Conference Report: Structure of scalar mesons and the Higgs sector of strong interaction
aa r X i v : . [ h e p - ph ] J u l Structure of scalar mesonsand the Higgs sector of strong interaction
Martin Schumacher Zweites Physikalisches Institutder Universität GöttingenD-37077 Göttingen, Germany
The σ meson may be considered as the Higgs boson of strong interaction. While the obser-vation of the electroweak Higgs boson is the primary goal in ongoing experiments at theLHC, the σ meson is by now well studied both as an on-shell particle and as a virtual par-ticle while being part of the constituent quark. This makes it timely to give an overview ofthe present status of the Higgs sector of strong interaction which includes the scalar mesons σ ( ) , κ ( ) , f ( ) and a ( ) together with the pseudo Goldstone bosons π , K and η . Scalar mesons below 1 GeV together with the pseudo Goldstone bosons π , K and η maybe considered as the Higgs sector of strong interaction. While the EW Higgs boson upto now appears to escape experimental observation in the ongoing LHC experiments [1]the strong counterpart, the σ meson is by now well studied both as on-shell particle andas a virtual particle while being part of the constituent quark. The latter observation hasbeen facilitated through Compton scattering by the proton in an experiment carried outat MAMI (Mainz) published in 2001 [2, 3]. In this experiment it has been shown that thescalar t -channel makes a strong contribution to the Compton scattering amplitude, beingsuccessfully represented in terms of a t -channel pole located at m σ where m σ is the baremass of the σ meson, determined in this experiment to be ∼
600 MeV. Inspite of this greatsuccess the physical interpretation of the experiment remained uncertain because an ex-plicit σ meson is a strongly unwanted particle in chiral perturbation theory. This led toan unnecessary delay, because a detailed theoretical investigation was required extendinguntil 2010, when it was shown that the t -channel pole at m σ is a well founded concept andthat the related t -channel amplitude may be understood as being due to Compton scatter-ing by the σ meson while being part of the constituent quark [4]. The findings in [4] wereextended to include the whole scalar nonet below 1 GeV in [5]. The present work is in partbased on this latter publication where more details may be found. [email protected] IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
The scalar nonet below 1 GeV cannot be understood in terms of flavor structures as pro-vided by SU ( ) f [6] because of the ordering of the meson masses. This problem was solvedby introducing tetraquarks ( qq ) [6]. The tetraquark model implies the possibility of a dis-sociation of the kind ( qq ) ⇄ ( qq + qq ) , leading to qq as a small structure component.In [5] this small qq structure component was interpreted in terms of a doorway state whichserves as the entrance channel in a two-photon fusion reaction and is in agreement withthe experimental two-photon widths of the mesons: σ ( ) : γγ → uu + dd √ → uudd → ππ ,(1) f ( ) : γγ → √ uu + dd √ − ss ! → ss ( uu + dd ) √ → ππ , KK ,(2) a ( ) : γγ → √ − uu + dd √ + ss ! → ss ( uu − dd ) √ → ηπ , KK .(3)The qq configuration of the a ( ) meson violates isospin conservation. This is of noproblem because we consider the qq configuration only as a small structure component.In t -channel nucleon Compton scattering the reaction chain(4) γγ → { σ ( ) , f ( ) , a ( ) } → NN is considered where the excitation of the NN pair is virtual. This leads to the consequencethat the masses of the scalar mesons entering into (4) are the bare masses, i.e. the massesfor the case of zero particle decay width. The validity of this concept has been shownin [7] where quantitative predictions of electromagnetic polarizabilities of the nucleon ledto excellent agreement with experimental data. In case of pseudoscalar