An effective Lagrangian approach to describe the dynamics of confinement and
symmetry breaking in the process of quark-gluon to hadron conversion is
proposed. The deconfined quark and gluon degrees of freedom of the perturbative
QCD vacuum are coupled to color neutral condensate fields representing the
non-perturbative vacuum with broken scale and chiral symmetry. As a first
application the evolution of gluons emitted by a fragmenting high energy $q\bar
q$ pair from the perturbative to the non-perturbative regime with confinement
is studied. For reasonable parameter choice the solution of the equations of
motion leads to flux tube configurations with a string tension $t \simeq 1$
GeV/fm.Comment: CERN-TH. 7409/94, 9 pages, 4 figures, postscript encoded with uufile

Geiger, K.

English

arXiv

An effective Lagrangian approach to describe the dynamics of confinement and symmetry breaking in the process of quark-gluon to hadron conversion is proposed. The deconfined quark and gluon degrees of freedom of the perturbative QCD vacuum are coupled to color neutral condensate fields representing the non-perturbative vacuum with broken scale and chiral symmetry. As a first application the evolution of gluons emitted by a fragmenting high energy q\bar q pair from the perturbative to the non-perturbative regime with confinement is studied. For reasonable parameter choice the solution of the equations of motion leads to flux tube configurations with a string tension t \simeq 1 GeV/fm.An effective Lagrangian approach to describe the dynamics of confinement and symmetry breaking in the process of quark-gluon to hadron conversion is proposed. The deconfined quark and gluon degrees of freedom of the perturbative QCD vacuum are coupled to color neutral condensate fields representing the non-perturbative vacuum with broken scale and chiral symmetry. As a first application the evolution of gluons emitted by a fragmenting high energy $q\bar q$ pair from the perturbative to the non-perturbative regime with confinement is studied. For reasonable parameter choice the solution of the equations of motion leads to flux tube configurations with a string tension $t \simeq 1$ GeV/fm

CERN Document Server

CERN-TH. 7409/94DYNAMIC GLUON CONFINEMENTIN HIGH ENERGY PROCESSESWITHIN EFFECTIVE QCD FIELD THEORYK. GeigerCERN TH-Division, CH-1211 Geneva 23, SwitzerlandAbstractAn eective Lagrangian approach to describe the dynamics of connement and symme-try breaking in the process of quark-gluon to hadron conversion is proposed. The deconnedquark and gluon degrees of freedom of the perturbative QCD vacuum are coupled to colorneutral condensate elds representing the non-perturbative vacuum with broken scale andchiral symmetry. As a rst application the evolution of gluons emitted by a fragmentinghigh energy qq pair from the perturbative to the non-perturbative regime with connementis studied. For reasonable parameter choice the solution of the equations of motion leadsto ux tube congurations with a string tension t ' 1 GeV/fm.e-mail: klaus@surya11.cern.chCERN-TH. 7409/94, August 19940The transformation of color charged quarks and gluons in high energy QCD processesinto colorless hadrons is commonly believed to be a local phenomenon. The universalityof parton fragmentation observed in hard processes as e+e annihilation, deep inelastic epscattering, Drell Yan, etc., is strongly supported by experimental observations. Thus, aconsistent description of the local hadronization mechanism must be independent of thedetails of the partons prehistory and should apply also to hadron-hadron, hadron-nucleus,or nucleus-nucleus collisions.To date most of the theoretical tools to study properties of QCD are inadequate todescribe the dynamics of the transformation from partonic to hadronic degrees of freedom:Perturbative techniques are limited to the deconned, short distance regime of high energypartons [1], QCD sumrules [2] and eective low energy models [3] are restricted to thelong distance domain of hadrons, and lattice QCD [4] lacks the