Inspired by the recent observation of the Higgs boson radiative decay into <math><msup><mi>Z</mi><mn>0</mn></msup></math> by ATLAS and CMS Collaborations, we investigate the next-to-leading-order (NLO) electroweak correction to this rare decay process in Standard Model (SM). Implementing the on shell renormalization scheme, we find that the magnitude of the NLO electroweak correction may reach 7% of the leading-order (LO) prediction, much more significant than that of the NLO QCD correction, which is merely about 0.3%. After incorporating the <math><mi>O</mi><mo>(</mo><mi>α</mi><mo>)</mo></math> correction, the predicted partial width from various <math><mi>α</mi></math> schemes tend to converge to each other. Including both NLO electroweak and QCD corrections, the SM prediction for the branching fraction shifts from the LO value of <math><mrow><mo>(</mo><mn>1.40</mn><mi>–</mi><mn>1.71</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math> to <math><mo>(</mo><mn>1.55</mn><mo>±</mo><mn>0.06</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>, considerably lower than the measured value <math><msub><mi>B</mi><mi>exp</mi></msub><mo>[</mo><mi>H</mi><mo>→</mo><msup><mi>Z</mi><mn>0</mn></msup><mi>γ</mi><mo>]</mo><mo>=</mo><mo>(</mo><mn>3.4</mn><mo>±</mo><mn>1.1</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>. Resolving this discrepancy clearly calls for further theoretical investigations, and, more importantly, experimental efforts from HL-LHC and the prospective Higgs factories such as CEPC and FCC-ee.

China

Wen-Long Sang

Feng Feng

Yu Jia

Next-to-leading-order electroweak correction to <math><mrow><mi>H</mi><mo>→</mo><msup><mrow><mi>Z</mi></mrow><mn>0</mn></msup><mi>γ</mi></mrow></math>

In this work, we investigate the inclusive photoproduction of the <math><mi>C</mi></math>-odd, <math><mi>S</mi></math>-wave fully charmed tetraquark at electron-ion colliders within the nonrelativistic QCD (NRQCD) factorization framework, at the lowest order in velocity and <math><msub><mi>α</mi><mi>s</mi></msub></math>. The value of the NRQCD long-distance matrix element is estimated from two phenomenological potential models. Our studies reveal that the photoproduction of the <math><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup></math> fully charmed tetraquark may be difficult to observe at HERA and the EicC; nevertheless, its observation prospects at the EIC appear to be bright.

United States

Yingsheng Huang

De-Shan Yang

Jia-Yue Zhang

Photoproduction of fully charmed tetraquark at electron-ion colliders

Within the framework of nonrelativistic QCD (NRQCD), we calculate the fragmentation function for a charm quark into an S-wave fully-charmed tetraquark, denoted as T 4c . The charm-to-T 4c  fragmentation function is expressed as a sum of products of the perturbatively calculable short-distance coefficients and the nonperturbative long-distance matrix elements (LDMEs). The short-distance coefficients are ascertained through the perturbative matching procedure at lowest order in α  s  expansion. The LDMEs are approximated using the T 4c  four-body wave functions at the origin, which have been evaluated by various phenomenological potential models in literature. Incorporating the celebrated QCD factorization and the charm-to-T 4c  fragmentation function, we predict the T 4c  production rate at high transverse momentum p  T  regime in colliders. Both the differential distribution over p  T  and the integrated cross sections are predicted at the LHC. The cross sections for T 4c  states production can reach several femtobarns to several hundreds femtobarns, suggesting a substantial potential for T 4c  event production at the LHC. Additionally, we estimate for the photoproduction of T 4c  in electron-proton (ep) collisions. It is observed that the cross sections for these processes are moderate at the HERA and EIC, and relatively small at the EicC. Given the luminosities of these colliders, the prospect of detecting these fully-charmed tetraquarks at ep colliders is somewhat challenging.

