The measurements of the total cross section of the <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="44.732pt" style="vertical-align:-3.429501pt" id="M2" height="14.2262pt" version="1.1" viewBox="-0.0498162 -10.7967 44.732 14.2262"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-102"/></g><g transform="matrix(.0091,0,0,-0.0091,5.382,-5.741)"><use xlink:href="#g54-36"/></g><g transform="matrix(.013,0,0,-0.013,11.461,0)"><use xlink:href="#g113-102"/></g><g transform="matrix(.0091,0,0,-0.0091,16.844,-5.741)"><use xlink:href="#g54-33"/></g><g transform="matrix(.013,0,0,-0.013,26.554,0)"><use xlink:href="#g117-149"/></g><g transform="matrix(.013,0,0,-0.013,32.33,0)"><use xlink:href="#g117-148"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="28.781pt" style="vertical-align:-3.429501pt" height="14.2262pt" version="1.1" viewBox="48.3921838 -10.7967 28.781 14.2262"><g transform="matrix(.013,0,0,-0.013,48.442,0)"><use xlink:href="#g113-225"/></g><g transform="matrix(.013,0,0,-0.013,54.958,0)"><use xlink:href="#g113-225"/></g><g transform="matrix(.013,0,0,-0.013,61.474,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.013,0,0,-0.013,65.972,0)"><use xlink:href="#g113-225"/></g><g transform="matrix(.013,0,0,-0.013,72.488,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg></span> reaction from the VENUS, TOPAS, OPAL, DELPHI, ALEPH, and L3 collaborations, collected between 1989 and 2003, are used to perform a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.6382pt" id="M3" height="15.2296pt" version="1.1" viewBox="-0.0498162 -11.5914 12.6404 15.2296" width="12.6404pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-244" d="M565 434L556 448L482 423L322 189L287 373C280 410 261 448 236 448C207 448 161 415 110 338L131 317C160 355 181 374 193 374C204 374 208 366 217 330C230 279 247 185 259 116L50 -175C18 -223 29 -260 41 -278H51C97 -198 207 -33 270 57L313 -165C328 -235 349 -261 376 -261S445 -244 510 -150L487 -127C444 -178 423 -186 411 -186C402 -186 398 -177 387 -130C371 -62 349 39 332 135C390 217 524 392 565 434Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.568,-5.741)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg> test to validate the current quantum electrodynamics (QED) theory and search for possible deviations with the direct contact term annihilation. By observing a deviation from the QED predictions on the total cross section of the <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="44.732pt" style="vertical-align:-3.429501pt" id="M4" height="14.2262pt" version="1.1" viewBox="-0.0498162 -10.7967 44.732 14.2262"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-102"/></g><g transform="matrix(.0091,0,0,-0.0091,5.382,-5.741)"><use xlink:href="#g54-36"/></g><g transform="matrix(.013,0,0,-0.013,11.461,0)"><use xlink:href="#g113-102"/></g><g transform="matrix(.0091,0,0,-0.0091,16.844,-5.741)"><use xlink:href="#g54-33"/></g><g transform="matrix(.013,0,0,-0.013,26.554,0)"><use xlink:href="#g117-149"/></g><g transform="matrix(.013,0,0,-0.013,32.33,0)"><use xlink:href="#g117-148"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="28.781pt" style="vertical-align:-3.429501pt" height="14.2262pt" version="1.1" viewBox="48.3921838 -10.7967 28.781 14.2262"><g transform="matrix(.013,0,0,-0.013,48.442,0)"><use xlink:href="#g113-225"/></g><g transform="matrix(.013,0,0,-0.013,54.958,0)"><use xlink:href="#g113-225"/></g><g transform="matrix(.013,0,0,-0.013,61.474,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,65.972,0)"><use xlink:href="#g113-225"/></g><g transform="matrix(.013,0,0,-0.013,72.488,0)"><use xlink:href="#g113-42"/></g></svg></span> reaction above <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="26.125pt" style="vertical-align:-2.636801pt" id="M5" height="13.1484pt" version="1.1" viewBox="-0.0498162 -10.5116 26.125 13.1484"><g transform="matrix(.013,0,0,-0.013,0,-.271)"><path id="g119-37" d="M782 731H740L476 -79H474L280 352L133 274L151 239L242 286L473 -219L782 731Z"/></g><rect height="0.65243" width="4.89327" x="9.96925" y="-9.80939"/><g transform="matrix(.013,0,0,-0.013,9.969,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"/></g><g transform="matrix(.013,0,0,-0.013,18.494,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="28.157pt" style="vertical-align:-2.636801pt" height="13.1484pt" version="1.1" viewBox="29.707183800000003 -10.5116 28.157 13.1484"><g transform="matrix(.013,0,0,-0.013,29.757,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.013,0,0,-0.013,35.997,0)"><path id="g113-57" d="M249 635C141 635 70 555 70 471C70 401 114 353 179 316C143 294 106 267 90 252C68 231 45 202 45 157C45 50 130 -12 237 -12C322 -12 435 52 435 169C435 256 372 304 303 343C349 374 375 398 383 407C401 429 411 458 411 487C411 569 344 635 249 635ZM238 603C285 