From string breaking to quarkonium spectrum
Marco Catillo, Marina Krstić Marinković, Pedro Bicudo, Nuno Cardoso
LLMU-ASC 21/20
From string breaking to quarkonium spectrum ∗ Marco Catillo † , Marina Krsti´c Marinkovi´c Arnold-Sommerfeld-Center for Theoretical Physics,Ludwig-Maximilians-Universit¨at, Theresienstr. 37, 80333 M¨unchen, Germany
Pedro Bicudo, Nuno Cardoso
CeFEMA, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa,PortugalWe present a preliminary computation of potentials between two staticquarks in n f = 2 QCD with O(a) improved Wilson fermions based on Wil-son loops. We explore different smearing choices (HYP, HYP2 and APE)and their effect on the signal to noise ratio in the computed static potentials.This is a part of a larger effort concerning, at first, a precise computationof the QCD string breaking parameters and their subsequent utilization forthe recent approach based on Born-Oppenheimer approximation (Bicudoet al. 2020 [1]) to study quarkonium resonances and bound states.
1. Introduction
The computation of quarkonium spectrum is one of the most challeng-ing problems in lattice QCD. Recent publications [1–4] provided a newand interesting method to study hadron resonances as well as exotic boundstates, which are currently posing a challenge for lattice QCD computations.The method is based on the Born-Oppenheimer approach, which approxi-mates the Hamiltonian for non-relativistic particles, and gives a Schr¨odingerequation that can be solved numerically. Recent studies [1–4] demonstratepromising results regarding energy levels, potentials and wave functions incase where heavy quarks are considered in the non-exotic or exotic quarko-nium spectrum. One of the necessary ingredients in this approach is theunderstanding of the string breaking phenomenon [5–9], which is commonlydescribed as the breaking of a flux tube formed due to a separation of theheavy quark-antiquark pair. At a large enough distance, the production oflight quark-antiquark pairs becomes more favorable than maintaining a fluxtube and systems of heavy-light mesons are formed. The transition from aquark-antiquark system to a meson-meson system in a n f = 2 QCD, can bedescribed by a 2 × ∗ Talk at “Excited QCD 2020”, Krynica Zdr´oj, Poland, February 2-8, 2020. † Presenter. (1) a r X i v : . [ h e p - l a t ] J un MarcoCatillo printed on June 23, 2020 involves correlators of different operators, namely heavy quark and mesonoperators, which are not orthogonal and the presence of mixing terms iscrucial for the comprehension of such a transition. We first focus on theupper left element of such a matrix, which is related to the static potentialof a quark-antiquark system.
2. Theoretical aspects
Given a system of two heavy quarks Q and ¯ Q with mass m Q [5, 7, 9], thefollowing matrix of correlators: C ( t ) = (cid:18) C QQ ( t ) C QB ( t ) C BQ ( t ) C BB ( t ) (cid:19) = e − m Q t C ( t ) = ( C QQ ( t ) C QB ( t ) C BQ ( t ) C BB ( t ) ) = e − m Q t √ n f √ n f − n f + C ( t ) = ( C QQ ( t ) C QB ( t ) C BQ ( t ) C BB ( t ) ) = e − m Q t √ n f √ n f − n f + C ( t ) = ( C QQ ( t ) C QB ( t ) C BQ ( t ) C BB ( t ) ) = e − m Q t √ n f √ n f − n f + C ( t ) = ( C QQ ( t ) C QB ( t ) C BQ ( t ) C BB ( t ) ) = e − m Q t √ n f √ n f − n f + (1)is the crucial tool for studying the string breaking from heavy quarks Q and¯ Q to two mesons B and ¯ B . The term C QQ ( t ) represents the correlator of twoheavy quarks, C BB ( t ) is the correlator of the two mesons of the system and C BQ ( t ) = C QB ( t ) are the terms denoting the mixing between the physicaleigenstates, which are relevant in the transition from a Q ¯ Q system to a B ¯ B system.In this proceeding, we concentrate on the first correlator of Eq. (1),which is basically a Wilson loop W ( r, t ) up to a prefactor, namely C QQ ( t ) = e − m Q t W ( r, t ) . (2)From its computation, we can obtain the static quark-antiquark potentialin the limit of large t , i.e. V QQ ( r ) = lim t →∞ a log (cid:18) C ( t ) C ( t + a ) (cid:19) = − m Q + 1 a V ( r ) . (3)However, for now we do not have access to the parameter m Q , therefore wefocus on V ( r ). In fact an additive constant is not relevant in the calculationof physical quantities and we can still fit the potential V ( r ) with an ansatz aV cont ( r ) = − αr + c + σr (4)where c remains unknown. arcoCatillo printed on June 23, 2020
3. Technical aspects
We consider a set of 79 CLS gauge configurations generated with n f = 2improved Wilson fermions [10]. The lattice parameters are summarized inTable 1, where we have indicated the Sommer parameter as R = r /a . V a m π R ×
64 0 . . , W ( r, t ) lm = (cid:68) Tr (cid:16) U ( x ) U i ( x + t ˆ4) ( m ) U ( x + r ˆ i ) † U i ( x ) ( l ) † (cid:17)(cid:69) , (5)where U µ ( x ) is a generic Wilson line at the point x in direction µ . W ( r, t )is the main ingredient for the computation of the static potential as showedin Eqs. (2) and (3). In Eq. (5) the labels l and m refer to differentsmearing levels, which are only applied on the Wilson lines in the spa-tial direction. The amount of smearing can be represented as an opera-tor S sm , namely U µ ( x ) ( l ) = ( S sm ) n l U µ ( x ). In our study every configura-tion is, at first, smeared with either HYP or HYP2 smearing. The differ-ence of these two is in the choice of the smearing parameters, i.e. HYP: α = 0 . , α = 0 . , α = 0 .
