PoS(Confinement X)064 http://pos.sissa.it/ ce. s and violate the Osterwalder- all is obtained using a generalized e resulting propagators are analytic plex momentum plane by numerical o this end, the gluon and ghost prop- er equation. in Landau gauge Yang-Mills theory within =++ ive Commons Attribution-NonCommercial-ShareAlike Licen ∗ [email protected] [email protected] [email protected] [email protected] solutions of the respective Dyson-Schwingerapart equations. from Th a cut structure on the real, timelike momentum axi the Dyson-Schwinger/ Bethe-Salpeter equation approach. T agators are determined non-perturbatively in the whole com Schrader positivity condition. The mass of the scalar glueb ladder truncation to the Poincare-covariant Bethe-Salpet We present results on the scalar glueball with PC Speaker. ∗ Copyright owned by the author(s) under the terms of the Creat c Xth Quark Confinement and the HadronOctober Spectrum, 8-12, 2012 TUM Campus Garching, Munich, Germany Stefan Strauss Christian S. Fischer Institut für theoretische Physik, Justus-Liebig-Universität, Heinrich-Buff-Ring 16, 35392 Gießen, E-mail: Christian Kellermann Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstraße 9, 64289 Darmstadt E-mail: Lorenz von Smekal Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstraße 9, 64289 Darmstadt E-mail: Glueballs in Landau Gauge Yang-Mills theoryDyson-Schwinger from equations Institut für theoretische Physik, Justus-Liebig-Universität, Heinrich-Buff-Ring 16, 35392 Gießen, E-mail:

PoS(Confinement X)064 (2.1) Stefan Strauss . These are deter- ) 2 p ( G ors are given by ropagator DSEs are the lowest , interesting in its own right. Color lving gluon and ghost degrees of ) e color singlet glueballs [1]. There or the higher vertex functions are a preliminary result for the lowest 2 on the Dyson-Schwinger equations 2 the one-particle irreducible Green’s ed set of Bethe-Salpeter equations. extended to complex momenta. Fi- p balls as bound states of two gluons ative propagators of the constituents otic states. Then a sufficient criterion on first results on this issue from the e needs to specify the dressed ghost- n of a two-gluon bound state with a p ( d antiquarks. The clarification of the r evidence from Schwinger functions e loops on the right hand side of the Z cal task. Glueballs in pure Yang-Mills ported and negative contributions in the vertex DSEs in the picture either from erent approaches [2, 3, 4, 5, 6], making sity. In a recent work [9] first numerical nterest since they reflect the mass gap in  f ansaetze for the vertices explored on the ν p 2 ly, however, these states are difficult to iden- p µ , p ) 2 − 2 2 p p ( µν G δ  − and the ghost dressing function ) ) = ) = 2 p p p ( ( ( G Z µν D D state. ++ The non-perturbative complex gluon propagator is of course In a linear covariant gauge, glueballs are bound states invo This contribution is organized as follows: in the next secti The physical excitations in pure SU(N) Yang-Mills theory ar In Landau gauge, the gluon and Faddeev-Popov ghost propagat with the gluon dressing function mined from their DSEsmembers given of diagrammatically an in infinite Fig. tower offunctions 1. of coupled a equations The theory. describing In p ordergluon, to three-gluon close the and system four-gluon atequations. vertices hand, on appearing Currently in there is th explicit great calculations activity [12] to or from include systematic the variations o freedom. In the simplestghost-antighost case component, this which amounts can to a beKey ingredients superpositio described to by such coupl kindat of general approach complex are momenta. the non-perturb are various predictions for the glueball spectrum from diff Dyson-Schwinger/ Bethe-Salpeter framework [7, 8]. confinement is understood to be the absencefor of colored color asympt confinement is a positivityresults for violating the spectral propagators den in the complex planespectral have been functions re have indeed been[10, found 11]. confirming earlie (DSEs) for the ghost andspecified. gluon In propagators section and 3 thenally, the ansaetze a well-known f system Euclidean solution of is or Bethe-Salpeter two equations ghosts (BSEs) is for sketchedglueball glue in mass, the the 0 following section. We report Glueballs 1. Introduction glueballs theoretically well-established. Experimental tify since they mix with states containing pairs of quarks an the ’physical’ spectrum of the theory. In this work we report unquenched glueball spectrum is thereforetheory a are, prime on theoreti the other hand, of considerable theoretical i 2. Yang-Mills Dyson-Schwinger equations PoS(Confinement X)064 5 (2.2) Lattice DSE Stefan Strauss p [GeV] (b) , momentum plane. In [10] ) 0 2 µνρ ( p Γ he following we focus on the ) n active discussion on the deep 2 ) nfrared vanishing or finite gluon uon propagator in Landau gauge. ) pes of solutions, the scaling and Euclidean space-time for positive 2 q ) plex ertex ansaetze from Ref. [16] which 20 percent level in the intermediate l ansatz for the ghost-gluon vertex q vertex [17, 18]. The ansatz for the me-like momenta requires either an- − left panel of Fig. 1). When compared neglected here (see however [19] for perturbation theory in the ultraviolet. agators have been limited to the space- − p e results of [13]. p (( host-gluon system agree perfectly in the 0 1 2 3 4 a (( 1 0 2