603 337 567 337 482C337 422 310 385 276 358C205 393 145 426 145 500C145 552 179 603 238 603ZM248 20C183 20 125 70 125 163C125 218 158 268 206 300C284 261 355 217 355 143C355 66 308 20 248 20Z"/></g><g transform="matrix(.013,0,0,-0.013,42.237,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g><g transform="matrix(.013,0,0,-0.013,48.477,0)"><path id="g113-47" d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"/></g><g transform="matrix(.013,0,0,-0.013,51.441,0)"><use xlink:href="#g113-49"/></g></svg></span> GeV, a non-QED direct contact term is introduced following the dimension 6 effective theory to explain the deviation. In the non-QED direct contact term, a threshold energy scale <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M6" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.05107 8.68572" width="9.05107pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-140" d="M671 0V28C608 36 593 47 562 130C495 309 432 486 368 665L336 656C270 481 202 311 134 139C98 47 84 39 20 28V0H241V28C167 38 156 47 178 115C217 236 276 383 327 530H329C375 398 437 223 484 92C499 47 489 37 424 28V0H671Z"/></g></svg> is included and explained to the finite interaction length in direct contact term and in consequence the size of the electron involved in the annihilation area. The experimental data of the total cross section is compared to the QED cross section by a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.6382pt" id="M7" height="15.2296pt" version="1.1" viewBox="-0.0498162 -11.5914 12.6404 15.2296" width="12.6404pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-244"/></g><g transform="matrix(.0091,0,0,-0.0091,7.568,-5.741)"><use xlink:href="#g50-51"/></g></svg> test, which gives a best fit of the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M8" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.05107 8.68572" width="9.05107pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-140"/></g></svg> to be <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="35.499pt" style="vertical-align:-0.3499298pt" id="M9" 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transform="matrix(.013,0,0,-0.013,50.885,0)"><use xlink:href="#g113-51"/></g></svg></span> GeV, corresponding to a finite interaction length of <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="20.517pt" style="vertical-align:-3.2911pt" id="M10" height="14.8825pt" version="1.1" viewBox="-0.0498162 -11.5914 20.517 14.8825"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g><g transform="matrix(.0091,0,0,-0.0091,4.901,3.132)"><path id="g50-102" d="M398 365C398 411 356 451 297 451C250 451 198 430 151 397C65 336 24 230 24 139C24 14 97 -12 148 -12C198 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style="vertical-align:-3.2911pt" height="14.8825pt" version="1.1" viewBox="103.34718380000001 -11.5914 27.902 14.8825"><g transform="matrix(.013,0,0,-0.013,103.397,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,109.637,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.0091,0,0,-0.0091,115.877,-5.741)"><use xlink:href="#g54-33"/></g><g transform="matrix(.0091,0,0,-0.0091,121.437,-5.741)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.0091,0,0,-0.0091,125.869,-5.741)"><path id="g50-56" d="M453 623H65C60 580 56 530 46 472H80C103 538 110 548 175 548H389C310 380 195 168 90 0L98 -12L175 -2C273 206 366 408 462 610L453 623Z"/></g></svg></span> (cm). In the direct contact term annihilation, this interaction length is a measure of the size of an electron <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29111pt" id="M11" height="9.25202pt" version="1.1" viewBox="-0.0498162 -5.96091 9.37172 9.25202" width="9.37172pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-115"/></g><g transform="matrix(.0091,0,0,-0.0091,4.901,3.132)"><use xlink:href="#g50-102"/></g></svg>.</span> By combining all the data results from the mentioned collaborations, we have at least 2 to 3 times more statistics than every single experiment at high <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.636801pt" id="M12" height="13.1484pt" version="1.1" viewBox="-0.0498162 -10.5116 14.9622 13.1484" width="14.9622pt"><g transform="matrix(.013,0,0,-0.013,0,-.271)"><use xlink:href="#g119-37"/></g><rect height="0.65243" width="4.89327" x="9.96925" y="-9.80939"/><g transform="matrix(.013,0,0,-0.013,9.969,0)"><use xlink:href="#g113-116"/></g></svg> region. This induces the best precision on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29111pt" id="M13" height="9.25202pt" version="1.1" viewBox="-0.0498162 -5.96091 9.37172 9.25202" width="9.37172pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-115"/></g><g transform="matrix(.0091,0,0,-0.0091,4.901,3.132)"><use xlink:href="#g50-102"/></g></svg> compared to the previous measurements.