Xiao-Wei Bai

Chang-Man Gan

Hong-Fei Zhang

Producing fully-charmed tetraquarks via charm quark fragmentation in colliders

We calculate the next-to-next-to-leading-order (NNLO) QCD radiative correction to the pion electromagnetic form factor with large momentum transfer. We explicitly verify the validity of the collinear factorization to two-loop order for this observable and obtain the respective IR-finite two-loop hard-scattering kernel in the closed form. The NNLO QCD correction turns out to be positive and significant. Incorporating this new ingredient of correction, we then make a comprehensive comparison between the finest theoretical predictions and numerous data for both space- and timelike pion form factors. Our phenomenological analysis provides a strong constraint on the second Gegenbauer moment of the pion light-cone distribution amplitude obtained from recent lattice QCD studies.

Long-Bin Chen

Wen Chen

Next-to-Next-to-Leading-Order QCD Corrections to Pion Electromagnetic Form Factors

In this study, we compute the electromagnetic and hadronic decay widths of the S-wave fully heavy tetraquark <math><msub><mi>T</mi><mrow><mn>4</mn><mi>Q</mi></mrow></msub></math> (<math><mi>Q</mi><mo>=</mo><mi>c</mi></math> or <math><mi>b</mi></math>) at lowest order in <math><msub><mi>α</mi><mi>s</mi></msub></math> and <math><mi>v</mi></math>, in the framework of nonrelativistic QCD. The short-distance coefficients are determined through the standard procedure of matching. The nonperturbative long-distance matrix elements are related to the phenomenological four-body Schrödinger wave functions at the origin, whose values are taken from literature. The branching fractions are predicted to be around <math><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> and <math><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math> for the <math><msub><mi>T</mi><mrow><mn>4</mn><mi>c</mi></mrow></msub></math> hadronic decay and electromagnetic decay, respectively. Combining our results with the <math><msub><mi>T</mi><mrow><mn>4</mn><mi>c</mi></mrow></msub></math> production cross sections at the LHC, we also predict the event numbers for various decay channels. With integrated luminosity <math><mi>L</mi><mo>=</mo><mn>100</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, it is expected that the event numbers can reach <math><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup><mi>–</mi><msup><mrow><mn>10</mn></mrow><mrow><mn>4</mn></mrow></msup></mrow></math> for <math><msub><mi>T</mi><mrow><mn>4</mn><mi>c</mi></mrow></msub><mo>→</mo><mi>γ</mi><mi>γ</mi></math>, and <math><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>5</mn></mrow></msup><mi>–</mi><msup><mrow><mn>10</mn></mrow><mrow><mn>6</mn></mrow></msup></mrow></math> for <math><msub><mi>T</mi><mrow><mn>4</mn><mi>c</mi></mrow></msub><mo>→</mo><mi>LH</mi></math>, at the LHC. The detecting prospect is promising. In addition, the decay widths of <math><msub><mi>T</mi><mrow><mn>4</mn><mi>b</mi></mrow></msub></math> are estimated based on simple dimensional analysis as well as velocity scaling rule.

Tao Wang

Yu-Dong Zhang

Electromagnetic and hadronic decay of fully heavy tetraquarks

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the long-awaited <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math> corrections for the exclusive double charmonium production process at B factories, i.e., <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></math> at <math><msqrt><mrow><mi>s</mi></mrow></msqrt><mo>=</mo><mn>10.58</mn></math> GeV. We confirm that NRQCD factorization does hold at next-to-next-to-leading-order (NNLO) for exclusive double charmonium production. It is found that including the NNLO QCD correction considerably reduces the renormalization scale dependence, and also implies the reasonable perturbative convergence behavior for this process. We also include both <math><mi>O</mi><mo>(</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> and <math><mi>O</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> corrections for this process. With all the available corrections lumped together, our most comprehensive prediction is consistent with both the BaBar and Belle measurement within errors.

