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Pramana – J. Phys. (2016) 87: 44 c Indian Academy of DOI 10.1007/s12043-016-1256-0

Mass generation via the Higgs and the 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 , recently discovered with a of 125.7 GeV is known to mediate the of elementary , but only 2% of the mass of the . Extending a previous investigation (Schumacher, Ann. Phys. (Berlin) 526, 215 (2014)) and including the strange-quark sector, masses are derived from the quark condensate of the QCD vacuum and from the effects of the Higgs boson. These calculations include the π , the nucleon and the scalar 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 , 0,± and 0,−.

Keywords. Higgs boson; sigma meson; ; 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 , 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 are the building blocks of the massless particles. When applied to the visible nucleon in a similar way as the are in the case of the , 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 space. The QCD vacuum contains a condensate of up- of 19.6 MeV and 20.5 MeV per for and down-quarks. Condensate means that the qq¯ pairs the and , respectively, again in analogy to the nuclear case where the binding energies are are correlated via interquark mediated by 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 , |σ=(|uu¯+ to which the effects of electroweak (EW) and strong |dd¯)/ 2, providing the σ . These two descriptions, interaction breaking combine in order to in terms of a vacuum condensate or a σ field, are essen- generate the masses of . 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 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 to a considerable increase of the vacuum or the Higgs field of the EW vacuum. As long constituent quark masses in the sector as we consider the two 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 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 decay constant in the chiral limit, serving as vac- = [11] and Higgs [12]. uum expectation value, vσ . The quantity vH 246 GeV is the of the Higgs field, - 0 = 0 = ing to the 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 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 , ,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 is para- ponents W of the weak vector bosons W. These lon- l metrized by the four- 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 = 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 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 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 of the Higgs boson. top-, 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 . 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 as we shall see later. 3.2 The 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 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– 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¯ 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 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 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- production of 0 a pion pair. The two 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 . 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 : , I3: 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

Table 3. Decouplet of scalar mesons including f0(1370) ≡ σ (1344). (a) Scalar meson masses in the chiral limit according to eq. (57), (b) scalar meson masses including the effects of EW interaction according to eq. (58), (c) scalar meson masses including the effects of EW interaction as predicted by the masses of pseudo-Goldstone bosons according to eqs (60), (61), 0 (d) current quark masses ms of the strange quark obtained by the adjustments to experimental data according to eq. (62).

cl 0 Meson Exp. mscalar Ref. fs mscalar (a) mscalar (b) mscalar (c) ms (d)

σ(600) 600 ± 70 [14] 0 652 666 666 – κ(800) 800 ± 100 [27] 1/4 720 836 806 139 f0(980) 980 ± 20 [27] 1/2 787 1005 929 191 a0(980) 990 ± 20 [27] 1/2 787 1005 929 191 f0(1370) 1368 ± 22 [28] 1 922 1344 – 244 Pramana – J. Phys. (2016) 87: 44 Page 9 of 11 44 pseudo-Goldstone bosons and, therefore, have the pro- Table 4. (a) Average experimental masses of the three perty that their masses tend to zero in the chiral limit, as groups of baryons, (b) sum of the masses of the three con- requested in eqs (60) and (61). The results of this cal- stituent quarks in the chiral limit, (c) sum of the masses culation are listed in column (c) of table 3. The agree- of the three constituent quarks including the effects of EW B ment of the two methods of calculation given in interaction and (d) binding energy ( ) per number of quarks (A = 3) (in units of MeV). columns (b) and (c) of table 3 is good for the κ(800) meson but shows the tendency of a deviation in case cl mexp. (a) mtheor. (b) mtheor. (c) B/A (d) of the f0(980) and a0(980) mesons. This deviation can be removed by replacing the K-meson mass by a larger p, n 939 978 999 20 ±, value of 590 MeV, showing that eq. (61) is a reasonable ,  0 1234 1113 1338 55 0,− but not a perfect approximation.  1318 1248 1677 120

