Roles of <inline-formula><tex-math id="M1">\begin{document}${\boldsymbol\Delta(\bf 1232)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M1.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}${\boldsymbol N^*(\bf 1520)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M2.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M2.png"/></alternatives></inline-formula>, and <inline-formula><tex-math id="M3">\begin{document}${\boldsymbol N^*(\bf 1650)}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M3.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M3.png"/></alternatives></inline-formula> resonances in <inline-formula><tex-math id="M4">\begin{document}${\boldsymbol\gamma \boldsymbol p\to \boldsymbol\pi^{\bf 0} \boldsymbol\pi^{\bf 0}\boldsymbol p}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M4.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/4047eb24-52ff-41fe-b919-159902fd536d/CPC-2025-0061_M4.png"/></alternatives></inline-formula> reaction within an effective Lagrangian approach -

[1]
S. Navaset al. (Particle Data Group), Phys. Rev. D110, 030001 (2024)
[2]
E. Klempt and J.-M. Richard, Rev. Mod. Phys.82, 1095 (2010), arXiv: 0901.2055[hep-ph]
[3]
V. Crede and W. Roberts, Rept. Prog. Phys.76, 076301 (2013), arXiv: 1302.7299[nucl-ex]
[4]
E. Wang, L.-S. Geng, J.-J. Wuet al., Chin. Phys. Lett.41, 101401 (2024), arXiv: 2406.07839[hep-ph]
[5]
N. Isgur and G. Karl, Phys. Rev. D18, 4187 (1978)
[6]
S. Capstick and N. Isgur, Phys. Rev. D34, 2809 (1986)
[7]
S. Capstick and W. Roberts, Phys. Rev. D49, 4570 (1994), arXiv: nucl-th/9310030
[8]
U. Loring, B. C. Metsch, and H. R. Petry, Eur. Phys. J. A10, 395 (2001), arXiv: hep-ph/0103289
[9]
S. Capstick and W. Roberts, Phys. Rev. D47, 1994 (1993)
[10]
M. Shrestha and D. M. Manley, Phys. Rev. C86, 055203 (2012), arXiv: 1208.2710[hep-ph]
[11]
B. S. Zou, Chin. Phys. C33, 1113 (2009), arXiv: 0907.4567[nucl-th]
[12]
H.-H. Zhong, M.-S. Liu, R.-H. Niet al., Phys. Rev. D110, 116034 (2024), arXiv: 2409.07998[hep-ph]
[13]
T. Sekihara, T. Arai, J. Yamagata-Sekiharaet al., Phys. Rev. C93, 035204 (2016), arXiv: 1511.01200[hep-ph]
[14]
C. An and B. Saghai, Phys. Rev. C84, 045204 (2011), arXiv: 1108.3282[nucl-th]
[15]
S. Abdel-Samadet al. (COSY-TOF), Phys. Lett. B632, 27 (2006)
[16]
S. Abd El-Samadet al. (TOF), Phys. Lett. B688, 142 (2010), arXiv: 1003.0603[nucl-ex]
[17]
Y. Li, S.-W. Liu, E. Wanget al., Phys. Rev. D110, 074010 (2024), arXiv: 2406.01209[hep-ph]
[18]
F. Huang, M. Doring, H. Haberzettlet al., Phys. Rev. C85, 054003 (2012), arXiv: 1110.3833[nucl-th]
[19]
N. Markovet al. (CLAS), Phys. Rev. C101, 015208 (2020), arXiv: 1907.11974[nucl-ex]
