aa r X i v : . [ h e p - ph ] S e p Proceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011 Probing Scalar Mesons Below and Above 1 GeV
Amir H. Fariborz
Department of Engineering, Science and Mathematics, State University of New York Institute of Technology,Utica, NY 13502, USA
Within the context of a generalized linear sigma model that includes two nonets of scalar mesons (a two-quarknonet and a four-quark nonet) and two nonets of pseudoscalar mesons (a two-quark nonet and a four-quarknonet) a collective description of scalar and pseudoscalar mesons below and above 1 GeV is studied. The quarkcontents of these states are probed; estimates on their quark components are extracted; and prediction for ππ scattering amplitude is made. An overview of these studies is presented here.
1. Introduction
Scalar mesons are important states in hadron spectroscopy [1] because they are related to the Higgs bosonsof QCD, induced spontaneous chiral symmetry breaking and probe QCD vacuum. They are also importantintermediate states in several low-energy processes such as ππ , πK , πη scatterings as well as several decayssuch as η and η ′ decays, as well as many heavier meson decays such as, for example, the semileptonic decays of D s meson. However, understanding their properties is known to be nontrivial: Experimentally, some of thesestates are very broad and therefore interfere with the nearby states, and theoretically, they do not fit into theconventional SU(3) multiplets, something that is know to work fairly well for other light hadrons (such as thevector mesons). As a result scalars mesons have been the topic of intense investigation in low-energy QCD[2, 3].Below 1 GeV the known states are listed in Fig. 1: the light and broad isosinglet f (600) or sigma, followedby isobublet K ∗ (800) or kappa meson and the two nearly degenerate states, isosinglet f (980) and isotriplet a (980). One can immediately see that these states do not quite follow a quark-antiquark spectroscopy: First,if they were quark-antiquark states, one would expect their masses to be close to the axial vector meson massesaround 1.2 GeV, but clearly these are all below 1 GeV. Second, if we attempt to collect them into a quark-antiquark nonet, we find that it does not quite work. In fact the mass ordering in a pure ideally mixed q ¯ q nonet is completely the opposite of that of scalars (see Fig. 1). Such ideally mixed q ¯ q nonet is known to workfor vector mesons that have the same natural mass ordering. As a fundamental solution to these problems,the four-quark model (i.e two-quark two-antiquark) for these states was proposed in MIT bag model [2]. Fig.1 shows the mass ordering for an ideally mixed four-quark nonet which agrees with the ordering of the lightscalars and therefore seems to be a natural template for these states. However, the ideally mixed four-quarkpicture, even though provides a consistent picture for the mass spectrum, but has deviations from some of theexperimental decay properties of scalar mesons.Above 1 GeV, the known scalar states are also listed in Fig. 1: The isosiglet state f (1370) followed by theisodoublet K ∗ (1430), the isotriplet a (1450) and the two isosinglet states f (1500) and f (1710). The f (1370)has a large uncertainty on its mass and decay width [1] and it could be anywhere in the 1.2 to 1.5 GeV range.The two isosinglet states f (1500) and f (1710) are speculated to contain a considerable glue component. Thescalar states above 1 GeV are generally believed to form a quark-antiquark nonet, even thought there are somedeviations from this picture: For example, if K ∗ (1430) and a (1450) belong to the same q ¯ q nonet, then whyshould a (that should not have a strange quark) be heavier than K ∗ which has one strange quark? Also theirdecay ratios do not quite follow SU(3) patterns: In Fig. 1 the SU(3) predictions for various decay ratios arecompared with the experimental estimates and we see that even though the order of magnitudes are consistentbut there are some deviations.To recap, the scalar states below 1 GeV seem to be close to four-quark states (with some deviations) andthose above 1 GeV seem be close to two-quark states (with some deviations). To generate such deviations