Zhewen Mo

Next-to-next-to-leading-order QCD corrections to → / At Factories

In the framework of nonrelativistic QCD (NRQCD) factorization, we compute both the polarized and the unpolarized decay widths for the processes <math><msub><mi>η</mi><mi>b</mi></msub><mo>(</mo><msub><mi>χ</mi><mrow><mi>b</mi><mi>J</mi></mrow></msub><mo>)</mo><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi></math>, accurate up to next-to-next-to-leading order (NNLO) in <math><msub><mi>α</mi><mi>s</mi></msub></math>. For the first time, we confirm that the NRQCD factorization does hold at NNLO for the process involving triple quarkonia. We find that the radiative corrections are considerable. In particular, for <math><msub><mi>χ</mi><mrow><mi>b</mi><mn>2</mn></mrow></msub></math>, both the <math><mi>O</mi><mo>(</mo><msub><mi>α</mi><mi>s</mi></msub><mo>)</mo></math> and <math><mi>O</mi><mo>(</mo><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup><mo>)</mo></math> corrections are sizable and negative, and both can significantly reduce the leading-order prediction. At NNLO, the branching fractions are <math><mn>8.2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, <math><mn>6.2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math>, <math><mn>7.2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, and <math><mn>2.7</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math> for <math><msub><mi>η</mi><mi>b</mi></msub></math>, <math><msub><mi>χ</mi><mrow><mi>b</mi><mn>0</mn></mrow></msub></math>, <math><msub><mi>χ</mi><mrow><mi>b</mi><mn>1</mn></mrow></msub></math>, and <math><msub><mi>χ</mi><mrow><mi>b</mi><mn>2</mn></mrow></msub></math> decay, respectively. Our theoretical predictions are consistent with the upper limits measured by the Belle Collaboration. Moreover, we investigate the dependence of the theoretical predictions on the ratio of the charm quark mass and the bottom quark mass—i.e., <math><mi>r</mi><mo>=</mo><msub><mi>m</mi><mi>c</mi></msub><mo>/</mo><msub><mi>m</mi><mi>b</mi></msub></math>. By fixing <math><msub><mi>m</mi><mi>b</mi></msub></math> and varying <math><msub><mi>m</mi><mi>c</mi></msub></math> from 1.25 to 1.9 GeV, we find that the branching fraction can change by factors of 2, 3, and 6 for <math><msub><mi>η</mi><mi>b</mi></msub></math>, <math><msub><mi>χ</mi><mrow><mi>b</mi><mn>0</mn></mrow></msub></math>, and <math><msub><mi>χ</mi><mrow><mi>b</mi><mn>1</mn></mrow></msub></math>, respectively. Although the branching fraction for <math><msub><mi>χ</mi><mrow><mi>b</mi><mn>2</mn></mrow></msub></math> decreases with the increase of <math><mi>r</mi></math> at leading order and next-to-leading order, it is almost independent of <math><mi>r</mi></math> at NNLO. In the phenomenological analysis, with the integrated luminosity <math><mi>L</mi><mo>=</mo><mn>100</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, we expect about <math><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>10</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup></mrow></mrow></math> <math><msub><mi>η</mi><mi>b</mi></msub><mo>(</mo><msub><mi>χ</mi><mrow><mi>b</mi><mi>J</mi></mrow></msub><mo>)</mo><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>→</mo><mo>ℓ</mo><mover><mo>ℓ</mo><mo>¯</mo></mover><mo>ℓ</mo><mover><mo>ℓ</mo><mo>¯</mo></mover></math> events to be produced at the Large Hadron Collider; thus, it might be helpful to search for these processes. On the other hand, there are fewer than 100 <math><msub><mi>η</mi><mi>b</mi></msub><mo>(</mo><msub><mi>χ</mi><mrow><mi>b</mi><mi>J</mi></mrow></msub><mo>)</mo><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi></math> signal events at the B Factory, so it seems that the experimental measurements on these channels are quite challenging based on the current dataset. Nevertheless, with the designed <math><mn>50</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>ab</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> integrated luminosity at Belle 2, the observation prospects of <math><msub><mi>η</mi><mi>b</mi></msub><mo>(</mo><msub><mi>χ</mi><mrow><mi>b</mi><mi>J</mi></mrow></msub><mo>)</mo><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi></math> may be promising in the foreseeable future.

