Z c (3900) as a Four-Quark State

Transcription

1 arxiv: v [hep-ph] 6 Aug 4 Z c (39) as a Four-Quark State Leonard S. Kisslinger and Steven Casper Department of Physics, Carnegie Mellon University, Pittsburgh, PA 53 Abstract Using the method of QCD Sum Rules, we derive the correlator Π Z for a state consisting of two charm quarks and two light quarks, c cu d, and carry out a Borel transform to find Π Z (M B ). From this we find the solution that M B 3.9 ±. GeV, showing that the Zc(39) is a tetra-quark state. PACS Indices:.5.-q,.38.Lg,4.4.Lb,4.65.Dw Keywords: Four quark state; charm states; QCD sum rules Introduction Recently e + e collision experiments by BESIII[] and Belle[] Collaborations have found a stateat about3,9mev, called thez c (39)[], that might beafour quark state, c cu d >[3] For many years states in the energy region of 3,9-4,5 MeV, near the D D threshold, there have been found possible tetra-quark states. See, e.g., Ref[4], for a theoretical study of the X(387) as a charm tetra-quark about one decade ago. Recently, there have been studies of the Z c (39) as a DD molecular state[5, 6], using the method of QCD Sum Rules as in the present study. See these references for references to earlier publications. In our study of the Z c (39) as possibly a c cu d > state we use the method of QCD sum rules[7]. Our approach differers from earlier studies[5, 6] in that our correlator corresponds to a four-quark (tetraquark) rather than a DD molecular state. First we find the correlator in momentum space, and then as a function of the Borel mass, M B, and see if it has a minimum near the value of 3.9 GeV, similar to our study of heavy quark hybrid meson states[8]. In Section II we briefly review the method of QCD sum rules, and derive the correlator for our four-quark model. In Section III we find the correlator as a function of the Borel mass in the region near 4, Mev; and then from a plot find the minimum value of M B. In Section IV we discuss the results and conclusions.

2 The c cu d > state and QCD Sum Rules The method of QCD sum rules[7] for finding the mass of a state A starts with the correlator, Π A (x) = T[J A (x)j A ()], () with the vacuum state and the current J A (x) creating the states with quantum numbers A: J A (x) = c A A + n c n n;a, () where A is the lowest energy state with quantum numbers A, and the states n;a are higher energy states with the A quantum numbers, which we refer to as the continuum. One then carries out a Borel transform to reduce the importance of the continuum and higher order diagrams. ForourtheoryoftheZ c (39)asa c cu d >, thecurrent isaproductofcurrents producing a D + (c d,j P = ) and a D ( cd,j P = ) J D + J D = i dγ 5 c = i cγ 5 u J Zc = J D +J D (3) Π Z (x) = T[J Zc (x)j Zc ()]. (4) Using Wick s Theorem to express Π Z (x) in terms of the quark propagators, and taking the Fourier transform, one obtains the correlator in momentum space: Π Z (p) = d 4 k d 4 k d 4 k 3 ((π) 4 ) 3 Tr[S d (k )γ 5 S c (k )γ 5 ]Tr[S c (k 3 )γ 5 S u (p+k k 3 +k 3 )γ 5 ], (5) with S q (k) a quark propagator, S q (k) = k +m q k m q where k = αγ α k α, with γ α a Dirac matrix. The correlator is illustrated in Fig., (6)

3 c d u c Figure : c, c are charm, anticharm quarks. u, d are up, antidown quarks Finally, we carry out a Borel Transform[7] BΠ Z (p) = Π Z (M B ) (7) 3 The Correlator Π Z (p) Π Z (M B ) From Eqs(5,6), carrying out the traces, one finds Π Z (p) 6 = d 4 k d 4 k d 4 k 3 ((π) 4 ) 3 mm(mm k k )+k 3 (p+k k +k 3 )(k k mm) (k m )(k M )(k 3 M )((p+k k +k 3 ) m ) (8). Since the k k term vanishes via the momentum integrals and the k 3 (p+k k +k 3 ) term vanishes via the Borel transform, one needs to evaluate two terms; d 4 k d 4 k d 4 k 3 m M I (p) = 6 ((π) 4 ) 3 (k m )(k M )(k3 M )((p+k k +k 3 ) m ).(9) d 4 k d 4 k d 4 k 3 k k k 3 (p+k k +k 3 ) I 4 (p) = 6 ((π) 4 ) 3 (k m )(k M )(k3 M )((p+k k +k 3 ) m ) (), with M = M(charm quark).5 GeV and m = m(u,d quark) 4 MeV. 3