frompure four-quark and pure two-quark pictures, it seems natural to investigate an underlying mixing among four-and two-quark states. This is the main objective of the present discussion. In the work of ref. [4] such a mixingwas studied within a nonlinear chiral Lagrangian framework and it was shown that it can provide a consistentpicture for the properties of scalar mesons below and above 1 GeV. The same framework was further extendedto the case of isosinglet scalar states in [5] and coherent results for these states were obtained. Similar mixingpatterns have also been studied by other investigators [6].The same types of mixing, similar to those of refs. [4, 5], has been also investigated within the context oflinear sigma model in refs. [7–10] which will be reviewed in some details in the present work. We see in Fig. 2 Proceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011
Figure 1: The two boxes on the left list the scalar mesons below and above 1 GeV. The box in the middle shows theisospin and the quark substructure of an ideally-mixed quark-antiquark nonet (which is the reverse of the scalar mesonspectrum below 1 GeV, but quite consistent with the vector meson nonet). The lowest box shows the isospin and thequark substructure of an ideally-mixed two quark-two antiquark nonet (which clearly has the same structure to that ofscalar states below 1 GeV). The top right box, provides some of the properties of the scalar mesons above 1 GeV whichare not quite consistent with the assumption that these states are pure quark-antiquark objects. the general idea of this mixing mechanism for the scalar sector: Starting out with two “bare” (unmixed) scalarmeson nonets, a pure four-quark nonet below 1 GeV ( S ′ ) and a pure two-quark nonet above 1 GeV ( S ) with theinternal isospins and mass spectrum given for each nonet in the figure. This mixing mechanism was first appliedin [4] to explain the mass spectrum and decay properties of the a (1450) and K ∗ (1430) mentioned above.Allowing the two nonets to mix with each other, the properties of a (1450) and K ∗ (1430) can be naturallyexplained in the following way. According to the general mixing property of two states with bare (unmixed)masses m and m , it can be easily seen that the mixing leads to splitting of the physical masses away fromthe “bare” masses and this splitting is inversely proportional to the “bare” mass difference. As Fig. 2 shows,the two bare isotriplet states are closer to each other (compared to the two isodoublet states), and as a result,when we allow these states to mix, the isotriplet states split more than the isodoublets and therefore there is alevel crossing that naturally explains why, for example, a (1450) is heavier than the K ∗ (1430). Moreover, someof the decay properties of these states can be naturally understood based on such a mixing scenario [4].In this article, a brief review of this mixing mechanism within the context of a generalized linear sigme modelis presented. We give a brief review of the Lagrangian in the next section, followed by a summary of thenumerical results in Sec. III and a short summary and discussion in Sec. IV.
2. The Lagrangian
The two scalar nonets S and S ′ are combined with their pseudoscalar partners to form the chiral nonets M and M ′ : M = S + iφ, (1) M ′ = S ′ + iφ ′ , (2)that transform under SU(3) L × SU(3) R × U(1) A as: M → e iν U L M U † R , (3) M ′ → e − iν U L M ′ U † R . (4) roceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011 Figure 2: Mixing mechanism between a two-quark scalar meson nonet S and a four-quark scalar meson nonet S ′ . These transformations can be easily verified in terms of the quark fields inside M and M ′ . The quark-antiquarksubstructure of M can be written as: M ba = ( q bA ) † γ γ q aA , (5)where a and A are respectively flavor and color indices. For M ′ there are three possibilities: First, to write M ′ as a molecule of two M ’s, i.e. M (2) ba = ǫ acd ǫ bef (cid:0) M † (cid:1) ce (cid:0) M † (cid:1) df . (6)The second and the third substructures for M ′ correspond to two different ways that two quarks and twoantiquarks can be combined to form a nonet. Depending on whether the two quarks and the two antiquarkshave spin 0 or 1, we have two possibilities. For spin 0: M (3) fg = (cid:0) L gA (cid:1) † R fA , (7)with L gE = ǫ gab ǫ EAB q TaA C − γ q bB ,R gE = ǫ gab ǫ EAB q TaA C − − γ q bB . (8)For spin 1: M (4) fg = (cid:16) L gµν,AB (cid:17) † R fµν,AB , (9)with L gµν,AB = L gµν,BA = ǫ gab q TaA C − σ µν γ q bB ,R gµν,AB = R gµν,BA = ǫ gab q TaA C − σ µν − γ q bB . (10)(See refs. [9] for a detailed description.) It can be shown that out of the three quark substructures for M ′ , onlytwo are independent. Proceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011