3, and HYP2: α = 1 . , α = 1 . , α = 0 . α = 0 . , α = 0 . , ,
10. We also consider an APEsmearing (always in the spatial direction) with two choices for the parameter α , i.e. α = 0 . α = 0 .
7. Furthermore we study the generalized eigen-value problem (GEVP), solving the system ˆ W ( r, t ) v = λ ( r, t ) ˆ W ( r, t ) v , withˆ W ( r, t ) = ( W ( r, t ) lm ), where t = a is kept fixed. Then the ground statepotential is extracted knowing that V ( r ) = lim t →∞ a log (cid:18) λ ( r, t ) λ ( r, t + a ) (cid:19) , (6)where λ ( r, t ) is the largest eigenvalue of the GEVP. The matrix ˆ W ( r, t ) ischosen to be a 5 × α = Coordinated Lattice Simulations, https://wiki-zeuthen.desy.de/CLS/ . Computed using B. Leder’s code ( https://github.com/bjoern-leder/wloop/ ). MarcoCatillo printed on June 23, 2020 . , α = 0 .
3) in the spatial direction, namely n l = 0 , , , ,
12; which arechosen according to the formula n l = ( l/ R , see Ref. [12]. Finally wecompare the described smearing choices with the case where no smearingis applied. The aim was to observe how sensitive our data are to differentsmearing strategies and we have chosen some of the most commonly usedtechniques in the literature.
4. Results
In Fig. 1, we present the results for the potential V ( r ) for differentsmearing choices, where the jackknife method is used for the first estimateof the errors. We observe that in the case without smearing, we only get afew points for small r , since for large r the signal to noise ratio deterioratesand the plateau cannot be reliably determined. However, already one levelof smearing (HYP or HYP2) is enough to get an acceptable signal and plot V ( r ) for all r . Furthermore, the curve with HYP smearing is shifted up with V (r) r/a no smearingHYP2+0HYP2+sHYP sm 4HYP2+sHYP sm 10HYP2+APE sm 10 α = 0.5HYP2+APE sm 10 α = 0.7GEVP HYP2 sm 12HYP+0HYP+sHYP sm 4HYP+sHYP sm 10HYP+APE sm 10 α = 0.5HYP+APE sm 10 α = 0.7GEVP HYP sm 12 Fig. 1: Potentials for different HYP2 and HYP smearing and the no-smearing case.respect the HYP2 smearing and this is evident for the range r/a ∈ [2 , r/a ∈ [1 : 6]. The plots with the fit curve are given in Fig. 2, where weshow the fit for “GEVP HYP(2) sm 12” case, where we applied a HYP (orHYP2) smearing step to all gauge configurations and then solved the GEVPproblem, described in the previous section, using a sHYP smearing in thespatial direction for the construction of the matrix ˆ W ( r, t ). The fit functionin this case is V ( r ) = aV cont ( r ) + δV ( r ), where δV ( r ) is a correction dueto the HYP(2) smearing, see Refs. [11, 13] for an explicit expression of this N. Cardoso’s code qfit is used for analysis (https://github.com/nmrcardoso/qfit). arcoCatillo printed on June 23, 2020 term. We did not attempt to fit the few obtained points in the no-smearingcase, as the points at small r/a can be affected by lattice artifacts, and thusan ansatz different than Eq. (4) should be taken in this case. V (r) r/a HYP+0HYP+sHYP sm 4HYP+sHYP sm 10HYP+APE sm 10 α = 0.5HYP+APE sm 10 α = 0.7GEVP HYP sm 12 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 V (r) r/a HYP2+0HYP2+sHYP sm 4HYP2+sHYP sm 10HYP2+APE sm 10 α = 0.5HYP2+APE sm 10 α = 0.7GEVP HYP2 sm 12 Fig. 2: Potentials for different HYP smearing (left) and HYP2 smearing(right). The fit curves are for the case of GEVP with HYP and HYP2smearing with 5 levels of smearing n l = 0 , , , ,
12, see section 3.From the fit results reported in Table 2 we can compare the string tension σ and the Coulomb-parameter α for different smearing choices. We observethat although in the same ballpark, the results for different smearing choicesshow slight inconsistencies among each other, which can be explained withsystematic effects that will not be addressed in this work. The separation ofthe two results for HYP and HYP2 smearing comes from an overall additiveconstant, as well as the correction term δV ( r ) between HYP and HYP2smearing, as discussed in Refs. [11, 14, 15]. Analyzing the signal to noiseratio and the χ of the fits, the use of “GEVP” smearing procedure seemsto give better results and in this case the potential for large r/a is betterapproximated by the continuum potential given in Eq. (4), see Fig. 2.Now we can also compute the Sommer parameter r from the relation r F ( r ) = 1 .
