Chinese Physics C

Searching for dark matter particles using Compton scattering

Shang Wang, Changbo Fu, De-Chang Dai, Hongwei Wang, Gongtao Fan, Xiguang Cao, Yugang Ma

2021, 45(9): 093001. doi: 10.1088/1674-1137/ac0c0f

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Abstract:
The dark matter puzzle is one of the most important fundamental physics questions in the 21st century. There is no doubt that solving the puzzle will be a new milestone for human beings in achieving a deeper understanding of nature. Herein, we propose the use of the Shanghai laser electron gamma source (SLEGS) to search for dark matter candidate particles, including dark pseudoscalar particles, dark scalar particles, and dark photons. Our simulations indicate that, with some upgrading, electron facilities such as SLEGS could be competitive platforms in the search for light dark matter particles with a mass below tens of keV.

Search for the doubly heavy baryons ${\boldsymbol \varOmega_{\boldsymbol{bc}}^{\bf 0}} $ and $ {\boldsymbol\varXi_{\boldsymbol{bc}}^{\bf 0} }$ decaying to ${ \boldsymbol \varLambda_c^+\pi^- }$ and $ {\boldsymbol\varXi_c^+\pi^- }$

R. Aaij, C. Abellán Beteta, T. Ackernley, B. Adeva, M. Adinolfi, H. Afsharnia, C.A. Aidala, S. Aiola, Z. Ajaltouni, S. Akar, J. Albrecht, F. Alessio, M. Alexander, A. Alfonso Albero, Z. Aliouche, G. Alkhazov, P. Alvarez Cartelle, S. Amato, Y. Amhis, L. An, L. Anderlini, A. Andreianov, M. Andreotti, F. Archilli, A. Artamonov, M. Artuso, K. Arzymatov, E. Aslanides, M. Atzeni, B. Audurier, S. Bachmann, M. Bachmayer, J.J. Back, P. Baladron Rodriguez, V. Balagura, W. Baldini, J. Baptista Leite, R.J. Barlow, S. Barsuk, W. Barter, M. Bartolini, F. Baryshnikov, J.M. Basels, G. Bassi, B. Batsukh, A. Battig, A. Bay, M. Becker, F. Bedeschi, I. Bediaga, A. Beiter, V. Belavin, S. Belin, V. Bellee, K. Belous, I. Belov, I. Belyaev, G. Bencivenni, E. Ben-Haim, A. Berezhnoy, R. Bernet, D. Berninghoff, H.C. Bernstein, C. Bertella, A. Bertolin, C. Betancourt, F. Betti, Ia. Bezshyiko, S. Bhasin, J. Bhom, L. Bian, M.S. Bieker, S. Bifani, P. Billoir, M. Birch, F.C.R. Bishop, A. Bitadze, A. Bizzeti, M. Bjørn, M.P. Blago, T. Blake

2021, 45(9): 093002. doi: 10.1088/1674-1137/ac0c70

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Abstract:
The first search for the doubly heavy

\begin{document}$ {{{{\varOmega}_{bc}^{0}}}} $\end{document}

baryon and a search for the

\begin{document}$ {{{{\varXi}_{bc}^{0}}}} $\end{document}

baryon are performed using

\begin{document}$ pp $\end{document}

collision data collected via the

\begin{document}$ {\rm{LHCb}} $\end{document}

experiment from 2016 to 2018 at a centre-of-mass energy of

\begin{document}$ 13 \;{\rm{TeV}} $\end{document}

, corresponding to an integrated luminosity of 5.2

\begin{document}$ \;{\rm{f}}{{\rm{b}}^{ - 1}} $\end{document}

. The baryons are reconstructed via their decays to

\begin{document}$ {{{{\varLambda}^+_c}}} {{{{\pi}^-}}} $\end{document}

and

\begin{document}$ {{{{\varXi}^+_c}}} {{{{\pi}^-}}} $\end{document}

. No significant excess is found for invariant masses between 6700 and 7300

\begin{document}$ \;{\rm{MeV}}/{c^2} $\end{document}

, in a rapidity range from 2.0 to 4.5 and a transverse momentum range from 2 to 20

\begin{document}$ \;{\rm{MeV}}/{c} $\end{document}

. Upper limits are set on the ratio of the

\begin{document}$ {{{{\varOmega}_{bc}^{0}}}} $\end{document}

and

\begin{document}$ {{{{\varXi}_{bc}^{0}}}} $\end{document}

production cross-section times the branching fraction to

\begin{document}$ {{{{\varLambda}^+_c}}}{{{{\pi}^-}}} $\end{document}

(

\begin{document}$ {{{{\varXi}^+_c}}}{{{{\pi}^-}}} $\end{document}

) relative to that of the

\begin{document}$ {{{{\varLambda}^0_b}}} $\end{document}

(

\begin{document}$ {{{{\varXi}_{b}^{0}}}} $\end{document}

) baryon, for different lifetime hypotheses, at 95% confidence level. The upper limits range from

\begin{document}$ 0.5\times10^{-4} $\end{document}

to

\begin{document}$ 2.5\times10^{-4} $\end{document}

for the

\begin{document}$ {{{{{{{{\varOmega}_{bc}^{0}}}}{{\rightarrow }}{{{{\varLambda}^+_c}}}{{{{\pi}^-}}}}}}} $\end{document}

(

\begin{document}$ {{{{{{{{\varXi}_{bc}^{0}}}}{{\rightarrow }}{{{{\varLambda}^+_c}}}{{{{\pi}^-}}}}}}} $\end{document}

) decay, and from

\begin{document}$ 1.4\times10^{-3} $\end{document}

to

\begin{document}$ 6.9\times10^{-3} $\end{document}

for the

\begin{document}$ {{{{{{{{\varOmega}_{bc}^{0}}}}{{\rightarrow }}{{{{\varXi}^+_c}}}{{{{\pi}^-}}}}}}} $\end{document}

(

\begin{document}$ {{{{{{{{\varXi}_{bc}^{0}}}}{{\rightarrow }}{{{{\varXi}^+_c}}}{{{{\pi}^-}}}}}}} $\end{document}

) decay, depending on the considered mass and lifetime of the

\begin{document}$ {{{{\varOmega}_{bc}^{0}}}} $\end{document}

(

\begin{document}$ {{{{\varXi}_{bc}^{0}}}} $\end{document}

) baryon.