Yutao Chen

Minghui Liu

Jürgen Ulbricht

Hint for a Minimal Interaction Length in <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.429501pt" height="14.2262pt" version="1.1" viewBox="-0.0498162 -10.7967 61.6558 14.2262" width="61.6558pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-102" d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" /></g><g transform="matrix(.0091,0,0,-0.0091,5.382,-5.741)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z" /></g><g transform="matrix(.013,0,0,-0.013,11.461,0)"><path id="g113-102" d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" /></g><g transform="matrix(.0091,0,0,-0.0091,16.844,-5.741)"><path id="g54-33" d="M556 236V289H56V236H556Z" /></g><g transform="matrix(.013,0,0,-0.013,26.554,0)"><path id="g117-149" d="M567 230V280H69V230H567Z" /></g><g transform="matrix(.013,0,0,-0.013,32.33,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z" /></g><g transform="matrix(.013,0,0,-0.013,48.442,0)"><path id="g113-225" d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" /></g><g transform="matrix(.013,0,0,-0.013,54.958,0)"><path id="g113-225" d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" /></g></svg> Annihilation in Total Cross Section of Center-of-Mass Energies 55-207 GeV

The analytical exact iteration method (AEIM) has been widely used to calculate <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M4" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g></svg>-</span>dimensional radial Schrodinger equation with medium-modified form of Cornell potential and is generalized to the finite value of magnetic field (eB) with quasiparticle approach in hot quantum chromodynamics (QCD) medium. In <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M5" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g></svg>-</span>dimensional space, the energy eigenvalues have been calculated for any states (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M6" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.6501 6.1673" width="6.6501pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg>,</span> <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M7" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 3.60972 9.49473" width="3.60972pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-109" d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z"/></g></svg>).</span> These results have been used to study the properties of quarkonium states (i.e, the binding energy and mass spectra, dissociation temperature, and thermodynamical properties in the <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M8" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g></svg>-</span>dimensional space). We have determined the binding energy of the ground states of quarkonium with magnetic field and dimensionality number. We have also determined the effects of magnetic field and dimensionality number on mass spectra for ground states of quarkonia. But the main result is quite noticeable for the values of dissociation temperature in terms of magnetic field and dimensionality number for ground states of quarkonia after using the criteria of dissociation energy. At last, we have also calculated the thermodynamical properties of QGP (i.e., pressure, energy density, and speed of sound) using the parameter eB with ideal equation of states (EoS). A preprint has previously been published (Solanki et al., 2023).

Siddhartha Solanki

Manohar Lal

Vineet Kumar Agotiya

Dissociation of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 19.9076 12.7178" width="19.9076pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-75" d="M429 650H175L169 622C251 614 259 612 244 532L174 152C163 91 152 37 135 -6C109 -72 67 -101 23 -113L33 -144C45 -143 73 -134 102 -119C186 -76 229 -1 254 133L327 532C341 609 347 613 423 622L429 650Z" /></g><g transform="matrix(.013,0,0,-0.013,5.879,0)"><path id="g113-48" d="M368 703H309L44 -163H104L368 703Z" /></g><g transform="matrix(.013,0,0,-0.013,11.24,0)"><path id="g113-245" d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z" /></g></svg> and <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.0498209pt" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 8.6074 8.68572" width="8.6074pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-90" d="M640 650H413L406 622L438 617C475 611 483 605 463 571C422 503 346 400 296 333C264 403 215 529 199 575C189 604 191 614 228 618L262 622L271 650H30L23 622C90 615 97 609 119 553L212 321C222 295 223 285 218 259L193 127C176 39 167 33 82 28L76 0H350L356 28C266 36 257 40 275 127L303 266C308 290 314 302 328 322C396 418 454 493 498 546C551 610 565 616 631 622L640 650Z" /></g></svg> Using Dissociation Energy Criteria in <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.2064009pt" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z" /></g></svg>-Dimensional Space