65 (7)where F ( r ) = V (cid:48) ( r ).This is an important crosscheck of the consistency of different smearingchoices. As we can observe in Fig. 3, the Sommer parameter is consistentwith the value from the literature r /a = 5 . MarcoCatillo printed on June 23, 2020
Type α σ [ GeV ] χ /ndf range r/a HYP2+0 0.346(7) 0.267(3) 1.01 [2:11]HYP2+sHYP sm 4 0.372(53) 0.256(13) 0.97 [2:11]HYP2+sHYP sm 10 0.430(42) 0.241(9) 1.08 [2:11]GEVP HYP2 sm 12 0.445(10) 0.235(2) 1.04 [4:12]HYP2+APE 0.5 0.346(8) 0.267(4) 0.65 [4:14]HYP2+APE 0.7 0.346(9) 0.267(4) 1.54 [3:11]HYP+0 0.318(7) 0.291(3) 1.31 [4:12]HYP+sHYP sm 4 0.368(97) 0.268(27) 0.28 [2:12]HYP+sHYP sm 10 0.468(38) 0.238(9) 0.27 [2:12]GEVP HYP sm 12 0.470(9) 0.234(2) 0.64 [4:16]HYP+APE 0.5 0.318(7) 0.291(3) 1.39 [3:8]HYP+APE 0.7 0.458(32) 0.241(8) 0.96 [4:12]Table 2: Parameters: α and σ , of the fit function V ( r ) in Eq. (4) fordifferent smearing choices.“APE” smearing is used instead of HYP, we obtain inconsistent results.It is important to note that the GEVP procedure already gives a goodestimation of r /a in combination with both HYP and HYP2 smearing. H Y P + H Y P + s H Y P s m H Y P + s H Y P s m G E V P H Y P s m H Y P + A P E s m α = . H Y P + A P E s m α = . H Y P + H Y P + s H Y P s m H Y P + s H Y P s m G E V P H Y P s m H Y P + A P E s m α = . H Y P + A P E s m α = . r /a=5.9 r / a Fig. 3: Sommer parameter r /a for different smearing choices. It is compat-ible with r /a = 5 .
9, given in Ref. [10].
5. Conclusions and outlook
We have reported on a preliminary study of static potentials between aquark-antiquark pair in a full QCD simulation with n f = 2 based on Wilsonloops. Different choices how to smear gauge configurations combined withthe GEVP procedure are deemed necessary to get reasonably good signals arcoCatillo printed on June 23, 2020 for our data. From such static potentials we got the value of string tensionand Sommer parameter and we compared them among different smearingprocedures. This work is still very preliminary and further studies are im-portant in order to get the remaining elements of the matrix of correlators inEq. (1) and then implement the Born-Oppenheimer approximation for thestudy of quarkonium states [1–4]. We plan to explore additional techniquesfor noise reduction and extend the calculation for different gauge ensemblesin order to study string breaking in the continuum limit. Acknowledgements
This work is supported by Physik-IT at Ludwig Maximilian University ofMunich. P.B. and N.C. thank CeFEMA, FCT contract UID/CTM/04540/2013.N.C. acknowledges the FCT contract SFRH/BPD/109443/2015.REFERENCES [1] P. Bicudo, M. Cardoso, N. Cardoso, and M. Wagner
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