The general structure of the Lagrangian is: L = −
12 Tr (cid:0) ∂ µ M ∂ µ M † (cid:1) −
12 Tr (cid:0) ∂ µ M ′ ∂ µ M ′† (cid:1) − V ( M, M ′ ) − V SB , (11)where V ( M, M ′ ) stands for a function made from SU(3) L × SU(3) R (but not necessarily U(1) A ) invariantsformed out of M and M ′ and V SB is the flavor symmetry breaking term. In dealing with this Lagrangian, twoapproaches have been considered:Approach 1 is based on the underlying chiral symmetry and the resulting generating equations [8]. In thisapproach no specific choice for the chiral invariant part of V is made, and only the axial anomaly, the V SB andthe condensates are modeled. The details of this approach can be found in ref. [8] and are not discussed here.The main conclusions are: (i) The light pseudoscalars, as expected from conventional phenomenology, remaindominantly close to quark-antiquark states; (ii) The kappa meson tends to become a dominantly four-quarkstate; and (iii) This approach does not provide any information on a and f systems. To study a and f systems specific choices for V should be made, and that leads to the second approach which is the topic of thepresent work.In approach 2, a specific choice for the chiral invariant part of V is made [9, 10] (in addition to modeling theaxial anomaly, the V SB and the condensates). The main conclusions are: (i) Again, light pseudoscalars have thetendency of becoming quark-antiquark states; (ii)Light scalars tend to become mainly four-quark states; and(iii)Predictions for various low-energy processes such as ππ , πK , πη scatterings, and decays such as η ′ decaysor semileptonic decays of D s can be made. The rest of this article is devoted to a review of approach 2.First, how is V modeled? Obviously, there are infinite number of terms that can be written down for V . Upto dimension 4, there are twenty one SU(3) L × SU(3) R invariant terms in V which can be made out of M and M ′ : V = − c Tr(
M M † ) + ˜ c (det M + h . c . ) + c a Tr(
M M † M M † ) + c b (cid:0) Tr(
M M † ) (cid:1) + d Tr( M ′ M ′† ) + d (det M ′ + h . c . ) + d a Tr( M ′ M ′† M ′ M ′† ) + d b (cid:0) Tr( M ′ M ′† ) (cid:1) + e (Tr( M M ′† ) + h . c . ) + e a ( ǫ abc ǫ def M ad M be M ′ cf + h . c . ) + e b ( ǫ abc ǫ def M ad M ′ be M ′ cf + h . c . )+ e a Tr(
M M † M ′ M ′† ) + e b Tr(
M M ′† M ′ M † ) + e c [Tr( M M ′† M M ′† ) + h . c . ] + e d [Tr( M M † M M ′† ) + h . c . ]+ e e [Tr( M ′ M ′† M ′ M † ) + h . c . ] + e f Tr(
M M † )Tr( M ′ M ′† ) + e g Tr(
M M ′† )Tr( M ′ M † )+ e h [(Tr( M ′ M ′† )) + h . c . ] + e i [Tr( M M † )Tr( M M ′† ) + h . c . ] + e j [Tr( M ′ M ′† )Tr( M ′ M † ) + h.c. ] . (12)Notice that among these terms, those with the coefficients c , d , c a , c b , d a , d b , e a , e a , e b , e f , e g and e h are U (1) A invariant. As we go down the list of terms in V , we see that the number of quark and antiquark linesincreases. To work with such a high number of terms, it seems reasonable to consider an approximation schemein which V is organized in terms of the number of quark and antiquark lines. We define a parameter to keeptrack of this counting: N = 2 n + 4 n ′ , (13)where n is the number of M or M † and n ′ is the number of M ′ or M ′† in a term. In our first attempt we workin N = 8 order which leads to: V ( N ≤ = − c Tr(
M M † ) + ˜ c (det M + h . c . ) + c a Tr(
M M † M M † ) + c b (cid:0) Tr(
M M † ) (cid:1) + d Tr( M ′ M ′† ) + e (Tr( M M ′† ) + h . c . ) + e a ( ǫ abc ǫ def M ad M be M ′ cf + h . c . ) . (14)The c b term has the structure of (Tr( · · · )) which can be shown to be inconsistent with OZI rule, therefore,we ignore this term at this level of investigation. In addition, we mock up the U (1) A anomaly exactly, whichimplies that the two terms ˜ c and e have to be combined in a nonlinear form in terms of natural logs in thefollowing form [11]: c (cid:20) γ ln (cid:18) det M det M † (cid:19) + (1 − γ )ln (cid:18) Tr (