The paper is devoted to the study of cosmological models with time-varying cosmological term (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M2" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.05107 8.68572" width="9.05107pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-140"/></g></svg>)</span> in the presence of creation field in the framework of Bianchi type III space-time. To obtain deterministic model of the universe, we have assumed <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="23.294pt" style="vertical-align:-2.1507pt" id="M3" height="13.7421pt" version="1.1" viewBox="-0.0498162 -11.5914 23.294 13.7421"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-140"/></g><g transform="matrix(.013,0,0,-0.013,12.543,0)"><path id="g113-29" d="M774 30C773 39 771 49 769 58C758 52 747 48 732 48C616 48 566 133 514 239C556 327 604 374 702 374C716 374 729 369 737 365C743 361 747 361 752 365C761 373 770 389 770 406C770 426 754 448 726 448C689 448 629 431 577 391C534 358 509 327 489 288C438 384 373 448 262 448C175 448 53 373 53 208C53 98 131 -12 262 -12C377 -12 426 52 472 147C516 51 579 -12 687 -12C713 -12 748 -2 774 30ZM451 198C415 110 363 21 272 21C195 21 145 114 145 240C145 353 194 415 250 415C357 415 409 304 451 198Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.596pt" style="vertical-align:-2.1507pt" height="13.7421pt" version="1.1" viewBox="26.8761838 -11.5914 11.596 13.7421"><g transform="matrix(.013,0,0,-0.013,26.926,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.013,0,0,-0.013,33.166,0)"><path id="g113-48" d="M368 703H309L44 -163H104L368 703Z"/></g></svg><span class="isep"/><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.347pt" style="vertical-align:-2.1507pt" height="13.7421pt" version="1.1" viewBox="38.4771838 -11.5914 13.347 13.7421"><g transform="matrix(.013,0,0,-0.013,38.527,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.0091,0,0,-0.0091,46.678,-5.741)"><path id="g50-52" d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"/></g></svg>,</span></span> where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M4" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 8.28119 8.8423" width="8.28119pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-83"/></g></svg> is the scale factor, for steady state cosmology and creation field, and shear (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063899pt" id="M5" height="6.34998pt" version="1.1" viewBox="-0.0498162 -6.14359 7.47218 6.34998" width="7.47218pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-240" d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"/></g></svg>)</span> is proportion to expansion (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M6" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.59789 9.49473" width="6.59789pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g></svg>)</span> which leads to <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="19.076pt" style="vertical-align:-0.2064095pt" id="M7" height="10.1628pt" version="1.1" viewBox="-0.0498162 -9.95639 19.076 10.1628"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.013,0,0,-0.013,11.445,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="14.049pt" style="vertical-align:-0.2064095pt" height="10.1628pt" version="1.1" viewBox="22.6581838 -9.95639 14.049 10.1628"><g transform="matrix(.013,0,0,-0.013,22.708,0)"><path id="g113-68" d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z"/></g><g transform="matrix(.0091,0,0,-0.0091,31.392,-5.741)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g></svg>,</span></span> where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M8" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 7.94191 8.68572" width="7.94191pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-67"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M9" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 8.8162 8.8423" width="8.8162pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-68"/></g></svg> are the metric potentials to explain small anisotropic behaviour of the universe and its isotropy. To obtain the results in terms of cosmic time <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M10" height="8.02022pt" version="1.1" viewBox="-0.0498162 -7.81382 4.54925 8.02022" width="4.54925pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-117" d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"/></g></svg>,</span> we have assumed <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="17.789pt" style="vertical-align:-0.2063904pt" id="M11" height="8.55521pt" version="1.1" viewBox="-0.0498162 -8.34882 17.789 8.55521"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.013,0,0,-0.013,10.158,0)"><use xlink:href="#g117-34"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="6.416pt" style="vertical-align:-0.2063904pt" height="8.55521pt" version="1.1" viewBox="21.3711838 -8.34882 6.416 8.55521"><g transform="matrix(.013,0,0,-0.013,21.421,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g></svg>.</span></span> The model satisfies conservation equations, and creation field increases with time. The present model is free from singularity, has particle horizon, and provides a natural explanation for inflationary scenario and isotropization. Creation field and Einstein field equations are derived using principle of least action and Lagrangian formulation of variable cosmological term. For illustrative purposes, evolutionary behaviour of some cosmological parameters are shown graphically. The other physical aspects like accelerating behaviour of the model are also discussed. Thus, the model represents not only expanding universe but also accelerating which matches with the results of present-day observations.