[20]
E. L. Isupovet al. (CLAS), Phys. Rev. C105, L022201 (2022), arXiv: 2112.07732[nucl-ex]
[21]
M. Mai, M. Döring, C. Granadoset al., Phys. Rev. C106, 015201 (2022), arXiv: 2111.04774[nucl-th]
[22]
X.-H. Zhong and Q. Zhao, Phys. Rev. C84, 045207 (2011), arXiv: 1106.2892[nucl-th]
[23]
J.-M. Suh, S.-H. Kim, and H.-C. Kim, Phys. Rev. D99, 074010 (2019), arXiv: 1810.05056[hep-ph]
[24]
F. Huang and K. Nakayama, Chin. Phys. C33, 1291 (2009)
[25]
X.-H. Zhong and Q. Zhao, Phys. Rev. C84, 065204 (2011), arXiv: 1110.5466[nucl-th]
[26]
F. Huang, H. Haberzettl, and K. Nakayama, Phys. Rev. C87, 054004 (2013), arXiv: 1208.2279[nucl-th]
[27]
M. Williamset al. (CLAS), Phys. Rev. C80, 065209 (2009), arXiv: 0908.2911[nucl-ex]
[28]
M. Williamset al. (CLAS), Phys. Rev. C80, 065208 (2009), arXiv: 0908.2910[nucl-ex]
[29]
I. Denisenko, A. V. Anisovich, V. Credeet al., Phys. Lett. B755, 97 (2016), arXiv: 1601.06092[nucl-ex]
[30]
N. C. Wei, F. Huang, K. Nakayamaet al., Phys. Rev. D100, 114026 (2019), arXiv: 1908.01139[hep-ph]
[31]
A. Kiswandhi, J.-J. Xie, and S. N. Yang, Phys. Lett. B691, 214 (2010), arXiv: 1005.2105[hep-ph]
[32]
S. N. Yang, A. Kiswandhi, and J.-J. Xie, AIP Conf. Proc.1388, 246 (2011), arXiv: 1108.1677[nucl-th]
[33]
B. Dey, C. A. Meyer, M. Belliset al. (CLAS), Phys. Rev. C89, 055208 (2014) [Addendum: Phys. Rev. C 90, 019901(2014)] , arXiv: 1403.2110
[34]
J.-J. Xie, Few Body Syst.59, 124 (2018)
[35]
S.-H. Kim, T. S. H. Lee, S.-i. Namet al., Phys. Rev. C104, 045202 (2021), arXiv: 2108.12039[nucl-th]
[36]
S.-M. Wu, F. Wang, and B.-S. Zou, Phys. Rev. C108, 045201 (2023), arXiv: 2306.15385[hep-ph]
[37]
A. de la Puente, O. V. Maxwell, and B. A. Raue, Phys. Rev. C80, 065205 (2009), arXiv: 0809.3805[nucl-th]
[38]
A. V. Anisovich, R. Beck, V. Burkertet al., Eur. Phys. J. A50, 129 (2014), arXiv: 1404.4587[nucl-ex]
[39]
A. V. Anisovichet al., Eur. Phys. J. A53, 242 (2017), arXiv: 1712.07537[nucl-ex]
[40]
B. C. Hunt and D. M. Manley, Phys. Rev. C99, 055204 (2019), arXiv: 1804.07422[nucl-ex]
[41]
S.-H. Kim and H.-C. Kim, Phys. Lett. B786, 156 (2018), arXiv: 1806.01992[hep-ph]
[42]
T. Mart and M. J. Kholili, J. Phys. G46, 105112 (2019)
[43]
D. Petrellis and D. Skoupil, Phys. Rev. C110, 065204 (2024)
[44]
Y. Oh and H. Kim, Phys. Rev. C73, 065202 (2006), arXiv: hepph/0602112
[45]
S.-H. Kim, S.-i. Nam, Y. Ohet al., Phys. Rev. D84, 114023 (2011), arXiv: 1110.6515[hep-ph]
[46]
W. Tanget al. (CLAS), Phys. Rev. C87, 065204 (2013), arXiv: 1303.2615[nucl-ex]
[47]
S.-H. Kim, A. Hosaka, and H.-C. Kim, Phys. Rev. D90, 014021 (2014), arXiv: 1402.1753[hep-ph]
[48]