capability of dynamicalcalculations concerning the quark-gluon to hadron conversion. On the other hand, phe-nomenological approaches to parton fragmentation [5], are mostly based on hadronizationmodels with adhoc prescriptions to simulate hadron formation from parton decays.In this letter I will propose a universal approach to the dynamic transition betweenpartons and hadrons based on an eective QCD eld theory description. The key element isto project out the relevant degrees of freedom for each kinematic regime and to embody themin an eective QCD Lagrangian which recovers QCD with its scale and chiral symmetryproperties at high momentum transfer, but yields at low energies the formation of symmetrybreaking gluon and quark condensates including excitations that represent the physicalhadrons. I will rst formulate the general eld theoretical framework. Subsequently I shalldemonstrate its applicability to the dynamics of parton-hadron conversion by studyingexemplarily the evolution of gluons produced by a fragmenting quark-antiquark pair. Theaim of this letter is to give a lucid presentation of the conceptual framework, details of thework reported here will be given elsewhere [6].The eective QCD Lagrangian is represented in the formL := L[ ;A] + L[; U ] + L[ ;A; ] : (1)HereL[ ;A] =  14F;aFa+  ii@  gsAaTa i+ Lgauge+ Lghost(2)1is the QCD Lagrangian in terms of the quark ( i) and gluon elds (Aa) and Fa= @Aa @Aa+ gsfabcAbAc. The eective Lagrangian [7]L[; U ] =12(@)(@) +14Tr(@U)(@Uy)  c03Trmq(U + Uy)  b1440+ 4lne1=40 1204m2020; (3)introduces a scalar gluon condensate eld  and a meson eld U = fexpiP8j=0jj=ffor the nine pseudoscalar meson elds j(f= 93 MeV), with non-vanishing vacuumexpectation values 0= h0jj0i 6= 0 and U0= ch0jU + Uyj0i 6= 0 that explicitly breakscale, respectively chiral symmetry. In (3), c is a constant, mq= diag(mu; md; ms) and bis related to the conventional bag constant B by B = b40=4. The key ingredient in (1) isthe connection between the scale and chiral symmetric, short distance regime of coloreductuations and the world of colorless hadrons with broken symmetries. It is mediated bythe coupling between the fundamental quark and gluon degress and the eective elds and U through coupling functions g() and (),L[ ;A; ] =()4F;aFa   ig() i: (4)The couplings are chosen [6] in the spirit of the Friedberg-Lee model [8] as g() = g0=( 1)and () = ()v3(3v   4) with v = =0which generate absolute connement of quarksand gluons [9], here intimately connected with the formation of the gluon condensate 0and the qq condensate U0. The non-linear couplings have the properties that  = g = 0 for = 0 at short distances, but in the long range regime  ! 1, g! 1 as ! 0.The specic form of L  L[; U ] + L[ ;A; ] is motivated by the dual QCD vacuumpicture [8]: High momentum, short distance quark-gluon uctuations (the perturbativevacuum) are embedded in a conning background eld  (the non-perturbative vacuum),in which by denition the low momentum, long range uctuations are absorbed. In analogyto a thermodynamic system in equilibriumwith a heat bath there can be a net ow of energybetween the system and the heat bath environment such that the bare energy of the systemis not conserved. However the free energy of the system, here high momentum quarks andgluons, is constant [10]. It corresponds to the conserved physical energy momentum tensorwith the well known trace anomaly [11]:= (1 + m)Xqmqqq +(s)4sFF; (5)2where mis the anomalous mass dimension and  the QCD beta function. This anomalyconstraint is modeled [7] by the third and fourth term in (3) with the correspondenceU0= h0jqqj0i and  B40= h0j(=2gs)FFj0i.The following remarks concerning the eective QCD Lagrangian (1) are important:First, at short distances the exact QCD Lagrangian is recovered, since  = U = 0 and = g = 0, whereas the long distance QCD properties emerge as =0; U=U0! 