R. Aaij, C. Abellán Beteta, T. Ackernley, B. Adeva, M. Adinolfi, H. Afsharnia, C.A. Aidala, S. Aiola, Z. Ajaltouni, S. Akar, J. Albrecht, F. Alessio, M. Alexander, A. Alfonso Albero, Z. Aliouche, G. Alkhazov, P. Alvarez Cartelle, S. Amato, Y. Amhis, L. An, L. Anderlini, A. Andreianov, M. Andreotti, F. Archilli, A. Artamonov, M. Artuso, K. Arzymatov, E. Aslanides, M. Atzeni, B. Audurier, S. Bachmann, M. Bachmayer, J.J. Back, P. Baladron Rodriguez, V. Balagura, W. Baldini, J. Baptista Leite, R.J. Barlow, S. Barsuk, W. Barter, M. Bartolini, F. Baryshnikov, J.M. Basels, G. Bassi, B. Batsukh, A. Battig, A. Bay, M. Becker, F. Bedeschi, I. Bediaga, A. Beiter, V. Belavin, S. Belin, V. Bellee, K. Belous, I. Belov, I. Belyaev, G. Bencivenni, E. Ben-Haim, A. Berezhnoy, R. Bernet, D. Berninghoff, H.C. Bernstein, C. Bertella, A. Bertolin, C. Betancourt, F. Betti, Ia. Bezshyiko, S. Bhasin, J. Bhom, L. Bian, M.S. Bieker, S. Bifani, P. Billoir, M. Birch, F.C.R. Bishop, A. Bitadze, A. Bizzeti, M. Bjørn, M.P. Blago, T. Blake, F. Blanc, S. Blusk, D. Bobulska, J.A. Boelhauve, O. Boente Garcia, T. Boettcher, A. Boldyrev, A. Bondar, N. Bondar, S. Borghi, M. Borisyak, M. Borsato, J.T. Borsuk, S.A. Bouchiba, T.J.V. Bowcock, A. Boyer, C. Bozzi, M.J. Bradley, S. Braun, A. Brea Rodriguez, M. Brodski, J. Brodzicka, A. Brossa Gonzalo, D. Brundu, A. Buonaura, C. Burr, A. Bursche, A. Butkevich, J.S. Butter, J. Buytaert, W. Byczynski, S. Cadeddu, H. Cai, R. Calabrese, L. Calefice, L. Calero Diaz, S. Cali, R. Calladine, M. Calvi, M. Calvo Gomez, P. Camargo Magalhaes, A. Camboni, P. Campana, A.F. Campoverde Quezada, S. Capelli, L. Capriotti, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, I. Carli, P. Carniti, L. Carus, K. Carvalho Akiba, A. Casais Vidal, G. Casse, M. Cattaneo, G. Cavallero, S. Celani, J. Cerasoli, A.J. Chadwick, M.G. Chapman, M. Charles, Ph. Charpentier, G. Chatzikonstantinidis, C.A. Chavez Barajas, M. Chefdeville, C. Chen, S. Chen, A. Chernov, V. Chobanova, S. Cholak, M. Chrzaszcz, A. Chubykin, V. Chulikov, P. Ciambrone, M.F. Cicala, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, J.L. Cobbledick, V. Coco, J.A.B. Coelho, J. Cogan, E. Cogneras, L. Cojocariu, P. Collins, T. Colombo, L. Congedo, A. Contu, N. Cooke, G. Coombs, G. Corti, C.M. Costa Sobral, B. Couturier, D.C. Craik, J. Crkovská, M. Cruz Torres, R. Currie, C.L. Da Silva, E. Dall'Occo, J. Dalseno, C. D'Ambrosio, A. Danilina, P. d'Argent, A. Davis, O. De Aguiar Francisco, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, M. De Serio, D. De Simone, P. De Simone, J.A. de Vries, C.T. Dean, D. Decamp, L. Del Buono, B. Delaney, H.-P. Dembinski, A. Dendek, V. Denysenko, D. Derkach, O. Deschamps, F. Desse, F. Dettori, B. Dey, P. Di Nezza, S. Didenko, L. Dieste Maronas, H. Dijkstra, V. Dobishuk, A.M. Donohoe, F. Dordei, A.C. dos Reis, L. Douglas, A. Dovbnya, A.G. Downes, K. Dreimanis, M.W. Dudek, L. Dufour, V. Duk, P. Durante, J.M. Durham, D. Dutta, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, S. Ek-In, L. Eklund, S. Ely, A. Ene, E. Epple, S. Escher, J. Eschle, S. Esen, T. Evans, A. Falabella, J. Fan, Y. Fan, B. Fang, S. Farry, D. Fazzini, M. Féo, A. Fernandez Prieto, J.M. Fernandez-tenllado Arribas, F. Ferrari, L. Ferreira Lopes, F. Ferreira Rodrigues, S. Ferreres Sole, M. Ferrillo, M. Ferro-Luzzi, S. Filippov, R.A. Fini, M. Fiorini, M. Firlej, K.M. Fischer, C. Fitzpatrick, T. Fiutowski, F. Fleuret, M. Fontana, F. Fontanelli, R. Forty, V. Franco Lima, M. Franco Sevilla, M. Frank, E. Franzoso, G. Frau, C. Frei, D.A. Friday, J. Fu, Q. Fuehring, W. Funk, E. Gabriel, T. Gaintseva, A. Gallas Torreira, D. Galli, S. Gambetta, Y. Gan, M. Gandelman, P. Gandini, Y. Gao, M. Garau, L.M. Garcia Martin, P. Garcia Moreno, J. García Pardiñas, B. Garcia Plana, F.A. Garcia Rosales, L. Garrido, C. Gaspar, R.E. Geertsema, D. Gerick, L.L. Gerken, E. Gersabeck, M. Gersabeck, T. Gershon, D. Gerstel, Ph. Ghez, V. Gibson, H.K. Giemza, M. Giovannetti, A. Gioventù, P. Gironella