Raj Bali

Seema Saraf

Creation Field Cosmological Model with Variable Cosmological Term (<svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.0498209pt" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.05107 8.68572" width="9.05107pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-140" d="M671 0V28C608 36 593 47 562 130C495 309 432 486 368 665L336 656C270 481 202 311 134 139C98 47 84 39 20 28V0H241V28C167 38 156 47 178 115C217 236 276 383 327 530H329C375 398 437 223 484 92C499 47 489 37 424 28V0H671Z" /></g></svg>) in Bianchi Type III Space-Time

In this study, we solved the Schrödinger wave equation by using effective potential in an artificial neural network (ANN) for the mass spectrum of different charmonium states, including <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M1" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 10.4717 9.39034" width="10.4717pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-229" d="M471 401C471 442 455 448 440 448C421 448 389 439 366 424C314 390 233 331 165 242H163L187 345C193 372 197 393 197 409C197 435 189 448 174 448C146 448 80 408 23 349L40 327C64 351 97 373 107 373C115 373 118 366 111 337L29 -4L36 -12C56 -4 83 3 110 9C123 69 136 126 150 172C227 283 336 381 377 381C392 381 396 370 379 298L326 74C290 -77 273 -172 273 -197C273 -219 316 -261 346 -261L353 -246C348 -227 352 -174 374 -66L457 310C467 352 471 381 471 401Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.097,3.132)"><path id="g50-100" d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z"/></g></svg>,</span> <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M2" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 12.9928 10.2124" width="12.9928pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-245" d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.917,3.132)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg>,</span> <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.63821pt" id="M3" height="9.59912pt" version="1.1" viewBox="-0.0498162 -5.96091 12.0532 9.59912" width="12.0532pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-244" d="M565 434L556 448L482 423L322 189L287 373C280 410 261 448 236 448C207 448 161 415 110 338L131 317C160 355 181 374 193 374C204 374 208 366 217 330C230 279 247 185 259 116L50 -175C18 -223 29 -260 41 -278H51C97 -198 207 -33 270 57L313 -165C328 -235 349 -261 376 -261S445 -244 510 -150L487 -127C444 -178 423 -186 411 -186C402 -186 398 -177 387 -130C371 -62 349 39 332 135C390 217 524 392 565 434Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.981,3.132)"><use xlink:href="#g50-51"/></g></svg>,</span> and <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.63821pt" id="M4" height="9.59912pt" version="1.1" viewBox="-0.0498162 -5.96091 14.2285 9.59912" width="14.2285pt"><g transform="matrix(.013,0,0,-0.013,2.175,0)"><use xlink:href="#g113-244"/></g><g transform="matrix(.0091,0,0,-0.0091,9.156,3.132)"><path id="g50-53" d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z"/></g></svg>.</span> The ANN approach provides an efficient, more general, and continuous solution-approximating strategy, thus eliminating the possibility of skipping any region of interest in mass spectroscopy. The close consistency of ANN results with the already-reported results from numerical and theoretical approaches and experimental ones shows the reliability and accuracy of the ANN approach.