1 [7] andno colored quanta survive. Second, the problem of double counting degrees of freedom hasto be carefully inspected. Although it does not arise in one-loop calculations (to whichI will restrict here), processes with e.g. two-gluon exchange could also be contained inthe exchange of a color singlet -quantum. A minimal possibility to avoid this problemis a rigid separation of high and low momentum modes, in a way that it is exclusive butcomplementary. Third, L[; U ] embodies the correct QCD scaling properties. It accountsfor the anomaly (5) without which Poincare invariance would be broken, and the massof the proton would come out wrong, since 2m2p= hpjjpi. Fourth, there is no need forrenormalization of L[; U ] since  and U are already interpreted as eective, compositeelds with loop corrections implicitely included.The eective QCD eld theory dened by (1) is readily applicable to describe thedynamic evolution from perturbative to non-perturbative vacuum in high energy processes.In the following I shall consider as an example the fragmentation of a qq jet system with itsemitted bremsstrahlung gluons and describe the evolution of the system as it converts fromthe parton phase to the hadronic phase. A time-like virtual photon in an e+e annihilationevent with large invariant mass Q2 2is assumed to produce a qq pair which initiates acascade of sequential gluon emissions. The early stage is characterized by emission of "hot"gluons far o mass shell in the perturbative vacuum. Subsequent gluon branchings yield"cooler" gluons with successively smaller virtualities, until they are within Q20, where Q0isof the order of m 4b20' 1 GeV. At this point condensation sets in, or loosely speaking,the "cool" gluons are eaten up by the color neutral scalar condensate eld  representingthe non-perturbative vacuum. Restricting the analysis to the gluonic sector, the equationsof motions that derive from (1) reduce to:@[()Fa] =  gs()fabcA;bFc(6)3@@ =  @V ()@ @()@F;aFa(7)@@U =   4 c03Tr[mq] ; (8)where  = 1    and V () is the potential evident from (3). The key problem is the rstequation, since the gluon elds drive the dynamics of the -eld which in turn feeds backvia (). Note that the U -eld does not couple directly to the gluon eld. The procedurein the following is therefore to 'solve' eq. (6) as a function of  and then to insert thesolution into (7), so that one is left (aside from the simple third equation) with a singlenon-linear equation for . As a simplication I shall work in the mean eld approximation,i.e. separating o the quantum uctuations of the -eld,  = + ^, where  is a c-numberand ^ a quantum operator (similarly for U).Solving the equation of motion (6) for the gluon eld is equivalent to the calculationof the complete Greens function with an arbitrary number of gluons, a formidable taskindeed. Instead, I will restrict to evaluate the 2-point Greens fuction only (Fig. 1), i.e.the full gluon propagator which includes both the gluon self-interaction through real andvirtual emission and absorption, and the eective interaction with the conning backgroundeld  [6]. In the framework of "jet calculus" [12], this gluon propagator describes howa gluon, produced with an invariant mass Q2, evolves in the variable x and the virtualityk2(or transverse momentum k2?) through these interactions. In one-loop approximationit is obtained by calculating the corresponding single loop cut diagrams. In the presentcase, one has in addition to the usual gluon branching and fusion processes, g ! gg andgg ! g, contributions from energy transfer and two gluon annihilation processes g ! gand gg! , respectively. This is illustrated in Fig. 1.The determining equation for the propagator corresponds an evolution equation for thegluon distribution, which is dened [13] as average number of gluons at 'light cone time'r+= 0 in the multi-gluon state jP i with light cone fractions x  k+=P+in a range dx andtransverse momenta in a range d2k?with respect to the jet axis:x g(x; k2?) =1P+Zdr d2r?(2)3e i(k+r  ~k?~r?)hP jF+a(0; r ; ~r?) Fa; +(0; 0;~0?) jP i :(9)Performing a standard Mellin transformation to the moment representation g(!; k2?) :=4R10dxx!g(x; k2?), the evolution equation reads:k2?