Exotic Tetraquark Mesons in Large- N c Limit: an Unexpected Great Surprise
aa r X i v : . [ h e p - ph ] A ug Exotic Tetraquark Mesons in Large- N c Limit:an Unexpected Great Surprise
Wolfgang Lucha , ∗ , Dmitri Melikhov , , , ∗∗ , and Hagop Sazdjian , ∗∗∗ Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18,A-1050 Vienna, Austria D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State University,119991 Moscow, Russia Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria Institut de Physique Nucléaire, CNRS-IN2P3, Université Paris-Sud, Université Paris-Saclay,91405 Orsay Cedex, France
Abstract.
Two-ordinary-meson scattering in large- N c QCD implies consistencycriteria for intermediate-tetraquark contributions. Their fulfilment at N c -leadingorder constrains the nature of the spectrum of genuinely exotic tetraquark states. N c -Leading First-Principles Approach to Tetraquarks Qualitative information on systems controlled by quantum chromodynamics may be collectedby considering large- N c QCD [1, 2], a quantum field theory generalizing QCD by enabling thenumber N c of colour degrees of freedom to di ff er from N c = N c α s of N c and the strong fine-structure coupling α s ≡ g π approaches a finite value in the large- N c limit. For the N c behaviour of α s , this demand implies α s ∝ N c for N c → ∞ . In an attempt to stay as close as possible to intuition, we adhere to the presumably very natural(but clearly not compulsory) assumption that the fermionic dynamical degrees of freedom, thequarks, continue to transform according to the N c -dimensional, fundamental representation ofthe gauge group SU( N c ) . By utilizing QCD’s large- N c limit ( N c → ∞ ) and 1 / N c expansion (inpowers of 1 / N c ) about this limit, we extract constraints on crucial features ( e.g. , decay widths)[3, 4] of tetraquarks, meson bound states of two quarks and two antiquarks predicted by QCD.With respect to their flavour degrees of freedom, tetraquarks can be classified (Table 1) byspecifying — for the two quarks and two antiquarks constituting the tetraquark bound state — • the number of di ff erent quark flavours encountered in such bound state, in combination with • the total number of open quark flavours, defined as the number of quark flavours that are notcounterbalanced by an antiquark of same flavour and hence carried by the observed mesons. ∗ e-mail: [email protected] ∗∗ e-mail: [email protected] ∗∗∗ e-mail: [email protected] able 1. Classification of tetraquark mesons by content of di ff erent and open quark flavour. The notion open-flavour number relates to the net sum of flavours not compensated by a corresponding antiflavour. number of di ff erent tentative quark configuration number of openquark flavours involved ¯ q (cid:3) q (cid:3) ¯ q (cid:3) q (cid:3) quark flavours involved4 ¯ q q ¯ q q
43 ¯ q q ¯ q q q q ¯ q q q q ¯ q q q q ¯ q q
22 ¯ q q ¯ q q q q ¯ q q q q ¯ q q q q ¯ q q q q ¯ q q
01 ¯ q q ¯ q q flavour-exotic tetraquarks T = ( ¯ q q ¯ q q ) with all four (anti-) quark flavours di ff erent;2. flavour-cryptoexotic tetraquarks T = ( ¯ q q ¯ q q ) involving three di ff erent (anti-) quarkflavours, by containing a quark–antiquark pair of same flavour di ff ering from the others.In order to work out — at least, at a qualitative level — some basic features of such kind oftetraquarks, we investigate, for two ordinary mesons of appropriate flavour quantum numbers,their possible scattering reactions into two ordinary mesons with regard to potential s -channelcontributions of intermediate poles interpretable as a manifestation of tetraquarks with narrowdecay width. For both sets of tetraquark in our focus of interest, as well as for the one with twodi ff erent flavours but no open flavour, we have to analyze two variants of scattering processes: • flavour-preserving ones, with identical flavour content of the initial- and final-state mesons; • flavour-rearranging ones, with unequal flavour content of the initial- and final-state mesons.We identify all the contributions to four-point correlation functions of quark bilinear operators j i j ≡ ¯ q i q j interpolating ordinary mesons M i j (notationally exploiting the actual irrelevance ofparity and spin therein) that are capable of supporting a pole related to a tetraquark built up byfour (anti-) quarks of masses m i , i = , . . . , , by imposing an unambiguous selection criterion[3, 4]: for the scattering of two mesons of momenta p and p , an allowable Feynman diagramnot only has to depend on the Mandelstam variable s ≡ ( p + p ) in a non-polynomial way buthas to admit a four-quark intermediate state with related branch cut starting at the branch point s = ( m + m + m + m ) . ⇐⇒ flavour-exotic tetraquark meson Surprisingly or not, for truly flavour-exotic tetraquark states, T = ( ¯ q q ¯ q q ) , the leading- N c dependence of all contributions to the correlation functions of four quark bilinear operators j i j b) s α s~ N c2 (c)(a) α j jj jj jj ~ N c jj jj ~ N c2 2 j Figure 1.
Flavour-preserving four-point correlation function of quark bilinear operators j ij : examples ofFeynman diagrams contributing at low order to this correlator’s 1 / N c expansion, viz. , at order N (a,b) or N (c) [3, Fig. 1]. Tetraquark-friendly contributions of lowest order in 1 / N c turn out to be of order N (c). s (a) α α s~ N c jj jj (b) ~ N c jj jj ~ N c 2 jj jj (c) Figure 2.