4.4 The masses of octet baryons 4.3.3 Effects of EW interaction taken in account by Scalar mesons and baryons differ by the fact that scalar adding current quark masses to the σ meson mass in mesons are loosely bound so that the mesons mass can the chiral limit. A further option for the calculation of be identified as the sum of constituent quark masses, the EW interaction part of scalar meson masses may be whereas in the case of baryon masses, binding energies obtained from the ansatz have to be taken into account. In table 4, a test of this concept is carried out. In column (a) the average exper- = cl + 0 mσ mσ 2 ms, (62) imental masses of the three groups of octet baryons are shown. In columns (b) and (c) the corresponding cl = where mσ 922 MeV, as shown in line 6 of column predicted masses in the chiral limit and including the (a) in table 3. This ansatz is an extension of the rela- effects of EW interaction are listed. Apparently, these = cl + 0 + 0 tion mσ mσ (mu md ) derived in eq. (25). Here predicted masses of column (c) are larger than the ex- 0 the current quark mass, ms , is an adjustable parameter perimental ones, confirming the expectation that bind- determined by adjusting the predicted mass according ing energies are involved in the formation of the to eq. (58) to the experimental scalar mass. total baryon mass. The binding energies B per num- The result of this adjustment procedure is shown in ber A = 3 of constituent quarks are listed in column column (d) of table 3. For the meson κ(800) the result (d). These binding energies increase with the fraction 0 = of strange quarks located in the baryon. This interesting ms 136 MeV is in good agreement with the result 0 = finding needs an explanation. ms 147 MeV obtained from the mass of K meson (see §4.2). At larger masses of the scalar mesons there is a drastic increase of the adjusted current quark mass 4.5 The magnetic moments of hyperons 0 0 = ms , but the results obtained remain in the range ms 133–232 MeV of values found in previous investiga- The magnetic moments of hyperons have been pre- tions (see §4.2). dicted using different assumptions about the masses of From this finding, the following conclusions may be the constituent quarks (see e.g. [29]). Within the quark drawn. First of all there should be a definite value for model, the masses of the constituent quarks are the only the current quark mass of the strange quark, viz. parameters in the prediction of magnetic moments. 0 ms (EW), which is the result of a genuine EW mass pro- duction process related to the Higgs boson. The differ- Table 5. Constituent mass of strange quark calculated from 0 magnetic moments. ent model-dependent values ms obtained by adjusting theor. theor. to experimental data contain additional contributions (9/mp) × μ μexp. μ from the qq¯ vacuum condensate which is not taken care of by the scalar mass calculated for the chiral  −3/ms −0.613 ± 0.004 −0.465 +  8/mu + 1/ms 2.458 ± 0.010 2.675 limit. These additional contributions may be quite siz- −  −4/md + 1/ms −1.160 ± 0.025 −1.090 able as shown by the increase of the values in column 0  −4/ms − 2/mu −1.250 ± 0.014 −1.250 (d) of table 3 with increasing mass of the scalar meson. −  −4/ms + 1/md −0.6507 ± 0.0025 −0.309 An explanation of this additional contribution would be −  −9/m −2.02 ± 0.05 −1.396 an interesting topic for further investigations. s 44 Page 10 of 11 Pramana – J. Phys. (2016) 87: 44

Mass[GeV]

1.4 σ

1.2 Mass[GeV]

f00a Ξ 1.0 1.5 ΛΛΣΣ κ p n 0.8 σ 1.0

0.6 0.5 0.4 0.0 0.2

0.0 0.5 (a) (b) Figure 2. Masses generated via strong interaction (green boxes) and EW interaction (red). (a) Scalar mesons and (b) octet baryons. The negative parts of the green boxes in figure 2a correspond to the binding energies observed in the octet baryons. Without the effects of the Higgs boson, only the masses represented by the green boxes would be present. The red boxes correspond to the effects of the Higgs boson.