Gironell, L. Giubega, C. Giugliano, K. Gizdov, E.L. Gkougkousis, V.V. Gligorov, C. Göbel, E. Golobardes, D. Golubkov, A. Golutvin, A. Gomes, S. Gomez Fernandez, F. Goncalves Abrantes, M. Goncerz, G. Gong, P. Gorbounov, I.V. Gorelov, C. Gotti, E. Govorkova, J.P. Grabowski, T. Grammatico, L.A. Granado Cardoso, E. Graugés, E. Graverini, G. Graziani, A. Grecu, L.M. Greeven, P. Griffith, L. Grillo, S. Gromov, B.R. Gruberg Cazon, C. Gu, M. Guarise, P. A. Günther, E. Gushchin, A. Guth, Y. Guz, T. Gys, T. Hadavizadeh, G. Haefeli, C. Haen, J. Haimberger, T. Halewood-leagas, P.M. Hamilton, Q. Han, X. Han, T.H. Hancock, S. Hansmann-Menzemer, N. Harnew, T. Harrison, C. Hasse, M. Hatch, J. He, M. Hecker, K. Heijhoff, K. Heinicke, A.M. Hennequin, K. Hennessy, L. Henry, J. Heuel, A. Hicheur, D. Hill, M. Hilton, S.E. Hollitt, J. Hu, J. Hu, W. Hu, W. Huang, X. Huang, W. Hulsbergen, R.J. Hunter, M. Hushchyn, D. Hutchcroft, D. Hynds, P. Ibis, M. Idzik, D. Ilin, P. Ilten, A. Inglessi, A. Ishteev, K. Ivshin, R. Jacobsson, S. Jakobsen, E. Jans, B.K. Jashal, A. Jawahery, V. Jevtic, M. Jezabek, F. Jiang, M. John, D. Johnson, C.R. Jones, T.P. Jones, B. Jost, N. Jurik, S. Kandybei, Y. Kang, M. Karacson, M. Karpov, F. Keizer, M. Kenzie, T. Ketel, B. Khanji, A. Kharisova, S. Kholodenko, T. Kirn, V.S. Kirsebom, O. Kitouni, S. Klaver, K. Klimaszewski, S. Koliiev, A. Kondybayeva, A. Konoplyannikov, P. Kopciewicz, R. Kopecna, P. Koppenburg, M. Korolev, I. Kostiuk, O. Kot, S. Kotriakhova, P. Kravchenko, L. Kravchuk, R.D. Krawczyk, M. Kreps, F. Kress, S. Kretzschmar, P. Krokovny, W. Krupa, W. Krzemien, W. Kucewicz, M. Kucharczyk, V. Kudryavtsev, H.S. Kuindersma, G.J. Kunde, T. Kvaratskheliya, D. Lacarrere, G. Lafferty, A. Lai, A. Lampis, D. Lancierini, J.J. Lane, R. Lane, G. Lanfranchi, C. Langenbruch, J. Langer, O. Lantwin, T. Latham, F. Lazzari, R. Le Gac, S.H. Lee, R. Lefèvre, A. Leflat, S. Legotin, O. Leroy, T. Lesiak, B. Leverington, H. Li, L. Li, P. Li, S. Li, Y. Li, Y. Li, Z. Li, X. Liang, T. Lin, R. Lindner, V. Lisovskyi, R. Litvinov, G. Liu, H. Liu, S. Liu, X. Liu, A. Loi, J. Lomba Castro, I. Longstaff, J.H. Lopes, G.H. Lovell, Y. Lu, D. Lucchesi, S. Luchuk, M. Lucio Martinez, V. Lukashenko, Y. Luo, A. Lupato, E. Luppi, O. Lupton, A. Lusiani, X. Lyu, L. Ma, R. Ma, S. Maccolini, F. Machefert, F. Maciuc, V. Macko, P. Mackowiak, S. Maddrell-Mander, O. Madejczyk, L.R. Madhan Mohan, O. Maev, A. Maevskiy, D. Maisuzenko, M.W. Majewski, J.J. Malczewski, S. Malde, B. Malecki, A. Malinin, T. Maltsev, H. Malygina, G. Manca, G. Mancinelli, D. Manuzzi, D. Marangotto, J. Maratas, J.F. Marchand, U. Marconi, S. Mariani, C. Marin Benito, M. Marinangeli, P. Marino, J. Marks, A.M. Marshall, P.J. Marshall, G. Martellotti, L. Martinazzoli, M. Martinelli, D. Martinez Santos, F. Martinez Vidal, A. Massafferri, M. Materok, R. Matev, A. Mathad, Z. Mathe, V. Matiunin, C. Matteuzzi, K.R. Mattioli, A. Mauri, E. Maurice, J. Mauricio, M. Mazurek, M. McCann, L. Mcconnell, T.H. Mcgrath, A. McNab, R. McNulty, J.V. Mead, B. Meadows, C. Meaux, G. Meier, N. Meinert, D. Melnychuk, S. Meloni, M. Merk, A. Merli, L. Meyer Garcia, M. Mikhasenko, D.A. Milanes, E. Millard, M. Milovanovic, M.-N. Minard, A. Minotti, L. Minzoni, S.E. Mitchell, B. Mitreska, D.S. Mitzel, A. Mödden, R.A. Mohammed, R.D. Moise, T. Mombächer, I.A. Monroy, S. Monteil, M. Morandin, G. Morello, M.J. Morello, J. Moron, A.B. Morris, A.G. Morris, R. Mountain, H. Mu, F. Muheim, M. Mukherjee, M. Mulder, D. Müller, K. Müller, C.H. Murphy, D. Murray, P. Muzzetto, P. Naik, T. Nakada, R. Nandakumar, T. Nanut, I. Nasteva, M. Needham, I. Neri, N. Neri, S. Neubert, N. Neufeld, R. Newcombe, T.D. Nguyen, C. Nguyen-Mau, E.M. Niel, S. Nieswand, N. Nikitin, N.S. Nolte, C. Nunez, A. Oblakowska-Mucha, V. Obraztsov, D.P. O'Hanlon, R. Oldeman, M.E. Olivares, C.J.G. Onderwater, A. Ossowska, J.M. Otalora Goicochea, T. Ovsiannikova, P. Owen, A. Oyanguren, B. Pagare, P.R. Pais, T. Pajero, A. Palano, M. Palutan, Y. Pan, G. Panshin, A. Papanestis, M. Pappagallo, L.L. Pappalardo, C. Pappenheimer, W. Parker, C. Parkes, C.J. Parkinson, B. Passalacqua, G. Passaleva, A. Pastore, M. Patel, C. Patrignani, C.J. Pawley, A. Pearce, A. Pellegrino, M. Pepe