = m/10). cases we also have continuity in the path crossing the transition border lim α→|j|− EII n(π 4) = EIII n(π 2) = limα→|j|+EIV n(π 2),(2.55) provided we choose the suitable values for the parameter of the boundary condition in the different regimes. In Figure (2.5) the αflow of energy levels of His displayed for EII n and EIV nfor different values of θ∈[0,1) for j=3 2. The top and the bottom curves correspond to two consecutive levels EII n(π 4), EIV n(π 2) and EII n+1(π 4), EIV n+1(π 2). Notice that the flow is unstable around these isolated 2.3. Bound states and the spectral flow 41 lines. The energy levels converge at α2=j2to energy levels of EIII n(π 2) (red dots). The flows of all the other energy levels reach one of the energy levels of EIII n(0), which correspond to the Hydrogenoid and meta-Hydrogenoid atomic spectrum (black dots). This situation is similar for every energy level of EIII n(0). All bound state energy levels, except the isolated ones are attracted by the levels Hydrogenoid levels of EIII n(0). In figures (2.6) 1.4995 1.5000 1.5005 1.5010 Α -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Em Figure 2.5: Flow of energy levels with angular momentum j= 3/2 (Λ = m/10) when the impurity charge crosses from subcritical regime to supercritical regime. and (2.7) we show the instability of the isolated flow lines. The central flux lines correspond, respectively to θ=π 4and θ=π 2, while the others correspond to small perturbations of these lines, respectively θ=π 4±0.001 and θ=π 2±0.005. We can see how, approaching to α2=j2, the perturbed curves follow the isolated lines flow but eventually they are attracted by two different eigenvalues of EIII n(0). In summary, the above analysis in terms of boundary conditions shows that in graphene we have infinite set of self adjoint Dirac operator for any α > 0 (for j=1 2) which are parameterized by an angle θ∈[0, π). Unitarity is guaranteed for any value of the charge impurity α. Even more, there is an extra parameter θwith observable consequences generated by 42 Coulomb impurities in graphene 1.49996 1.49998 1.5 Α 0.0 0.1 0.2 0.3 0.4 0.5 Em Figure 2.6: Instability properties of the flow of EII n(θ) for θ=π 4and θ±=π 4±0.001 (up/down) and j= 3/2 (Λ = m/10). 1.5 1.5002 1.5004 Α 0.0 0.1 0.2 0.3 0.4 0.5 Em Figure 2.7: Instability properties of the flow of EIV n(θ) for θ=π 2and θ±=π 2±0.005 (up/down) and j= 3/2 (Λ = m/10). the renormalization of the singular UV of the impurity. The dependence on the choice of boundary conditions at the singularity defines a RG flow of energy levels. The analysis of the boundary flow points out interest- 2.3. Bound states and the spectral flow 43 ing physical properties. Changes of the θparameter which characterizes the self adjoint extension we can bring each Hydrogenoid level in the next one after a recursive loop in the parameter space. All energy levels in the Hydrogenoid spectrum, except the fundamental one, are degenerate, but the introduction of the parameter θbreaks breaks down this degeneracy. Moreover, it is possible to move, varying α, the energy levels from the subcritical to the supercritical region in a continue way; This is a consequence of the interesting properties of the RG flow for the subcritical and supercritical region. Near the critical charge the energy levels are attracted by the points of the spectra of the Hamiltonian at the critical charge α2=j2 and the particular value of the boundary conditions θ= 0. The attracting Hamiltonian corresponds to the Hydrogenoid atom spectrum at the critical charge. Only few levels remain isolated in a unstable way. This point out in particular that the critical charge α2=j2of the Hydrogenoid case is not singular, the theory is well defined below and above this critical charge in the subcritical and supercritical regimes. The transition from the subcritical to the supercritical regime does not imply a change in the physical description of the system. However, the apparent stability of the vacuum pointed out by the careful analysis of the boundary conditions of the Hamiltonian does not hide that the physical behaviour of graphene is quite special in the supercritical phase. The fact that Hydrogenoid energy levels become complex in the supercritical regime implies that the presence of resonances in the spectral density of the scattering matrix in the positron (hole) channel. These resonances are also the root of bound states levels which emerge from the continuum negative spectrum E < −m(see Fig. (2.3)). In the supercritical regime there is an infinite number of quasi-bound states embedded in the lower continuum E < −mwhich are visible in the spectral density. If they are not filled as we cross the supercritical value some normal electrons will jump into these empty levels generating particle/hole pairs. The positive charge will move to infinite and disappears whereas the negative sticks localized near the impurity giving rise to a screening of the impurity charge. We have assumed a positively charged impurity but due to the CP invariance of the theory a similar phenomenon occurs for negative charged 44 Coulomb impurities in graphene impurities. In QED we expect the same phenomenon to occur. In a superheavy nucleus with high enough Z, the hydrogenoid bound levels can merge with the positron continuum. When that happens, the bound energy level becomes degenerate with the positron states and by the Schwinger mechanism an electron-positron pair can be created since there is no energy cost involved in creating an electron-positron pair. The electron occupies the bound level close to the nucleus, while the positron escapes to infinity. Since the positron continuum constitutes the vacuum of QED, the spontaneous creation of an electron in a level below E=−mleads to a restructuration of the vacuum and an effective screening of Coulomb charge. The main difference of the same phenomenon in graphene and QED is that the value of the critical charge in graphene is α=jwhereas in QED is Z= 137 which is very hard to realize in Nature. The screening phenomenon due to supercritical pair creation has been recently observed in graphene and in QED a similar phenomenon has been observed in the lead-lead collisions. There is another remarkable difference with the QED case. Graphene for any value of α > 0 is a subcritical regime which requires an extra parameter to fix the boundary condition at the origin. However, in QED for Z < 118, e.g. for the Hydrogen atom the Hamiltonian is essentially selfadjoint. Thus there is no need to fix the boundary condition at the origin. In particular a δlike perturbation has no effect in the spectrum. This means that the relativistic interpretation of the Lamb effect requires a full field theoretical analysis, unlike in the non-relativistic approach. Thus, although the basics of the Gribov hypothesis is confirmed the emergence of quark confinement in QCD requires a stronger mechanism going beyond the simple (anti)screening of the color charge of the quarks. 3 Coulomb Instabilities in heavy quark backgrounds In the previous chapter we analyzed the problem of supercritical charges in Coulomb backgrounds. In spite of the unitary time evolution governed by Dirac Hamiltonian we found some traces of instability when particle creation is permitted. This was just the hint that motivated Gribov to raise his conjecture about quark confinement. In this chapter we address the derivation of Gribov’s confinement scenario from first principles, i.e. from Euclidean functional integral defined by the QCD Lagrangian. In particular we consider a heavy static quark background. The Coulomb potential is a saddle point configuration of the partition function and we analyze its stability properties under gauge field fluctuations. 46 Coulomb Instabilities in heavy quark backgrounds 3.1. Euclidean functional integral in one quark background The basic dynamical fields of QCD are a SU(3) gauge field and three dynamical quarks in the fundamental representation of SU(3) fluctuating a Minkowski space-time with metric gµν = diag(+ −−−). The SU(N) Yang Mills action is given by SY M =1 2g2 sZd4x Tr(FµνFµν),(3.1) where Fµν =∂µAν−∂νAµ+ [Aµ, Aν],(3.2) gsis the coupling constant and Aµ(x) the gauge field with values on a traceless anti-hermitian N×Nmatrix. The theory is invariant under gauge transformations U(x) A0 µ(x) = U(x)AµU−1(x)−(∂µU(x))U−1(x),(3.3) with values on SU(N). The transformation rule of the field streght Fµν is covariant F0 µν =U(x)FµνU†(x), which guarantees the gauge invariance of (3.1). It is convenient to use a basis of N−1 generators Taof the Lie algebra su(N) of SU(N), i.e a basis of anti-hermitian, traceless matrices that satisfy the commutation relations [Ta, Tb] = fabcTc,(3.4) with fabc antisymmetric, and the normalization condition Tr[TaTb] = −1 2δab.