Pion phenomenology from the thermal soft-wall model of holographic QCD

Within the framework of the thermal soft-wall model of AdS/QCD, we investigate phenomenological properties of pions at finite temperature. This includes the electromagnetic form factor <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>F</a:mi> <a:mi>π</a:mi> </a:msub> <a:mo stretchy="false">(</a:mo> <a:msup> <a:mi>Q</a:mi> <a:mn>2</a:mn> </a:msup> <a:mo>,</a:mo> <a:mi>T</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> , the thermal mass <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>M</e:mi> <e:mi>π</e:mi> </e:msub> <e:mo stretchy="false">(</e:mo> <e:mi>T</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> , charge radius <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>r</i:mi> <i:mi>π</i:mi> </i:msub> <i:mo stretchy="false">(</i:mo> <i:mi>T</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> , the generalized parton distribution (GPD) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>H</m:mi> <m:mi>π</m:mi> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>Q</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:math> , the charge density <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>ρ</q:mi> <q:mi>π</q:mi> </q:msub> <q:mo stretchy="false">(</q:mo> <q:mi>b</q:mi> <q:mo>,</q:mo> <q:mi>T</q:mi> <q:mo stretchy="false">)</q:mo> </q:math> of the pion, the pion-nucleon coupling constant <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msub> <u:mi>g</u:mi> <u:mrow> <u:mi>π</u:mi> <u:mi>N</u:mi> <u:mi>N</u:mi> </u:mrow> </u:msub> <u:mo stretchy="false">(</u:mo> <u:mi>T</u:mi> <u:mo stretchy="false">)</u:mo> </u:math> , and pion- <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi mathvariant="normal">Δ</y:mi> </y:math> baryon coupling constant <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:msub> <bb:mi>g</bb:mi> <bb:mrow> <bb:mi>π</bb:mi> <bb:mi mathvariant="normal">Δ</bb:mi> <bb:mi mathvariant="normal">Δ</bb:mi> </bb:mrow> </bb:msub> <bb:mo stretchy="false">(</bb:mo> <bb:mi>T</bb:mi> <bb:mo stretchy="false">)</bb:mo> </bb:math> coupling constant at finite temperature. The thermal pion form factor is extrapolated from the zero-temperature case. Subsequently, the GPD is obtained at finite temperature from this form factor. The above-mentioned quantities were analyzed using a thermal dilaton field in the five-dimensional anti–de Sitter action. Moreover, we determine the theoretical expression for the temperature-dependent pion-nucleon coupling constant, and the pion- <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi mathvariant="normal">Δ</hb:mi> </hb:math> baryon coupling constant using thermal profile functions of the nucleon, <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi mathvariant="normal">Δ</kb:mi> </kb:math> baryon and the pion. Our results show that the values of these quantities decrease with increasing temperature and vanish near the critical temperature <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:msub> <nb:mi>T</nb:mi> <nb:mi>c</nb:mi> </nb:msub> </nb:math> , except for the pion radius, which diverges at <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:msub> <pb:mi>T</pb:mi> <pb:mi>c</pb:mi> </pb:msub> </pb:math> . Our results reproduce expected features of low-energy hadron dynamics, thus validating the phenomenological utility of the thermal soft-wall model.