@@k2?g(!; k2?) = (!; k2?) g(!; k2?) ; (10)where (!; k2?) plays the role of a generalized anomalous dimension ( ()2=4),(!; k2?) =s(k2?)2(k2?  Q20)Ag!gg(!)  k2?Agg!g(!)+(k2?)2Ag!g(!)  k2?Agg!(!): (11)In Leading Log approximation one nds:Ag!gg(!) = 2 CA1112+1! 2! + 1+1! + 2 1! + 3  S(!)Agg!g(!) =2!Ag!gg(!)Ag!g(!) =1432 1! + 1 1! + 2  2S(!)Agg!(!) =22!1! + 1 2! + 2 2! + 3: (12)Here CA= Nc,  = O(1) is a parameter, S(!) =  (! + 1)   (1) with the Psi (Digamma)function  (z) = d[ln (z)]=dz and   (1) = E= 0:5772 the Euler constant. Fig. 2 shows(!; k2?) for dierent values of k2?.The formal solution of eq. (10) is obtained by complex integration in the !-plane,x g(x; k2?) =12iZCd! x !exp(ZQ2k2?dk02?k0 2?(!; k02?))(13)which yields the x-distribution of gluons as a function of the evolution variable k2?. Forthe full anomalous dimension (11) this inversion must be done numerically [6]. However,the main features of the evolution of the gluon distribution in the presence of the -eldbecome already evident when making some simple analytic estimates. For simplicity I willdivide the kinematic range of the k2?-evolution into two distinct domains: (i) Q2 k2?> Q20and k2?=2 1 so that ' 0 and @=@ ln k2?' 0, (ii) Q20 k2? 2and k2?=2' 1with ! (4) 1and @=@ ln k2?> 0. In this case, the gluon distribution g(x; k2?) can beevaluated analytically [6] which yields for k2?>Q20the well known Leading Log solution ofQCD [14], but for k2?< Q20a substantial suppression at small x is found to set in, reectingthe "condensation" of gluons into the conning background eld . The corresponding k2?-dependence of the total gluon multiplicity Ng(k2?) =R10dx g(x; k2?) can also be estimated5in the above approximation with the result:(i) Ng(k2?) = Ng(Q2)ln(Q2=2)ln(k2?=2) 1=4exph2p(CA=b) ln(Q2=2)iexph2p(CA=b) ln(k2?=2)i(14)(ii) Ng(k2?) = Ng(Q20) exp 22k2?1 + [1 + ln(k2?=Q20)]2  2k2?=Q20ln(Q20=2): (15)In the region (i) Q2 k2?> Q20, the multiplicity coincides with the QCD result [14], char-acterized by a logarithmic growth as the gap between the hard scale Q2and k2?increases.On the other hand, in the region (ii) Q20 k2?> 2, the exponent is always negative so thatthe number of gluons rapidly decreases and vanishes at k2?= 0, ensuring that no gluonsand therefore no color uctuations exist at distances R> 1. This behaviour is evident inFig. 3.Returning to eqs. (6)-(8), the coupled system of equations can now be solved numer-ically by utilizing the denition (9) together with the solution for the gluon distribution(10) and (13). Since  is taken to be equal the mean eld , the multiplication of eq. (7)by hP j and jP i from the left and right, respectively, aects only the FFterm whichgives the gluon distribution by virtue of (9). With the solution of the gluon distributioninserted, the remaining task is to solve eq. (7) for the -eld. An interesting phenomeno-logical application [15] is to calculate the string constant, which characterizes the linearlyrising potential between the qq-pair due to the gluon interactions. Considering the frag-mentation of a heavy qq pair, an adiabatic (Born-Oppenheimer) treatment is a reasonableapproximation, because one can view the ithgluon as being emitted from the qq pair plusgluons g1; g2; : : : ; gi 1, with the spatial coordinates of these "sources" being frozen duringthe emission of the gluon i [16]. Using the separation Rqqof the receding qq as a measure ofthe typical gluon transverse momenta k2? R 2qqand minimizing the energy per unit lengthof this system, one obtains in the adiabatic approximation for each gluon conguration at agiven Rqqthe form of  and the string tension t. The total energy per unit length t receivescontributions from both the -eld in the low energy regime and the gluon eld in the highenergy domain, which on account of (5) is given by [6],t ERqq=ZdA12jrj2+ V ()+ZdA () hP j(s)4sF;aFajP i ; (16)where A is the cross-sectional area of the ux tube of the gluons between the qq, perpen-6dicular to Rqq. In one loop order  =  b2swith b = 33=(12) for Nf= 0. Using eq. (9),assuming axial symmetry, and minimizing t with respect