4 and For the evaluation of Eqs(9,) one uses d 4 k I H (p) = (π) 4 (k M )((p k) M ) = (4π) (5 p 4M ) dα α( α)p M, () I Hh (p) = = d 4 k (π) 4 (k M )((p k) m ) M +p ) +m ) (4π) [( (m p dα α( α)p ( α)m αm ] + 5 3M m p ln(m /m ). () Carrying out the integrals in Eqs(9,), one finds I(p) = 6m M (4π) 6 dα 5(.8α( α)) (α( α)) [ m 3M α( α) ln(m /m )I Hh (p) m= + m +M α( α) α( α) β( β)α( α) I Hh(p) ma (3) m ( (β( β)) + m +α( α)m )I m Hh (p) mb + m +α( α)m I α( α) m Hh (p) m= ], α( α) with ma = ( β)m +βm and mb = m /(β( β)) ; and I4(p) = 3 (4π) m dα dγ α( α) (5α( α) 7/) γ [M 5m 7 m /4 I Hh (p) m γ +(p m )(m 4 m 3)( 3γ)I Hh (p) m 4 ], (4) with m 3 = (M + ( γ)m )/(γ( γ)), m 4 = (M +(γ )m )/(γ( γ)), and m = m /(α( α)). Taking the Borel transform one obtains (after removing common factors) BI(p) = 5. dα dβ(.8α( α))[((m 3α( α)m )ln(m /m ) +(m +α( α)m )/(( β)m βα( α)m )) (M ( α β( β) )e αm /β( β)mb + dγ(/)[ (+β)m +( α)(α+β)m (m +α( α)m )α( α) M ( β)m βα( α)m ) (3m M +(( β)m α( α)m (m M )γ( γ) )/)+ (( β)m α( α)m ) ] e (( β)m α( α)m )/[α( α)γ( γ)m B ] (5) 4

5 BI4(p) = dα dγ (5α( α) 7/)[(5α( α)γ( γ) 7/4)m (m M α( α)γ( γ)+ ((m M α( α)γ( γ)) λ( λ) (( λ)m +λα( α)γ( γ)m ) + (( λ)m λα( α)γ( γ)m )) e (( λ)m +λα( α)γ( γ)m )/α( α)γ( γ)λ( λ)m B ((3(γ )m +α( α)( γ( γ)) M ) M ((γ )m + α( α)( γ( γ))m ) e M /[α( α)γ( γ)λ( λ)m B ] (6) BΠ Z (p)(p) = Π Z (M B ) = BI(p)+BI4(p). (7) The correlator as a function of the Borel mass, Π Z (M B ) is shown in Fig. below. Note that the result for the mass is given by M B at the minimum in the plot of Π Z (M B ), and the error by the shape of the plot near the minimum[7], as is discussed in detail in Ref[8]. Π(M B ) (M ) B Figure : Π(M B ) is the 4-quark correlator,a function of the Borel Mass M B, in units of MeV 5

6 4 Results and Conclusions From Figure, the mass of the c cu d > state is 39 ± MeV, in agreement with the state found in the recent BESIII[] and Belle[] experiments. From this we conclude that the conjecture of these two collaboretions is correct, that the Z c (39) is a four-quark state. For decades experimentalists and theorists have attempted to find tetra-quark states, so this is a very important discovery. Acknowledgements This work was supported in part by a grant from the Pittsburgh Foundation. LSK thanks LANL experimentalists for suggesting that the BESIII and Belle experiments could be an important discovery in high energy particle physics, and for a discussion with Prof. Eric Swanson. References [] M. Ablikim et al., BESIII Collaboration, Phys. Rev. Lett., 5 (3). [] Z.Q. Liu et al., Belle Collaboration, Phys. Rev. Lett., 5 (3). [3] Eric Swanson, Physics 6, 69 (3). [4] L. Maiani, F. Piccinini, V. Riquer, and A.D. Polosa, Phys. Rev. D 7, 48 (5). [5] Jian-Rong Zhang, Phys. Rev. D 87, 64 (3) [6] Zhi-Gang Wang, Tao Huang, Phys. Rev. D 89, 549 (4) [7] M.A. Shifman, A.I. Vainstein, and V.I. Zakharov, Nucl. Phys. B47, 385 (979); B47, 448 (979). [8] Leonard S. Kisslinger, Phys. Rev. D 79, 46 (9) 6