Exploration of a physical picture for the QCD scalar channel
aa r X i v : . [ h e p - ph ] O c t SU-4252-879
Exploration of a physical picture for the QCD scalar channel
Amir H. Fariborz a ‡ , Renata Jora b † , and Joseph Schechter c § a Department of Mathematics/Science, State University of New York Institute of Technology, Utica, NY 13504-3050, USA. and b , c Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA, (Dated: October 30, 2018)A generalized linear sigma model is employed to study the quark structure of low lying scalar aswell as pseudoscalar states. The model allows the possible mixing of quark anti-quark states withothers made of two quarks and two antiquarks but no a priori assumption is made about the quarkcontents of the predicted physical states. Effects of SU(3) symmetry breaking are included. Thelighter conventional pseudoscalars turn out to be primarily of two quark type whereas the lighterscalars turn out to have very large four quark admixtures.
PACS numbers:
I. INTRODUCTION
Recently the Belle collaboration [1] provided strong evidence for the Z(4430) resonance in the Ψ ′ - π channel. Ithas the quantum numbers of c ¯ cn ¯ n ,where n stands for either a u or d quark, and would thus seem to be a “smokinggun” candidate for a meson containing two quarks and two antiquarks. Are there others? Many people believe thatthe mass ordering of the candidates for the lightest scalar nonet suggests such a picture: I = 0 : m [ f (600)] ≈
500 MeV n ¯ nn ¯ nI = 1 / m [ κ ] ≈
800 MeV n ¯ nn ¯ sI = 0 : m [ f (980)] ≈
980 MeV n ¯ ns ¯ sI = 1 : m [ a (980)] ≈
980 MeV n ¯ ns ¯ s (1)Here the postulated four quark content is displayed for each state. This level ordering, obtained by simply countingthe number of strange (s) type quarks, is seen to be flipped [2] compared to that of the standard vector meson nonet: I = 1 : m [ ρ (776)] ≈
776 MeV n ¯ nI = 0 : m [ ω (783)] ≈
783 MeV n ¯ nI = 1 / m [ K ∗ (892)] ≈
892 MeV n ¯ sI = 0 : m [ φ (1020)] ≈ s ¯ s (2)Note that the level inversion of four quark states would hold either for “molecular” or diquark- antidiquark pictures.There is another side to this story. Why are the experimental candidates for a “normal” p-wave q ¯ q scalar nonete.g., a (1450) , K (1430) etc., (3)somewhat heavier than expected? A possible answer, based on the repulsion of“two quark” and “four quark” stateswhich mix with each other, was proposed some time ago [3], see also [4]. This level repulsion also would explain whythe lower scalars seem to be unusually light. II. TOY MODEL TO CHECK MIXING PICTURE
Note that QCD with massless light quarks obeys SU(3) L x SU(3) R symmetry spontaneously broken to SU(3) V .We wish to realize this in a Lagrangian model [5] with linearly transforming chiral nonet fields (There are necessarily ‡ Email: [email protected] † Email: [email protected] § Speaker;Email: [email protected] two scalar nonets as well as two pseudoscalar nonets present in this approach). We input some physical particlemasses and predict the two vs. four quark contents of each state. The non-inputed masses are also predicted. Tocheck the stability of the approach we have carried out the calculations for the zero mass [6] quark case, the non-zeroequal quark mass case [7] and finally the non equal quark mass case. The method is conceptually straightforward butcomplicated in detail.We employ the 3 × M = S + iφ, M ′ = S ′ + iφ ′ . (4)The matrices M and M ′ transform in the same way under chiral SU(3)x SU(3) transformations but may be dis-tinguished by their different U(1) A transformation properties. M desribes the “bare” quark antiquark scalar andpseudoscalar nonet fields while M ′ describes “bare” scalar and pseudoscalar fields containing two quarks and twoantiquarks. At the symmetry level with which we are working, it is is unnecessary to further specify the four quarkfield configuration.The Lagrangian density 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 , (5)where V ( M, M ′ ) stands for a function made from SU(3) L × SU(3) R (but not necessarily U(1) A ) invariants formedout of M and M ′ . The leading choice of terms corresponding to eight or fewer quark plus antiquark lines at eacheffective