Altarelli, S. Perazzini, D. Pereima, P. Perret, M. Petric, K. Petridis, A. Petrolini, A. Petrov, S. Petrucci, M. Petruzzo, T.T.H. Pham, A. Philippov, L. Pica, M. Piccini, B. Pietrzyk, G. Pietrzyk, M. Pili, D. Pinci, F. Pisani, P.K Resmi, V. Placinta, J. Plews, M. Plo Casasus, F. Polci, M. Poli Lener, M. Poliakova, A. Poluektov, N. Polukhina, I. Polyakov, E. Polycarpo, G.J. Pomery, S. Ponce, D. Popov, S. Popov, S. Poslavskii, K. Prasanth, L. Promberger, C. Prouve, V. Pugatch, H. Pullen, G. Punzi, W. Qian, J. Qin, R. Quagliani, B. Quintana, N.V. Raab, R.I. Rabadan Trejo, B. Rachwal, J.H. Rademacker, M. Rama, M. Ramos Pernas, M.S. Rangel, F. Ratnikov, G. Raven, M. Reboud, F. Redi, F. Reiss, C. Remon Alepuz, Z. Ren, V. Renaudin, R. Ribatti, S. Ricciardi, K. Rinnert, P. Robbe, A. Robert, G. Robertson, A.B. Rodrigues, E. Rodrigues, J.A. Rodriguez Lopez, A. Rollings, P. Roloff, V. Romanovskiy, M. Romero Lamas, A. Romero Vidal, J.D. Roth, M. Rotondo, M.S. Rudolph, T. Ruf, J. Ruiz Vidal, A. Ryzhikov, J. Ryzka, J.J. Saborido Silva, N. Sagidova, N. Sahoo, B. Saitta, D. Sanchez Gonzalo, C. Sanchez Gras, R. Santacesaria, C. Santamarina Rios, M. Santimaria, E. Santovetti, D. Saranin, G. Sarpis, M. Sarpis, A. Sarti, C. Satriano, A. Satta, M. Saur, D. Savrina, H. Sazak, L.G. Scantlebury Smead, S. Schael, M. Schellenberg, M. Schiller, H. Schindler, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M. Schubiger, S. Schulte, M.H. Schune, R. Schwemmer, B. Sciascia, S. Sellam, A. Semennikov, M. Senghi Soares, A. Sergi, N. Serra, L. Sestini, A. Seuthe, P. Seyfert, Y. Shang, D.M. Shangase, M. Shapkin, I. Shchemerov, L. Shchutska, T. Shears, L. Shekhtman, Z. Shen, V. Shevchenko, E.B. Shields, E. Shmanin, J.D. Shupperd, B.G. Siddi, R. Silva Coutinho, G. Simi, S. Simone, N. Skidmore, T. Skwarnicki, M.W. Slater, I. Slazyk, J.C. Smallwood, J.G. Smeaton, A. Smetkina, E. Smith, M. Smith, A. Snoch, M. Soares, L. Soares Lavra, M.D. Sokoloff, F.J.P. Soler, A. Solovev, I. Solovyev, F.L. Souza De Almeida, B. Souza De Paula, B. Spaan, E. Spadaro Norella, P. Spradlin, F. Stagni, M. Stahl, S. Stahl, P. Stefko, O. Steinkamp, O. Stenyakin, H. Stevens, S. Stone, M.E. Stramaglia, M. Straticiuc, D. Strekalina, F. Suljik, J. Sun, L. Sun, Y. Sun, P. Svihra, P.N. Swallow, K. Swientek, A. Szabelski, T. Szumlak, M. Szymanski, S. Taneja, F. Teubert, E. Thomas, K.A. Thomson, V. Tisserand, S. T'Jampens, M. Tobin, L. Tomassetti, D. Torres Machado, D.Y. Tou, M.T. Tran, E. Trifonova, C. Trippl, G. Tuci, A. Tully, N. Tuning, A. Ukleja, D.J. Unverzagt, E. Ursov, A. Usachov, A. Ustyuzhanin, U. Uwer, A. Vagner, V. Vagnoni, A. Valassi, G. Valenti, N. Valls Canudas, M. van Beuzekom, M. Van Dijk, E. van Herwijnen, C.B. Van Hulse, M. van Veghel, R. Vazquez Gomez, P. Vazquez Regueiro, C. Vázquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, A. Venkateswaran, M. Veronesi, M. Vesterinen, D. Vieira, M. Vieites Diaz, H. Viemann, X. Vilasis-Cardona, E. Vilella Figueras, P. Vincent, G. Vitali, D. Vom Bruch, A. Vorobyev, V. Vorobyev, N. Voropaev, R. Waldi, J. Walsh, C. Wang, J. Wang, J. Wang, J. Wang, J. Wang, M. Wang, R. Wang, Y. Wang, Z. Wang, Z. Wang, H.M. Wark, N.K. Watson, S.G. Weber, D. Websdale, C. Weisser, B.D.C. Westhenry, D.J. White, M. Whitehead, D. Wiedner, G. Wilkinson, M. Wilkinson, I. Williams, M. Williams, M.R.J. Williams, F.F. Wilson, W. Wislicki, M. Witek, L. Witola, G. Wormser, S.A. Wotton, H. Wu, K. Wyllie, Z. Xiang, D. Xiao, Y. Xie, A. Xu, J. Xu, L. Xu, M. Xu, Q. Xu, Z. Xu, Z. Xu, D. Yang, S. Yang, Y. Yang, Z. Yang, Z. Yang, Y. Yao, L.E. Yeomans, H. Yin, J. Yu, X. Yuan, O. Yushchenko, E. Zaffaroni, M. Zavertyaev, M. Zdybal, O. Zenaiev, M. Zeng, D. Zhang, L. Zhang, S. Zhang, Y. Zhang, Y. Zhang, A. Zhelezov, Y. Zheng, X. Zhou, Y. Zhou, X. Zhu, V. Zhukov, J.B. Zonneveld, Q. Zou, S. Zucchelli, D. Zuliani and G. Zunica. Search for the doubly heavy baryons