M M ′† )Tr ( M ′ M † ) (cid:19)(cid:21) . (15)Similarly, there are infinite number of terms in V SB . Up to dimension 4, terms linear in the flavor symmetry roceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011 A = diag . ( A , A , A ), are: V SB = + k [Tr( AM ) + h . c . ] + k [Tr( AM ′ ) + h . c . ] + k [Tr( AM M † M ) + h . c . ] + k [Tr( AM M ′† M ′ ) + h . c . ]+ k [Tr( AM M † M ′ ) + h . c . ] + k [Tr( AM M ′† M ) + h . c . ] + k [Tr( AM ′ M ′† M ′ ) + h . c . ]+ k [Tr( AM ′ M † M ) + h . c . ] + k [Tr( AM ′ M ′† M ) + h . c . ] + k [Tr( AM ′ M † M ′ ) + h . c . ]+ k [Tr( AM ) + h . c . ]Tr( M M † ) + k [Tr( AM ) + h . c . ]Tr( M ′ M ′† ) + k [Tr( AM )Tr( M M ′† ) + h . c . ]+ k [Tr( AM )Tr( M ′ M † ) + h . c . ] + k [Tr( AM ′ ) + h . c . ]Tr( M M † ) + k [Tr( AM ′ ) + h . c . ]Tr( M ′ M ′† )+ k [Tr( AM ′ )Tr( M M ′† ) + h . c . ] + k [Tr( AM ′ )Tr( M ′ M † ) + h . c . ] + k [ A ba ǫ bcd ǫ aef M ce M df + h . c . ]+ k [ A ba ǫ bcd ǫ aef M ′ ce M ′ df + h . c . ] + k [ A ba ǫ bcd ǫ aef M ce M ′ df + h . c . ] . (16)In our first approximation, we consider only the first term in the right hand side of Eq. (16) that is consistentwith the quark mass term in the QCD Lagrangian. Therefore, at this level of approximation, the potential is: V = − c Tr(
M M † ) + c a Tr(
M M † M M † ) + d Tr( M ′ M ′† ) + e a ( ǫ abc ǫ def M ad M be M ′ cf + h . c . )+ c (cid:20) γ ln (cid:18) det M det M † (cid:19) + (1 − γ )ln (cid:18) Tr (
M M ′† )Tr ( M ′ M † ) (cid:19)(cid:21) + k [Tr( AM ) + h . c . ] . (17)We assume isospin limit which implies: A = A = A ,α = α = α ,β = β = β , (18)where α ’s and β ’s are the two- and four-quark condensates of fields S and S ′ , respectively: h S ba i = δ ba α a , h S ′ ba i = δ ba β a . (19)
3. Some Numerical Results
At the level of the approximate potential of Eq. (17), altogether, there are 12 unknown parameters c , c a , d , e a , c , γ , α , α , β , β , A , A that need to be determined using 12 inputs. We take the following eightexperimental inputs: m [ a (980)] = 984 . ± . ,m [ a (1450)] = 1474 ±
19 MeV ,m [ π (1300)] = 1300 ±
100 MeV ,m π = 137 MeV ,F π = 131 MeV ,A A = 20 → , det( M η ) = det( M η ) exp . , Tr( M η ) = Tr( M η ) exp . , (20)together with four minimum conditions: (cid:28) ∂V∂S (cid:29) = (cid:28) ∂V∂S (cid:29) = (cid:28) ∂V∂S ′ (cid:29) = (cid:28) ∂V∂S ′ (cid:29) = 0 . (21)These inputs and minimum conditions allow a determination of the Lagrangian parameters and the detailednumerical analysis of this study in given in [10]. The main uncertainties are clearly on m [ π (1300)] as well as theratio A A , and therefore, the results will have some variations that reflect these two uncertainties. The predictionof the model for a typical solution is given in the Tables I and II. We see in Table I that the underlying mixing Proceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011
State ¯ qq % ¯ q ¯ qqq % m (GeV) π
85 15 0.137 π ′
15 85 1.215 K
86 14 0.515 K ′
14 86 1.195 η
89 11 0.553 η
78 22 0.982 η
32 68 1.225 η qq percentage (2nd column), ¯ q ¯ qqq (3rd column) and masses(last column). The two underlined masses in the last column are inputs and the rest of the numbers are predictions.The results are obtained for m [Π(1300)] = 1.215 GeV.State ¯ qq % ¯ q ¯ qqq % m (GeV) a
24 76 0.984 a ′
76 24 1.474 κ κ ′
92 8 1.624 f
40 60 0.742 f f
63 37 1.493 f
93 7 1.783Table II: Typical predicted properties of scalar states: ¯ qq percentage (2nd column), ¯ q ¯ qqq (3rd column) and masses (lastcolumn). The two underlined masses in the last column are inputs and the rest of the numbers are predictions. Theresults are obtained for m [Π(1300)] = 1.215 GeV. among two- and four-quark pseudscalars does not change the well-established picture of the light pseudoscalarsas dominantly being quark-antiquark states. The situation is of course reversed for the heavy pseudoscalars.The present level of this investigation predicts four etas: The two below 1 GeV ( η and η ) are clearly close tothe physical states η (547) and η (958). The two