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 . The result is given by = cl = and the second√ to the LσM. With vσ fσ 89.8 MeV ¯ 8π 3 3 and g = 2π/ 3 = 3.63 this leads to |ψψ | = √ (fπ ) = (225 MeV) , 3 2 cl 8π fπ = (92.43 ± 0.26) MeV. (9) m = 652 MeV and λσ = = 26.3. (16) σ 3 This equation shows that a representation of the QCD For the Higgs boson, we have quark condensate through the vacuum expectation value of the σ field is possible and leads to a prediction mH = 2λH vH,mH = 125.7GeV for a value of the vacuum condensate which is tested and and found valid in the next subsection. Figure 1 suggests a formal similarity of strong interac- vH = 246 GeV (17) tion and EW interaction symmetry breaking. This for- leading to mal similarity is incomplete because at the present state of our knowledge we have to consider the Higgs boson λH = 0.130. (18) as elementary, i.e. without a fermion–antifermion sub- We see that the strong interaction self-coupling is a fac- structure. Formerly, substructures of the Higgs field in tor λ /λ = 202 larger than the EW self-coupling. terms of techniquark–antitechniquark pairs or top–anti- σ H But except for this, there indeed is a formal similarity top quark pairs have been discussed. In the techniquark between the two versions of symmetry breaking. This model, strong self-coupling has been discussed leading may help to get a better understanding of the underly- to a predicted Higgs boson mass of ∼1TeV.Inthe ing physics of the Higgs boson. top-quark model, the predicted Higgs boson mass is expected to be ∼2mt . With a comparatively small expe- rimental mass of mH = 125.7 GeV of the Higgs boson 2.1 Prediction of the masses of the π and the σ meson these models seem to be excluded. As for figure 1a, we π may write down a potential in the form [13] The mass of the meson is given by the Gell-Mann– Oakes–Renner (GOR) relation in the form V(φ) =−μ2φ†φ + λ(φ†φ)2,μ2 > 0,λ>0. (10) m2 f 2 = (m0 + m0 )|ψψ¯ |, The potential V(φ) then has its minimum at a finite π π u d value of |φ| where ¯ 1 ¯ 8π 3 |ψψ | = |uu ¯ + dd | = √ (fπ ) (19) 2 1 μ2 3 φ†φ = (φ2 + φ2 + φ2 + φ2) = . (11) 2 1 2 3 4 2λ (see eq. (9)) leading to We can choose, say, π 2 = 0 + 0 √8 mπ (mu md ) fπ . (20) μ2 3 φ1 = φ2 = φ4 = 0, φ3 = ≡ vH, (12) = λ The numerical value derived from eq. (20) is mπ 137.0 MeV. This predicted value may be compared where v = 246 GeV is the vacuum expectation value H with the experimental values m0 = 135.0 MeV and of the Higgs field. In the case of strong interaction, the π m± = 139.6 MeV. Apparently, the predicted value of corresponding relations are π the π meson mass is quite satisfactory when the current 1 μ2 quark masses from eqs (2) and (3) is used. Further- φ†φ = (σ 2 + π2) = (13) 2 2λ more, in the neutral π 0 we find a Coulomb attraction 44 Page 4 of 11 Pramana – J. Phys. (2016) 87: 44 between the quark and the antiquark, leading to an scale not only for the mass of the nucleon but also for experimental value smaller than the predicted value, such sophisticated structure constants as the magnetic whereas in the charged π ± we find a Coulomb repul- moment and the polarizability. This is an important sion between the quark and the antiquark leading to an finding because it implies that the underlying models experimental value larger than the predicted value. are confirmed in different and complementary ways. In From eqs (1)–(3) we can see that the constituent the present article we return to this problem and make quark in the chiral limit and the current quarks gen- some necessary amendments. erate masses independently. But these masses are not the ones observed in low-energy QCD. The procedure 3.1 The magnetic moment of the nucleon to arrive at predictions for the observable masses has been derived from arguments given by the LσMaswell The magnetic moments of the nucleon are given by as NJL models. First, we calculate the mass of the σ 4 1 μp = μu − μd , (27) meson in the chiral limit via the Nambu relation 3 3 π cl = = = √2 cl = 4 1 mσ 2M 652 MeV with M fπ 326 MeV μn = μd − μu, (28) 3 3 3 (21) in units of the nuclear magneton μN = eh/¯ 2mp. and then use the relation Constituent quark masses enter through the relations 2 2 1/2 mσ = (4M +ˆm ) = 666 MeV. (22) 2 mp 1 mp π μu = ,μd =− , (29) 3 m 3 m In eq. (22) the effects of the Higgs boson enter via the u d where m = 331 MeV and m = 335 MeV. This leads average pion mass mˆ π . It is interesting to