\begin{document}$ \varOmega_{bc}^0 $\end{document}

and

\begin{document}$ \varXi_{bc}^0 $\end{document}

decaying to

\begin{document}$ \Lambda_c^+\pi^- $\end{document}

and

\begin{document}$ \varXi_c^+\pi^- $\end{document}

[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac0c70.

Inclusive production of fully-charmed 1⁺⁻ tetraquark at B factory

Yingsheng Huang, Feng Feng, Yu Jia, Wen-Long Sang, De-Shan Yang, Jia-Yue Zhang

2021, 45(9): 093101. doi: 10.1088/1674-1137/ac0b38

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Abstract:
Inspired by the recent discovery of the

\begin{document}$ X(6900) $\end{document}

meson in the

\begin{document}$ {\mathsf{ LHCb}}$\end{document}

experiment, we investigate the inclusive production rate of the

\begin{document}$ C $\end{document}

-odd fully-charmed tetraquarks associated with light hadrons at the

\begin{document}$ B $\end{document}

factory within the nonrelativistic QCD (NRQCD) factorization framework. The short-distance coefficient is computed at the lowest order of velocity and

\begin{document}$ \alpha_s $\end{document}

. Employing two different kinds of phenomenological models to approximately estimate the long-distance NRQCD matrix element, we predict the rate for the inclusive production of the

\begin{document}$ 1^{+-} $\end{document}

\begin{document}$ T_{4c} $\end{document}

state and discuss the observation prospect of the

\begin{document}$ {{\mathsf{Belle}}\; {\mathsf{2}}} $\end{document}

experiment.

Exotic ${\bar{\boldsymbol D}_{\boldsymbol s}^{({\bf *})}{\boldsymbol D}^{({\bf *})}}$ molecular states and ${ {\boldsymbol{sc}}\bar {\boldsymbol q}\bar {\boldsymbol c} }$ tetraquark states with J^P=0⁺, 1⁺, 2⁺

Qi-Nan Wang, Wei Chen, Hua-Xing Chen

2021, 45(9): 093102. doi: 10.1088/1674-1137/ac0b3b

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Abstract:
We have calculated the mass spectra for the

\begin{document}$\bar{D}_s^{(*)}D^{(*)}$\end{document}

molecular states and

\begin{document}$sc\bar q\bar c$\end{document}

tetraquark states with

\begin{document}$J^P=0^+, 1^+, 2^+$\end{document}

. The masses of the axial-vector

\begin{document}$\bar{D}_sD^{*}$\end{document}

,

\begin{document}$\bar{D}_s^{*}D$\end{document}

molecular states and

\begin{document}${\bf 1}_{\boldsymbol{[sc]}}\boldsymbol \oplus {\bf 0}_{\boldsymbol{[\bar q \bar{c}]}}$\end{document}

,

\begin{document}${\bf 0}_{\boldsymbol{[sc]}} \oplus {\bf 1}_{\boldsymbol{[\bar q \bar{c}]}}$\end{document}

tetraquark states are predicted to be approximately 3.98 GeV, in good agreement with the mass of

\begin{document}$Z_{cs}(3985)^-$\end{document}

from BESIII. In both the molecular and diquark-antidiquark scenarios, our results suggest that there may exist two almost degenerate states, as the strange partners of

\begin{document}$X(3872)$\end{document}

and

\begin{document}$Z_c(3900)$\end{document}

. We propose to carefully examine

\begin{document}$Z_{cs}(3985)$\end{document}

in future experiments to verify this. One may also search for more hidden-charm four-quark states with strangeness in not only the open-charm

\begin{document}$\bar{D}_s^{(*)}D^{(*)}$\end{document}

channels but also the hidden-charm channels

\begin{document}$\eta_c K/K^\ast$\end{document}

,

\begin{document}$J/\psi K/K^\ast$\end{document}

.

NLO effects for Ω_QQQ baryons in QCD Sum Rules

Ren-Hua Wu, Yu-Sheng Zuo, Ce Meng, Yan-Qing Ma, Kuang-Ta Chao

2021, 45(9): 093103. doi: 10.1088/1674-1137/ac0b3c

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Abstract:
We study the triply heavy baryons

\begin{document}$\Omega_{QQQ}$\end{document}

\begin{document}$(Q=c, b)$\end{document}

in the QCD sum rules by performing the first calculation of the next-to-leading order (NLO) contribution to the perturbative QCD part of the correlation functions. Compared with the leading order (LO) result, the NLO contribution is found to be very important to the

\begin{document}$\Omega_{QQQ}$\end{document}

. This is because the NLO not only results in a large correction but also reduces the parameter dependence, making the Borel platform more distinct, especially for the

\begin{document}$\Omega_{bbb}$\end{document}

in the

\begin{document}$\overline{\rm{MS}}$\end{document}

scheme, where the platform appears only at NLO but not at LO. Particularly, owing to the inclusion of the NLO contribution, the renormalization schemes (

\begin{document}$\overline{\rm{MS}}$\end{document}

and On-Shell) dependence and the scale dependence are significantly reduced. Consequently, after including the NLO contribution to the perturbative part in the QCD sum rules, the masses are estimated to be

\begin{document}$4.53^{+0.26}_{-0.11}$\end{document}

GeV for

\begin{document}$\Omega_{ccc}$\end{document}

and

\begin{document}$14.27^{+0.33}_{-0.32}$\end{document}

GeV for

\begin{document}$\Omega_{bbb}$\end{document}

, where the results are obtained at

\begin{document}$\mu=M_B$\end{document}

with errors including those from the variation of the renormalization scale μ in the range

\begin{document}$(0.8-1.2) M_B$\end{document}

. A careful study of the μ dependence in a wider range is further performed, which shows that the LO results are very sensitive to the choice of μ whereas the NLO results are considerably better. In addition to the

\begin{document}$\mu=M_B$\end{document}

result, a more stable value, (4.75-4.80) GeV, for the

\begin{document}$\Omega_{ccc}$\end{document}

mass is found in the range of

\begin{document}$\mu=(1.2-2.0) M_B$\end{document}

, which should be viewed as a more relevant prediction in our NLO approach because of

\begin{document}$ \mu $\end{document}

dependence.

X(2900) in a chiral quark model

Yue Tan, Jialun Ping

2021, 45(9): 093104. doi: 10.1088/1674-1137/ac0ba4

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Abstract:
Recently, the LHCb Collaboration reported their observation of the first two fully open-flavor tetraquark states named

\begin{document}$ X_0 $\end{document}

(2900) and

\begin{document}$ X_1 $\end{document}

(2900) with unknown parity. Inspired by the report, we consider all the possible four-quark candidates for X(2900), which include the molecular structure, diquark structure, and their coupling in a chiral quark model via the Gaussian expansion method. To identify the genuine resonances, the real-scaling method (stabilization method) was employed. Our results show that five possible resonances,

\begin{document}$ R_0(2914) $\end{document}

with

\begin{document}$ \Gamma = 42 $\end{document}

MeV,

\begin{document}$ R_1(2906) $\end{document}

with

\begin{document}$ \Gamma = 29 $\end{document}

MeV,

\begin{document}$ R_1(2912) $\end{document}

with

\begin{document}$ \Gamma = 10 $\end{document}

MeV,

\begin{document}$ R_J(2920) $\end{document}

with

\begin{document}$ \Gamma = 9 $\end{document}

MeV, and

\begin{document}$ R_J(2842) $\end{document}

with

\begin{document}$ \Gamma = 24 $\end{document}

MeV, originate in the

\begin{document}$ cs\bar{q}\bar{q} $\end{document}

system. Compared with experimental data,

\begin{document}$ R_0(2914) $\end{document}

with

\begin{document}$ \Gamma = 42 $\end{document}

MeV may be an optimal

\begin{document}$ X_0(2900) $\end{document}

candidate. However, none of the resonances have a similar width for

\begin{document}$ X_1(2900) $\end{document}

. Hence, further study is required.

Precise evaluation of ${\boldsymbol{h\to c\bar{c}}}$ and axion-like particle production

Shi-Yuan Li, Zhen-Yang Li, Peng-Cheng Lu, Zong-Guo Si

2021, 45(9): 093105. doi: 10.1088/1674-1137/ac0c0d

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Abstract:
We study the decay of the SM Higgs boson to a massive charm quark pair at the next-to-next-to-leading order QCD and next-to-leading order electroweak. At the second order of QCD coupling, we consider the exact calculation of flavour-singlet contributions where the Higgs boson couples to the internal top and bottom quark. Helpful information on the running mass effects related to Yukawa coupling may be obtained by analyzing this process. High precision production for

\begin{document}$ h\to c\bar{c}$\end{document}

within the SM makes it possible to search for new physics that may induce relatively large interactions related to the charm quark. As an example, we evaluate the axion-like particle associate production with a charm quark pair in the Higgs decay and obtain some constraints for the corresponding parameters under some assumptions.

Scalar neutrino dark matter in the BLMSSM

Ming-Jie Zhang, Shu-Min Zhao, Xing-Xing Dong, Zhong-Jun Yang, Tai-Fu Feng

2021, 45(9): 093106. doi: 10.1088/1674-1137/ac0c0e

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Abstract:
The BLMSSM is an extension of the minimal supersymmetric standard model (MSSM). Its local gauge group is

\begin{document}$SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_B \times U(1)_L$\end{document}

. Supposing the lightest scalar neutrino is a dark matter candidate, we study the relic density and the spin independent cross section of sneutrino scattering off a nucleon. We calculate the numerical results in detail and find a suitable parameter space. The numerical discussion can confine the parameter space and provide a reference for dark matter research.

Probing top-philic new physics via four-top-quark production

Qing-Hong Cao, Jun-Ning Fu, Yandong Liu, Xiao-Hu Wang, Rui Zhang

2021, 45(9): 093107. doi: 10.1088/1674-1137/ac0c6f

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We explore constraints on various new physics resonances from four top-quark production based on current experimental data. Both light and heavy resonances are studied in this work. A comparison of the full width effect and narrow width approximation is also presented.

Probing magnetic moment operators in ${\boldsymbol H \gamma }$ production and ${\boldsymbol H \to \tau^+ \tau^- \gamma }$ rare decay

Qing-Hong Cao, Hao-Ran Jiang, Bin Li, Yandong Liu, Guojin Zeng

2021, 45(9): 093108. doi: 10.1088/1674-1137/ac0e88

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Abstract:
The magnetic moment (

\begin{document}$ a_\gamma $\end{document}

) and weak magnetic moment (

\begin{document}$ a_W $\end{document}

) of charged leptons and quarks are sensitive to quantum effects of new physics heavy resonances. In effective field theory,

\begin{document}$ a_\gamma $\end{document}

and

\begin{document}$ a_W $\end{document}

are induced by two independent operators. Therefore, one has to measure both

\begin{document}$ a_\gamma $\end{document}

and

\begin{document}$ a_W $\end{document}

to shed light on new physics. The

\begin{document}$ a_W $\end{document}

's of the SM fermions are measured at the LEP. In this work, we analyze the contributions from magnetic and weak magnetic moment operators in the processes of

\begin{document}$ pp\to H \gamma $\end{document}

and

\begin{document}$ gg\to H \to \tau^+ \tau^- \gamma $\end{document}

at the High-Luminosity Large Hadron Collider. We demonstrate that the two processes can cover most of the parameter space that cannot be probed at the LEP.

New physics and two boosted W-jets plus missing energy

Qing-Hong Cao, Nuo Chen, Hao-Ran Jiang, Bin Li, Yandong Liu

2021, 45(9): 093109. doi: 10.1088/1674-1137/ac0e8a

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Abstract:
We show that the signature of two boosted W-jets plus substantial missing energy is very promising for probing heavy charged resonances (

\begin{document}$X^\pm$\end{document}

) through the process of

\begin{document}$pp\to X^+X^-\to W^+W^- X^0 X^0$\end{document}

, where

\begin{document}$X^0$\end{document}

denotes the dark matter candidate. The hadronic decay mode of the W boson is considered to maximize the number of signal events. When the mass split between

\begin{document}$X^\pm$\end{document}

and

\begin{document}$X^0$\end{document}

is large, the jet-substructure technique must be utilized to analyze the boosted W-jet. Here, we consider the process of chargino pair production at the LHC, i.e.,

\begin{document}$pp\to \chi_1^+\chi^-_1 \to W^+W^-\chi_1^0\chi_1^0$\end{document}

, and demonstrate that the proposed signature is able to cover more parameter space of

\begin{document}$m_{\chi_1^\pm}$\end{document}

and

\begin{document}$m_{\chi_1^0}$\end{document}

than the conventional signature of multiple leptons plus missing energy. More importantly, the signature of interest is not sensitive to the spin of heavy resonances.