a

0.5 1.5 −

) Z(p Z 2 G ) 2 ) , 3 q 2 µ ( q a ( iq 2 a is related to the (perturbative) anomalous dimension − Z = a G ) 0 1 µ ( 1 Z Γ ) = ) = q q , , p p ( ( µ Γ µνρ Γ (a) are the tree-level vertices and ) 0 ( Γ Traditionally, non-perturbative results of the gluon prop where Glueballs is a good qualitativethree-gluon approximation vertex to reproduces, the by full construction,Furthermore, ghost-gluon resummed contributions from thefirst four-gluon results vertex on are the inclusion ofwith the lattice sunset calculations, diagram the (b) ininfrared resulting the and solutions ultraviolet of momentum the region g momentum and regime, deviate on see the the rightinfrared panel behavior of of Fig. the 1.decoupling gluon one There and have is ghost been also propagators.propagator a identified, [11, which Two 20, correspond ty 21, todecoupling 22, an solution. i 23, 24, 25, 26, 27, 28, 29,3. 30]. Analytic In Structures of t Gluons and Ghosts like momentum region, i.e.real all squared computation momenta. have been Recovering done thealytic in propagator continuation also or for solving ti the Yang-Mills DSE in the com of the ghost propagator. It is well known that this tree-leve Figure 1: The Dyson-SchwingerEuclidean equations gluon dressing for function the compared ghost with and the gl lattic level of the propagator DSEs [14, 15].explicitly In read this work we use the v PoS(Confinement X)064 15 10 5 0 15 10 5 0 15 10 5 0 15 10 5 0 -5 -10 -15 -5 -10 -15 -5 -10 -15 -5 -10 -15 . (Panel b) ) Stefan Strauss 2 p 2 2 2 2 ( p p p p 2 2 2 2 D ℑ ℑ ℑ ℑ 1 1 1 1 omplex ghost dressing [9]. The figures include 0 0 0 0 ) 2 p ( -1 -1 -1 -1 D the Yang-Mills DSEs for the first n propagators haven been contin- agator function mentum plane have been devised in -2 -2 -2 -2 ically simpler fermion DSE, several uon propagator is restricted in order ion ave branch cuts, starting at the origin along 2 2 2 2 (a) (b) 4 1 1 1 1 0 0 0 0 2 2 2 2 p p p p ℜ ℜ ℜ ℜ -1 -1 -1 -1 -2 -2 -2 -2 5 0 5 0 5 0 5 0

-5 -5 -5 -5 15 10 15 10 15 10 15 10

-10 -15 -10 -15 -10 -15 -10 -15 lo rpgtr-IaiayPart Part Imaginary Imaginary - - Propagator Propagator Gluon Gluon lo rpgtr-Ra Part Part Real Real - - Propagator Propagator Gluon Gluon The Figures 2 and 3 show the complex gluon propagator and the c the first method has been used and fit functions to the Euclidea colored contour maps and lines.to The resolve displayed smaller range structures. of the gl ued analytically to complex momenta. Inmethods the for case obtaining of a the numerical solution techn the in past the [31, complex 32]. mo Intime. [9] the former method has been applied to function, respectively [9]. Notice that both propagators h Figure 2: (Panel a) Results for the real part of the gluon prop Results for the imaginary part of the gluon propagator funct Glueballs PoS(Confinement X)064 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 2.5 2 1.5 1 0.5 0 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -0.5 -1 -1.5 -2 -2.5 Stefan Strauss 2 2 2 2 p p p p 2 2 2 2 . (Panel b) Results ℑ ℑ ℑ ℑ ) 2 p 1 1 1 1 ( G 0 0 0 0 -1 -1 -1 -1 r these assumptions. Our result tain regions of the complex plane nphysical particles into unphysical -2 -2 -2 -2 rtex couplings. Due to the positivity onal singularities besides the branch opagator the branch cut is singular at he radiation of (massless) ghosts and s in the complex plane away from the sing function tically forms for the gluon propagator [9]. The figures include colored contour ) 2 p ( G 2 2 2 2 (a) (b) 5 1 1 1 1 0 0 0 0 2 2 2 2 p p p p ℜ ℜ ℜ ℜ -1 -1 -1 -1 -2 -2 -2 -2

