Mass of the H-dibaryon
Recent lattice QCD calculations have reported evidence for the existence of a bound state with strangeness and baryon number 2 at quark masses somewhat higher than the physical values. By developing a description of the dependence of this binding energy on the up, down and strange quark masses that allows a controlled chiral extrapolation, we explore the hypothesis that this state is to be identified with the -dibaryon. Taking as input the recent results of the HAL and NPLQCD Collaborations, we show that the -dibaryon is likely to be unbound by MeV at the physical point.
pacs:12.38.Gc, 13.75.Ev, 14.20.Pt
Our understanding of quantum chromodynamics has been challenged for decades by the apparent absence of multi-quark states. The outstanding candidate for such a state has been the -dibaryon, ever since it was suggested that it should be very deeply bound with respect to the threshold Jaffe:1976yi (). Extensive experimental efforts to find this new particle Yoon:2007aq (); Takahashi:2001nm (); Stotzer:1997vr (); Iijima:1992pp () have led to the conclusion that it does not appear to be bound. However, the issue has been given new life in the past few months by reports from the HAL and NPLQCD Collaborations, whose lattice simulations find that the -particle is indeed bound at quark masses somewhat above the physical range Inoue:2010es (); Beane:2010hg ().
Early spectroscopic studies of the baryon and di-baryon spectrum within the bag model Mulders:1982da () showed that the pion cloud contribution was phenomenologically very important in determining whether or not the was indeed bound. A preliminary analysis of some of the lattice data using a fit linear in suggests that it is bound at the physical quark mass, while an extrapolation linear in , while also consistent with binding, does allow that the may be slightly unbound Beane:2011xf (). We consider it important that the extrapolation to the physical quark masses should respect the constraints of chiral symmetry, such as ensuring the correct leading non-analytic behavior. Given the rather large range of quark masses over which current lattice simulations have been made, the technique which offers the best opportunity for a quantitative fit, while preserving the correct non-analytic behavior, is finite range regularization (FRR) Leinweber:2003dg (); Young:2002ib (); Young:2002cj ().
In this Letter we explore the possibility that the is a compact, multi-quark state that may be bound with respect to the threshold. We apply the FRR technique to describe the quark-mass dependence of both the octet-baryon masses and the binding energy of the -dibaryon. With few lattice results available for the dibaryon binding energy, it is essential to utilise the fit to the hyperon masses to determine the dependence on the non-singlet combination of quark masses (i.e., ), which plays a critical role in describing the variation of the (non-singlet) mass. Extrapolated to the physical quark masses, we conclude that the -dibaryon is most likely unbound, with its mass being MeV above the threshold.
Following the technique described in Ref. Young:2009zb (), we fit the data for octet masses recently published by the PACS-CS Collaboration Aoki:2008sm (), using an expansion about the SU(3) limit for the light and strange quark masses:
Here the leading term, , denotes the degenerate mass of the baryon octet in the SU(3) chiral limit, and
with the coefficients given in Table 1, is the correction linear in the quark masses.
At next order, after the linear mass insertions, one finds quantum corrections associated with the chiral loops involving the pseudo-Goldstone bosons, , , . While our formal assessment of the order of a given diagram treats the intermediate octet and decuplet baryons as degenerate, in order to more accurately represent the branch structure near , we retain the octet-decuplet mass difference () in the numerical evaluations. These loops take the form:
The meson loops involve the integrals:
where the subtraction constants, , are defined so that the parameters and are renormalized (explicit expressions may be found in Ref. Leinweber:2003dg (), or can be readily evaluated numerically by Taylor expanding the integrand in ). The loop contribution parameters are taken to be 1.27, , , GeV and GeV. Within the framework of FRR, we introduce a mass scale, , through a dipole regulator, . is related to the scale (typically for a dipole) beyond which a formal expansion in powers of the Goldstone boson masses breaks down Lepage:1997cs (); Young:2004tb (). However, rather than growing uncontrollably as some power of , in this regime the Goldstone loops are actually suppressed, decreasing as powers of Thomas:2002sj (); Stuckey:1996qr (); Donoghue:1998bs (); Leinweber:1998ej (). In practice, we choose this mass parameter by fitting the lattice data itself. Extensive studies of the FRR technique have established that the extrapolation is independent of the functional form chosen for the regulator Leinweber:2003dg () – essentially because of the rapid decrease of the loop contributions which we just explained.
In order to describe the mass of the , treated as a compact, multi-quark state (rather than a loosely bound molecular state), we note that because it is an SU(3) singlet, at the equivalent order of the quark mass expansion its mass can be expressed as:
As indicated in Eq. (5), because it is a flavour-singlet, at leading order in the quark masses depends only on the sum of the quark masses. We therefore set:
where and are parameters determined by the fit to the lattice data for Inoue:2010es (); Beane:2010hg (). Of course, and are determined by the fit to the baryon octet described above. For simplicity, we keep the regulator mass for the octet and the the same, while varying the chiral coefficient for the , , to fit the lattice data.