vertex reads: 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( det M det M † ) + (1 − γ ) Tr( M M ′† )Tr( M ′ M † ) (cid:21) . (6)All the terms except the last two (more discussion of them is given in [8]) possess the U(1) A invariance. The symmetrybreaking term which models the QCD mass term takes the form: V SB = − A S ) (7)where A = diag ( A , A , A ) is proportional to the three light quark mass matrix, diag ( m u , m d , m s ). The modelallows for two quark condensates, α a = h S aa i as well as four quark condensates β a = h S ′ aa i . These are assumed toobey isotopic spin symmetry: α = α = α , β = β = β (8)We also need the “minimum” conditions, (cid:28) ∂V ∂S (cid:29) + (cid:28) ∂V SB ∂S (cid:29) = 0 , (cid:28) ∂V ∂S ′ (cid:29) = 0 . (9)There are twelve parameters describing the Lagrangian and the vacuum. These include the six coupling constantsgiven in Eq.(6), the two quark mass parameters, ( A = A , A ) and the four vacuum parameters ( α = α , α , β = β , β ).The four minimum equations reduce the number of needed input parameters to eight.Five of these eight are supplied by the following masses together with the pion decay constant: m [ a (980)] = 984 . ± . m [ a (1450)] = 1474 ±
19 MeV m [ π (1300)] = 1300 ±
100 MeV m π = 137 MeV F π = 131 MeV (10)Because m [ π (1300)] has such a large uncertainty, we will, as previously, examine predictions depending on the choiceof this mass within its experimental range. The sixth input will be taken as the light “quark mass ratio” A /A ,which will be varied over an appropriate range. The remaining two inputs will be taken from the masses of the four(mixing) isoscalar, pseudoscalar mesons. This mixing is characterized by a 4x4 matrix M η . A practically convenientchoice is to consider Tr M η and det M η as the inputs. Note that the presence of the last two terms in Eq.(6)- whichexactly mock up the QCD U(1) A anomaly - decouples the initial treatment of the other particles from that of thecomplicated pseudoscalar singlet sector.Given these inputs there are a very large number of predictions. At the level of the quadratic terms in theLagrangian, we predict all the remaining masses and decay constants as well as the angles describing the mixingbetween each of ( π, π ′ ), ( K, K ′ ), ( a , a ′ ), ( κ, κ ′ ) multiplets and each of the 4 × III. BRIEF SUMMARY OF RESULTS
A detailed report on the results when the SU(3) flavor symmetry breaking, which yields m K = m π , is taken intoaccount will be given elsewhere (in preparation). Here we just mention some main features.It is comforting to first note that the appropriate predicted results do not change much as one proceeds from zeroquark masses (and hence, by spontaneously broken chiral symmetry, zero masses for the lighter pseudoscalar octet) tonon-zero but degenerate light quark masses (and hence, the lighter pseudoscalar masses all taking the value, m π ) andfinally to the realistic case where m π , m K and m η all differ from each other. The zero quark mass case is an especiallyimportant “touchstone” since it was noted in the second of ref.[5] that there are actually twenty one different allowedterms which might replace the symmetry breaker, Eq.(7). Hence, without such a check, one might worry that thesomewhat surprising results obtained could be an artifact of a particular choice of symmetry breaking terms.In the zero light mass case the only change in the inputs of Eq.(10) is to set m π = 0. A suitable choice for the“adjustable” parameter, m[ π (1300)] turned out [6] to be 1215 MeV. Then the masses of the two scalar SU(3) singletsare predicted as, m σ ≈ M eV m σ ′ ≈ M eV. (11)Clearly the lighter SU(3) singlet is the abnormally light f (600) candidate. These two states turn out to be roughlythe linear combinations, 1 √ S + S + S ) ± ( S ′ + S ′ + S ′ )] , (12)so the “sigma” appears to be 50 per cent two quark and 50 percent four quark in nature. As for the other SU(3)multiplets, the model predicts the 2 quark [2] and four quark [4] contents as roughly : lighter + octet : 0 . . lighter − octet : 0 . . − octet and the mainly four quark lighter 0 + octet is evident.Since, for example, the lighter and heavier 0 + octets