Motivated by the observation of the doubly charmed tetraquark $ T_{cc}(3875)^+ $ , we present a systematic study of doubly heavy tetraquarks ( $ T_{QQ'\bar{q}\bar{q}'} $ ) using heavy antiquark-diquark symmetry (HADS) within a constituent quark model. By calibrating model parameters to known hadron spectra and incorporating the effective mass formula, we predict the masses for 38 ground-state tetraquarks with $ cc $ , $ bb $ , and $ bc $ heavy quark pairs, including the non-strange, single-strange, and double-strange configurations with quantum numbers $ J^P = 0^+, 1^+ $ and $ 2^+ $ . Notably, we identify several stable states below the relevant meson-meson thresholds, particularly in the $ bb\bar{q}\bar{q}' $ sector. The explicit connection between the doubly heavy tetraquark and the heavy baryon spectra through HADS reduces model dependence and reveals the fundamental systematics in the heavy-quark hadron landscape.

China

Liu-Yu Zhang

Tian-Wei Wu

Yong-Liang Ma

Bridging doubly heavy tetraquark mass spectrum with heavy baryons utilizing heavy antiquark-diquark symmetry * The work of Tian-Wei Wu is supported by the National Natural Science Foundation of China (12405108). Yong-Liang Ma is supported in part by the National Science Foundation of China (NSFC) (12347103) and Gusu Talent Innovation Program (ZXL2024363)

Inspired by the experimental measurement of the charmed hadronic state X(6900), we calculate the mass spectra of tetraquark hybrid states with configuration of $$[8_{c}]_{Q\bar{Q}} \otimes [8_{c}]_{G} \otimes [8_{c}]_{Q\bar{Q}}$$ in color, by virtue of QCD sum rules. The two feasible types of currents with quantum numbers $$J^{PC} = 0^{++}$$ and $$0^{-+}$$ are investigated, in which the contributions from operators up to dimension six are taken into account in operator product expansion (OPE). In the end, we find that, in charm sector, the tetracharm hybrid states with quantum number $$0^{++}$$ has a mass of about $$6.98^{+0.16}_{-0.14} \, \text {GeV}$$ , while $$0^{-+}$$ state mass is about $$7.26^{+0.16}_{-0.15} \, \text {GeV}$$ . The results overlap with the experimental observations, suggesting potential tetracharm hybrid interpretations. In bottom sector, calculation shows that the masses of tetrabottom hybrid states with quantum numbers $$0^{++}$$ and $$0^{-+}$$ are $$19.30^{+0.16}_{-0.17} \, \text {GeV}$$ and $$19.50^{+0.17}_{-0.17} \, \text {GeV}$$ , respectively, which are left for future experimental confirmation.

Chun-Meng Tang

Chun-Gui Duan

Liang Tang

Cong-Feng Qiao

A novel configuration of gluonic tetraquark state

Stimulated by the recent experimental results on the fully-charm tetraquark states, we systematically calculate the mass spectra of the fully-charm tetraquark states in  $$8_{[c\bar{c}]}\otimes 8_{[c\bar{c}]}$$  <math> <mrow> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> <mo>⊗</mo> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> </mrow> </math>   color configuration via QCD sum rules. By constructing nine  $$8_{[c\bar{c}]}\otimes 8_{[c\bar{c}]}$$  <math> <mrow> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> <mo>⊗</mo> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> </mrow> </math>   type currents with quantum numbers  $$J^{PC}=0^{-+},0^{--},1^{-+},1^{+-},1^{--}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mo>+</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mo>-</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mo>-</mo> <mo>+</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mo>+</mo> <mo>-</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mo>-</mo> <mo>-</mo> </mrow> </msup> </mrow> </math>   and  $$2^{++}$$  <math> <msup> <mn>2</mn> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </math>  , we perform analytic calculation up to dimension six in the operator product expansion (OPE). We find the fully-charm tetraquark states with  $$J^{PC}=1^{+-},2^{++}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow> <mo>+</mo> <mo>-</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>2</mn> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </mrow> </math>   lie around 6.48  $$\sim $$  <math> <mo>∼</mo> </math>   6.62 GeV while the fully-charm tetraquark states with  $$J^{PC}=0^{-+},0^{--},1^{--},1^{-+}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mo>+</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mo>-</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mo>-</mo> <mo>-</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mo>-</mo> <mo>+</mo> </mrow> </msup> </mrow> </math>   are about 6.85  $$\sim $$  <math> <mo>∼</mo> </math>   7.02 GeV. Notably, the mass predictions for the  $$c\bar{c}c\bar{c}$$  <math> <mrow> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   tetraquarks, specifically those with  $$J^{PC}=2^{++}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </mrow> </math>  , align with the broad structure identified by LHCb. Moreover, the masses of fully-charm tetraquarks with  $$J^{PC}=0^{-+}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mo>+</mo> </mrow> </msup> </mrow> </math>   and  $$1^{-+}$$  <math> <msup> <mn>1</mn> <mrow> <mo>-</mo> <mo>+</mo> </mrow> </msup> </math>   are anticipated to match closely with the mass of X(6900), considering the margin of error. Such findings hint at the presence of some  $$8_{[c\bar{c}]}\otimes 8_{[c\bar{c}]}$$  <math> <mrow> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> <mo>⊗</mo> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> </mrow> </math>   components within the di- $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   structures observed by LHCb. The predictions for tetraquark states with  $$J^{PC}=0^{--},1^{+-},1^{--}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mo>-</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mo>+</mo> <mo>-</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mo>-</mo> <mo>-</mo> </mrow> </msup> </mrow> </math>   may be accessible in the future BelleII, Super-B, PANDA, and LHCb experiments.