Flavour-rearranging four-point correlation function of quark bilinear currents j ij : examples ofFeynman diagrams contributing at low order to this correlator’s 1 / N c expansion, that is, at order N c (a,b)or N − (c) [3, Fig. 2]. Tetraquark-phile contributions of lowest order in 1 / N c prove to be of order N − (c). (exemplified in Figs. 1 and 2) potentially capable of developing this tetraquark pole di ff ers , forthe flavour-preserving (Fig. 1(c)) and flavour-rearranging (Fig. 2(c)) cases, by one order of N c : h j † j † j j i T = O ( N ) , h j † j † j j i T = O ( N ) , h j † j † j j i T = O ( N − ) . The resulting N c -leading tetraquark–two-ordinary-meson amplitudes A imply the presence of,at least, two tetraquarks T A , B with, however, decay rates Γ ( T A , B ) of similar large- N c decrease: A ( T A ←→ M M ) = O ( N − ) | {z } = ⇒ Γ ( T A ) = O ( N − ) N c > A ( T A ←→ M M ) = O ( N − ) , A ( T B ←→ M M ) = O ( N − ) N c < A ( T B ←→ M M ) = O ( N − ) | {z } = ⇒ Γ ( T B ) = O ( N − ) . In such a situation, their conclusions some people in the form may phrase “always two there are, . . . , no less” [5].
Two open quark flavours ≡ flavour-cryptoexotic tetraquark meson In the case of flavour-cryptoexotic tetraquark mesons built from three di ff erent quark flavours, T = ( ¯ q q ¯ q q ) , the large- N c behaviour of flavour-preserving (Fig. 3) and flavour-reshu ffl ing(Fig. 4) subcategories of those contributions to the correlation functions of four quark bilinearcurrents j i j which might support the development of a tetraquark pole turns out to be identical: h j † j † j j i T = O ( N ) , h j † j † j j i T = O ( N ) , h j † j † j j i T = O ( N ) . (b) s j j α s (a) α jj jj
12 12 ~ N c2 2
23 23
23 2 2 jj ~ N c2 2 Figure 3.
Flavour-preserving four-point correlation function of quark bilinear operators j ij : examples oftetraquark-phile Feynman diagrams contributing at the N c -leading order N to this correlation function’s1 / N c expansion [3, Fig. 3]. (Purple crosses indicate quarks potentially contributing to a tetraquark pole.)
33 2 j j j j α s~ N c 2 (a) α s~ N c2 2 (b) jj jj
23 1322
Figure 4.
Flavour-rearranging four-point correlation function of quark bilinear currents j ij : examples oftetraquark-phile Feynman diagrams contributing at low order to such correlator’s 1 / N c expansion, i.e. , atorder N − (a) or N (b) [3, Fig. 4]; among these, the N c -leading contributions prove to be of order N (b). So, a single tetraquark T C (which, due to its cryptoexotic nature, can mix with the meson M )satisfies all constraints induced by the N c -leading tetraquark–two-ordinary-meson amplitudes A ( T C ←→ M M ) = O ( N − ) N c = A ( T C ←→ M M ) = O ( N − ) | {z } = ⇒ Γ ( T C ) = O ( N − ) . Insights: N c -Leading Conclusions for (Crypto-) Exotic Tetraquarks In summary, we find that self-consistency conditions arising from the inspection of tetraquarkcontributions to the scattering amplitudes of two ordinary mesons into two ordinary mesons inthe 1 / N c expansion of large- N c QCD provide rigorous constraints on the features of tetraquarkstates [3, 4]. Demanding these constraints to be satisfied at N c -leading order implies [3, 4] that • genuinely exotic tetraquarks need to appear in pairs ( T A , T B ) the members of which di ff er inthe large- N c behaviour of their dominant or preferred decay modes to two ordinary mesons; • both genuinely exotic ( T A , B ) and cryptoexotic ( T C ) tetraquarks exhibit narrow decay widths Γ ( T ) ∝ / N −−−−−→ N c →∞ T = T A , T B , T C . Table 2 confronts these findings for the rates of the large- N c decrease of the total decay widthsof exotic and cryptoexotic tetraquarks with corresponding outcomes of earlier analyses [6–8].Imposition of additional requirements clearly may strengthen the predicted large- N c decrease.Di ff erences to our results arise from misidentifying the actually N c -leading contribution to thetetraquark pole or from consideration of merely a single (say, the flavour-reshu ffl ing) channel. Table 2.
Comparison: predictions of upper bounds on the large- N c behaviour of tetraquark decay rates. Author Collective Decay Width Γ ReferenceExotic Tetraquarks Cryptoexotic TetraquarksLucha et al. O (1 / N ) O (1 / N ) [3, 4]Knecht and Peris O (1 / N ) O (1 / N c ) [6]Cohen and Lebed O (1 / N ) — [7]Maiani et al. O (1 / N ) O (1 / N ) [8] Acknowledgements
D. M. is grateful for support by the Austrian Science Fund (FWF) under project P29028-N27.
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