(3.5) In terms of these generators any gauge transformation U(x) can be given as as U= exp[waTa], 3.2. Gluonic instabilities of the Coulomb phase 53 where Dµτν=∂µτν+ [Aµ, τν]. Keeping only second order terms in the variation of the Lagrangian Tr(F0 µνF0 µν) = T r(FµνFµν)+2Tr(Fµν(Dµτν−Dντµ)) +2Tr(Fµν[τµ, τν]) + Tr((Dµτν−Dντµ)2). we get for the action δ(2)SE=−1 2Zd4x Tr(Dµτν−Dντµ)2+ 2Fµν[τµ, τν]. By integrating by parts and using the cyclic property of the trace T r(Fµν[τµ, τν]) = (τν[Fµν, τµ]), it can be rewritten as δ(2)SE=−Zd4x Trτµ(−δµνD2+DµDν−2adFµν )τν,(3.26) where adFµν mean [Fµν,·]. Let us now considerthe imaginary Coulomb background (3.2): D2=(∂0+ [A0,·])(∂0+ [A0,·]) + (∂i+ [Ai,·])(∂i+ [Ai,·]) =∂2 0+ [∂0A0,·] + 2[A0, ∂0·]+[A0,[A0,·]] + ∂2 i+ [∂iAi,·] + 2[Ai, ∂i·]+[Ai,[Ai,·]] =∂2 0−Φ2ad2 T3+ 2iΦadT3∂0+M, where M=P3 i=1 ∂i∂i, DµDν= (∂µ+ [Aµ,·])(∂ν+ [Aν,·]) =∂µ∂ν+ [∂µAν,·]+[Aν, ∂µ·]+[∂νAµ,·]+[Aµ,[Aν,·]] =∂µ∂ν+i(δν0∂µΦ + δν0Φ∂µ+δµ0Φ∂ν)adT3−δµ0δν0Φ2ad2 T3, Fa µν =δa3i(δν0∂µΦ−δµ0∂νΦ) and ⇒adFµν =i(δν0∂µΦ−δµ0∂νΦ)adT3. 54 Coulomb Instabilities in heavy quark backgrounds Finally, we obtain δ(2)SE=−Zdx4Trτµ[−δµν(∂2 0+M) + ∂µ∂ν+δµjδνjΦ2ad2 T3(3.27) +i(−2δµνΦ∂0+ 2δν0∂µΦ+Φδν0∂µ+ Φ δµ0∂ν−δµ0∂νΦ)adT3]τν The existence of possible negative quadratic contributions generating instabilities of the Coulomb potential background, is reduced to the existence of negative eigenvalues of the second order variations operator [−δµν(∂2 0+M) + ∂µ∂ν+δµjδνjΦ2ad2 T3(3.28) +i(−2δµνΦ∂0+ 2δν0∂µΦ+Φδν0∂µ+ Φ δµ0∂ν−δµ0∂νΦ)adT3] In presence of negative eigenvalues the saddle point method contribution in the Gaussian approximation might lead to a complex contribution to the partition function. Using the expansion in Cartan components (3.11) for the variations and remembering the relations (??), the eigenvalue equations of τ±read ∂µ∂ν−δµν(∂2 0+M)τ± ν−δµjδνjΦ2τ± ν(3.29) ±(−2δµνΦ∂0+ 2δν0∂µΦ+Φδν0∂µ+ Φ δµ0∂ν−δµ0∂νΦ)τ± ν=−λ2τ± µ. Now using the facts ∂0Φ=0,and ∇Φ = −α r3r, and restricting ourselves to static fluctuations (∂0τµ= 0), we get −Mτ± 0±α r[∇·τ±+1 r2τ±·r] = −λ2τ± 0, ∇(∇·τ±)−Mτ±−α r2τ±±α r[−2 r2τ± 0r+∇τ± 0] = −λ2τ±.(3.30) 3.2. Gluonic instabilities of the Coulomb phase 55 3.2.1. Critical coupling constant Considering only pure magnetic perturbations 1 τ±=x׈ n |x|2φ±(|x|), τ± 0= 0,(3.31) where ˆ nis a unit vector, instabilities are defined the following eigenvalues equation, in spherical coordinates: h−d2 d r2+2−α2 r2iφ±(r) = −λ2φ±(r),(3.32) where r=|x|.φ±satisfy the same equation. In order to restrict the fluctuation to real gauge fields we have to impose the condition (φ+)∗=φ− (3.12). The index ±will omitted from now on to simplify the notation. The singularity introduced by the Coulomb potential has a very special behavior. The corresponding operator within the bracket in eq. is symmetric acting on functions vanishing at the origin but it is not self-adjoint. In order to have real eigenvalues, it must be a self adjoint operator. This situation is well known and it has been reviewed in Appendix A, where we introduce an novel way of renormalizing the singularity. The results are the following. In general, the self adjoint extensions are defined in terms of boundary conditions at the singularity. There are four regimes, depending on the value of the coupling constant. i) For α2≤5 4the operator is essentially self-adjoint and has a unique self-adjoint extension with boundary Dirichlet conditions at the singularity. In this case the spectrum is continuum and positive. The absence of negative eigenvalues means that in this region of the coupling constant the Coulomb potential is stable. For α > 5 4there is a one parameter family of boundary conditions which give rise selfadjoint extensions. This parameter Λ ∈[0,∞) has dimension 1In Appendix B we consider the general solution 56 Coulomb Instabilities in heavy quark backgrounds of mass [L]−1. ii) For 5 4< α2<9 4there is a one parameter family of selfadjoint extensions given by the boundary conditions lim r→02rφ0(r)−(1 + 2νcoth[νlog(Λr)])φ(r)= 0, where ν=q9 4−α2. Apart from the positive continuum spectrum, in this case there is also one negative eigenvalue −λ2=−4Λ2Γ[1 + ν] Γ[1 −ν]1 ν , corresponding to the eigenstate φ(r) = r1 2Kν(λr). iii) For α2=9 4the self adjoint boundary boundary conditions are lim r→02r(log(Λr)−1)φ0(r)−(log(Λr) + 1)φ(r)= 0, Also in this case there is one negative eigenvalue −λ2=−4Λ2e−2−2γ, corresponding to the eigenstate φ(r) = r1 2K0(λr). For 5 4< α2≤9 4the arbitrary parameter Λ, introduced by the boundary conditions in order to guarantee the Hermiticity of the second-order variations operator, breaks conformal invariance which is crucial for having negative eigenvalues. This region characterizes a regime of weakly instability for the Coulomb background potential. For 5 4< α2<2 the unstable mode diverges at the singularity, for α2= 2 it approaches to a constant, while 2 < α2≤9 4it goes to zero. 3.2. Gluonic instabilities of the Coulomb phase 57 iv) Finally for α2>9 4the family of self adjoint extension with boundary boundary conditions lim r→02rφ0(r)−(1 + 2|ν|cot[|ν|log(Λr)])φ(r)= 0. give rise to operators with an infinity negative eigenvalues. −λ2 n=−4Λ2exp 2iπn ν+1 νlog Γ[1 + ν] Γ[1 −ν], corresponding to the eigenstates φn(r) = r1 2Kν(λnr). with n= 0,±1,±2. . .. Thus, in this case we have a strong instability of the Coulomb potential for the heavy quark background and again the parameter Λ introduced by the boundary condition breaks conformal invariance, but not completely since a discrete conformal symmetry is preserved. The boundary conditions and the spectrum are invariant under the discrete rescaling Λ →Λe2πi ν. The unstable modes in this region oscillate approaching the singularity. In figure 3.1 the unstable modes of the different regimes of the coupling constant are displayed. The picture emerging from this analysis is that for small αthe Coulomb potential is the leading contribution to the partition function. In that regime the Coulomb background is a good approximation for the system, but when we reach a critical value of the coupling constant α=q5 4, six degenerate unstable modes appear: one for each axis and each color different of the quark color. The partition function calculated by the steepest descent method acquires a factor i6=−1. As consequence the free energy, being the logarithm of the partition function, becomes complex: this can be seen as the smoking gun of the instability of the Coulomb phase. In other words the for large values of the coupling constant, the Coulomb solution is no more a relevant configuration for the system, opening the window to confinement. 58 Coulomb Instabilities in heavy quark backgrounds 0 1 2 3 4 5 6 -2 2 4 6 Figure 3.1: Unstable modes for α= 1.4 (green), α= 1.49 (red) and α= 3 (blue) for Λ = 5 The instability of the Coulomb phase is intrinsically associated to the breaking of conformal symmetry. In perturbation theory this emerges from the renormalization of the coupling constant α. In this picture it arises from the need of fixing the boundary conditions of the singularity of quark potentials. The novelty is that in this case it implies the instability of the Coulomb vacuum background for large enough coupling constant. A similar critical coupling, precisely α=3 2, was founded from previous analysis of the stability of the Coulomb solution in classical Yang Mills field theory in presence of external source. That critical coupling coincides with our value marking the transition from the weakly to the strong instability regime of the Coulomb potential. However, the connection of the picture with real confinement is not yet clear because one quark background alone does not match the neutrality of color. For this reason in the next chapters will be analyzed what happen in presence of a quark-antiquark pair. 4 Two quarks in one dimension Until now we have analyzed the instability of the Coulomb potential generated by a single point charge. However, a single quark background does not correspond to a consistent physical system since it does not match the color neutrality condition. For this reason it is convenient to introduce an antiquark, in order to get a neutral system with the same quantum numbers than a meson. An important novelty of the introduction of an antiquark will be the presence of a new parameter in the theory, the distance between the q−¯qpair. Before dealing with this problem in three dimensions, we will analuze a simpler system: a pair q−¯qin one space dimension, expecting to shed light and understand how the distance between the quarks can affect the stability of the system. The relevance of the analysis of this simplified model stands in the fact that it can be solved analytically and many of its properties can hold in the generalization to three dimensions. In order to analyze the situation with neutral quark-antiquark pair, we consider the equation (3.2.1) in the presence of two external sources 60 Two quarks in one dimension of opposite charge located at x=Land x=−L. What we are going to study is then the bound states of the following one dimensional Schr¨odinger equation "−d2 d x2−α21 |x−L|−1 |x+L|2#φ(x) = −λ2φ(x).