A. C. Wang, W. L. Wang, F. Huanget al., Phys. Rev. C96, 035206 (2017), arXiv: 1704.04562[hep-ph]
[49]
A.-C. Wang, F. Huang, W.-L. Wanget al., Phys. Rev. C102, 015203 (2020), arXiv: 1902.05267[nucl-th]
[50]
S.-H. Kim, S.-i. Nam, A. Hosakaet al., Phys. Rev. D88, 054012 (2013), arXiv: 1211.6285[hep-ph]
[51]
A.-C. Wang, W.-L. Wang, and F. Huang, Phys. Rev. C98, 045209 (2018), arXiv: 1810.07012[nucl-th]
[52]
D. Ben, A.-C. Wang, F. Huanget al., Phys. Rev. C108, 065201 (2023), arXiv: 2302.14308[nucl-th]
[53]
M. Niiyamaet al., Phys. Rev. C78, 035202 (2008), arXiv: 0805.4051[hep-ex]
[54]
S.-H. Kim, S.-i. Nam, D. Jidoet al., Phys. Rev. D96, 014003 (2017), arXiv: 1702.08645[hep-ph]
[55]
Y. Zhang and F. Huang, Phys. Rev. C103, 025207 (2021), arXiv: 2102.10404[hep-ph]
[56]
E. Wang, J.-J. Xie, W.-H. Lianget al., Phys. Rev. C95, 015205 (2017), arXiv: 1610.07117[hep-ph]
[57]
Y. Oh, C. M. Ko, and K. Nakayama, Phys. Rev. C77, 045204 (2008), arXiv: 0712.4285[nucl-th]
[58]
P. Gao, J.-J. Wu, and B. S. Zou, Phys. Rev. C81, 055203 (2010), arXiv: 1001.0805[nucl-th]
[59]
Y.-H. Chen and B.-S. Zou, Phys. Rev. C88, 024304 (2013), arXiv: 1302.5466[nucl-th]
[60]
J. He, Phys. Rev. C89, 055204 (2014), arXiv: 1311.0571[nucl-th]
[61]
X.-Y. Wang, J. He, and H. Haberzettl, Phys. Rev. C93, 045204 (2016), arXiv: 1511.04262[nucl-th]
[62]
A.-C. Wang, W.-L. Wang, and F. Huang, Phys. Rev. D101, 074025 (2020), arXiv: 2002.04213[hep-ph]
[63]
H. Kohriet al. (LEPS), Phys. Rev. Lett.104, 172001 (2010), arXiv: 0906.0197[hep-ex]
[64]
S.-i. Nam and C.-W. Kao, Phys. Rev. C81, 055206 (2010), arXiv: 1003.0700[hep-ph]
[65]
J.-J. Xie and J. Nieves, Phys. Rev. C82, 045205 (2010), arXiv: 1007.3141[nucl-th]
[66]
J. He and X.-R. Chen, Phys. Rev. C86, 035204 (2012), arXiv: 1209.4138[nucl-th]
[67]
J.-J. Xie, E. Wang, and J. Nieves, Phys. Rev. C89, 015203 (2014), arXiv: 1309.7135[nucl-th]
[68]
J.-J. Xie, E. Wang, B.-C. Liuet al., Int. J. Mod. Phys. Conf. Ser.29, 1460244 (2014), arXiv: 1311.0439[nucl-th]
[69]
E. Wang, J.-J. Xie, and J. Nieves, Phys. Rev. C90, 065203 (2014), arXiv: 1405.3142[nucl-th]
[70]
J. He, Nucl. Phys. A927, 24 (2014), arXiv: 1405.1145[nuclth]
[71]
N.-C. Wei, Y. Zhang, F. Huanget al., Phys. Rev. D103, 034007 (2021), arXiv: 2101.08988[hep-ph]
[72]
N. Muramatsuet al. (LEPS2/BGOegg), Phys. Rev. C107, L042201 (2023)
[73]
A. Braghieriet al., Phys. Lett. B363, 46 (1995)
[74]
F. Harter, J. Ahrens, R. Becket al., Phys. Lett. B401, 229 (1997)
[75]
M. Wolfet al., Eur. Phys. J. A9, 5 (2000)
[76]
M. Battaglieriet al. (CLAS), Phys. Rev. D80, 072005 (2009), arXiv: 0907.1021[hep-ex]
[77]
Sokhoyanet al. (CBELSA/TAPS), Eur. Phys. J. A51, 95 (2015) [Erratum: Eur.Phys.J.A 51, 187 (2015)], arXiv: 1507.02488