Tariq Mahmood

Jumanah Ahmed Darwish

Talab Hussain

Maqsood Ahmed

Rehan Ahmad Khan Sherwani

Solving Schrödinger Wave Equation for the Charmonium Spectrum Using Artificial Neural Networks

We study the transverse momentum (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M1" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 13.9009 10.2124" width="13.9009pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><path id="g50-85" d="M655 676H628C608 656 600 650 571 650H156C118 650 114 651 102 676H77C63 615 45 548 24 485L60 486C81 531 97 562 114 580C134 603 148 611 252 611H308L217 127C201 41 196 38 105 32L98 0H384L393 32C300 38 288 41 304 127L393 611H466C553 611 571 604 582 580C592 559 597 530 595 485L630 489C636 548 645 624 655 676Z"/></g></svg>)</span> spectra of neutral pions and identified charged hadrons produced in proton–proton (pp), deuteron–gold (d–Au), and gold–gold (Au–Au) collisions at the center of mass energy <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="40.131pt" style="vertical-align:-4.16998pt" id="M2" height="13.1484pt" version="1.1" viewBox="-0.0498162 -8.97842 40.131 13.1484"><g transform="matrix(.013,0,0,-0.013,0,1.262)"><path id="g119-37" d="M782 731H740L476 -79H474L280 352L133 274L151 239L242 286L473 -219L782 731Z"/></g><rect height="0.65243" width="18.8994" x="9.96925" y="-8.27617"/><g transform="matrix(.013,0,0,-0.013,9.969,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"/></g><g transform="matrix(.0091,0,0,-0.0091,14.779,3.132)"><path id="g190-79" d="M719 650H483V622C553 619 576 606 581 560C585 532 588 487 588 394V148H585L167 650H20V622C67 618 88 610 109 585C127 562 129 557 129 484V264C129 173 126 124 123 93C118 44 92 31 35 28V0H272V28C204 32 181 44 176 95C173 124 170 172 170 264V515H172L598 -9H629V394C629 487 631 532 635 563C639 606 663 619 719 622V650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,21.54,3.132)"><use xlink:href="#g190-79"/></g><g transform="matrix(.013,0,0,-0.013,32.5,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.919pt" style="vertical-align:-4.16998pt" height="13.1484pt" version="1.1" viewBox="43.713183799999996 -8.97842 18.919 13.1484"><g transform="matrix(.013,0,0,-0.013,43.763,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.013,0,0,-0.013,50.003,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g><g transform="matrix(.013,0,0,-0.013,56.243,0)"><use xlink:href="#g113-49"/></g></svg></span> GeV. The study is made in the framework of a multisource thermal model used in the partonic level. It is assumed that the contribution to the <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M3" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 13.9009 10.2124" width="13.9009pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-113"/></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><use xlink:href="#g50-85"/></g></svg>-</span>value of any hadron comes from two or three partons with an isotropic distribution of the azimuthal angle. The contribution of each parton to the <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M4" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 13.9009 10.2124" width="13.9009pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-113"/></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><use xlink:href="#g50-85"/></g></svg>-</span>value of a given hadron is assumed to obey any one of the standard (Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein) distributions with the kinetic freeze-out temperature and average transverse flow velocity. The <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M5" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 13.9009 10.2124" width="13.9009pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-113"/></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><use xlink:href="#g50-85"/></g></svg> spectra of the final-state hadrons can be fitted by the superposition of two or three components. The results obtained from our Monte Carlo method are used to fit the experimental results of the PHENIX and STAR Collaborations. The results of the present work serve as a suitable reference baseline for other experiments and simulation studies.

Jia-Yu Chen

Mai-Ying Duan

Fu-Hu Liu

Khusniddin K. Olimov

Extracting Kinetic Freeze-Out Properties in High-Energy Collisions Using a Multisource Thermal Model