mix with each other, the heavier 0 + octet would have a 24percent four quark content and a 76 percent two quark content.These results are not essentially changed in the case [7] when the light 0 − octet has the mass, m π instead of masszero.To trace what happens in the scalar isosinglet sector when the SU(3) symmetry breaking is turned on, we notethat all four such states (i.e. including the appropriate SU(3) octet members) will mix with each other. We use theconvenient basis fields: S + S √ , S , S ′ + S ′ √ , S ′ , (14)and label the four scalar isosinglets, which are linear combinations of the above, in order of increasing mass as f , f , f , f . The lightest, f is identified with the “sigma” and is predicted to have a mass about 730 MeV which isqualitatively similar to that of the previous lighter scalar SU(3) singlet state. It is predicted to have percentages ofthe basis states in Eq.(14) 0 . , . , . , . . (15)Thus we may give the 2 quark and 4 quark percentages of the “sigma” as f : 0 . , . , (16)which is similar to the previous 50-50 split. The predicted two quark vs. four quark percentages for some other lighterparticles in this model are, π : 0 . , . K : 0 . , . κ : 0 . , . . (17)These are again similar to those in Eq.(13). It seems as though, large four quark content for the lighter scalar statesis a stable result of the present model. IV. PHYSICAL INTERPRETATION OF SCALAR MASSES
The masses obtained above appear as tree level quantities in the effective Lagrangian under discussion. Especiallyin the case of the scalars the physical states are rather broad and will appear as poles in the scattering of twopseudoscalar mesons. A simple way to estimate the scattering amplitude is to first compute the tree level scalarpartial wave scattering amplitude and then unitarize it by using the K-matrix method. This is equivalent to an earlierapproach [9] and amounts to replacing the tree level amplitude, T tree by, T = T tree − iT tree . (18)For example, the sigma pole at 730 MeV discussed in the last section appears in the unitarized pion scatteringamplitude at z ≡ M − iM Γ with M = 473 M eV,
Γ = 473
M eV. (19)This is of the same order as usual estimates for the sigma. Such scattering calculations should be performed to findthe “actual” mass and width parameters for all the scalars in the present model.
Acknowledgments
One of us (JS) would like to thank the organizers of QCD08 for arranging such a stimulating and friendly conference.We are happy to thank A. Abdel-Rehim, D. Black, M. Harada, S. Moussa, S. Nasri and F. Sannino for many helpfulrelated discussions. This work was supported in part by the U. S. DOE under contract no. DE-FG-02-85ER 40231. [1] S.-K.Choi et al, arXiv:0708.1790.[2] R.L. Jaffe, Phys. Rev. D , 267 (1977);D. Black, A.H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D59 , 074026(1999).[3] D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D , 1391 (2002); ibid , 663 (2004); F. Close and N. Tornqvist, ibid. , R249 (2002); A.H. Fariborz, Int. J. Mod. Phys. A , 2095 (2004); 5417 (2004); Phys. Rev. D , 054030 (2006); M.Napsuciale and S. Rodriguez, Phys. Rev. D , 094043 (2004); F. Giacosa, Th. Gutsche, V.E. Lyubovitskij and A. Faessler,Phys. Lett. B , 277 (2005); J. Vijande, A. Valcarce, F. Fernandez and B. Silvestre-Brac, Phys. Rev. D , 034025 (2005);S. Narison, Phys. Rev. D , 114024 (2006); L. Maiani, F. Piccinini, A.D. Polosa and V. Riquer, hep-ph/0604018; J.R.Pelaez, Phys. Rev. Lett. , 102001 (2004); J.R. Pelaez and G. Rios, Phys. Rev. Lett. , 242002 (2006); F. Giacosa, Phys.Rev. D ,054007 (2007); G. ’t Hooft, G. Isidori, L. Maiani, A.D. Polosa and V. Riquer, arXiv:0801.2288[hep-ph].[5] Section V of D. Black, A.H. Fariborz, S. Moussa, S. Nasri and J. Schechter, Phys. Rev. D , 014031 (2001); A.H. Fariborz,R. Jora and J. Schechter, Phys. Rev. D , 034001 (2005).[6] A.H.Fariborz, R.Jora and J.Schechter, Phys. Rev. D , 034006 (2008), arXiv:0707.0843 [hep-ph].[7] A.H.Fariborz, R.Jora and J.Schechter, Phys. Rev. D , 114001 (2007), arXiv:0708.3402 [hep-ph].[8] A.H.Fariborz,R.Jora and J.Schechter, Phys. Rev. D , 094004 (2008), arXiv:0801.2552 [hep-ph].[9] N.N. Achasov and G.N. Shestakov, Phys. Rev. D49