Fully charmed tetraquark states in  $$8_{[c\bar{c}]}\otimes 8_{[c\bar{c}]}$$  <math> <mrow> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> <mo>⊗</mo> <msub> <mn>8</mn> <mrow> <mo>[</mo> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>]</mo> </mrow> </msub> </mrow> </math>   color structure via QCD sum rules

In this article, we study the mass spectrum of the low-lying triply heavy hybrid baryon, which consists of three valence heavy quarks in a color octet and one valence gluon, with spin-parity  $$J^P=(\frac{1}{2})^+$$  <math> <mrow> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> </msup> </mrow> </math>   via QCD sum rules. This is the first study on the triply heavy hybrid baryons in the framework of QCD sum rules. After performing the QCD sum rule analysis, we find that the mass of cccg hybrid baryon lies in  $$M_{cccg}= $$  <math> <mrow> <msub> <mi>M</mi> <mrow> <mi>cccg</mi> </mrow> </msub> <mo>=</mo> </mrow> </math>   5.91–6.13 GeV. As a byproduct, the mass of the triply bottom hybrid baryon state is extracted to be around  $$M_{bbbg}=$$  <math> <mrow> <msub> <mi>M</mi> <mrow> <mi>bbbg</mi> </mrow> </msub> <mo>=</mo> </mrow> </math>   14.62–14.82 GeV. The contributions up to dimension eight at the leading order of  $$\alpha _s$$  <math> <msub> <mi>α</mi> <mi>s</mi> </msub> </math>   (LO) in the operator product expansion are taken into account in the calculation. The triply charmed hybrid baryon predicted in this work can decay into one doubly charmed baryon and one charmed meson. Especially, we propose to search for cccg hybrid baryon with  $$J^{P}= (1/2)^+$$  <math> <mrow> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>+</mo> </msup> </mrow> </math>   in the P-wave decay channels  $$\Xi _{cc}^{++} D^0$$  <math> <mrow> <msubsup> <mi>Ξ</mi> <mrow> <mi>cc</mi> </mrow> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msubsup> <msup> <mi>D</mi> <mn>0</mn> </msup> </mrow> </math>  ,  $$\Xi _{cc}^{+} D^+$$  <math> <mrow> <msubsup> <mi>Ξ</mi> <mrow> <mi>cc</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>D</mi> <mo>+</mo> </msup> </mrow> </math>  , and  $$\Xi _{ccs}^{+} D_s^+$$  <math> <mrow> <msubsup> <mi>Ξ</mi> <mrow> <mi>ccs</mi> </mrow> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> </mrow> </math>  , which may be accessible in future BelleII, Super-B, PANDA, and LHCb experiments.

Yi-Cheng Zhao

Mass predictions of triply heavy hybrid baryons via QCD sum rules

We present the full next-to-leading order (NLO) prediction for the jet-gap-jet cross section at the LHC within the BFKL approach. We implement, for the first time, the NLO impact factors in the calculation of the cross section. We provide results for differential cross sections as a function of the difference in rapidity and azimuthal angle betwen the two jets and the second leading jet transverse momentum. The NLO corrections of the impact factors induce an overall reduction of the cross section with respect to the corresponding predictions with only LO impact factors.