Therefore, within this model information on the con- moments of octet baryons are also predicted leading stituent quark masses is obtained by comparing the to results being in line with the experimental values. predicted magnetic moments with the experimental The masses of constituent quarks are composed of values. the masses Mq predicted for the chiral limit and the = 0 In the present investigation we use mu 331 MeV, mass of the respective current quark mq provided by md = 335 MeV and ms = 672 MeV. With these con- the Higgs boson (EW interaction) alone. For scalar me- 0 stituent quark masses, the predicted magnetic moments sons the sum of Mq and mq leads to a zero-order theor. μ given in column 4 of table 5 are obtained. The approximation for the constituent quark mass mq,but value predicted for the 0 hyperon is in perfect agree- there are dynamical effects described by the NJL model = + 0 ment with experimental value. In other cases the agree- which modify the simple relation mq Mq mq, ment is less pronounced. However, by using a different except for the non-strange sector where this relation choice for the constituent mass of the strange quark no is a good approximation. Similar results are obtained improvement of the general agreement is obtained. The for the octet baryons. A difference between the scalar conclusion we have to draw is that our set of constituent mesons and the octet baryons is that for scalar mesons quark masses is in line with the experimental magnetic binding energies do not play a role whereas they are moments, though there is no strong support for the spe- important in the case of octet baryons. cific value adopted for the s quark. The information In figures 2a and 2b, graphical representations of obtained from the masses of scalar mesons on the con- mass generation are given. The green boxes correspond stituent mass of the s quark appears to be more reliable to masses in a world without the Higgs boson, whereas than the one obtained from magnetic moments. the red boxes represent the effects of the Higgs boson on the mass generation process. 5. Summary and conclusions References In the forgoing it has been shown that the prediction of scalar meson and baryon masses can be extended [1] Y Nambu and G Jona-Lasinio, Phys. Rev. 122, 345 (1961); from the SU(2) sector to the SU(3) sector by using a 124, 246 (1961) | ¯ [2] D Lurié and A J MacFarlane, Phys. Rev.B136, 816 (1964) second σ meson, viz. σ (1344),havingthe ss struc- [3] T Eguchi, Phys. Rev.D14, 2755 (1976); 17, 611 (1978) ture. This second σ meson implies that the constituent [4] U Vogl and W Weise, Prog.Part.Nucl.Phys.27, 195 (1991) quark mass of the s quark when including the effects [5] S P Klevansky, Rev. Mod. Phys. 64, 649 (1992) [6] T Hatsuda and T Kunihiro, Phys. Rep. 247, 221 (1994) of the Higgs boson is ms = 672 MeV. This large con- [7] J Bijnens, Phys. Rep. 265, 369 (1996) stituent quark mass leads to reasonable predictions of [8] M Gell-Mann and M Levy, Nuovo Cimento 16, 705 (1960) the masses of scalar mesons below 1 GeV and of the V De Alfaro, S Fubini, G Furlan and C Rosetti, Currents in masses of octet baryons. Furthermore, the magnetic hadron physics (North Holland, Amsterdam, 1973) Pramana – J. Phys. (2016) 87: 44 Page 11 of 11 44

[9] M Schumacher, Ann. Phys. (Berlin) 526, 215 (2014), [20] R Delbourgo and M Scadron, Mod. Phys. Lett. A 10, 251 arXiv:1403.7804 [hep-ph] (1995), arXiv:hep-ph/9910242 [10] J Schwinger, Ann. Phys. (NY) 2, 407 (1957) R Delbourgo, M D Scadron and A A Rawlinson, Mod. Phys. [11] F Englert, Ann. Phys. (Berlin) 526, 201 (2014) Lett. A 13, 1893 (1998), arXiv:hep-ph/980750 [12] P W Higgs, Ann. Phys. (Berlin) 526, 211 (2014) [21] R Delbourgo and M Scadron, Int. J. Mod. Phys. A 13, 657 [13] F Halzen and A D Martin, Quarks and : An introduc- (1998), arXiv:hep-ph/9807504] tory course in modern (John & Sons, [22] M Schumacher, Eur. Phys. J. A 30, 413 (2006); A 32, 121 New York, 1984) (2007) (E), arXiv:hep-ph/0609040 [14] M Schumacher and M D Scadron, Fortschr. Phys. 61, 703 [23] S Narison, Riv.Nuovo Cimento 10(2), 1 (1987) (2013), arXiv:1301.1567 [hep-ph] [24] J Gasser and H Leutwyler, Phys. Rep. 87, 77 (1982) [15] G Galler et al, Phys. Lett. B 503, 245 (2001) [25] C A Dominguez and E De Rafael, Ann. Phys. 174, 372 (1987) [16] S Wolf et al, Eur. Phys. J. A 12, 231 (2001) [26] R J Jaffe, Phys. Rev.D15, 267 (1977); D 15, 281 (1977); D [17] M Schumacher, Eur. Phys. J. C 67, 283 (2010), 17, 1444 (1978) arXiv:1001.0500 [hep-ph] [27] : K A Olive et al, Chin. Phys. C 38, [18] M Schumacher, J. Phys. G 38, 083001 (2011), 090001 (2014) arXiv:1106.1015 [hep-ph] [28] E Klempt, Nucl. Phys. A 629, 131c (1998) [19] A H Fariborz, A Azizi and A Asrar, arXiv:1503.05041 [29] M D Scadron, R Delbourgo and R Rupp, J. Phys. G 32, 735 [hep-ph] (2006), arXiv:hep-ph/0603196