[78]
J. A. Gomez Tejedor and E. Oset, Nucl. Phys. A600, 413 (1996), arXiv: hep-ph/9506209
[79]
J. C. Nacher and E. Oset, Nucl. Phys. A697, 372 (2002), arXiv: nucl-th/0106005
[80]
C.-R. Ji, R. Kaminski, L. Lesniaket al., Phys. Rev. C58, 1205 (1998), arXiv: hepph/9710510
[81]
A. Donnachie and Y. S. Kalashnikova, Phys. Rev. C93, 025203 (2016), arXiv: 1507.07408[hep-ph]
[82]
J.-H. Lee, H.-C. Kim, S.-H. Kimet al., PTEP2017, 093D05 (2017), arXiv: 1608.01508[hep-ph]
[83]
H. Xing, C.-S. An, J.-J. Xieet al., Phys. Rev. D98, 094007 (2018), arXiv: 1807.11151[hep-ph]
[84]
N.-C. Wei, A.-C. Wang, and F. Huang, Phys. Rev. C110, 025207 (2024), arXiv: 2408.11352[hep-ph]
[85]
J.-J. Xie, B.-S. Zou, and B.-C. Liu, Chin. Phys. Lett.22, 2215 (2005)
[86]
J.-J. Xie and B.-S. Zou, Phys. Lett. B649, 405 (2007), arXiv: nucl-th/0701021
[87]
J.-J. Xie, B.-S. Zou, and H.-C. Chiang, Phys. Rev. C77, 015206 (2008), arXiv: 0705.3950[nucl-th]
[88]
J.-j. Xie, C. Wilkin, and B.-S. Zou, Phys. Rev. C77, 058202 (2008), arXiv: 0802.2802[nucl-th]
[89]
J.-J. Xie and C. Wilkin, Phys. Rev. C82, 025210 (2010), arXiv: 1005.2957[nucl-th]
[90]
B.-C. Liu and J.-J. Xie, Phys. Rev. C85, 038201 (2012), arXiv: 1112.5327[nucl-th]
[91]
B.-C. Liu and J.-J. Xie, Phys. Rev. C86, 055202 (2012), arXiv: 1211.4375[hep-ph]
[92]
J.-J. Xie and B.-C. Liu, Phys. Rev. C87, 045210 (2013), arXiv: 1302.0468[nucl-th]
[93]
J.-J. Xie, B.-C. Liu, and C.-S. An, Phys. Rev. C88, 015203 (2013), arXiv: 1307.3707[nucl-th]
[94]
Q.-F. Lü, X.-H. Liu, J.-J. Xieet al., Mod. Phys. Lett. A29, 1450012 (2014), arXiv: 1312.5803[hep-ph]
[95]
Q.-F. Lü, J.-J. Xie, and D.-M. Li, Phys. Rev. C90, 034002 (2014), arXiv: 1404.6868[hep-ph]
[96]
J.-J. Xie, E. Wang, and B.-S. Zou, Phys. Rev. C90, 025207 (2014), arXiv: 1405.5586[nucl-th]
[97]
J.-J. Xie, L.-S. Geng, and X.-R. Chen, Phys. Rev. C90, 048201 (2014), arXiv: 1407.7403[nucl-th]
[98]
J.-J. Xie, J.-J. Wu, and B.-S. Zou, Phys. Rev. C90, 055204 (2014), arXiv: 1407.7984[nucl-th]
[99]
C.-Z. Wu, Q.-F. Lü, J.-J. Xieet al., Commun. Theor. Phys.63, 215 (2015), arXiv: 1409.7242[nucl-th]
[100]
X. Cao and J.-J. Xie, Chin. Phys. C40, 083103 (2016), arXiv: 1411.1493[nucl-th]
[101]
X.-Y. Wang, X. Cao, J.-J. Xieet al., Phys. Rev. C92, 015202 (2015), arXiv: 1412.4895[nucl-th]
[102]
Q.-F. Lü, R. Wang, J.-J. Xieet al., Phys. Rev. C91, 035204 (2015), arXiv: 1412.6272[nucl-th]
[103]
Y. Huang, J. He, X.-R. Chenet al., Phys. Rev. C91, 065202 (2015), arXiv: 1412.7947[nucl-th]
[104]
L.-Y. Xiao, Q.-F. Lü, J.-J. Xieet al., Eur. Phys. J. A51, 130 (2015), arXiv: 1504.07535[nucl-th]
[105]
J.-J. Xie, Y.-B. Dong, and X. Cao, Phys. Rev. D92, 034029 (2015), arXiv: 1506.01133[hep-ph]