note that in u d the limit of small current-quark masses, the result of to the magnetic moments of the constituent quarks eq. (22) can also be derived by simply adding the con- μu = 1.890,μd =−0.934 (30) tributions from the QCD quark condensate and from and to the predicted magnetic moments of the nucleon the Higgs boson, leading to theor. = theor. =− μp 2.831,μn 1.875. (31) = + 0 = mu M mu 331 MeV, (23) Comparing these values with the experimental mag- 0 netic moments of the nucleon md = M + m = 335 MeV, (24) d exp. exp. = cl + 0 + 0 = μp = 2.79285,μn =−1.91304 (32) mσ mσ mu md 666 MeV. (25) we arrive at very small differences μ = μexp. −μtheor. The arguments leading to an equivalence of eqs (22) =− =− and (25) are as follows. Equation (22) can be written μp 0.038, μn 0.038. (33) in the form Apparently, the necessary corrections to the quark- model predictions of the magnetic moments for the 1 mˆ 2 m = mcl + π +···=mcl + (m0 + m0 ). proton and neutron are the same. This may help to find σ σ cl σ u d (26) 2 mσ an explanation for these corrections. The most proba- Here, eqs (20) and (21) are used to show that the term ble explanation may be found in terms of meson ex- ˆ 2 1 mπ 0 + 0 change currents though available calculations lead to 2 cl of eq. (26) is equal to (mu md ) if higher-order mσ too large values. cl terms amounting to 4% and the deviation of fπ /f π In the present work, we are mainly interested in the from 1 amounting to 3% are neglected. small sizes of 1.4–2.0% of the differences showing that The result shown in eq. (25) is in line with the expec- the predictions obtained on the basis of the NJL model tation that the σ meson is a loosely bound object where are very precise. no additional term, viz. the binding energy B,hasto be taken into account. This is different in the case of baryons as we shall see later. 3.2 The polarizabilities of the nucleon A nucleon in an electric field E and magnetic field 3. The fundamental structure constants: Magnetic H obtains an electric dipole moment d and magnetic moment, polarizability and mass of the nucleon dipole moment m given by = In the previous paper [9] it has been shown that it is d 4παE (34) possible to make precise predictions on an absolute m = 4πβH (35) Pramana – J. Phys. (2016) 87: 44 Page 5 of 11 44 in a unit system where the electric charge e is given by with the mass mσ = 666 MeV gives an excellent agree- 2 e /4π = αem = 1/137.04. The quantities α and β are ment of the predicted and the experimental polarizabil- the electric and magnetic polarizabilities belonging to ities of the nucleon as shown in table 1. Furthermore, the fundamental structure constants of the nucleon. It there is a Compton scattering experiment on the pro- is of importance that these quantities are composed of ton where the σ meson as part of the constituent quark two components structure is directly visible in the differential cross- α = αs + αt , (36) section for Compton scattering in the energy range from 400 to 700 MeV and at large scattering angles s t β = β + β , (37) [15–17]. This latter experiment leads to a σ meson mass = ± where the superscript s denotes the s-channel contri- of mσ 600 70 MeV [14] which is in good = bution and the superscript t denotes the t-channel con- agreement with the standard value mσ 666 MeV. tribution. The s-channel contribution is related to the meson-photoproduction amplitudes of the nucleon via 3.3 The masses of the nucleons the optical theorem whereas the t-channel contribution is related to the σ meson as part of the constituent As shown above, the constituent quark masses includ- quark structure. Therefore, it is possible to use the polar- ing the effects of the Higgs boson are izabilities as a tool to test the predicted mass and mu = 331 MeV and md = 335 MeV. structure of the σ meson. This is summarized in the following equations [14]: This leads to the nucleon masses 1 m0 = 2m + m = 997 MeV, (42) |σ =√ (|uu¯ +|dd¯ ), p u d 2 0 = + = mn 2md mu 1001 MeV. (43) α N 2 − 2 M → = em c 2 + 1 (σ γγ) , (38) The difference of these quantities from the experimen- πfπ 3 3 tal values gσNNM(σ → γγ) (α − β)t = = 15.2, p,n 2 m = . , 2πmσ p 938 27 MeV (44) + t = = (α β)p,n 0, (39) mn 939.57 MeV, (45) may be interpreted in terms of the binding energy B, αt =+7.6,βt =−7.6 (40) p,n p,n leading to s =+ s =+ s =+ = 0 − = αp 4.5,αn 5.1,βp 9.4, Bp mp mp 59 MeV, (46) s = = 0 − = βn 10.1 (41) Bn mn mn 61 MeV. (47) −4 