heavier etas above 1 GeV ( η and η ) may be identified withtwo of the several physical eta states above 1 GeV. In the work of [10] various scenarios for this identificationis studied in detail and the closest agreement corresponds to identifying η with η (1294) and η with η (1617).The results for scalars are given in Table II and overall are the inverse of those of pseudoscalars: Lightscalars below 1 GeV tend to gain a large four-quark component (and vice versa for heavy scalars above 1 GeV).The predicted “Lagrangian masses” for the scalars are higher than their expected values. However, they receivecorrections when probed in appropriate unitarized scattering amplitudes. For example, the Lagrangian squared-masses for the four isosinglet scalar states ( f , f , f and f ) appear as poles in the ππ scattering amplitudeand differ from the poles in the K-matrix unitarized ππ scattering amplitude. When physical masses and decaywidths are extracted from the poles in the unitarized amplitudes, it is found that these masses and decay widthsare considerably closer to their expected values. Table III presents the four physical masses and decay widthsfound from the poles of the K-matrix unitarized ππ scattering amplitude in ref. [12]. The first physical mass andwidth is clearly consistent with the property of sigma meson; the second mass and width is qualitatively closeto those of f (980) and the remaining two masses and widths may be compared with the masses and widths oftwo of the several isosinglet states above 1 GeV. However, at this level of investigation we do not expect theproperties of the two physical states above 1 GeV to be accurate due to the neglect (for simplicity) of severalimportant factors, including K ¯ K threshold and mixing with glueballs. The predicted K-matrix unitarized ππ scattering amplitude is given in Fig. 3 and we see a reasonable agreement up to about 1 GeV. roceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011 Pole Mass (MeV) Width (MeV) Mass (MeV) Width (MeV)1 483 455 477 5042 1012 154 1037 843 1082 35 1127 644 1663 2.1 1735 3.5Table III: The physical mass and decay width of the isosinglet scalar states, with m [Π(1300)] = 1 .
215 GeV and with A /A = 20 (the first two columns) and with A /A = 30 (the last two columns). (GeV)-0.4-0.200.20.4 R e { T ) m [ Π (1300)] = 1.2 GeVm [ Π (1300)] = 1.3 GeVm [ Π (1300)] = 1.4 GeV A / A = 30 Figure 3: Prediction of the present mixing approximation for the real part of the I=J=0 ππ scattering amplitudeunitarized by K-matrix method.
4. Summary and Conclusion
This review presented a mixing picture for scalar and pseudoscalar mesons below and above 1 GeV. We sawthat the generalized linear sigma model provides an appropriate framework for studying this mixing and allowsan estimate of the substructure of scalar and pseudoscalar mesons. Allowing a mixing between a two- and afour-quark chiral nonets, the scalars below 1 GeV come out close to four-quark states (whereas those above 1GeV come out closer to two-quark states) and vice versa for pseudoscalar mesons. A summary of this chiralmixing model in presented in Fig. 4 in which it is shown how at the present level of approximation two- andfour-quark nonets are formed out of “bare” states. Further extensions of this model is desirable: Higher order N values are expected to improve the estimates; inclusion of glueballs will allow a more reliable determinationof the isosinglet states above 1 GeV (both scalars and pseudoscalars); and the effect of K-matrix unitarizationon πK channel is expected to reveal the properties of kappa meson. Acknowledgments
The author wishes to thank the organizers of DPF 2011 for a very productive conference. This work is basedon an ongoing collaboration with R. Jora, J. Schechter and M.N. Shahid, and the author thanks collaboratorsfor many helpful discussions. This work has been partially supported by the NSF Grant 0854863 and by a 2011grant from the Office of the Provost, SUNYIT.
Proceedings of the DPF-2011 Conference, Providence, RI, August 8-13, 2011
Figure 4: The “bare” mass spectrum obtained in the present order of approximation ( N ≤
8) of the MM ′ mixing model(with m [Π(1300)] = 1 .
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