Holographic drag force with translational symmetry breaking

Abstract In order to investigate how the drag force is affected by translational symmetry breaking (TSB), we utilize a holographic model in which the background metric remains translational symmetric while a graviton mass or other fields in the theory break this symmetry. We calculate analytically the drag force, considering an asymptotic $$\hbox {AdS}_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>AdS</mml:mtext> <mml:mn>5</mml:mn> </mml:msub> </mml:math> in which parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> arises from TSB. This parameter can be intuitively understood as a measure of TSB strength and we anticipate that non-zero values of it will affect the drag force. In this asymptotic $$\hbox {AdS}_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>AdS</mml:mtext> <mml:mn>5</mml:mn> </mml:msub> </mml:math> background, we will demonstrate that a decrease in $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> results in a reduction of the drag force. Moreover, we study the diffusion constant, which falls with increasing $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . It will eventually be shown that at lower values of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> or $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> (chemical potential), the transverse diffusion coefficient is larger than the longitudinal one, and the speed of the heavy quark has minimal impact on the ratio.