(4.1) The aim is to analyze the dependence of the bound states spectrum on the separation 2Lbetween the q−¯qpair. Due to the singularities of the quarks sitting at x=±L, we have to fix proper boundary conditions at x=±Lto have a well defined self adjoint operator. They can be given in terms of asymptotic zero modes near the singularities. The equations of these zero modes near x=±L, are "−d2 d x2−α21 |x−L|2−1 L|x+L|2#φ0(x)=0. The solutions are φ± 0>(x) = p|x−L|I±2ν2α √Lp|x−L|, for x=Land φ± 0<(x) = p|x+L|I±2ν2α √Lp|x+L|, for x=−L, where ν=q1 4−α2. We recall the expansion of the Bessel I function near x±L: I±2ν2α √Lp|x∓L|≃α2 L|x∓L|±ν1 Γ[1 ±2ν]+α2x LΓ[2 ±2ν]+. . . . (4.2) In the vicinity of each singularity, both ±νsolutions are normalizable at the origin for α > 0. Since the term x3 2−νand its derivative go to zero as 4.1. Critical size of q−¯qpairs 61 x→0 for α≥0, only the first term is relevant. The most general boundary conditions which leads to a self adjoint Hamiltonian is given by (A.11): lim x→Lφ0(x)φΛ>(x)−φ(x)φ0 Λ>(x)= 0, lim x→−Lφ0(x)φΛ<(x)−φ(x)φ0 Λ<(x)= 0,(4.3) where φΛ>(x) = p|x−L|(Λ|x−L|)ν−(Λ|x−L|)−ν, φΛ<(x) = p|x+L|(Λ|x+L|)ν−(Λ|x+L|)−ν(4.4) for 0 < α26=1 4, and φΛ>(x) = p|x−L|(1 −log (Λ|x−L|)) , φΛ<(x) = p|x+L|(1 −log (Λ|x+L|)) (4.5) for α2=1 4. Notice that in principle it is possible to choose different parameters approaching to the left or to the right at each singularity. For symmetry reasons, it has been chosen the same for the four cases. 4.1. Critical size of q−¯qpairs The singularities introduced by the q−¯qpair at the points x=±Lact like infinite barriers between the left and right domains. For this reason, the complete problem in x∈(−∞,+∞) can be split into three different problems defined in the subdomains: x < −L,−L < x < L and x > L. These three problems can be analyzed separately: the spectrum of the complete problem will be the union of the spectra of these three regions. First we analyze the region x > L. Due to parity symmetry, the results obtained in this region can be translated immediately to the region x < −L. It is convenient to introduce a change of variable, x→x+L, such that the equation becomes: "−d2 d x2−α21 x−1 x+ 2L2#φ(x) = −λ2φ(x),(4.6) 62 Two quarks in one dimension defined for x > 0. The boundary conditions (4.3) reduce to: lim x→0+φ0(x)φΛ(x)−φ(x)φ0 Λ(x)= 0,(4.7) where φΛ(x) = x1 2(Λx)ν−(Λx)−ν.(4.8) for α26=1 4, and φΛ(x) = x1 2(1 −log(Λx)) .(4.9) The situation gets very much simplified in the limit L→ ∞ where equation (4.6) reduces to the case of a single quark. This is natural, if we move one of the quarks to infinity there is only one quark left in the space. The spectral equation then becomes h−d2 d x2−α2 x2iφ(x) = −λ2φ(x),(4.10) As it is expected this correspond to the situation already analyzed of one quark (3.2.1), with the only replacement of α2→α2+ 2. For 0 < α < 1 2 and α=1 2there is one bound state with energy respectively −λ2=−4Λ2Γ[1 + ν] Γ[1 −ν]1 ν ,(4.11) and −λ2=−4Λ2e−2−2γ,(4.12) while for α > 1 2there are infinite bound states, characterized by the energies −λ2 n=−4Λ2exp 2iπn ν+1 νlog Γ[1 + ν] Γ[1 −ν].(4.13) with ν=q1 4−α2. Another limit where we can obtain analytic results is in the analysis of zero modes (λ= 0), i.e. solutions of (4.6) with λ2= 0, h−d2 d x2−α21 x−1 x+ 2L2iφ(x)=0,(4.14) 4.3. Symmetric and antisymmetric bound states 69 with µ=q1 4−4α2and ν=q1 4−α2, where Lη,λ(x) are the associated Legendre functions of the first kind. In figure 4.4 these solutions are plotted for α= 0.4. Expanding near the singularities and keeping only the diver- -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 Figure 4.4: Symmetric (Red) and antisymmetric (blue) zero mode for α= 0.4 gent terms, it is possible to see that the boundary conditions are satisfied if Lc s=1 2Λ Γ(2ν+ 1)Γ 1 2−µ−2νΓ1 2+µ−2ν(cos(πµ) + sin(2πν)) πΓ(1 −2ν)!−1 2ν (4.21) for the symmetric solution and Lc a=1 2Λ Γ(2ν+ 1)Γ 1 2−µ−2νΓ1 2+µ−2ν(cos(πµ)−sin(2πν)) πΓ(1 −2ν)!−1 2ν (4.22) for the antisymmetric one. Lc sand Lc arepresent respectively the critical distance where the symmetric and antisymmetric bound states disappear. The dependence of these two critical distances on the coupling constant α is displayed in Figure 4.5. We remark that the antisymmetric one is always 70 Two quarks in one dimension 0.1 0.2 0.3 0.4 0.5 Α 0.10 0.15 0.20 0.25 0.30 Lc Figure 4.5: Red and blue lines represent the dependence of the critical distances on αfor α2<1 4and Λ = 5 for the symmetric and antisymmetric bound states on the domain between the two quarks bigger than the symmetric one. If we begin at large separations between the two quarks and start to decrease L, the antisymmetric bound state will disappear before the symmetric one does it. On the other hand if we begin with short distances between the quarks and star to move them appart the symmetric bound state will appear before the antisymmetric one . This suggests that the energies of the symmetric bound state are always less negative respect the energies of the antisymmetric one. In Figure 4.6 we display the values of both energy levels for α= 0.4 and Λ = 5 Even if in this case it does not work a fit of the type −λ2(L) = c11+c2arctan(c3L), from the numerics results that the eigenvalue at L= 103is −λ2=−29.219 in agreement with the analytic value for infinite separation 29.225, while for L= 0.126 and L= 0.309 respectively the symmetric and the antisymmetric eigenvalues are very close to zero (-0.058 and -0.09) in agreement with the analytic critical distance Ls= 0.126 and La= 0.308. In the region α2>1 4there are instead an infinity of symmetric −λ2 s,n and antisymmetric −λ2 a,n bound states, labeled by an integer n. For large Lthese states become degenerate, −λ2 s,n =−λ2 a,n =−λ2 n, and the quotient 4.3. Symmetric and antisymmetric bound states 71 2 4 6 8 10 L -25 -20 -15 -10 -5 -Λ2 Figure 4.6: Negative energy of symmetric (red dots) and antysymmetric (blue dots) bound states locate in the domain between the q−¯qpair for α= 0.4 and Λ = 5 between two successive is λn+1/λn= exp[2π νi] is constant. Decreasing the separation between the quarks increases the gap between the eigenvalues, which never cross each other until both reach their zero values at two different critical distances. The analytic formulas for the symmetric Lc s,n and Lc a,n critical distances can be obtained in the same way as before and the results are plotted in figure 4.7. In the region α2<1 4, for each couple of symmetric and antisymmetric degenerate bound states, the corresponding symmetric critical distance is always lower than the antisymmetric critical distance (Ls,n < La,n, for each n). Again it is possible to deduce the same analytic properties between the critical distances of different eigenvalues: Lc s,n Lc s,n−1 =Lc a,n Lc a,n−1 = exp h−iπ νi Lc s,n −Lc a,n Lc s,n−1−Lc a,n−1 = exp h−iπ νi Lc s,n Lc a,n =2 sin (2πν) sec (πµ)+1−11 2ν . 72 Two quarks in one dimension 1.0 1.5 2.0 2.5 3.0 Α 5 10 15 20 Lc Figure 4.7: Critical distance dependence on the coupling constant for symmetric (red lines) and antysymmetric (blue lines) bound states locate in the domain between the q−¯qpair in the supercritical regime α2>1 4(Λ = 5. In summary we have analysys of the problem of a q−¯qpair in one-dimension show us that is the introduction of a new parameter, the separation between the sources, affects the structure of the spectrum in a very interesting way. The fact that the problem is one-dimensional allow us to split it into three subdomains: the external regions respect the singularities and the central region between the two quarks. In the exterior regions, when the two sources are separated by infinite distance the spectrum coincides with the case of one source, studied in the previous chapter, with one bound state for small (α2<1/4) coupling constant or infinite bound states for large (α2>1/4) coupling constant. Reducing the separation between the two quarks all energy levels start to increase until at a particular distance, called critical distance, when they cross the zero. This is the principal novelty: the existence of a distance where the bound states disappear. This critical distance can be found analytically. The regime of small coupling is the 4.3. Symmetric and antisymmetric bound states 73 more interesting: for small separation there are not bound states, for large distance there is four bound states in the spectrum, two symmetric and two antisymmetric. For large coupling constant, even if some bound states disappears, at each distance there are always infinite number of bound states. The importance of this model stands in the fact that it is quite similar to the dynamics of gluonic fluctuations in a q−¯qbackground which points out to a similarity between the negative energy bound states of this problem and unstable modes of the second order variation operator around the Coulomb saddle point. The above results suggest that similar phenomena could occur in the behavior of the Coulomb phase in the quark antiquark system. 74 Two quarks in one dimension 5 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background The analysis of the effects of a q−¯qpair in a simple but very interesting one a dimensional model raise the question of whether similar effects can occur or not in three-dimensional real QCD. A colorless q−¯qpair having the same quantum numbers than a meson can be described by the current J0=iδ3(x+L)T3−δ3(x−L)T3, Ji= 0, i = 1,2,3.