We investigate wormhole solutions using the modified gravity model <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="24.939pt" style="vertical-align:-3.42947pt" id="M2" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 24.939 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.013,0,0,-0.013,8.352,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.85,0)"><use xlink:href="#g113-82"/></g><g transform="matrix(.013,0,0,-0.013,21.975,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.015pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="27.0681838 -9.28833 13.015 12.7178"><g transform="matrix(.013,0,0,-0.013,27.118,0)"><use xlink:href="#g113-85"/></g><g transform="matrix(.013,0,0,-0.013,35.404,0)"><use xlink:href="#g113-42"/></g></svg></span> with viscosity and aim to find a solution for the existence of wormholes mathematically without violating the energy conditions. We show that there is no need to define a wormhole from exotic matter and analyze the equations with numerical analysis to establish weak energy conditions. In the numerical analysis, we found that the appropriate values of the parameters can maintain the weak energy conditions without the need for exotic matter. Adjusting the parameters of the model can increase or decrease the rate of positive energy density or radial and tangential pressures. According to the numerical analysis conducted in this paper, the weak energy conditions are established in the whole space if <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.648pt" style="vertical-align:-0.6370001pt" id="M3" height="8.98582pt" version="1.1" viewBox="-0.0498162 -8.34882 18.648 8.98582"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g><g transform="matrix(.013,0,0,-0.013,11.017,0)"><path id="g117-91" d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"/></g></svg><span class="irelop"></span><svg 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id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.013,0,0,-0.013,12.48,0)"><path id="g113-47" d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"/></g><g transform="matrix(.013,0,0,-0.013,15.444,0)"><path id="g113-54" d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"/></g><g transform="matrix(.013,0,0,-0.013,21.684,0)"><path id="g113-55" d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"/></g><g transform="matrix(.013,0,0,-0.013,31.556,0)"><use xlink:href="#g117-91"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.817pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="42.7691838 -9.28833 18.817 12.7178"><g transform="matrix(.013,0,0,-0.013,42.819,0)"><path id="g113-224" d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"/></g><g transform="matrix(.013,0,0,-0.013,54.005,0)"><use xlink:href="#g117-91"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="28.317pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="65.2181838 -9.28833 28.317 12.7178"><g transform="matrix(.013,0,0,-0.013,65.268,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,71.508,0)"><use xlink:href="#g113-54"/></g><g transform="matrix(.013,0,0,-0.013,77.748,0)"><use xlink:href="#g113-47"/></g><g transform="matrix(.013,0,0,-0.013,80.712,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,86.952,0)"><use xlink:href="#g113-51"/></g></svg></span> or <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.648pt" style="vertical-align:-0.6370001pt" id="M5" height="8.98582pt" version="1.1" viewBox="-0.0498162 -8.34882 18.648 8.98582"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-223"/></g><g transform="matrix(.013,0,0,-0.013,11.017,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="6.418pt" style="vertical-align:-0.6370001pt" height="8.98582pt" version="1.1" viewBox="22.230183800000002 -8.34882 6.418 8.98582"><g transform="matrix(.013,0,0,-0.013,22.28,0)"><use xlink:href="#g113-49"/></g></svg></span> and <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.817pt" style="vertical-align:-3.42947pt" id="M6" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 18.817 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-224"/></g><g transform="matrix(.013,0,0,-0.013,11.186,0)"><use xlink:href="#g117-92"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="28.184pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="22.399183800000003 -9.28833 28.184 12.7178"><g transform="matrix(.013,0,0,-0.013,22.449,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,28.689,0)"><use xlink:href="#g113-54"/></g><g transform="matrix(.013,0,0,-0.013,34.929,0)"><use xlink:href="#g113-47"/></g><g transform="matrix(.013,0,0,-0.013,37.893,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,44.133,0)"><use xlink:href="#g113-51"/></g></svg>.</span></span> The analysis also showed that the supporting matter of the wormhole is near normal matter, indicating that the generalized <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="24.939pt" style="vertical-align:-3.42947pt" id="M7" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 24.939 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.013,0,0,-0.013,8.352,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.85,0)"><use xlink:href="#g113-82"/></g><g transform="matrix(.013,0,0,-0.013,21.975,0)"><use xlink:href="#g113-45"/></g></svg><span class="ipunc"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.015pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="27.0681838 -9.28833 13.015 12.7178"><g transform="matrix(.013,0,0,-0.013,27.118,0)"><use xlink:href="#g113-85"/></g><g transform="matrix(.013,0,0,-0.013,35.404,0)"><use xlink:href="#g113-42"/></g></svg></span> model with viscosity has an acceptable parameter space to describe a wormhole without the need for exotic matter.

S. Davood Sadatian

S. Mohamad Reza Hosseini

Modified Gravity Model <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 40.1331 12.7178" width="40.1331pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z" /></g><g transform="matrix(.013,0,0,-0.013,8.352,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" /></g><g transform="matrix(.013,0,0,-0.013,12.85,0)"><path id="g113-82" d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z" /></g><g transform="matrix(.013,0,0,-0.013,21.975,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" /></g><g transform="matrix(.013,0,0,-0.013,27.118,0)"><path id="g113-85" d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z" /></g><g transform="matrix(.013,0,0,-0.013,35.404,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" /></g></svg> and Wormhole Solution