5 4 3 2 1 5 4 3 2 1 2 1 0 2 1 0

-1 -2 -1 -2

hs rsig-Ra Part Part Real Real - - Dressing Dressing Ghost Ghost hs rsig-IaiayPart Part Imaginary Imaginary - - Dressing Dressing Ghost Ghost Figure 3: (Panel a) Results for the real part of the ghost dres for the imaginary part of the ghost dressing function the negative real axis and extend to infinity. For the ghost pr maps and lines. zero. Moreover, the gluonbut, propagator within is the highly present peakedcuts. numerical accuracy, at This shows cer observation no is additi in in the contrast literature to [33, the 34, suggestedreal 35, analy 36, momentum 37], axis. which all Fromhas assumed our pole a result simple we interpretation: seegluons the no from branch the justification cuts gluon fo correspond via to theviolations ghost-gluon t in and the three-gluon ve gluons and ghosts this radiation is one of u Glueballs PoS(Confinement X)064 (4.1) (4.7) (4.6) (4.2) (4.3) (4.5) (4.4) , ] Stefan Strauss G D , D [ 2 Γ , , + l one finds the following l lueball ghost amplitude G , 1 + + D are their tree-level counter- − 1 S 0 − 0 g functional derivatives with γ G G 1 D − sed in Section 2. Applying two is the Bethe-Salpeter amplitude. 2). Additionally, once the phys- , S γ 0 sentation the dressed vertices. Note that the , TrD = D , µν G − ) ther adjustment of scales or coupling G g χ 1 P ) · − G 1 s 2 space of the theory. This part is occupied P ection. es as explicit input) and demanding the fective action (2PI EA) [38]. The general p · D , 0 + p , 2 2 G , 0 p + 2 ( − p = G ( γ A D 6 S A Trln D Γ δ 2 )= − S δ p δ )= , D p D 1 P , D ( − 0 P 1 ( 12 ++ 0 − ++ χ TrD µν 0 diagrammatically as 2 1 χ 2 ]= = = Γ + G 1 , − D [ D 2 0 D G Γ Trln D 1 2 are the full gluon and ghost propagators and G ]= D G denotes the dressing function. While the decomposition of g D , and stands for either the gluon or the ghost propagator and D D A [ S D Γ Both DSEs and Bethe-Salpeter equations can be derived takin The scalar glueball gluon amplitude has the covariant repre where parts. Assuming the interaction Glueballs particles leaving no traces in theby physical part the of color-singlet the glueball state states discussed in the next s 4. Scalar Glueball respect to the propagators of a two-particle irreducible ef form of the 2PI EA is where one finds the one loopfunctional DSEs derivatives for to the (4.1) ghoststwo-particle and (while onshell gluons, treating conditions as provides the discus the vertic glueball BSE After introducing the modified Bethe-Salpeter amplitudes where where the double arrow meansdressed symmetrization vertices with in respect the to BSEsical are scale the in same the as propagator inconstants equations the in is the DSEs fixed BSEs Eq. there possible. (2. is no fur coupled system of BSEs for ghost and gluon bound states reads PoS(Confinement X)064 (4.8) nd solving Stefan Strauss ator Grant VH-NG-332 and gluon and ghost propagators (2006) 081601 96 mass of the scalar glueball with oaches like lattice gauge theory nta. The complex dressing func- (2012) 252001 [arXiv:1208.6239 eball spectrum based on the BSE. balls as bound state of two gluons potential models [6]. E program of the State of Hesse. pagator that are expected from the lueball was determined in agreement gative real axis. We do not observe 109 (1993) 378 [hep-lat/9304012]. h propagators have positivity violating tt. unt of our approach will be subject of a ev. Lett. gluons or ghosts as asymptotic states of . (2003) 61 [hep-ph/0308268]. 309 (1999) 034509 [hep-lat/9901004]. 577 60 8 GeV (1975) 393. . 1 7 30 = (2001) 281. [hep-ph/0007355]. ++ 0 353 (2004) 014017 [hep-ph/0412173]; V. Mathieu, C. Semay and m 70 (2008) 094009 [arXiv:0803.0815 [hep-ph]]. 77 (2006) R253 [hep-ph/0605173]. 32 being the glueball mass. Putting all ingredients together a G M [UKQCD Collaboration], Phys. Lett. B with 2 G et al. M of the order of = 2 P [hep-ph]]. [hep-ph/0507205]. B. Silvestre-Brac, Phys. Rev. D This work has been supported by the Helmholtz Young Investig In this contribution we have presented an approach to the glu Furthermore we summarized the results of [9] for the complex = ++ [1] H. Fritzsch and P. Minkowski, Nuovo Cim. A [7] R. Alkofer, L. von Smekal, Phys. Rept. [9] S. Strauss, C. S. Fischer and C. Kellermann, Phys. Rev. Le [8] C. S. Fischer, J. Phys. G G [6] F. Brau and C. Semay, Phys. Rev. D [2] G. S. Bali [4] A. P. Szczepaniak and E. S. Swanson, Phys. Lett. B [5] F. J. Llanes-Estrada, P. Bicudo and S. R. Cotanch, Phys. R [3] C. J. Morningstar and M. J. Peardon, Phys. Rev. D Glueballs where the system of BSEs numericallyPC we find a preliminary value for obtained by solving the correspondingtions DSEs have for no complex mome singularities besidespole branch singularities cuts on along theGribov-Zwanziger the first approach Riemann ne or its sheet modern in counterparts.spectral the Bot functions gluon and pro allow nothe particle theory. interpretation of 6. Acknowledgments This result is in[2, good 3], Coulomb agreement gauge with Hamiltonians [4], findings Regge from theory [5] other and appr 5. Conclusions These BSEs are a coupledand system two of ghosts. 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