(5.1) The two quarks are located along the z axis: L= (0,0, L). The current induced by the heavy pair can be seen as the abelian projection of two Wilson lines extending in the time direction at a distance 2L. Inserting the current into the equations of motion (3.22) DµFµν =g2 sJν, it easy to show the existence of a Coulomb like solution A0(x) = iΦT3,A= 0, 76 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background with Φ = α 1 |x+L|−1 |x−L|!, α =g2 s 4π,(5.2) corresponding to an imaginary chromo-electric dipole potential. This is the classical background where the Euclidean action has be expanded. 5.1. Gluonic instabilities The search of instabilities lead to the study of the real fluctuations on this background. Negative contributions coming from second order variations can give rise to instabilities. The separation between the quarks introduce a new scale in the theory. We have already studied the appearance of instabilities of the Coulomb regime in the case of single external heavy quark. By continuity, these instabilities should persist for large separations, but it is not clear if they hold at all the distances. The technical drawback is that the dipole structure of the q−¯qpair breaks the spherical symmetry and cylindrical symmetry is not enough to have analytical solutions. Gluonic instabilities require the existence of negative eigenvalues of the second order operator which governs the quadratic terms in equation (3.29) ∂µ∂ν−δµν(∂2 0+M)τ± ν−δµjδνjΦ2τ± ν ±(−2δµνΦ∂0+ 2δν0∂µΦ+Φδν0∂µ+ Φ δµ0∂ν−δµ0∂νΦ)τ± ν=−λ2τ± µ, which split as −Mτ± 0±[Φ∇·τ±−τ±·∇Φ] = −λ2τ± 0, ∇(∇·τ±)−Mτ±−Φ2τ±±[2τ± 0∇Φ+Φ∇τ± 0] = −λ2τ±,(5.3) In cylindrical coordinates, ∇Φ=Φ0 ρˆuρ+ Φ0 zˆuz, 5.1. Gluonic instabilities 77 with Φ0 ρ=−α ρ (ρ2+ (z−L)2)3 2−ρ (ρ2+ (z+L)2)3 2!, Φ0 z=−α (z−L) (ρ2+ (z−L)2)3 2−(z+L) (ρ2+ (z+L)2)3 2! Eventually, possible instabilities can arise from static magnetic modes τ±=x׈ ez ρ3 2 φ±(ρ, z), τ± 0= 0,(5.4) where ˆ ezis a unit vector in the direction of the axis connecting the two quarks. We postpone to appendix B the discussion of more general roots of instability. The functions φ±(ρ, z) satisfy the following eigenvalues equation, in cylindrical coordinates: h−∂2 ∂ ρ2−∂2 ∂ z2+3 4ρ2−V(ρ, z)iφ±(ρ, z) = −λ2φ±(ρ, z),(5.5) where V(ρ, z) = α2 1 pρ2+ (z+L)2−1 pρ2+ (z−L)2!2 . Again, to have real gauge fields(3.12), we must require that (φ+)∗=φ−. The equation (5.5) is invariant under the replacement z→ −z: for this reason, eventual solutions will be either symmetric or antisymmetric with respect reflections with respect to the plane z= 0. The form of the unstable modes (5.4) induce a field lying on the plane perpendicular to the axis connecting the quarks and it has non-vanishing component only in the direction perpendicular to the radial direction of this plane. For simplicity we omit from now on the indices ±on the function φ. The potential in (5.5) presents two singularities, situated at the positions of the quarks, and in order to have a self adjoint operator, both must be renormalized by the choice of proper boundary conditions. As usual, 78 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background the self adjoint boundary conditions are given in terms of the zero modes around the singularities ρ≈0 and z≈ ±L. The equations that define the asymptotic zero modes are: h∂2 ∂ ρ2+∂2 ∂ z2−3 4ρ2+ α2 ρ2+ (z±L)2−α2 Lpρ2+ (z±L)2!iφ0(ρ, z)=0. The solutions are φ± 0>(ρ, z) = ρ3 2 (ρ2+ (z−L)2)3/4I±2ν2α √Lpρ2+ (z−L)2 near ρ≈0 and z≈L, and φ± 0<(ρ, z) = ρ3 2 (ρ2+ (z+L)2)3/4I±2ν2α √Lpρ2+ (z+L)2 near ρ≈0 and z≈ −L, with ν=q9 4−α2. i) For α2≤5 4the solutions φ− 0>and φ− 0<are not normalizable near the singularities. The operator is essentially self adjoint with boundary conditions lim ρ→0, z→Lˆ n·∇φ(ρ, z)φ>(ρ, z)−φ(ρ, z)ˆ n·∇φ>(ρ, z)= 0, lim ρ→0, z→−Lˆ n·∇φ(ρ, z)φ<(ρ, z)−φ(ρ, z)ˆ n·∇φ<(ρ, z)= 0,(5.6) where ˆ n·∇is the normal derivative on a cylindric cut-off around the singularities and φ>(ρ, z) = ρ pρ2+ (z−L)2!3 2pρ2+ (z−L)2ν, φ<(ρ, z) = ρ pρ2+ (z−L)2!3 2pρ2+ (z+L)2ν.(5.7) 5.1. Gluonic instabilities 85 8 10 12 14 16 18 LL -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -Λ2L2 Figure 5.5: Fit of the antisymmetric mode by the curve −λ2(L) = −λ2 1Λ21 + c2arctan(c3ΛL)with c2= 1.15 and c3=−0.11 for α= 1.495 the symmetric and the antisymmetric unstable modes are shown. The configuration of the symmetric mode, the one that drive the instability, has a shape of a thick string connecting the two quarks. As example of coupling constant near α=√2 the result obtained for the estimated critical distances for α= 1.42 are Lc sΛ = 67.687 and Lc aΛ = 67.689, respectively, showing the increasing of the critical distance with the coupling constant. In the regime 5 4< α2≤√2 for all Λ except for Λ 6=±∞, by reducing the q−¯qdistance, the two negative eigenvalues start to decrease, and they become more and more negative. For these boundary conditions there is not a critical distance, because for each separation, both symmetric and antisymmetric bound states are negative modes. In summary, in the regime α2≤5 4, as a consequence of the essentially self-adjointness and the conformal symmetry of the second order variation operator, there are not negative modes. For √2< α2≤9 4, breaking the conformal symmetry, a critical distance exists that tends to infinity for 86 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background Figure 5.6: Density plot for symmetric unstable mode at α= 1.495 Figure 5.7: Density plot for antisymmetric unstable mode at α= 1.495 α2= 2. For sizes smaller than a critical distance there are not unstable 5.1. Gluonic instabilities 87 Figure 5.8: 3D plot for symmetric unstable mode at α= 1.495 Figure 5.9: 3D plot for antisymmetric unstable mode at α= 1.495 modes, while for distances larger than the critical distance a symmetric and an antisymmetric unstable modes appear. In order to match in a continue way these two regimes of the coupling constant, in the regime 5 4< α2≤√2 the parameter Λ = +∞is chosen. This is equivalent to do not break 88 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background the conformal symmetry also in this regime and as consequence there are not negative modes at any distance. The infinity of the critical distance approaching to α2= 2 from the right matches perfectly this condition. The full dependence of the critical distance on the coupling constant will considered later. The picture emerging is that for α2≤2 there are not negative modes, while for α2>2 it exist a critical distance which marks the existence of unstable mode. If the conformal symmetry had been broken also in the regime 5 4< α2≤2, it would be a non natural match between the different regimes: for α2≤5 4no negative modes, crossing the critical coupling α2=5 4 there are negative modes at all the distance for 5 4< α2≤2, while for √2< α2≤9 4there are negative modes only if the separation between the quarks is bigger than a critical distance that becomes infinite for α2= 2. 5.1.2. Supercritical regime In the strong coupling regime α2>9 4, for each distance there is an infinite number of unstable modes, half of them symmetric while the other half are antisymmetric. From this viewpoint they can be organized by symmetric-antismmetric pairs. In each pear the symmetric state is s always more negative. The modes of each pair become degenerate for large distances, converging to the unstable modes of the case of one quark in the supercritical regime. They value increases with decreasing the distances, until they become null at the critical distance. Even if each unstable mode disappears by reducing the distance between the quarks, there is always an infinite number of negative modes, marking the strong instability of the Coulomb potential in this regime. Let us consider a particular case with coupling constant α= 2.5 and boundary condition with Λ = 12. In this case, two consecutive eigenvalues of one quark are −λ2=−28.337 and −λ2=−655.733. In the supercritical case the simulation converge faster that in the subcritical case. The reason of this stability is due to the fact that for subcritical αthe solution tends to infinity at the singularity, while for supercritical αthe solution even if oscillates it goes to zero. In this case the simulation are performed by 5.1. Gluonic instabilities 89 setting the cut off at r0= 0.001, the radius of the domain at r∞= 2 and six refinement of the grid. The precision of the eigenvalue (0.1% error) is reached in computation time of 1 minute per point. In figures 5.10 and 5.11 we display the numerical values found for two consecutive pairs of symmetric and antisymmetric eigenvalues. Again, the 8 10 12 14 16 18 20 LL -0.10 -0.08 -0.06 -0.04 -0.02 -Λ2L2 Figure 5.10: Symmetric (red disks) and antisymmetric (blue disks) pair eigenvalues for the supercritical case α= 2.5 points can be fitted with the curves −λ2(L) = −λ2 1Λ21 + c2arctan(c3ΛL), where, for the first pair of eigenvalues, −λ2 1Λ2=−28.337, is fitted by c2=−1.115 (c2=−1.215), c3= 0.103 (d3= 0.115) for the symmetric (antisymmetric) modes, while for the second pair plotted in figure 5.12 and 5.13, −λ2 1Λ2=−655.733 the parameters of the fit are c2=−1.1017 (s2=−1.198), c3= 0.489,0.541 respectively for the symmetric and antisymmetric modes. From the fits we can extract the values of the critical distances. They are given by Lc s,1Λ=7.76 and Lc a,1Λ=8.09 for the symmetric and antisymmetric modes of the first pair and by Lc s,2Λ=1.59 and Lc a,2Λ=1.67 for the symmetric and antisymmetric modes of the second 90 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background 5 10 15 LL -4 -3 -2 -1 -Λ2L2 Figure 5.11: Symmetric (red disks) and antisymmetric (blue disks) pair eigenvalues for the subcritical case α= 2.5 pair of eigenvalues. It is possible to see that the symmetric mode always reaches the zero level at larger distances. It is also interesting to check the relation obtained analytically in the toy model. The ratio between two successive eigenvalues of the one quark case, that are the limit value for infinite separation of two quarks, in the supercritical case is λ2 n+1 λ2 n = exp 2π i ν. For the critical distance we find from the numerical stimations Lc s,1 Lc s,2 = 4.88 and Lc a,1 Lc a,2 = 4.84, that practically coincide with the value exp 2π i ν≃4.81. In figures 5.14, 5.15, 5.16 and 5.17 the symmetric and the antisymmetric unstable modes are shown. It is possible to notice the oscillations near the singularities. 