In this study, we investigated the impact of a topological defect (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M1" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g></svg>)</span> on the properties of heavy quarkonia using the extended Cornell potential. We solved the fractional radial Schrödinger equation (SE) using the extended Nikiforov-Uvarov (ENU) method to obtain the eigenvalues of energy, which allowed us to calculate the masses of charmonium and bottomonium. One significant observation was the splitting between nP and nD states, which attributed to the presence of the topological defect. We discovered that the excited states were divided into components corresponding to <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="20.275pt" style="vertical-align:-0.3499298pt" id="M2" height="9.63826pt" version="1.1" viewBox="-0.0498162 -9.28833 20.275 9.63826"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path id="g113-109" d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z"/></g><g transform="matrix(.013,0,0,-0.013,12.644,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g></svg><span class="ibiop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="6.427pt" style="vertical-align:-0.3499298pt" height="9.63826pt" version="1.1" viewBox="23.1301838 -9.28833 6.427 9.63826"><g transform="matrix(.013,0,0,-0.013,23.18,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g></svg>,</span></span> indicating that the gravity field induced by the topological defect interacts with energy levels like the Zeeman effect caused by a magnetic field. Additionally, we derived the wave function and calculated the root-mean radii for charmonium and bottomonium. A comparison with the classical models was performed, resulting in better results being obtained. Furthermore, we investigated the thermodynamic properties of charmonium and bottomonium, determining quantities such as energy, partition function, free energy, mean energy, specific heat, and entropy for P-states. The obtained results were found to be consistent with experimental data and previous works. In conclusion, the fractional model used in this work proved an essential role in understanding the various properties and behaviors of heavy quarkonia in the presence of topological defects.

M. Abu-shady

H. M. Fath-Allah

Properties and Behaviors of Heavy Quarkonia: Insights through Fractional Model and Topological Defects

The phase diagram of quantum chromodynamics (QCD) and its associated thermodynamic properties of quark-gluon plasma (QGP) are studied in the presence of time-dependent magnetic field. The study plays a pivotal role in the field of cosmology, astrophysics, and heavy-ion collisions. In order to explore the structure of quark-gluon plasma to deal with the dynamics of quarks and gluons, we investigate the equation of state (EoS) not only in the environment of static magnetic field but also in the presence of time-varying magnetic fields. So, for determining the equation of state of QGP at nonzero magnetic fields, we revisited our earlier model where the effect of time-varying magnetic field was not taken into consideration. Using the phenomenological model, some appealing features are noticed depending upon the three different scales: effective mass of quark, temperature, and time-independent and time-dependent magnetic fields. Earlier the effective mass of quark was incorporated in our calculations, and in the current work, it is modified for static and time-varying magnetic fields. Thermodynamic observables including pressure, energy density, and entropy are calculated for a wide range of temperature- and time-dependent as well as time-independent magnetic fields. Finally, we claim that the EoS are highly affected in the presence of a magnetic field. Our results are notable compared to other approaches and found to be advantageous for the measurement of QGP equation of state. These crucial findings with and without time-varying magnetic field could have phenomenological implications in various sectors of high-energy physics.

Yogesh Kumar

Poonam Jain

Pargin Bangotra

Vinod Kumar

D. V. Singh

S. K. Rajouria

Role of Time-Varying Magnetic Field on QGP Equation of State

We investigate thick brane solutions in the Horndeski gravity. In this setup, we found analytical solutions, applying the first-order formalism to two scalar fields where the first field comes from the nonminimal scalar-tensor coupling and the second is due to the matter contribution sector. With these analytical solutions, we evaluate the symmetric thick brane solutions in Horndeski gravity with four-dimensional geometry. In such a setup, we evaluate the gravity fluctuations to find “almost massless modes,” for any values of the Horndeski parameters. These modes were used to compute the corrections to the Newtonian potential and evaluate the limit four-dimensional gravity.

Fabiano F. Santos

F. A. Brito

Thick Branes in Horndeski Gravity

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<p><em>Advances in High Energy Physics</em> publishes the results of theoretical and experimental research on the nature of, and interaction between, energy and matter.</p>


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<p>Chief Editor, Professor Seidel, is a professor in the Department of Physics and Astronomy at the University of New Mexico. She is a collaborator on the ATLAS experiment at the Large Hadron Collider, researching high-energy collider physics.</p>


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Advances in High Energy Physics publishes the results of theoretical and experimental research on the nature of, and interaction between, energy and matter.

Advances in High Energy Physics