[106]
E. J. Garzon and J.-J. Xie, Phys. Rev. C92, 035201 (2015), arXiv: 1506.06834[hep-ph]
[107]
J.-J. Xie, Phys. Rev. C92, 065203 (2015), arXiv: 1509.06196[nucl-th]
[108]
Q.-F. Lü, X.-Y. Wang, J.-J. Xieet al., Phys. Rev. D93, 034009 (2016), arXiv: 1510.06271[hep-ph]
[109]
Y. Huang, J.-J. Xie, J. He,et al., Chin. Phys. C40, 124104 (2016), arXiv: 1604.05969[nucl-th]
[110]
C. Cheng, J.-J. Xie, and X. Cao, Commun. Theor. Phys.66, 675 (2016), arXiv: 1609.00442[nucl-th]
[111]
Y. Huang, J. He, J.-J. Xieet al., Phys. Rev. D99, 014045 (2019), arXiv: 1610.06994[hep-ph]
[112]
X. Zhang and J.-J. Xie, Commun. Theor. Phys.70, 060 (2018), arXiv: 1712.05572[nucl-th]
[113]
X. Zhang and J.-J. Xie, Chin. Phys. C43, 064104 (2019), arXiv: 1812.04242[hep-ph]
[114]
H. Zhou and J.-J. Xie, Commun. Theor. Phys.72, 045202 (2020), arXiv: 1911.06536[hep-ph]
[115]
H.-N. Wang, Q. Wang, and J.-J. Xie, Phys. Rev. D106, 056022 (2022), arXiv: 2206.14456[hep-ph]
[116]
Y.-Y. Wang, L.-J. Liu, E. Wanget al., Phys. Rev. D95, 096015 (2017), arXiv: 1701.06007[hep-ph]
[117]
C.-G. Zhao, G.-Y. Wang, G.-N. Liet al., Phys. Rev. D99, 114014 (2019), arXiv: 1904.08569[hep-ph]
[118]
V. E. Tarasov, W. J. Briscoe, W. Gradlet al., Phys. Rev. C88, 035207 (2013), arXiv: 1306.6618[hep-ph]
[119]
Y.-s. Oh and T. S. H. Lee, Phys. Rev. C69, 025201 (2004), arXiv: nucl-th/0306033
[120]
Y.-s. Oh, A. I. Titov, and T. S. H. Lee, Phys. Rev. C63, 025201 (2001), arXiv: nucl-th/0006057
[121]
B. S. Zou and F. Hussain, Phys. Rev. C67, 015204 (2003), arXiv: hep-ph/0210164
[122]
Q.-F. Lü and D.-M. Li, Chin. Phys. C39, 113104 (2015), arXiv: 1501.06266[nucl-th]
[123]
A. Matsuyama, T. Sato, and T. S. H. Lee, Phys. Rept.439, 193 (2007), arXiv: nucl-th/0608051
[124]
D. O. Riska and G. E. Brown, Nucl. Phys. A679, 577 (2001), arXiv: nucl-th/0005049
[125]
A. Xu, R. Li, X. Luoet al., Eur. Phys. J. C84, 1107 (2024), arXiv: 2404.17088[hep-ph]
[126]
K. Tsushima, A. Sibirtsev, A. W. Thomaset al., Phys. Rev. C59, 369 (1999), [Erratum: Phys.Rev.C 61, 029903 (2000)] , arXiv: nucl-th/9801063
[127]
G. Penner and U. Mosel, Phys. Rev. C66, 055211 (2002), arXiv: nucl-th/0207066
[128]
H. Haberzettl, K. Nakayama, and S. Krewald, Phys. Rev. C74, 045202 (2006), arXiv: nucl-th/0605059
[129]
V. Shklyar, H. Lenske, and U. Mosel, Phys. Rev. C72, 015210 (2005), arXiv: nucl-th/0505010
[130]
T. Feuster and U. Mosel, Phys. Rev. C58, 457 (1998), arXiv: nucl-th/9708051
[131]
R. Brockmann and R. Machleidt, Phys. Rev. C42, 1965 (1990)
[132]
R. Machleidt, Adv. Nucl. Phys.19, 189 (1989)
[133]
R. Machleidt, K. Holinde, and C. Elster, Phys. Rept.149, 1 (1987)