5.2. Analytic appoximation 91 5 10 15 LL -4 -3 -2 -1 -Λ2L2 Figure 5.12: Fit of the symmetric mode by the curve −λ2(L) = −λ2 1Λ21+ c2arctan(c3ΛL)with c2=−1.115 and c3= 0.103 for α= 2.5 5.2. Analytic appoximation Since the results of numerical simulations are very relevant for understanding the mechanism of confinement we would like to have an analytic understanding of the main effects. The behavior of the eigenvalues of the relevant operator are determined by two conditions, the exponential decay at infinity and the boundary conditions at the quark singularities. Once assumed that a normalizable solution at infinity exists, the possible eigenvalues will depend uniquely on the behavior of the solution near the singularities. As the boundary conditions are the same at both quarks, one can reduce to investigate the behavior of the solution near one of them. In the proximity of one quark the effect of the other can be approximated by a 92 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background 5 10 15 LL -4 -3 -2 -1 -Λ2L2 Figure 5.13: Fit of the anti-symmetric mode by the curve −λ2(L) = −λ2 1Λ21 + c2arctan(c3ΛL)with c2=−1.215 and c3= 0.115 for α= 2.5 Figure 5.14: Density plot for the symmetric unstable mode at α= 2.5 5.2. Analytic appoximation 93 Figure 5.15: Density plot for the antisymmetric unstable mode at α= 2.5 Figure 5.16: 3D plot for symmetric unstable mode at α= 2.5 constant, so that the equation (5.5) reduce to  −∂2 ∂ ρ2−∂2 ∂ w2+3 4ρ2− α pρ2+w2−α 2L!2 φ(ρ, z) = −λ2φ(ρ, z), (5.13) 94 Gluonic Instabilities in a Coulomb Heavy Quark-Antiquark background Figure 5.17: 3D plot for antisymmetric unstable mode at α= 2.5 where w=z±L. This introduces a new via to analytically address the spectral problem. It is possible to understand the approximation also from another viewpoint: the standard procedure to find the eigenvalues is to find the a normalizable solution and then to expand it near the singularity in order to match the boundary condition. The strategy is to interchange these two steps, making first the expansion of the equation near the singularity, and taking the limit ρ→0 and z→ ±Lwithout discarding any term, and then to find a solution of this simplified equation. In general it is possible to find an analytic solution of this simplified equation, on which the boundary condition can be imposed. In the intermediate regime 5 4< α2≤9 4and for α < 2Lλ there is one real solution decaying at infinity, given in terms of Whittaker function: φ(ρ, w) = ρ3 2 ρ2+z2W −α √4L2λ2−α2, ν, √4L2λ2−α2 Lpρ2+w2!. (5.14) When the the real solution is concentrated near the quark, more this approximation is more reliable. The approximation is more accurate then for large eigenvalues. But even if when the eigenvalue start to increase and the 6 Light quark instabilities The critical value of coupling constant, αc=√2 where the gluon instabilities arise is larger than Gribov’s critical value, αc= 3π(1 −p2/3)/4≃ 0.43. In the Gribov scenario of confinement a fundamental role is played by light quarks in destabilization of the Coulomb phase. In simple terms, the Coulomb phase instability would imply a vacuum decay into light quarkantiquark pairs. In order to derive this picture from first principles one has to consider dynamical quarks with small masses. In this chapter we analyze the source of possible instabilities coming from the fermionic sector of the action. The contribution of dynamical quarks to vacuum energy is given by the determinant of Dirac operator i/ D+m: Z=ZDA exp[−SY M E(A)−SY E(A)] det i/ D+m. The operator i/ D+mis not selfadjoint because it is the sum of an antiselfadjoint operator i/ Dand a selfadjoint one m. Thus, all eigenvalues are of the of the form iλ +mwith λreal. Now, since i/ Danticommutes with 102 Light quark instabilities γ5, i.e. {γµ, γ5}= 0, for any eigenvalue iλ of i/ Dthere is another one of the form −iλ. Therefore, the fermionic determinant can be expressed as det(i/ D+m) = Y λ (iλ +m)(−iλ +m) = Y λ (λ2+m2)>0, which is strictly positive. In the saddle point approximation, including one loop corrections, the partition function is given by Z(1) = exp[−S0]det δ(2)SY M E−1 2det i/ Dc+m, where Ac µ=iΦ(x)δµ0T3, Φ(x) being the Coulomb potential generated by the external heavy quarks, and / Dcdenotes the covariant derivative in a Coulomb background / Dc=γµ(∂µ+iA0 µ).(6.1) The imaginary Coulomb background might break the stability properties. In fact some eigenvalues iλ can become real, implying that the Coulomb phase would become unstable when iλ =−m. This would introduce a new source of vacuum instability generated by the fluctuations of the dynamical quarks. The root of possible instabilities are the singularities of Coulomb potentials which require the introduction of consistent boundary conditions, similar to those introduced in chapter 2 for graphene. 6.1. Quarks instabilities in a Coulomb heavy quark background Let us multiply the Dirac operator in Coulomb background byiγ0on the left, i.e. iγ0(i/ Dc+m) (6.2) Notice that the only color dependent term of the operator / Dcis the Coulomb potential term. Stationary spinor eigenvalues of the operator (6.2) are of the form Ψ = (Ψ+(x),Ψ−(x))eiEx0, 6.1. Quarks instabilities in a Coulomb heavy quark background 103 where the two SU(2) color components of Ψ±(x) satisfy the following equations iEΨ±(x) + ±Φ(r) 2−γ0γi∂i+m iγ0Ψ±(x) = 0. Since the operator within the brackets is hermitian with real eigenvalues η, considering both positive and negative energies we get det iγ0(i/ Dc+m)=Y η (iE +η)(−iE +η) = (E2+η2)>0, unless E=η= 0. Due to the identity det(iγ0) det(i/ Dc+m) = det iγ0(i/ Dc+m), it follows that det(i/ Dc+m) is well defined (not imaginary nor negative). The only possible pathologies can occur when E=η= 0, i.e. when there is a zero mode of the Dirac operator iγ0(i/ Dc+m). Let us consider only the second color Ψ−(x) component of the spinor Ψ and omit the color subindex to simplify the notation. A parallel analysis of the other component will give similar results. The zero modes are then solutions of the −iγ0 Mγi∂i−Φ(r) 2+mγ0Ψ = HcΨ=0,(6.3) where we reintroduce the Minkowski γ0 M=−iγ0matrix, to show that the zero mode problem is reduced to find out the zero-modes of the Dirac Hamiltonian Hcwith Coulomb potential in Minkowski space-time. Then, the techniques used to find the Dirac spectrum in the graphene can reintroduced. However, we know that the Hydrogen spectrum does not have zero energy bound states, but in the strong coupling regime when we need to introduce extra the boundary condition around the Coulomb singularities it can be possible. Let us consider the case of a single heavy quark. Then, Φ(r) = α r. Because to the spherical symmetry the total angular momentum and the parity operators commute with the Hamiltonian. To built such a eigenstates 104 Light quark instabilities of definite energy we can use the appropriate combinations of states with definite angular momentum and spin. A stationary Dirac spinor can be defined in terms of spherical spinors Ωj l mjfrom the following ansatz Ψjm = Gj(r)Ωj l mj iFj(r)Ωj l0mj,!(6.4) with l0= 2j−l:l0=l+ 1 for j=l+1 2and l0=l−1 for j=l−1 2. Explicitly: Ωl+1 2 l mj=1 √2j pj+mjYl mj−1 2 pj−mjYl mj+1 2!,(6.5) for j=l+1 2and Ωl−1 2 l mj=1 √2j+ 2 −pj+ 1 −mjYl mj−1 2 pj+1+mjYl mj+1 2!,(6.6) for j=l−1 2. Let us introduce the orbital angular momentum L=−iˆ r×∇ and spin-orbit K=I+σ·Loperators. Now using the following relations σ·ˆ rΩj j±1 2mj=−Ωj j∓1 2mj,(6.7) and (2 + L·σ)Ωj l mj= (1 −k)Ωj l mj(2 + L·σ)Ωj l0mj= (1 + k)Ωj l0mj(6.8) where k=∓(j+1 2) for j=l±1 2it is possible to show that σi∂iG(r)Ωj l mj=−Ωj l0mjd dr +k+ 1 rG(r),(6.9) and σi∂iF(r)Ωj l0mj=−Ωj l mjd dr −k−1 rF(r).