3 = in units of 10 fm , where use is made of gπNN The larger binding energy Bn of the neutron compared = ± = gσNN 13.169 0.057 and mσ 666 MeV as to Bp of the proton, Bn − Bp ≈ 2 MeV, has previously predicted by the NJL model. [9] been interpreted in terms of a Coulomb attraction, The polarizability components listed in eq. (40) cor- leading to zero in the case of proton and a non-zero respond to the t-channel and have been calculated value of the right order of magnitude in the case of from eqs (38) and (39). The polarizability components neutron. The arguments were as follows. The electro- in eq. (41) correspond to the s-channel and have been magnetic potential acting between three constituent calculated from high-precision meson photoproduction quarks may be written in the form amplitudes [14]. eiej The purpose of this subsection is to show that the qq¯ U = α hc,¯ (48) r em structure of the σ meson as given in eq. (38) together i Table 1. Total predicted polarizabilities and experimental results (unit 10−4 fm3) αp βp αn βn Total predicted +12.1 +1.8 +12.7 +2.5 Experimental result +(12.0 ± 0.6) +(1.9 ∓ 0.6) +(12.5 ± 1.7) +(2.7 ∓ 1.8) 44 Page 6 of 11 Pramana – J. Phys. (2016) 87: 44 where the denominator has been replaced by an edu- may be expected that these two mesons have different cated guess for the average interquark distance [9], viz. quark–meson coupling constants g for the u and d rij ≈0.3 fm [9]. This tentative consideration leads quarks coupling to the σ meson and the coupling con- to Up = 0 MeV and Un =−1.6MeVorBn − Bp = stant gs for the s quark to the σ meson. This difference 1.6 MeV. The difference Bn − Bp = 2.0MeV has been investigated by Delbourgo and Scadron contained in eqs (46) and (47) would lead to rij = [20,21] (see also [22]) on the basis of a diagrammatic 0.24 fm which is in reasonable agreement with the approach. Using dimensional regularization, the graphs educated guess. sum up in the chiral limit to [21] It may be expected that a calculation of the hadronic 4p M2 binding energy of the nucleon leads to interesting cl 2 = 2 d − 1 (mσ ) 16iNcg insights into the constituent quark structure of the (2π)4 (p2−M2)2 p2−M2 nucleon. At the present point of research we leave this N g2 M2 = c , (49) as an open problem for further investigations. π 2 d4p M2 1 (mcl )2 = 8iN g2 s − σs c s 4 2− 2 2 2− 2 4. Hadron masses in the SU(3) sector (2π) (p Ms ) p Ms N g2 M2 = c s s . (50) In the SU(2) sector we have the π mesons serving as 2π 2 Nambu–Goldstone boson and the σ(666) meson serv- Here, a function identity (2 − l) + (1 − l) →−1 ing as Higgs boson of strong interaction. In the SU(3) has been used in 2l = 4 dimensions. The conclusions sector we expect an octet π, K and η of Nambu– drawn from these considerations are Goldstone bosons and a nonet σ(666), κ(800) and 2π √ √ f0(980), a0(980) of Higgs bosons of strong interac- = √ = cl = = g ,gs 2g, ms 2Ms and Ms 2M. tion. This latter case has been investigated in a previous 3 paper [18]. Since the√σ(666) meson is given by the (51) |nn¯ =|(uu¯ + dd)¯ / 2 state one should expect that The last of the relations in eq. (51) implies that the vac- the f (980) meson is given by the related |ss¯ state. 0 uum expectation values of the non-strange and strange This, however cannot be the case because the f (980) 0 quark sigma fields are the same in the chiral limit. and a (980) mesons have equal masses and, therefore, 0 Keeping this in mind we arrive at must have an equal fraction fs of strange quarks in the = cl = meson structure. There are arguments that the missing Ms 461 MeV and mσs 922 MeV. (52) |ss¯ scalar meson may be identified with the f0(1370) state. This has previously been pointed out by Hatsuda and Kunihiro [6] and recently by Fariborz et al [19]. 4.2 Current quark mass of the s quark According to Hatsuda and Kunihiro, f0(1370) may be In the SU(3) sector, the effects of the Higgs boson enter considered as a second sigma meson, σ ,whichtakes into the mass generation process via the current quark | ¯ over the role of the σ meson when we replace nn by masses of the u, d and s quarks. In low-energy QCD | ¯ ss . The mesons σ and σ differ by the fact that the the current quark masses of the u and d quarks are well wave function of the σ meson contains two flavours, known to be m0 = 5 MeV and m0 = 9 MeV, as already = u d Nf 2, whereas the wave function of σ contains only stated above. The current quark mass of