Italy

Dimitri Colferai

United States

Federico Deganutti

Timothy Raben

Christophe Royon

First computation of Mueller Tang processes using a full NLL BFKL approach

The advent of laser-driven high-intensity <math><mi>γ</mi></math>-photon beams has opened up new opportunities for designing advanced photon-photon colliders. Such colliders have the potential to produce a large yield of linear Breit-Wheeler (LBW) pairs in a single shot, which offers a unique platform for studying the polarized LBW process. In our recent work [Q. Zhao et al., Signatures of linear Breit-Wheeler pair production in polarized <math><mi>γ</mi><mi>γ</mi></math> collisions, Phys. Rev. D 105, L071902 (2022)], we investigated the polarization characteristics of LBW pair production in circularly-polarized (<math><mrow><mi>C</mi><mi>P</mi></mrow></math>) <math><mrow><mi>γ</mi></mrow></math>-photon collisions. To fully clarify the polarization effects involving both <math><mrow><mi>C</mi><mi>P</mi></mrow></math> and linearly-polarized <math><mrow><mi>γ</mi></mrow></math>-photons, here we further investigate the LBW process using the polarized cross section with explicit azimuthal-angle dependence due to the base rotation of photon-polarization vectors. We accomplished this by defining a new spin basis for positrons and electrons, which enables us to decouple the transverse and longitudinal spin components of <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>±</mo></mrow></msup></mrow></math>. By means of analytical calculations and Monte Carlo simulations, we find that the linear polarization of photon can induce the highly angle-dependent pair yield and polarization distributions. The comprehensive knowledge of the polarized LBW process will also open up avenues for investigating the higher-order photon-photon scattering, the laser-driven quantum electrodynamic plasmas and the high-energy astrophysics.

Qian Zhao

Yan-Xi Wu

Mamutjan Ababekri

Zhong-Peng Li

Jian-Xing Li

Angle-dependent pair production in the polarized two-photon Breit-Wheeler process

Motivated by the newly observed resonance <math><mi>X</mi><mo>(</mo><mn>2600</mn><mo>)</mo></math> by BESIII Collaboration, we examine the trigluon glueball interpretation for it in the framework of QCD sum rules. We evaluate the mass spectra of the trigluon glueballs with quantum numbers <math><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> and <math><msup><mn>2</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> up to dimension 8 condensate in the operator product expansion. Our numerical results indicate that the mass of the <math><msup><mn>2</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> trigluon glueball is about <math><mn>2.66</mn><mo>±</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, which is consistent with the mass of the X(2600) within the uncertainties, while <math><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> has a mass of <math><mn>2.01</mn><mo>±</mo><mn>0.14</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. The possible decay channels of the <math><msup><mn>2</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> state are analyzed, which are crucial in decoding <math><mi>X</mi><mo>(</mo><mn>2600</mn><mo>)</mo></math>’s internal structure and are hopefully measurable in BESIII, BEllEII, PANDA, and LHCb experiments.

Sheng-Qi Zhang

Bing-Dong Wan

Gluonic nature of the newly observed state <math><mrow><mi>X</mi><mo>(</mo><mn>2600</mn><mo>)</mo></mrow></math>

In this work, we study the <math><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> double-gluon charmonium (<math><mover><mi>c</mi><mo>¯</mo></mover><mi>g</mi><mi>g</mi><mi>c</mi></math>) and bottomonium (<math><mover><mi>b</mi><mo>¯</mo></mover><mi>g</mi><mi>g</mi><mi>b</mi></math>) hybrids in terms of QCD sum rules. We find that the mass of <math><mrow><mover><mrow><mi>c</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>g</mi><mi>g</mi><mi>c</mi></mrow></math> hybrid lies in <math><mrow><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub><mo>=</mo><mn>5.33</mn><mi>–</mi><mn>5.90</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>, while in the bottom sector the mass of <math><mover><mi>b</mi><mo>¯</mo></mover><mi>g</mi><mi>g</mi><mi>b</mi></math> hybrid may be situated in <math><mrow><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></msub><mo>=</mo><mn>11.20</mn><mi>–</mi><mn>11.68</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. The contributions up to dimension eight at leading order of <math><msub><mi>α</mi><mi>s</mi></msub></math> (LO) in the operator product expansion are taken into account in the calculation. The double-gluon charmonium hybrid meson predicted in this work can decay into a pair of charmed mesons or a pair of charmed mesons together with a light meson. Especially, we propose to search for <math><mover><mi>c</mi><mo>¯</mo></mover><mi>g</mi><mi>g</mi><mi>c</mi></math> hybrid with <math><mrow><msup><mrow><mi>I</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mo>−</mo></mrow></msup><mo>)</mo></mrow></math> in their decay channels <math><mrow><mi>D</mi><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>/</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>/</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> with P wave and <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><mi>π</mi><mo>/</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><mi>η</mi><mo>/</mo><mi>D</mi><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>ρ</mi><mo>/</mo><mi>D</mi><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>ω</mi></mrow></math> with S wave, which may be accessible in Belle II, PANDA, Super-B, GlueX, and LHCb experiments.