(6.10) 6.1. Quarks instabilities in a Coulomb heavy quark background 105 From (6.3) we find a relation between G(r) and F(r) G0(r) + 1 + k rG(r)−α 2r+mF(r) = 0 (6.11) F0(r) + 1−k rF(r) + α 2r−mF(r)=0,(6.12) where we omitted the index j for simplicity. Self adjoint Boundary conditions Because the UV singularities introduced by the external quarks in the Dirac Hamiltonian Hcoperator defined in (6.3), we have to introduce boundary conditions that the dynamical quark fields have to satisfy at the singularities. To obtain a well defined self adjoint operator Dirac operator we have to follow the steps carried out in the two dimensional case of chapter 2. First we introduce an spherical cut-off of radius r0around each singularity. Let us assume that the quark is located at the origin. The most general boundary condition is given by a unitary operator U(r0) defined on the S2cut-off sphere of radius r0 (1 + ˆ n/)Ψ(r0) = U(r0)γ0(1 −ˆ n/)Ψ(r0), where ˆ n/=r0/r0is the unit normal vector to the cut-off boundary S2sphere. If the boundary condition has to preserve rotation invariance U(r0) has to be an infinite diagonal matrix of U(1) phases in the angular momentum decomposition of the boundary spinor space, i.e. (1 + ˆ n/)Ψj(r0)=eiβ0 jγ0(1 −ˆ n/)Ψ(r0).(6.13) Now, to define the ultraviolet limit by removal of the cut-off r0we follow the the same prescription as in chapter 2. First we analyze the asymptotic zero modes of Dirac Hamiltonian Hc. Depending on the strength of the coupling constant αwe have three different regimes. 106 Light quark instabilities Case I) α2<2(j+1 2)2−1 For α2<4(j+1 2)2−1 there is just one asymptotic zero mode given by G0(r) = C+r−1+ν F0(r) = r−1+ν(6.14) with C+=2(k−ν) αand ν=qk2−α2 4. Case II) α2>4(j+1 2)2−1, α26= 4(j+1 2)2 In this case there is a family of asymptotic zero modes parameterized by Λ with the dimension of the inverse of the distance 1 G0(r) = C+(Λr)−1+ν±C+(Λr)−1−ν F0(r) = (Λr)−1+ν±(Λr)−1−ν(6.15) C±=2(k∓ν) αand ν=qk2−α2 4. Case III) α2= 4(j+1 2)2 In that case G0(r) = r−1k |k| 1±log(Λr)−1 k! F0(r) = r−11±log(Λr).(6.16) In the case I the existence of only one asymptotic zero-mode implies that the operator Hcis essentially selfadjoint and we do not need to impose an extra boundary condition. In fact, in this case the spectrum coincide 1In fact there are more posibilities if α2>4(j+1 2)2−1 because we can introduce a complex phase eiθ instead of ±sign flip as we considered in the graphene case of chapter 2. Here we assume that θ=nπ for simplicity and then we allows to merge the cases II and IV of chapter 2 6.1. Quarks instabilities in a Coulomb heavy quark background 107 with the Hydrogen one, where we already know that there are not solution with zero energy. Thus, the Coulomb phase is stable under light quark fluctuation in this regime. In the other two case we need to remove the cut-off, sending r0to zero. We follow the RG flow shown by the asymptotic zero modes. We can associate a zero mode to the boundary condition defined by β0 jin (6.13) by means of the relation ei β0 j=Fj 0(r0) + iGj 0(r0) Fj 0(r0)−iGj 0(r0).(6.17) If we consider a smaller cut-off r< r0we can use boundary condition which is given by a running phase βj defined following the flow of zero mode (Fj 0, Gj 0) associated to the initial phase βj 0at r0by the relations (6.17), ei βj =Fj 0(r) + iGj 0(r) Fj 0(r)−iGj 0(r).(6.18) More explicitly, we define the boundary condition by the following relation lim r→0Fj(r)Gj 0(r)−Gj(r)Fj 0(r)= 0,(6.19) derived from (6.17) and (6.17). Once that we have fixed the boundary conditions which ensure that Hc is a selfadjoint operator we can search for zero modes on its spectrum. The existence of non trivial boundary conditions might make possible of normalizable zero modes in the regime α2>4(j+1 2)2−1 of (6.11) and (6.12), for very special values of the parameter Λ labeling the boundary condition. Let us introduce a new variable x= 2mr and following ansatz G(x) = 1 rexp h−x 2iφ1(x) + φ2(x), F(x) = 1 rexp h−x 2iφ1(x)−φ2(x).(6.20) 108 Light quark instabilities This allows to decouple the two equations and give rise to a second order equation for φ1(x) φ00 1(x) + 1 x−1φ0 1(x) + α2−4k2 4x2−1 xφ1(x)=0 and a constrain for φ2(x) in terms of φ1(x): φ2(x) = 2x α+ 2kφ1(x)−φ0 1(x). Let us fix the total angular momentum to its lower value j=1 2, so that k=±1 There localize solutions in terms of the Hypergeometric function U: φ1(x) = xνU(1 + ν, 1+2ν, x), φ2(x) = 2 α+ 2kxνU(ν, 1+2ν, x), Using the asymptotic expansion of the Hypergeometric function for x≃0: xνU(1 + ν, 1+2ν, x)≃Γ[−2ν] Γ[1 −ν]xν+Γ[2ν] Γ[1 + ν]xν xνU(ν, 1+2ν, x)≃Γ[−2ν] Γ[−ν]xν+Γ[2ν] Γ[ν]x−ν, it is easy to show that in the regime II for 3 < α2<4 the solution matches the boundary conditions only for one value of Λ Λc=m 2 ±(α+ 2ν+k)Γ[1 2−ν] (α−2ν+k)Γ[1 2+ν]!1 2ν , for k=±1, respectively. In n the regime III (α2= 4) there is only on solution for k= 1 and Λc= 2m e−1 2+γ. 6.2. Quarks instabilities in heavy quark-antiquark background 109 Finally, for α2>4 in the regime II there is an infinity discrete series of values of Λ Λn c=m 2 ±(α+ 2ν+k)Γ[1 2−ν] (α−2ν+k)Γ[1 2+ν]!1 2ν exp πi νn, with n= 0,±1,±2, . . . and both signs of k=±1. The light quark fluctuations only make Coulomb regime is only unstable on this cases and for those particular boundary conditions. For any types of the boundary conditions parameter the Coulomb background of one heavy quark is stable under dynamical quark fluctuations. However, as pointed out above, one quark background does not correspond to a physical quantum state because global gauge invariance is not preserved in such a background. For this reason we shall consider next a more realistic background of a pair of heavy quark antiquark. 6.2. Quarks instabilities in heavy quark-antiquark background In the Coulomb background generated by a quark and antiquark, the potential in the operator (6.3) becomes, according (3.2): Φ = α1 |x+L|−1 |x−L|, with L= (0,0, L). The search of instabilities is reduced to find solutions zero modes of the Dirac Hamiltonian (6.3) in such a background. Even if they cannot be found analytically there is an analytic argument which shows the existence of these zero-modes. As in the case of gluon fluctuations a good approximation of the spectral problem is to reduce, in a neighborhood of one quark (ρ≃0, z ≃ ±L), the effect of the other to a constant, so the background potential becomes: Φ = α 1 pρ2+w2−1 2L!,(6.21) 110 Light quark instabilities where w=z±L. Equation (6.3) than becomes: −iγ0γi∂i−α 2r+α 4L+mγ0Ψ = 0, which is equivalent to the problem of finding a bound state with energy E=−α 4Lof the Dirac equation in a Coulomb potential with α0=α 2. Again, we know that such a negative energy solutions do not exist in the hydrogen atom spectrum, but the existence of new types of boundary conditions opens new posibilities. Using the same techniques as in the one quark backgrounds we can start with the same ansatz for the spinors (6.4). A similar analysis, leads to the following equations for G(r) and F(r) G0(r) + 1 + k rG(r)−α 2r−α 4L+mF(r) = 0 (6.22) F0(r) + 1−k rF(r) + α 2r−α 4L−mF(r)=0.(6.23) The singularity at r= 0 is renormalized with the self adjoint boundary condition (??), where G0(r) and F0(r) are the asymptotic zero modes obtained by neglecting the non diverging terms of (6.22) and (6.23). In fact, they are the same as those of one quark background (6.14), (6.15) and (6.16). This is nice because the structure of the quarks should not be dependent on how many other quarks are in the space. The interesting observation is that since now we have a new scale given by the distance between the quarks 2Lit might be possible that for any boundary condition one can find a critical distance where the Dirac Hamiltonian present a zero mode. In the regime I (α2≤2(j+1 2)2−1) this is impossible, as the problem is equivalent to find bound states with negative energies of the hydrogen spectrum. The search of bound states with negative energies must be then focused to the regimes II and III for α2>2(j+1 2)2−1. Again we introduce the change of variable x= 2r,=qm2−α2 16L2and 6.3. Alternative approach to search instabilities 117 In the regime II α2>3 the normalizable solutions are given in terms of Hypergeometric functions: φ±(r) = e− rr±νUα2 8L + 1 ±ν, 2±2ν, 2r,(6.31) where =√16m2L2−α2/4L. Once Ψ+is found, Ψ−can be calculated following the same steps from (6.29) to (6.30). We can write Ψ+in a different way Ψ+=O1(r)Ωj j−1 2mj−i O2(r)Ωj j+1 2mj, where O1(r) = A φ+(r) + B φ−(r), O2(r) = 2 αA c+φ+(r) + B c−φ−(r), with φ±now given by (6.31). Finally, we get Ψ−=f O1(r)Ωj j−1 2mj−if O2(r)Ωj j+1 2mj, where now f O1(r) = 1 m −α 4L+α 2rO1(r)− d dr +(j+1 2)+1 r!O2(r)!, and f O2(r) = 1 m −α 4L+α 2rO2(r) + d dr +−(j+1 2)+1 r!O1(r)!, Requiring again that the solution has definite parity O1(r) = ±f O1(r) and O2(r) = ∓f O2(r), leads fixes the ratio B A=−16m2L2−α2 4L2−να2±4αLm −(1 + 2j+ 2ν)√16m2L2−α2 α2±4αLm −(1 + 2j−ν)√16m2L2−α2 118 Light quark instabilities and implies that G(r) = C O1(r),and F(r) = −C O2(r), with k=−(j+1 2), while in the second one F(r) = C0O1(r),and G(r) = C0O2(r), with k= (j+1 2) (Cand C0are constant). Thus, the solution found by this method does coincide with (6.24) founded with the other approach, and there are no extra solutions. The derivation also shows that both methods are completely equivalent. 