the s quark is = one flavour, Nf 1. This may lead to the consequence less well known and, therefore, requires some further that the coupling constant of a u or d quark to the σ investigation. Here we first attempt to exploit an ana- meson may be different from the coupling strength of a log of eq. (20) given for π meson and write down for s quark to the σ meson. This point will be investigated the K+ in the next subsection. π 2 = 0 + 0 √8 m + (mu ms ) fK . (53) K 3 4.1 Properties of the strange quark σ meson in the This equation allows to calculate the current quark chiral limit 0 + + mass ms from the mass of the K meson, the K meson 0 The non-strange and the strange quark σ mesons dif- decay constant fK and the current quark mass mu. fer by the fact that we have flavour numbers Nf = Then with mK+ = 493.67 MeV, fK = 110.45 MeV, = 0 = 0 = 2forσ and Nf 1forσ . Due to this difference it mu 5 MeV we arrive at ms 147 MeV. Pramana – J. Phys. (2016) 87: 44 Page 7 of 11 44 For low-energy QCD, the following values may be The tetraquark structure of the σ(600) meson needs 0 = ± found in the literature: ms (161 28) MeV [6,23], a special consideration. For this purpose we study the 0 = ± 0 = ± ms (175 55) MeV [24] and ms (199 33) two reaction chains given in eqs (54) and (55) 0 = √ MeV [6,25]. These data span the range from ms 133 ¯ ¯ 0 = 0 = γγ → (uu¯ + dd)/ 2 → uddu¯ → ππ, (54) to ms 232 MeV with the value ms 147 MeV √ following from eq. (53) being close to the lower limit. γγ → (uu¯ + dd)/¯ 2 → NN.¯ (55) The following considerations appear to be justified. EW interaction alone should lead to a definite value, Equation (54) describes the two-photon production of 0 a pion pair. The two photons first excite the qq¯ struc- ms (EW), of the current quark mass of the strange quark and deviations from this value may be due to ture component of the σ meson which is simpler than an incomplete decomposition of the effects of EW and the tetraquark structure component and therefore has strong interaction. Equation (53) contains the effects a larger transition matrix element. Thereafter, a rear- of strong interaction only due to the decay constant fK rangement of the structure leads to the tetraquark struc- which is well determined experimentally. Furthermore, ture which then decays into two pions. In this reaction ¯ the relation eq. (53) is well justified through its close chain the qq structure component serves as a doorway similarity with the corresponding eq. (20) derived and state for the two-photon excitation of the tetraquark found valid for the π meson. This consideration leads structure component [18]. Equation (55) describes 0 = Compton scattering via the t-channel. In this case only to the supposition that a value around ms 147 MeV 0 the qq¯ structure plays a role. For kinematical reasons, may be identified with ms (EW). the σ meson described in eq. (55) shows up as a narrow resonance having a definite mass of mσ = 666 MeV, 4.3 The masses and structures of scalar mesons whereas the σ meson described in eq. (54) corresponds to a pole on the second Riemann sheet. In the SU(3) sector it has become customary to dis- It is apparent that table 2 does not contain a scalar tinguish between scalar mesons with masses below meson having a ss¯ structure. This leads to the expecta- 1 GeV and scalar mesons with masses above 1 GeV. tion that one of the scalar mesons located above 1 GeV The properties of the scalar mesons with masses below should have this structure. This expectation has been 1 GeV have been investigated and described in a previ- confirmed by Hatsuda and Kunihiro [6] who applied ous paper [18]. There is a nonet of scalar mesons with RPA techniques to the mass relation of the NJL model. a (qq)¯ 2 tetraquark structure component coupled to a qq¯ In this way it has been predicted that two σ mesons component. The reason for the assumption of a tetra- exist, viz. quark structure component is that in a qq¯ model the mσ = 668.0 MeV and mσ = 1344 MeV, (56) electrically neutral a0(980) meson√ should have a quark √ ¯ ¯ structure in the form (−uu¯ + dd)/ 2andthef0(980) where |σ =|(uu¯+dd)/ 2 and |σ =|ss¯ . The mass meson a quark structure in the form ss¯. This would lead of the σ meson is in close agreement with the mass to the consequence that the masses should be differ- mσ = 666 MeV derived above, showing that our method ent, whereas in reality they are equal to each other. On and the