Ming-Zhen Zhou

Two-loop QCD corrections to <math><mi>C</mi></math>-even bottomonium exclusive decays to double <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math>

We compute the <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math> corrections to the exclusive channels <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> (<math><mi>J</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></math>) at <math><msqrt><mrow><mi>s</mi></mrow></msqrt><mo>=</mo><mn>10.58</mn></math> GeV within the nonrelativistic QCD (NRQCD) factorization framework. The validity of NRQCD factorization at <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math> has been confirmed for these double-charmonium exclusive production processes. We analyze the impact of the <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math> corrections on the polarized and unpolarized cross sections, as well as the <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> angular distributions, which largely reduce the renormalization scale dependence but increase the <math><mi>O</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math> NRQCD predictions to some extent for <math><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></math>. With high numerical accuracy, our predictions for <math><mi>σ</mi><mo>(</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></math> through <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math> are compatible with the upper limit of the Belle measurement. Although the theoretical prediction for <math><mi>σ</mi><mo>(</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>0</mn></mrow></msub><mo>)</mo></math> is consistent with both Belle and BaBar measurements within uncertainties, there still exists serious tension between the predicted and the measured profiles for the <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> angular distribution. Regarding the bright observation prospect of the <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math> channels in Belle 2 experiment, it is interesting to compare the future measurements with our NRQCD predictions. The more accurate measurement of <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>0</mn></mrow></msub></math> at Belle 2 will also help to clarify the long-standing puzzle of <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> angular distribution.

Corrections to / + Production at Factories

The failure of observing the <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> events at <math><mi>B</mi></math> factories to date is often attributed to the significant negative order-<math><msub><mi>α</mi><mi>s</mi></msub></math> correction. In this work we compute the <math><mi>O</mi><mo>(</mo><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup><mo>)</mo></math> correction to this process for the first time. The magnitude of the next-to-next-to-leading order (NNLO) perturbative correction is substantially negative so that the standard nonrelativistic QCD prediction would suffer from an unphysical, negative cross section. This dilemma may be traced in the fact that the bulk contribution of the fixed-order radiative corrections stems from the perturbative corrections to the <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> decay constant. We thus implement an improved nonrelativistic QCD factorization framework, by decomposing the amplitude into the photon-fragmentation piece and the nonfragmentation piece. With the measured <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> decay constant as input, which amounts to resumming a specific class of radiative and relativistic corrections to all orders, the fragmentation-induced production rate can be predicted accurately and serves a benchmark prediction. The nonfragmentation type of the amplitude is then computed through NNLO in <math><msub><mi>α</mi><mi>s</mi></msub></math> and at lowest order in velocity. Both the <math><mrow><mi>O</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></mrow></math> and <math><mi>O</mi><mo>(</mo><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup><mo>)</mo></math> corrections in the interference term become positive and exhibit a decent convergence behavior. Our finest prediction is <math><mrow><mi>σ</mi><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>)</mo><mo>=</mo><mn>2.1</mn><msubsup><mrow><mn>3</mn></mrow><mrow><mo>−</mo><mn>0.06</mn></mrow><mrow><mo>+</mo><mn>0.30</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>fb</mi></mrow></math> at <math><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>10.58</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. With the projected integrated luminosity of <math><mrow><mn>50</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>ab</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math>, the prospect to observe this exclusive process at Belle 2 experiment appears to be bright.

Jichen Pan

Optimized <math><mrow><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></math> Correction to Exclusive Double-<math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> Production at <math><mi>B</mi></math> Factories

The <math><mi>X</mi><mo>(</mo><mn>6900</mn><mo>)</mo></math> resonance, originally discovered by the LHCb Collaboration and later confirmed by both the ATLAS and CMS experiments, has sparked broad interest in the fully charmed tetraquark states. Relative to the mass spectra and decay properties of fully heavy tetraquarks, our knowledge on their production mechanism is still rather limited. In this work, we investigate the inclusive production of fully charmed <math><mi>S</mi></math>-wave tetraquarks at the LHC within the nonrelativistic QCD factorization framework. The partonic cross sections are computed at lowest order in <math><msub><mi>α</mi><mi>s</mi></msub></math> and velocity, while the long-distance nonrelativistic QCD matrix elements are estimated from phenomenological potential models. We predict the differential <math><msub><mi>p</mi><mi>T</mi></msub></math> spectra of various fully charmed <math><mi>S</mi></math>-wave tetraquarks at the LHC and compare with the results predicted from the fragmentation mechanism at the large-<math><msub><mi>p</mi><mi>T</mi></msub></math> end.