6.4. The role of light quark fluctuations The Gribov approach to confinement is based on the instability in heavy quarks Coulomb backgrounds for values of the effective coupling constant larger than αs≃0.43. The instability would imply a vacuum decay into light quark-antiquark pairs. In the previous chapters, from first principles, it has been shown the existence of an instability of the Coulomb phase in pure gauge theories for α > √2, much beyond the Gribov critical value. However, Gribov assigned a leading role to lighter dynamical quarks in the confinement mechanism. For this reason, the last chapter has been dedicated to analyze the effects of dynamical quarks in the instability of the Coulomb phase. The results show the appearance of new instabilities in the fermionic sector beyond a new value of the critical coupling constant α=√3. The mechanism driving the instability is the spectral flow of the eigenvalues induced by the introduction of a dimensional parameter Λ in the self adjoint boundary conditions. Considering first one heavy quark Coulomb background, for each α > 3. We have been shown the existence of a discrete number of parameters at which zero modes of the light quarks determinant appear. In the presence of zero modes, the partition function becomes zero and this implies an infinite amount of energy in the system, the signal of the creation of a quark antiquark pair, inducing the instability of Coulomb 6.4. The role of light quark fluctuations 119 phase. In the heavy quark antiquark Coulomb background it has been shown that for each α∈(√3,2] and for each value of the parameter of the boundary conditions, there exists a critical distance between the pair where the zero modes appear. The situation is different respect the instabilities due to gluonic fluctuation, where the instabilities hold for an interval of distances. In the case of fermionic fluctuations only at particular values of the separation between the pair. In particular, the critical distance decreases at large coupling constant, in agreement with the idea of permanent confinement for all distances for large value of the coupling constant. For α > 2 also there is an infinite discrete values of distances at which the zero modes appear. However, the value of the coupling constant where the fermionic instabilities begin to arise α=√3 is larger than the critical value α=√2 where the pure gauge fluctuations start to develop negative modes. This means that the fundamental instability is due to the boson sector instead of light quark as it was advocated by Gribov. These results provides further consistency to the picture where quark confinement is mainly driven by gluon fluctuation instabilities. 120 Light quark instabilities A Boundary conditions for singular potentials Let us consider the differential operator H=−d2 dx2−a x2.(A.1) His a very special Hamiltonian with a conformal invariant singular potential. Let us analyze the theory of boundary conditions which induce self adjoint extension of (A.1) over the whole half line (0,∞). Due to the singularity, the self adjoint extensions have to be given in terms of asymptotic boundary conditions. 122 Boundary conditions for singular potentials A.1. Regularization by a space cut off x0 Let us consider a space cut-off x0>0 is introduced near the singularity. On the half line (x0,+∞) any self adjoint extension of Hβis defined by the restriction of H†to a domain of functions ψsatisfying the following boundary conditions eiβ0=ψ(x0)−iψ0(x0) ψ(x0) + iψ0(x0),(A.2) where β0∈[0, π) is an arbitrary angle. Notice that even if the wave functions ψare complex the value ψ0 ψ(x0) = tanβ/2 is always real. This is a requirement due to hermiticity. To define an asymptotic selfadjoint Hamiltonian in the whole half line we have to renormalize the initial phase β0in the process of removing the UV cut-off x0. An convenient ingredient in that limit is given by the asymptotic zero modes that encode the divergent behavior of the operator H. They are defined by −d2 dx2−a x2ψ0(x)=0.(A.3) The key observation is that as the cut-off goes to zero, the function belonging to the domain of the self adjoint extension behaves as zero mode near the singularity. In the limit, the boundary condition (A.2) must be then satisfied by the zero modes. The solutions of (A.3) are ψ0(x) = x1 2±ν for a6=1 4, and ψ0(x) = x1 2and ψ(x) = x1 2log(Λx) for a=1 4, with ν=q1 4−aand Λ a fixed scale parameter with dimension [L]−1. There are different regime depending on the value of the parameter a. Notice that νis real for a≤1 4and imaginary for a > 1 4. A.2. Renormalization of the phase 123 i) For a≤ −3 4the solution ψ0(x) = x1 2−νis not normalizable at x= 0 and must be excluded. The most general zero mode is simply given by ψ0(x) = Nx1 2+ν(A.4) with N arbitrary constant. For a > −3 4both solutions are normalizable and the zero mode is in general a linear combination of the two behaviors. The general zero mode, such that ψ0 0(x) ψ0(x)is real, is given by: ii) for −3 4<a<1 4 ψ0(x) = N x1 2cos θ(Λx)ν−sin θ(Λx)−ν,(A.5) iii) for a=1 4 ψ0(x) = N x1 2(cos θ−sin θlog(Λx)) ,(A.6) iv) for a > 1 4 ψ0(x) = N x1 2exp(−iθ)(Λx)ν+ exp(iθ)(Λx)−ν,(A.7) where N is an arbitrary constant, Λ fixed parameter with dimension [L]−1 and t∈[0,1). A.2. Renormalization of the phase The idea is, once fixed the cut-off x0, to associate to the phase β0a zero mode defined by (A.4), (A.5), (A.6) and (A.7) (β0, x0)→ψ0, through the definition ψ0(x0)−iψ0 0(x0) ψ0(x0) + iψ0 0(x0):= eiβ0,(A.8) 124 Boundary conditions for singular potentials It is easy to see that the left term is a well defined phase. More precisely, the relation (A.8) defines a class of equivalence of zero mode, as the normalization constant is factorized. The explicit relation depends on the constant a. i) For a≤ −3 4there is just one family of zero mode. This means that at each cut-off there is not the freedom to choose the phase. The phase and the cut off are connected by the following formula. x0=−1+2ν 2Λy This is telling us that there is a unique self adjoint extension for a≤ −3 4. For a > −3 4as a family of zero modes exists, there is the freedom to choose β0 fixed x0. For simplicity we define ζ= tan θ, the parameter characterizing the zero mode. Fixed x0, to each β0∈[0,2π) is assigned a zero mode through the following relations: ii) for −3 4<a<1 4 ζ= 1+2ν+ 2 tan β0 2Λx0 1−2ν+ 2 tan β0 2Λx0 (Λx0)2ν, iii) for a=1 4 ζ= 1 + 2 tan β0 2Λx0 2 + (1 + 2 tan β0 2Λx0) log(Λx0) , iv) for a > 1 4 ζ=−|ν|tan(|ν|log(Λx0)) + Λx0(1 + tan β0 2) |ν|+ Λx0(1 + tan β0 2) tan(|ν|log(Λx0)) . After having defined the boundary condition in a cut-off domain through a phase at which we have assigned a zero mode, the next step is to remove A.2. Renormalization of the phase 125 the cut-off. At a different cut-off the boundary conditions are determined by a new phase β: it must be given a procedure how to change the phase reducing the cut-off in terms of the initial phase. Being ψ0the zero mode, defined by t, associated to the initial phase β0at x0, the phase βat different is renormalized according eiβ():= ψ0()−iψ0 0() ψ0() + iψ0 0().(A.9) The idea is then that the initial phase define a zero mode and the phase run keeping fix the zero mode. Schematically: (β0, x0)→ψ0→β(). The explicit form of the renormalization function for βdepends by the value of the charge. We define y= tan β 2, the parameter to be renormalized. The renormalization function are: i) for a≤1 4 y=−1+2ν 2Λ ii) for 3 4<a<1 4 y=−1 + ν 2Λ2 + 4 tan θ (Λ)2ν−tan θ iii) for a=1 4 y=−1 2Λ1 + 2 tan θ zlog(Λ)−1 iv) for a > 1 4 y=−1+2|ν|tan π t +|ν|log(Λ) 126 Boundary conditions for singular potentials A.3. Boundary conditions and bound states Let us define Has a self adjoint operator on all (0,∞). Once started with the boundary condition (A.2) on the cut-off x0, decreasing it is imposed that it must be satisfied eiβ =ψ()−iψ0() ψ() + iψ0(),(A.10) with βrunning according (A.9). Explicitly, in the limit →0, the boundary conditions (A.10) that define the domain of the self adjoint extensions are: lim x→0ψ(x)ψ0 0(x)−ψ0(x)ψ0(x)= 0.(A.11) Resuming, the self adjoint extensions on (0,∞) are defined by boundary conditions in terms of zero mode. For a > −3 4the family of self adjoint extensions is parameterized by the parameter t∈[0, π): (β0, x0)→ψ0→β()→s.a. H. Explicitly: i) for a≤1 4 lim x→0(1 2+ν)ψ(x)−xψ0(x)= 0, ii) for 3 4< a < 1 4 lim x→0(Λ)2ν(µ+ψ(x)−xψ0(x)) + tan(π t)(µ−ψ(x) + xψ0(x))= 0 where µ±=1 2±ν, iii) for a=1 4 lim x→0 1−tan θ(2 + log(Λx))ψ(x)+2xtan θlog(Λx)−1ψ0(x)!= 0 133 with n= 1,...j. It iss easy to show that the previous unstable mode (3.31) correspond to j= 1. In fact in that case τ+ 1=τ− 1=b(r)sin θsin ϕˆex−sin θcos ϕˆey τ+ 2=−τ− 2=i b(r)sin θsin ϕˆex−sin θcos ϕˆey τ+ 3=τ− 3=b(r)cos θˆex−sin θcos ϕˆez τ+ 4=τ− 4=i b(r)cos θˆex−sin θcos ϕˆez τ+ 5=−τ− 5=b(r)−cos θˆey+ sin θcos ϕˆez τ+ 6=−τ− 6=i b(r)−cos θˆey+ sin θcos ϕˆez. The modes proportional to Y220 are, in the approach (3.31), τ±=x×ez |x|2cos θ φ±(|x|), τ± 0= 0, Let us to show that, however, it is not possible to have unstable modes with lower angular momentum. For j= 0 the ansatz is instead τ± 0=f±(r)Y00 τ±=b±(r)Y010.(B.10) We arrive at the two following coupled equations: 4f±(r)∓α rb0±(r) + 3 rb(r)=−λ2f(r) b(r) = ∓r2 α2−r2λ2f0(r) + 2 rf(r). These two equation are incompatible: notice the singularity of b±for r=α r that make the function not normalizable. This shows the impossibility of having unstable modes with j=0. 