to  yields following non-linearintegro-dierential equation (r is the radial coordinate perpendicular to Rqq): d2dr2+1rddr(r) +@V ()@+b s2(P+)2Zd2k?Zdx x g(x; k2?)Zdr r () = 0 :(17)This equation is to be solved subject to the constraint [15] that the total color electricux through a plane between the q and q equals the color charge on one of them, and theboundary conditions 0(0) = 0; (1) = 0. The solutions for t,  and  are shown in Fig. 4for the reasonable parameter choice 0= fand bag constant B = (150MeV)4[6]. As thequarks separate with increasing Rqq, the gluons rst multiply which results in a growingstring tension, but then gluon condensation sets in, yielding a saturating behaviour of t(Fig. 4a). For Rqq 1 fm the gluon eld is completely conned within a ux tube of radiusr ' 1 fm (Fig. 4b).In perspective, the Lagrangian approach presented here to describe the connementdynamics of high energy partons in the framework of eective QCD eld theory has thepotential to be extended to a systematic quantum description of hadronization. With theinclusion of quark degrees and quantum uctuations of the - and U -elds, one couldcalculate e.g. the mass spectrum of glueball and meson excitations as physical hadrons.Ultimately one would like to address the microscopic dynamics in full 6-dimensional phase-space, with explicit inclusion of the color ow throughout the evolution. The possibleapplications are manifold. One particular interest is the expected (non-)equilibrium QCDphase transition in high energy systems as in heavy ion collisions or the early universe, anissue which could be addressed along the lines of Campbell, Ellis and Olive [7].AcknowledgementsI would like to thank J. Ellis and H.-T. Elze for valuable suggestions.References[1] Yu. L. Dokshitzer, D. I. Dyakonov, and S. I. Troyan, Phys. Rep. 58, 269 (1980).7[2] M. A. Shifman, A. I. Vainshtein, and V. A. Zakharov, Nucl. Phys. B147, 448 (1979).[3] E.g.: E. Witten, Nucl. Phys. B223, 422, 433 (1983); J. Gasser and H. Leutwyler, Ann.Phys. 158, 142 (1984); U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195 (1991).[4] F. Karsch, Nucl. Phys. (Proc. Suppl.) B9, 357 (1989).[5] B. Webber, Ann. Rev. Nucl. Part. Sci. 36, 253 (1986).[6] K. Geiger, in preparation.[7] J. Lanik, Phys. Lett. B144, 439 (1984); B. A. Campbell, J. Ellis, and K. A. Olive,Phys. Lett. B235, 325 (1990); Nucl. Phys. B345, 57 (1990).[8] R. Friedberg and T. D. Lee, Phys. Rev. D15, 1694 (1977); D16, 1096 (1977); D18,2623 (1978).[9] L. Wilets, Nontopological Solitons (World Scientic, Singapore 1989), and referencestherein.[10] S. D. Bass, Cavendish Preprint HEP 94/6 (1994).[11] M. Shifman, Phys. Rep. 209, 341 (1991).[12] K. Konishi, A. Ukawa, and G. Veneziano, Nucl. Phys. B157, 45 (1979).[13] J. C. Collins and D. E. Soper, Nucl. Phys. B194, 445 (1982).[14] See e.g.: A. Basetto, M. Ciafaloni, and G. Marchesini, Phys. Rep. 100, 201 (1983).[15] M. Bickeboller, M. C. Birse, H. Marshall and L. Wilets, Phys. Rev. D31, 2892 (1984);M. Grabiak and M. Gyulassy, J. Phys. G17, 583 (1991).[16] A. H. Mueller, Nucl. Phys. B415, 373 (1994).8FIGURE CAPTIONSFigure 1: Diagrammatic representation of the two-point Greens function of gluons, in-cluding both the gluon (self) interactions and the eective interaction with the conningbackground eld  (indicated by the dashed lines). This gluon propagator describes theevolution of a gluon from a chosen cascade branch in x and k2, starting from x0and Q2.Figure 2: The anomalous dimension (!; k2?) of eq. (11) versus ! for dierent values ofk2?.Figure 3: Evolution of the gluon multiplicity Ng(k2?) from Q2down to 2. At rst thegluons multiply, but at k2 Q20a rapid condensation sets which is complete at 2(Q0= 1GeV,  = 0:23 GeV).Figure 4: a) String tension t versus separation Rqqof the qq pair, and b) the solutionsfor  and  at Rqq= 1 fm versus r which is the radial coordinate perpendicular to Rqq(0= fand B1=4= 150MeV).9

Dynamic Gluon Confinement in High Energy Processes within Effective QCD Field Theory

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