Pramana – J. Phys. (2016) 87: 44 c Indian Academy of Sciences DOI 10.1007/s12043-016-1256-0 Mass generation via the Higgs boson and the quark condensate of the QCD vacuum MARTIN SCHUMACHER II. Physikalisches Institut der Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany E-mail: [email protected] Published online 24 August 2016 Abstract. The Higgs boson, recently discovered with a mass of 125.7 GeV is known to mediate the masses of elementary particles, but only 2% of the mass of the nucleon. Extending a previous investigation (Schumacher, Ann. Phys. (Berlin) 526, 215 (2014)) and including the strange-quark sector, hadron masses are derived from the quark condensate of the QCD vacuum and from the effects of the Higgs boson. These calculations include the π meson, the nucleon and the scalar mesons σ(600), κ(800), a0(980), f0(980) and f0(1370). The predicted second σ meson, σ (1344) =|ss¯, is investigated and identified with the f0(1370) meson. An outlook is given on the hyperons , 0,± and 0,−. Keywords. Higgs boson; sigma meson; mass generation; quark condensate. PACS Nos 12.15.y; 12.38.Lg; 13.60.Fz; 14.20.Jn 1. Introduction adds a small additional part to the total constituent- quark mass leading to mu = 331 MeV and md = In the Standard Model, the masses of elementary parti- 335 MeV for the up- and down-quark, respectively [9]. cles arise from the Higgs field acting on the originally These constituent quarks are the building blocks of the massless particles. When applied to the visible matter nucleon in a similar way as the nucleons are in the case of the Universe, this explanation remains unsatisfac- of nuclei. Quantitatively, we obtain the experimental tory as long as we consider the vacuum as an empty masses of the nucleons after including a binding energy space. The QCD vacuum contains a condensate of up- of 19.6 MeV and 20.5 MeV per constituent quark for and down-quarks. Condensate means that the qq¯ pairs the proton and neutron, respectively, again in analogy to the nuclear case where the binding energies are are correlated via interquark forces mediated by gluon 3 exchanges. As part of the vacuum structure, the qq¯ pairs 2.83 MeV per nucleon for 1H and 2.57 MeV per 3 have to be in a scalar–isoscalar configuration. This nucleon for 2He. suggests that the vacuum condensate may be described In the present work we extend our previous [9] inves- tigation by exploring in more detail the rules according in terms√ of a scalar–isoscalar particle, |σ=(|uu¯+ to which the effects of electroweak (EW) and strong |dd¯)/ 2, providing the σ field. These two descriptions, interaction symmetry breaking combine in order to in terms of a vacuum condensate or a σ field, are essen- generate the masses of hadrons. As a test of the con- tially equivalent and are the bases of the Nambu–Jona- cept, the mass of the π meson is precisely predicted on σ σ Lasinio (NJL) model [1–7] and the linear model (L M) an absolute scale. In the strange-quark sector the Higgs [8], respectively. Furthermore, it is possible to write boson is responsible for about 1/3 of the constituent down a bosonized version of the NJL model where quark mass, so that effects of the interplay of the two the vacuum condensate is replaced by the vacuum components of mass generation become essential. expectation value of the σ field. Progress is made by taking into account the predicted In the QCD vacuum the largest part of the mass M second σ meson, σ (1344) =|ss¯ [6]. It is found that of an originally massless quark, up (u) or down (d), is the coupling constant of the s-quark coupling to the σ generated independent of the presence of the Higgs field meson is larger than the corresponding quantity of the and amounts to M = 326 MeV [9]. The Higgs field only u and d quarks coupling to the σ meson by a factor 1 44 Page 2 of 11 Pramana – J. Phys. (2016) 87: 44 √ of 2. This leads to a considerable increase of the vacuum or the Higgs field of the EW vacuum. As long constituent quark masses in the strange quark sector as we consider the two symmetry breaking processes in comparison with the ones in the non-strange sector separately we can write down [9] already in the chiral limit, i.e. without the effects of the M = gvcl, (1) Higgs boson. There is an additional sizable increase of σ the mass generation mediated by the Higgs boson due 0 = × −5 mu 2.03 10 vH, (2) to ∼20 times stronger coupling of the s quark to the 0 = × −5 Higgs boson in comparison to the u and d quarks. md 3.66 10 vH. (3) In addition to the progress made in [9] as described above, this paper contains a history of the subject from The quantity g is the quark-σ coupling√ constant which = Schwinger’s seminal work of 1957 [10] to the dis- has been derived to be g 2π/ 3. This quantity leads = covery of the Brout–Englert–Higgs (BEH) mechanism, via eq. (1) to the constituent quark mass M 326 MeV with emphasis on the Nobel prize awarded to Nambu in in the chiral limit (cl), i.e. without the effects of the cl ≡ cl = 2008. This is the reason why paper [9] has been pub- Higgs boson. The quantity vσ fπ 89.8 MeV is the lished as a supplement of the Nobel lectures of Englert pion decay constant in the chiral limit, serving as vac- = [11] and Higgs [12]. uum expectation value, vσ . The quantity vH 246 GeV is the vacuum expectation value of the Higgs field, lead- 0 = 0 = ing to the current quark masses mu 5 MeV and md 2. Symmetry breaking in the non-strange sector 9 MeV. These values are the well-known current quark masses entering into low-energy QCD via explicit sym- In figure 1 the symmetry breaking process is illustrated. metry breaking. The coupling constants in eqs (2) and Figure 1a corresponds to the LσM and figure 1b to the (3) are chosen such that these values are reproduced. bosonized NJL model, together with their EW counter- In the following we study the laws according to parts. In figure 1a symmetry breaking provides us with which the two sources of mass generation combine in v H v a vacuum expectation H of the Higgs field and σ order to generate the observable particle masses. For σ of the -field. Without the effects of the Higgs field, the this purpose, we write down the well-known NJL equa- π strong-interaction Nambu–Goldstone bosons, ,are tion and refer to [9] for more details massless. The π mesons generate mass via the interac- tion with the Higgs field in the presence of the QCD G L = ψ(i/¯ ∂ −m )ψ + [(ψψ)¯ 2 +(ψiγ¯ τψ)2]. (4) quark condensate, as will be outlined below. The EW NJL 0 2 5 counterparts of the π mesons are the longitudinal com- In eq. (4) the interaction between the fermions is para- ponents W of the weak vector bosons W. These lon- l metrized by the four-fermion interaction constant G. gitudinal components are transferred into the originally Explicit symmetry breaking as mediated by the Higgs massless weak vector bosons W via the Brout–Englert– m = m0 + m0 Higgs (BEH) mechanism. In figure 1b, the view of the boson is represented by the trace 0 u d of bosonized NJL model is presented. The originally mass- the current quark mass matrix [4]. From eq. (4) the less quarks interact via the exchange of a σ meson or constituent quark mass in the chiral limit (cl) may be a Higgs boson with the respective σ field of the QCD derived via the relation M = G|ψψ¯ |cl. (5) V(Φ) σ, Η The bosonization of eq. (4) is obtained by replacing G by a propagator g2 G → ,q2 → 0, (6) cl 2 − 2 (mσ ) q σ,Η σ Η cl π, v where m is the mass of the σ meson in the chiral limit WL q σ (a) (b) and q is the momentum carried by the σ meson, and by Figure 1. Strong-interaction and EW interaction symme- introducing the σ and π fields via try breaking. (a)TheLσM together with the EW counterpart G ¯ G ¯ and (b) the bosonized NJL model together with the EW σ =− ψψ, π =− ψiγ5 τψ. (7) counterpart. g g Pramana – J. Phys. (2016) 87: 44 Page 3 of 11 44 cl = with Using the Nambu relation√mσ 2M, the quark-σ g = π/ M = gf cl f cl = coupling constant 2 3, π and π μ2 89.8 MeV we arrive at π = 0, σ= ≡ vcl ≡ f cl. (14) λ σ π π | ¯ |cl = √8 cl 3 = 3 ψψ (fπ ) (219 MeV) , 3 It is of interest to compare the self-coupling strengths of strong interaction symmetry breaking with the one 1 −5 −2 G = = 3.10 × 10 MeV . (8) of EW symmetry breaking. The σ meson mass in the 4(f cl)2 π chiral limit may be expressed in two ways [9]: In the present case of small current quark masses it is cl = cl cl = cl straightforward to arrive at a version which includes mσ 2gvσ ,mσ 2λσ vσ , (15) cl the effects of the Higgs boson by replacing fπ with where the first version corresponds to the NJL model fπ .