134 General solution for gluonic instabilities B.1. Quark-antiquark Coulomb background Let us understand the origin of the instability of the Coulomb regime in the case of a heavy q−¯qpair. The most general ansatz for the variations of the fields, compatible with the cylindrical symmetry of the problem is: τ± 0=T± m(ρ, z)eimφ τ±=R± m(ρ, z)eimφ ˆuρ+F± m(ρ, z)eimφ ˆuφ+Z± m(ρ, z)eimφ ˆuz We insert the variations in the equations (5.3): using the identity ∇×(∇× τ) = ∇(∇·τ)−Mτ, we get 4τ± 0=1 ρ ∂T± m ∂ρ +∂2T± m ∂ρ2−m2 ρ2T± m+∂2T± m ∂z2eimφ, ∇τ± 0=∂T± m ∂ρ eimφ ˆuρ+im ρT± meimφ ˆuφ+∂T± m ∂z eimφ ˆuz, ∇·τ±=∂R± m ∂ρ +R± m ρ+im ρF± m+∂Z± m ∂z eimφ, ∇×(∇×τ±) = im ρ2∂(ρF± m) ∂ρ −imR± m−∂ ∂z ∂R± m ∂z −∂Z± m ∂ρ eimφ ˆuρ+ ∂ ∂z im ρZ± m−∂F± m ∂z −∂ ∂ρ 1 ρ∂(ρF± m) ∂ρ −imR± meimφ ˆuφ+ 1 ρ∂ ∂ρ ρ∂R± m ∂z −∂Z± m ∂ρ −im 1 ρimZ± m−∂F± m ∂z eimφ ˆuz, The eigenvalues of these gluonic fluctuations are solutions of the four equations 1.1 ρ ∂T± m ∂ρ +∂2T± m ∂ρ2−m2 ρ2T± m+∂2T± m ∂z2±φ∂R± m ∂ρ +R± m ρ+ +im ρF± m+∂Z± m ∂z −R± mφ0 ρ−Z± mφ0 z=−λ2T± m B.1. Quark-antiquark Coulomb background 135 2.im ρ2∂(ρF± m) ∂ρ −imR± m−∂ ∂z ∂R± m ∂z −∂Z± m ∂ρ −φ2R± m± 2T± mφ0 ρ+φ∂T± m ∂ρ =−λ2R± m 3.∂ ∂z im ρZ± m−∂F± m ∂z −∂ ∂ρ 1 ρ∂(ρF± m) ∂ρ −imR± m− φ2F± m±φim ρT± m=−λ2F± m 4.1 ρ∂ ∂ρ ρ∂R± m ∂z −∂Z± m ∂ρ −im 1 ρimZ± m−∂F± m ∂z − φ2Z± m±2T± mφ0 z+φ∂T± m ∂z =−λ2Z± m Setting T±=R±=Z±= 0, m= 0 and F±=ρ−1/2φ±(ρ, z) we obtain the unstable modes (5.4) and from 3) we obtain a decoupled equation which is exactly (5.5). This is the only possible decoupled equation. Any attempt to to obtain an equation for the other component always fails. Thus all unstable moes in the q−¯qbackground are generalizations of the magnetic field perturbation (5.4). 136 General solution for gluonic instabilities Conclusions In the search of an analytic proof of confinement in QCD, based on Gribov conjecture, we have found: 1. The original Gribov motivation based on the unitarity loss of QED in a Coulomb supercritical regime is not completely correct. The Coulomb phase is stable even in the supercritical regime. The Dirac Hamiltonian is selfadjoint. The only remarkable feature is that for large values of the Coulomb charge (Z > 118) one needs to select a boundary condition near the Coulomb singularity. The choice of boundary condition is labeled by a dimensionful parameter Λθ, which introduces an anomalous breaking of conformal symmetry. 2. The same physical phenomena occurs in 2-dimensional condensed matter systems with lower values of the critical charge. In particular, in the analysis of graphene impurities the critical charge can attain as low values as Z= 1. A detailed analysis of the supercritical regime including many body interactions gives rise to a screening phenomenon of the charge impurity. The theoretical predictions are in agreement with recent experimental data concerning this phenomenon. 3. In the saddle point approximation to Yang-Mills theory in the presence of external quarks, we found that the Coulomb solutions become unstable for values of the strong coupling constant αslarger than √5/2. The instability is due to the appearance of negative modes in the gluon fluctu- 138 General solution for gluonic instabilities ations around the Coulomb gauge field configuration. 4. The same analysis shows that the Coulomb regime is also unstable in a background of heavy q−¯qquarks. In that case for coupling constants αs>√2 the same phenomenon occurs, but it becomes strongly dependent on the distance between the quarks of the q−¯qpair. For any choice of boundary condition there is a critical distance Lcsuch that, the Coulomb background is stable for q−¯qpairs with size lower than 2Lcand become unstable for larger sizes L>Lcof the q−¯qpair. 5. In the intermediate regime √2< αs<3/2 the compatibility of asymptotic freedom and confinement is explicit. The critical distance approaches at infinite for αs=√2. At small couplings α < √2 the Coulomb phase is always stable and at large couplings α > 3 2it is always unstable, no matter what the size of q−¯qpair is. In the intermediate regime the stability properties depend on the distance between the q−¯qpair, pointing out the existence of a smooth interpolation from an asymptotic freedom regime to a confinement regime. These are the first analytic indications derived from first principles that QCD does not undergo a phase transition at intermediate energy scales. 6. It is remarkable that the fluctuations leading to the instability of the Coulomb regime exhibit a prominent thick string connecting the two quarks which points out to a picture of QCD where confinement would be driven by thick strings rather than by fundamental strings. 7. In Gribov’s confinement picture a prominent role is assigned to dynamical light quarks in the confinement mechanism. Our analysis of the contribution of light quarks fluctuations in a Coulomb background pointed our the existence of new type of fermionic instabilities. They only appear for couplings larger than αs=√3. For each coupling bigger than this new critical value (√3< αs≤2) there exists a distance where the vacuum energy of the system becomes infinite. This is the signal of quark antiquark pair creation, that breaks the chromoelectric string and induces the insta- B.1. Quark-antiquark Coulomb background 139 bility of Coulomb phase. 8. However, the fact that fermionic instabilities appear at larger values of the coupling constant than in the bosonic case. points out that the confinement mechanism is mostly driven by gluons rather than light quarks, unlike in the scenario advocated by Gribov. 9. The new results confirm most of the features of the Gribov scenario and our derivation from first principles opens a new avenue to the analytic proof of quark confinement. 140 General solution for gluonic instabilities Conclusiones En la b´usqueda de una demostraci´on anal´ıtica del confinamiento en QCD basada en la conjetura de Gribov hemos encontrado: 1. La motivaci´on original de Gribov basada en la p´erdida de unitariedad de QED en un r´egimen de Coulomb supercr´ıtico no es del todo correcta. La fase de Coulomb es estable incluso en el r´egimen supercr´ itico. El hamiltoniano de Dirac es autoadjunto. El ´unico detalle relevante es que para valores grandes de la carga de Coulomb (Z > 118) es necesario seleccionar una condici´on de contorno cerca de la singularidad de Coulomb. La elecci´on de la condici´on de contorno se puede etiquetar por un par´ametro con dimensiones λque introduce una ruptura an´omala de la simetr´ıa conforme. 2. El mismo fen´omeno f´ısico se produce en sistemas bidimensionales de materia condensada, pero con valores m´as bajos de la carga cr´ıtica. En particular, en el an´alisis de impurezas en el grafeno, la carga cr´ıtica puede alcanzar valores tan bajos como Z= 1. Un an´alisis detallado del r´egimen supercr´ıtico teniendo en cuenta interacciones de muchos cuerpos da lugar a un fen´omero de apantallamiento de la carga de la impureza. Las predicciones te´oricas sobre este fen´omeno coinciden con datos experimentales recientes. 3. En la aproximaci´on del punto de silla a la teor´ıa de Yang-Mills en presencia de quarks externos, encontramos que las soluciones de Coulomb se vuelven inestables para valores constantes del acoplamiento fuerte αsmay- 142 General solution for gluonic instabilities ores que √5/2. La inestabilidad se debe a la aparici´on de modos negativos en las fluctuaciones del glu´on alrededor de la configuraci´on de Coulomb del campo gauge. 4. El mismo an´alisis muestra que el r´egimen de Coulomb tambi´en es inestable en un fondo de quarks q−¯qpesados. En ese caso, para constantes de acoplamiento αs>√2 ocurre el mismo fen´omeno, pero se vuelve fuertemente dependiente de la distancia entre los quarks del par q−¯q. Para cualquier elecci´on de la condici´on de contorno hay una distancia cr´ıtica Lc tal que el trasfondo de Coulomb es estable para pares q−¯qcon tama˜no m´as peque˜no que 2Lcy se vuelve inestable para tama˜nos mayores L > Lc. 5 . En el r´egimen intermedio √2< αs<3/2, la compatibilidad entre libertad asint´otica y confinamiento es expl´ıcita. La distancia cr´ıtica se aproxima al infinito para αs=√2. Para acoplamiento m´as peque˜no αs<√2 y acoplamiento m´as grande αs>3 2, la fase de Coulomb es respectivamente estable e inestable para todas las separaciones. As´ı, el r´egimen intermedio interpola suavemente desde un r´egimen de libertad asint´otica a un r´egimen de confinamiento. ´ Estos son los primeros indicios anal´ıticos derivados de primeros principios de que QCD no atraviesa una transici´on de fase a escalas de energ´ıa intermedias. 6. Resulta sorprendente que las fluctuaciones que conducen al r´egimen de inestabilidad de Coulomb exhiban una cuerda gruesa prominente que conecta los dos quarks, lo que apunta a una imagen de QCD en la que el confinamiento ser´ıa debido a cuerdas gruesas en lugar de a cuerdas fundamentales. 7. En la imagen de confinamiento de Gribov, se asigna un papel destacado a los quarks ligeros en el mecanismo de confinamiento. Nuestro an´alisis de la contribuci´on de las fluctuaciones de los quarks ligeros al trasfondo de Coulomb se˜nala la existencia de un nuevo tipo de inestabilidades fermi´onicas. ´ Estas aparecen s´olo para acoplamiento mayor que αs=√3. Para cada acoplamiento mayor que este nuevo valor cr´ıtico (√3< αs≤2)