Single top quark production with and without a Higgs boson

Qing-Hong Cao, Hao-ran Jiang, Guojin Zeng

2021, 45(9): 093110. doi: 10.1088/1674-1137/ac0e8b

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Abstract:
One method of probing new physics beyond the Standard Model is to check the correlation among higher-dimensional operators in the effective field theory. We examine the strong correlation between the processes

\begin{document}$ pp\rightarrow tHq $\end{document}

and

\begin{document}$ pp\rightarrow tq $\end{document}

, which both depend on the same three operators. The correlation indicates that, according to the data of

\begin{document}$ pp\rightarrow tq $\end{document}

,

\begin{document}$ \sigma_{tHq} = \big[106.8 \pm 64.8\big]\; {\rm fb} $\end{document}

, which is significantly below the current upper limit

\begin{document}$ \sigma_{tHq}\leqslant 900\; {\rm fb} $\end{document}

.

Color halo scenario of charmonium-like hybrids

Yunheng Ma, Wei Sun, Ying Chen, Ming Gong, Zhaofeng Liu

2021, 45(9): 093111. doi: 10.1088/1674-1137/ac0ee2

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The internal structures of

\begin{document}$J^{PC} = 1^{--}, (0,1,2)^{-+}$\end{document}

charmonium-like hybrids are investigated under lattice QCD in the quenched approximation. We define the Bethe-Salpeter wave function (

\begin{document}$ \Phi_n(r) $\end{document}

) in the Coulomb gauge as the matrix element of a spatially extended hybrid-like operator (

\begin{document}$ \bar{c}{c}g $\end{document}

) between the vacuum and n-th state for each

\begin{document}$ J^{PC} $\end{document}

, with r being the spatial separation between a localized

\begin{document}$ \bar{c}c $\end{document}

component and the chromomagnetic strength tensor. These wave functions exhibit some similarities for states with the aforementioned different quantum numbers, and their r-behaviors (no node for the ground states and one node for the first excited states) imply that r can be a meaningful dynamical variable for these states. Additionally, the mass splittings of the ground states and first excited states of charmonium-like hybrids in these channels are obtained for the first time to be approximately 1.2-1.4 GeV. These results do not support the flux-tube description of heavy-quarkonium-like hybrids in the Born-Oppenheimer approximation. In contrast, a charmonium-like hybrid can be viewed as a “color halo” charmonium for which a relatively localized color octet

\begin{document}$ \bar{c}c $\end{document}

is surrounded by gluonic degrees of freedom, which can readily decay into a charmonium state along with one or more light hadrons. The color halo picture is compatible with the decay properties of

\begin{document}$ Y(4260) $\end{document}

and suggests LHCb and BelleII to search for

\begin{document}$ (0,1,2)^{-+} $\end{document}

charmonium-like hybrids in

\begin{document}$ \chi_{c0,1,2}\eta $\end{document}

and

\begin{document}$ J/\psi \omega (\phi) $\end{document}

final states.

Lax connections in $ {\boldsymbol{T\bar{T}}} $-deformed integrable field theories

Bin Chen, Jue Hou, Jia Tian

2021, 45(9): 093112. doi: 10.1088/1674-1137/ac0ee4

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In this work, we attempt to construct the Lax connections of

\begin{document}$ T\bar{T} $\end{document}

-deformed integrable field theories in two different ways. With reasonable assumptions, we make an ansatz and find the Lax pairs in the

\begin{document}$ T\bar{T} $\end{document}

-deformed affine Toda theories and the principal chiral model by solving the Lax equations directly. This method is straightforward, but it may be difficult to apply for general models. We then make use of a dynamic coordinate transformation to read the Lax connection in the deformed theory from the undeformed one. We find that once the inverse of the transformation is available, the Lax connection can be read easily. We show the construction explicitly for a few classes of scalar models and find consistency with those determined using the first method.

Effect of pairing correlation on low-lying quadrupole states in Sn isotopes

Shuai Sun, Shi-Sheng Zhang, Zhen-Hua Zhang, Li-Gang Cao

2021, 45(9): 094101. doi: 10.1088/1674-1137/ac0b39

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Abstract:
We examined the low-lying quadrupole states in Sn isotopes in the framework of fully self-consistent Hartree-Fock+BCS plus QRPA. We focus on the effect of the density-dependence of pairing interaction on the properties of the low-lying quadrupole state. The SLy5 Skyrme interaction with surface, mixed, and volume pairings is employed in the calculations, respectively. We find that the excitation energies and the corresponding reduced electric transition probabilities of the first 2⁺ state are different, given by the three pairing interactions. The properties of the quasiparticle state, two-quasiparticle excitation energy, reduced transition amplitude, and transition densities in ¹¹²Sn are analyzed in detail. Two different mechanisms, the static and dynamical effects, of the pairing correlation are also discussed. The results show that the surface, mixed, and volume pairings indeed affect the properties of the first 2⁺ state in the Sn isotopes.

Cross section of the Coulomb excitation of ${^{\boldsymbol{229m}}{\bf{Th}}} $ by low energy muons

E. V. Tkalya

2021, 45(9): 094102. doi: 10.1088/1674-1137/ac0b3a

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The inelastic scattering cross section for muons,

\begin{document}$ \mu^- $\end{document}

, with energies

\begin{document}$ E $\end{document}

= 9–100 eV from the

\begin{document}$ ^{229} $\end{document}

Th nuclei is calculated in the framework of the second order of the perturbation theory for quantum electrodynamics. The dominant contribution to the excitation of the low energy isomer

\begin{document}$ ^{229m} $\end{document}

Th

\begin{document}$ (3/2^+,8.19\pm0.12 $\end{document}

eV) originates from the

\begin{document}$ E2 $\end{document}

multipole. The excitation cross section reaches the value of

\begin{document}$ 10^{-21} $\end{document}

cm

\begin{document}$ ^2 $\end{document}

in the range

\begin{document}$ E\approx $\end{document}

10 eV. This value is four to five orders of magnitude larger than the electron excitation cross section and is sufficient for the efficient excitation of

\begin{document}$ ^{229m} $\end{document}

Th on the muon beam at the next generation of muon colliders.