In order to fully maintain the correlations between the errors associated with all of the fitting parameters, we carried out a simultaneous analysis by minimizing for the fit to both the masses of the nucleon octet and to the difference in mass of the and two hyperons. Of course, the parameters , , , , as well as the regulator mass , were primarily determined by the fit to the PACS-CS data, which is shown in Fig. 1, with the corresponding parameters given in Table 3. We note that, as explained in Ref. Young:2009zb (), the octet data was corrected for small, model independent finite volume effects before fitting. In the figure we show the lattice data after applying both the finite volume correction and a correction (based upon our fit parameters) arising because the PACS-CS simulations used values of the strange quark mass that were somewhat larger than the empirical values. The per degree of freedom for the octet data was (7.3 divided by ). This is lower than unity as, without access to the original data, we cannot incorporate the effect of correlations between the lattice data. Nevertheless, the fit is clearly very satisfactory over the entire range of quark masses explored in the simulations and should provide an excellent basis for the study of the possible binding of the -dibaryon. Indeed, the masses of the and baryons at the physical point are GeV, where all the errors include the correlated uncertainties of all the fit parameters, including the regulator mass, . For comparison we note that the physical octet masses are GeV.
|best fit value||1.02||0.861||-1.71||-1.20||-0.51||0.019||-2.36||5.65|
With respect to the -dibaryon we have retained only the data from the HAL Collaboration Inoue:2010es () which was generated on the largest lattice volume, namely fm. These data points correspond to large (degenerate) pseudoscalar masses, , and GeV, for which the finite volume corrections are expected to be very small. Accordingly we used the reported values without applying any finite volume correction. In the case of the NPLQCD Collaboration Beane:2010hg (), where the calculation was performed at MeV and MeV, we include in the fit only the value for the binding of the determined after their extrapolation to infinite volume.
We chose the mass splitting between the -dibaryon and the other dibaryon states appearing in its chiral loop corrections to be the same as the octet-decuplet mass splitting used earlier, namely GeV. This is compatible with the estimates of Aerts et al. Aerts:1977rw () calculated within the MIT bag model, as well as with the experimental absence of other nearby states. The sensitivity of our fit to is quite small, with an increase (decrease) of by 100 MeV increasing (decreasing) the mass difference by only 4 MeV. This small shift is combined in quadrature with the error found from our chiral fit to yield the final, quoted error in the binding of the .
The best fit parameters describing the binding energy of the -dibaryon are also given in Table 3 and the actual fit is shown in Fig. 2. As explained above, the data shown in Fig. 2 are from NPLQCD (lowest mass point) and HAL (three largest mass points). In each case the curve nearest the data point illustrates the extrapolation as a function of the light quark mass implied by our fit at the value of the strange quark mass corresponding to that lattice data point. The errors shown are the result of combining in quadrature the statistical and systematic errors quoted by the collaborations. The shaded error bands incorporate the effect of correlations between the fit parameters, including the uncertainty on the regulator mass. We note the remarkable result that the best fit value of the chiral coefficient for the -dibaryon, (which for convenience is normalized with respect to the chiral coefficient for loops on the hyperon in Eq. (6)), is within 20% of the theoretical value reported by Mulders and Thomas Mulders:1982da (), who calculated it using SU(6) symmetry. If instead we retain the Mulders-Thomas coefficient, the -dibaryon is unbound by MeV but the quality of the fit is significantly reduced, with a per degree of freedom of almost 2.
It is clear from Figs. 1 and 2 that both the octet data and the data on the binding energy of the -dibaryon are very well described. The binding of the is reduced by a decrease in the masses of the and quarks and the , with significant chiral curvature for below 0.4 GeV. It is important to note that our analysis does not include the effect of the coupling of the -dibaryon to the open channel. Given where we find the multi-quark state, this might be either attractive or repulsive, depending on the range of momenta which dominate the continuum channel. It is also true that it would be extremely valuable to have new simulations at lower pion mass and large volume which would further constrain the extrapolation. Nevertheless, the conclusion of our study is quite clear. Even though the is bound at larger quark masses, we see in Fig. 2 that chiral physics leads to a more rapid decrease of the mass of the as approaches its physical value than we find for the and as a result one must conclude that at the physical values of the quark masses the -dibaryon is most likely unbound. Our estimate, including the effect of correlations between all the fit parameters, is that the is unbound by a mere MeV at the physical point. That this is so close to the threshold will undoubtedly spur investigations into the consequences for doubly strange hypernuclei as well as the equation of state of dense matter.
This work was supported by the University of Adelaide and the Australian Research Council through grant FL0992247 (AWT) and DP110101265 (RDY).
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