Mass predictions of vector (<math><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mo>−</mo></mrow></msup></mrow></math>) double-gluon heavy quarkonium hybrids from QCD sum rules

The polarization characteristics of the linear Breit-Wheeler (LBW) pair-production process in polarized <math><mi>γ</mi><mi>γ</mi></math> colliders have been investigated via our developed spin-resolved binary collision simulation method. We find that the polarization of <math><mi>γ</mi></math>-photons modifies the kinematics of scattering particles and induces the correlated energy-angle shift of LBW pairs, and the latter’s polarization characteristic depends on the helicity configures of scattering particles. We confirm that the polarized <math><mi>γ</mi><mi>γ</mi></math> collider with an asymmetric setup can be performed with currently achievable laser-driven high-density x rays and high-brilliance <math><mrow><mi>γ</mi></mrow></math>-photon beams to produce abundant polarized LBW pairs, fulfilling the detection power of polarimetries. Our method and results on the polarized LBW process have plenty of significant applications in strong-field physics, high-energy physics and astrophysics, such as calibrating and monitoring the polarized <math><mi>γ</mi><mi>γ</mi></math> collider and challenging the current understanding of astrophysical objects through enhancing the opacity of <math><mi>γ</mi></math>-photons to exacerbate the inconsistency between some observations and standard models.

Feng Wan

Bo-Chao Liu

Ruo-Yu Liu

Rui-Zhi Yang

Jin-Qing Yu

Xue-Guang Ren

Zhong-Feng Xu

Yong-Tao Zhao

Yong-Sheng Huang

Signatures of linear Breit-Wheeler pair production in polarized <math><mrow><mi>γ</mi><mi>γ</mi></mrow></math> collisions

In this work, we determine up/down-quark mass  $$m_{q=u/d}$$  <math> <msub> <mi>m</mi> <mrow> <mi>q</mi> <mo>=</mo> <mi>u</mi> <mo>/</mo> <mi>d</mi> </mrow> </msub> </math>   in the isoscalar scalar channel from both the Shifman–Vainshtein–Zakharov (SVZ) and the Monte-Carlo-based QCD sum rules. The relevant spectral function, including the contributions from the  $$f_0(500)$$  <math> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>500</mn> <mo>)</mo> </mrow> </mrow> </math>  ,  $$f_0(980)$$  <math> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>980</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$f_0(1370)$$  <math> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1370</mn> <mo>)</mo> </mrow> </mrow> </math>   resonances, is determined from a sophisticated U(3) chiral study. Via the traditional SVZ QCD sum rules, we give the prediction to the average light-quark mass  $$m_q(2 ~\text {GeV})=\frac{1}{2}(m_u(2 ~\text {GeV}) + m_d(2 ~\text {GeV}))=(3.46^{+0.16}_{-0.22} \pm 0.33) ~\text {MeV}$$  <math> <mrow> <msub> <mi>m</mi> <mi>q</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mspace width="3.33333pt"></mspace> <mtext>GeV</mtext> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mspace width="3.33333pt"></mspace> <mtext>GeV</mtext> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>m</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mspace width="3.33333pt"></mspace> <mtext>GeV</mtext> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>.</mo> <msubsup> <mn>46</mn> <mrow> <mo>-</mo> <mn>0.22</mn> </mrow> <mrow> <mo>+</mo> <mn>0.16</mn> </mrow> </msubsup> <mo>±</mo> <mn>0.33</mn> <mo>)</mo> </mrow> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>  . Meanwhile, by considering the uncertainties of the input QCD parameters and the spectral functions of the isoscalar scalar channel, we obtain  $$m_q (2~\text {GeV}) = (3.44 \pm 0.14 \pm 0.32) ~\text {MeV}$$  <math> <mrow> <msub> <mi>m</mi> <mi>q</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mspace width="3.33333pt"></mspace> <mtext>GeV</mtext> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>3.44</mn> <mo>±</mo> <mn>0.14</mn> <mo>±</mo> <mn>0.32</mn> <mo>)</mo> </mrow> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>   from the Monte-Carlo-based QCD sum rules. Both results are perfectly consistent with each other, and nicely agree with the Particle Data Group value within the uncertainties.

Fang-Hui Yin

Wen-Ya Tian

Zhi-Hui Guo

Determination of the up/down-quark mass within QCD sum rules in the scalar channel