Symmetric and asymmetric structural evolutions of Te isotopes across the N = 82 shell closure

Hui Jiang, Yi-jie Zhou, Yang Lei, Jia-Jie Shen, Man Bao

2021, 45(9): 094103. doi: 10.1088/1674-1137/ac0ce1

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Systematic calculations of low-lying energy levels,

\begin{document}$B(E2)$\end{document}

transitions, and g factors of even-even tellurium isotopes with mass numbers from 128 to 140 are performed via the nucleon-pair approximation (NPA) of the shell model with phenomenological multipole-multipole interactions. An optimal agreement is obtained between the calculated results and experimental data. The yrast band structures of nuclei below and above the

\begin{document}$N=82$\end{document}

shell closure are compared and discussed. In particular, the evolutionary differences of

\begin{document}$B(E2;2_1^{+}\rightarrow 0_1^{+})$\end{document}

values and

\begin{document}$g(2_1^{+})$\end{document}

factors, with respect to the symmetry of

\begin{document}$N=82$\end{document}

, are attributed to the dominant contribution differences in neutron and proton excitations, respectively.

On the structure in the ΛN cross section at the ΣN threshold

Johann Haidenbauer, Ulf-G. Meißner

2021, 45(9): 094104. doi: 10.1088/1674-1137/ac0e89

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The complexity of threshold phenomena is exemplified on a prominent and long-known case - the structure in the

\begin{document}$\Lambda p$\end{document}

cross section (invariant mass spectrum) at the opening of the

\begin{document}$\Sigma N$\end{document}

channel. The mass splitting between the

\begin{document}$\Sigma$\end{document}

baryons together with the angular momentum coupling in the

\begin{document}$^3S_1$\end{document}

-

\begin{document}$^3D_1$\end{document}

partial wave imply that, in principle, up to six channels are involved. Utilizing hyperon-nucleon potentials that provide an excellent description of the available low-energy

\begin{document}$\Lambda p$\end{document}

and

\begin{document}$\Sigma N$\end{document}

scattering data, the shape of the resulting

\begin{document}$\Lambda p$\end{document}

cross section is discussed and the poles near the

\begin{document}$\Sigma N$\end{document}

threshold are determined. Evidence for a strangeness

\begin{document}$S=-1$\end{document}

dibaryon is provided, in the form of a (unstable)

\begin{document}$\Sigma N$\end{document}

bound state in the vicinity of the

\begin{document}$\Sigma N$\end{document}

threshold. Predictions for level shifts and widths of

\begin{document}$\Sigma^-p$\end{document}

atomic states are given.

The entrance channel effect on the synthesis of a superheavy element ²⁹⁶119

D. Naderi, B. Sharifi

2021, 45(9): 094105. doi: 10.1088/1674-1137/ac0ee3

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Abstract:
In this study, we investigated the entrance channel effect on the evaporation residue cross section of a superheavy element ²⁹⁶119. Using 29 projectile-target combinations, we investigated the role of the entrance channel on the 3n and 4n evaporation channels in hot combinations. This effect can be evaluated based on the entrance channel asymmetry and Q value of complete fusion. We calculated the variation of the maximum evaporation residue cross sections (

\begin{document}$\sigma_{3n}^{\rm max}$\end{document}

and

\begin{document}$\sigma_{4n}^{\rm max}$\end{document}

) with

\begin{document}$|Q|$\end{document}

for the reactions

\begin{document}$^{49-47}{\rm{Ti}}+^{247-249}{\rm{Bk}}$\end{document}

,

\begin{document}$^{60-57}{\rm{Fe}}+^{236-239}{\rm{Np}}$\end{document}

,

\begin{document}$^{44-42}{\rm{Ca}}+^{252-254}{\rm{Es}}$\end{document}

, and

\begin{document}$^{55,54,52}{\rm{Mn}}+^{241,242,244}{\rm{Pu}}$\end{document}

. With an increase in

\begin{document}$|Q|$\end{document}

,

\begin{document}$\sigma_{3n}^{\rm max}$\end{document}

and

\begin{document}$\sigma_{4n}^{\rm max}$\end{document}

increase. In addition, we studied the role of asymmetry and mean fissility parameters in the synthesis of the superheavy element. The obtained results in this study can be utilized in future studies.

Core breaking and possible magnetic rotation in the semimagic nucleus ⁹⁰Zr

Hao Wang, Ke-Yan Ma, Si-Ying Liu, Jing-Bin Lu

2021, 45(9): 094106. doi: 10.1088/1674-1137/ac0fd2

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The semimagic nucleus ⁹⁰Zr, with Z = 40 and N = 50, is investigated in terms of large scale shell model calculations. A logical agreement is obtained between the available experimental data and predicted values. The calculated results indicate that the low-lying states are primarily dominated by the proton excitations from the fp orbitals across the Z = 38 or 40 subshell into the high-j

\begin{document}$1g_{9/2}$\end{document}

orbital. For the higher-spin states of ⁹⁰Zr, the breaking of the N = 50 core plays a crucial role, and the contribution of different orbitals to each state are discussed in this article. The evolution from neutron core excitations to proton excitations is systematically studied along the neighboring N = 50 isotones. Furthermore, the strong

\begin{document}$\Delta I$\end{document}

= 1 sequence demonstrates an abrupt backbend attributed to the alignment of the valence nucleons in fp proton orbitals and is proposed to have a

\begin{document}$\pi(fp)^{-2}(1g_{9/2})^{2} \otimes $\end{document}

\begin{document}$ \nu(1g_{9/2})^{-1}(2d_{5/2}/1g_{7/2})^{1}$\end{document}

configuration before the backbend, based on the shell model calculations. The properties of this sequence before the backbend indicate a general agreement with the fingerprints of magnetic rotation; hence, the sequence with the

\begin{document}$\pi(fp)^{-2}(1g_{9/2})^{2} \otimes \nu(1g_{9/2})^{-1}(2d_{5/2}/1g_{7/2})^{1}$\end{document}

configuration is suggested as a magnetic rotational band arising from shears mechanism.

Note on gauge invariance of second order cosmological perturbations

Zhe Chang, Sai Wang, Qing-Hua Zhu

2021, 45(9): 095101. doi: 10.1088/1674-1137/ac0c74

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Abstract:
We study the gauge invariant cosmological perturbations up to the second order. We demonstrate that there are infinite families of gauge invariant variables at both the first and second orders. The conversion formulae among different families are verified to be described by a finite number of bases that are gauge invariant. For the second order cosmological perturbations induced by the first order scalar perturbations, we explicitly represent their equations of motion in terms of the gauge invariant Newtonian, synchronous and hybrid variables, respectively.

2021 Vol. 45, No. 9