one of Hatsuda and Kunihiro [6] are essentially the other hand, in a tetraquark model, the fraction of the same. This gives us confidence that it is appropriate strange quarks fs is equal in the two mesons as can be to use mσ = 1344 MeV as one basis for predictions of seen in table 2. masses of scalar mesons containing strange quarks, in Table 2. Summary of scalar mesons in the (qq)¯ 2 representation according to [26]. Y : hypercharge, I3: isospin component, fs: fraction of strange and/or antistrange quarks in the tetraquark structure. Y/I3 −1 −1/20+1/2 +1Mesonfs +1 dsu¯ uu¯ sd¯ dκ(¯ 800) 1/4 ¯ 0 udduσ(¯ √ 600) 0 ¯ ¯ ¯ ¯ − ¯ ¯ ¯ 0 dussss(uu dd)/√2 udssa0(980) 1/2 ¯ 0 ss(u¯ u¯ + dd)/ 2 f0(980) 1/2 −1 sud¯ ds¯ du¯ uκ(¯ 800) 1/4 44 Page 8 of 11 Pramana – J. Phys. (2016) 87: 44 case the effects of the current quarks are included. The three experiments via the pp¯ → 5π reaction [28], lead- other basis is the predicted constituent quark mass in ing to the weighted average listed in line 6 of column the chiral limit given in eq. (52), i.e. for the case that 2 in table 3. It is straightforward to identify the pre- the effects of the current quarks are not included. In the dicted scalar meson σ (1344) with the observed scalar following we discuss three models for the mass gener- meson f0(1370). One argument for this identification ation of scalar mesons differing by the procedure of is the agreement of the values obtained for the masses. combining the effects of strong interaction and EW Another argument is that f0(980) is excluded because interaction. These models are extensions of the corres- of its tetraquark structure. A third argument is based ponding models used in the SU(2) sector. on calculations of Fariborz et al [19] leading to argu- ments in favour of a ss¯ structure. One consequence of these findings is that the constituent mass of the s quark 4.3.1 First overview on the masses of scalar mesons including the effects of EW interaction is rather large, without and with the effects EW interaction. The masses viz. of scalar mesons may be composed of the masses of 1 the two σ mesons making contributions in proportion ms = × 1344 MeV = 672 MeV. (59) to the fraction of non-strange quarks and strange quarks, 2 respectively. The appropriate mass formulae are 4.3.2 Effects of EW mass generation calculated using cl = − cl + cl mscalar (1 fs)mσ fsmσ , (57) the masses of pseudo-Goldstone bosons. In the pre- mscalar = (1 − fs)mσ + fsmσ , (58) ceeding subsection, the masses of scalar mesons are constructed for the two cases where the effects of EW where eq. (57) refers to the chiral limit where the interaction are not included (column (a)) and included effects of EW interaction are disregarded and eq. (58) (column (b)). The considerations presented have the refers to the case where these effects are included. advantage that the effect of EW interaction on the cl = Equation (57) is evaluated using mσ 652 MeV and mass of the scalar meson are clearly demonstrated and cl = mσ 922 MeV as predicted in §4.1. Equation (58) is quantitative values of the masses for the two cases are evaluated using the standard value of the mass of the predicted. = = σ meson mσ 666 MeV and mσ 1344 MeV as Another independent method which takes the effects predicted by Hatsuda and Kunihiro [6]. The results of of EW interaction into account may be obtained from these mass predictions are given in columns (a) and (b) the supposition that the masses of the scalar mesons of table 3. By comparing the masses in column (a) with κ(800), f0(980) and a0(980) may be calculated using those of column (b) we see that the largest part of the mass formulae analogous to eq. (22) which was written mass is already present in the chiral limit, i.e. before down for the σ meson. These formulae are EW interaction is taken into account. The values in col- cl 2 1 2 2 1/2 umn (b) have to be compared with the experimental mκ = ((m (κ)) + (m + m )) , (60) scalar 2 π K values and should show a reasonable agreement. These m = ((mcl (f ,a ))2 + (m )2)1/2. (61) experimental values given in table 3 have been obtained f0,a0 scalar 0 0 K as follows. The experimental masses of the κ(800), In eqs (60) and (61) the masses of pseudoscalar mesons a0(980) and f0(980) mesons are taken from [27]. The π and K are used in order to take the effects of mass of the f0(1370) meson has been determined in the EW interaction into account. These mesons are