and scalar mesons the following phenomena contribute to the gen-eration of the masses of the mesons:(i) The U ( ) A anomaly,(ii) spontaneous or dynamical symmetry breaking,(iii) explicit symmetry breaking leading to non-zero current-quark masses.The U ( ) A anomaly is a gluonic (instanton [8]) effect which works on SU ( ) f flavor stateswhich are completely symmetric in the chiral limit. For pseudoscalar and scalar mesonsthis is only the case for the η flavor state and has the consequence that η has a mass in2 IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany the chiral limit whereas all the other pseudoscalar mesons are massless. These latter pseu-doscalar mesons form the octet of Goldstone bosons as depicted in the left panels of Figures1 and 2. The left panel of Figure 1 shows the mexican-hat potential where the Goldstone
U(M, f ) M/ g s p K h a) b) G g
Figure 1:
Left panel: Spontaneous symmetry breaking in the chiral limit (cl) illustrated by the L σ M:In the SU ( ) sector there is one “strong Higgs boson”, the σ meson having a mass of m cl σ =
652 MeVtaking part in spontaneous symmetry breaking, accompanied by an isotriplet of massless π mesonsserving as Goldstone bosons. In the SU ( ) sector there are 8 massless Goldstone bosons π , K , η ,and nine scalar mesons σ , κ , f and a , all of them having the same mass as the σ meson in thechiral limit. The mass degeneracy is removed by explicit symmetry breaking. Right panel: Tadpolegraphs of dynamical symmetry breaking. a) Four fermion version of the Nambu-Jona–Lasinio(NJL) model, b) bosonized NJL model. bosons correspond to the minimum of the potential. The mexican-hat potential describesspontaneous symmetry breaking in terms of a mass parameter µ and a self-coupling pa-rameter λ . Since these parameters are unknown no quantitative prediction of the massesof the constituent quark and of the scalar mesons is possible. This is different in the quark-level linear σ model (QLL σ M) where the graphs shown in Figure 1 a) and b) are taken intoaccount. In this way the Delbourgo-Scadron relation [9](5) M = g f is obtained with g = π / √ σ -quark coupling constant and f = m cl = M =
652 MeV as given inthe caption of Figure 2. Explicit symmetry breaking is described by generalizing the massformula valid for the σ meson(6) m σ = π f π + m π by taking into account the larger fraction of strange quarks in the κ ( ) and the ( f ( ) , a ( )) mesons in their tetraquark structures. This leads to m κ = π ( f π + f K ) + ( m π + m K ) (7) m a , f = π f K + m η (8) 3 IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany pp pp ,,,, K, hh hhhh hh U(1) explicitsym. br. pp pp hh hh hh hh K A
2 M=m ss ss f ss ss cl kk kk a f = f = f = ss s Figure 2:
Left panel: Pseudoscalar mesons after U(1) A symmetry breaking (left column) and afteradditional explicit symmetry breaking (right column). Right panel: Masses of the members of thescalar nonet. In the chiral limit all the scalar mesons have the same mass amounting to 2 M = m cl σ =
652 MeV, where M is the mass of the constituent quark in the chiral limit and m cl σ the mass of the σ mesons in the chiral limit (cl). Explicit symmetry breaking shifts the masses upward with thefraction f s of strange quarks in the tetraquark structure being the parameter determining the sizeof the shift. where f π = ± f K = ± m σ =
685 MeV, m κ =
834 MeV and m a , f =
986 MeV in close agreement with theexperimental data.
References [1] Markus Schumacher, for the ATLAS collaboration, arXiv:1106.2496 [hep-ex].[2] G. Galler, et al., Phys. Lett. B 501 (2011) 245.[3] S. Wolf, et al., Eur. Phys. J. A 12 (2001) 231.[4] Martin Schumacher, Eur. Phys. J. C 67 (2010) 283, arXiv:1001.0500 [hep-ph][5] Martin Schumacher, J. Phys. G: Nucl. Part. Phys. 38 (2011) 083001, arXiv:1106.1015[hep-ph].[6] R.L. Jaffe, K. Johnson, Phys. Lett. B 60 (1976) 201, R.J. Jaffe, Phys. Rev. D 15 (1977) 267, ibid. ibid.ibid.