Holographic Schwinger effect in spinning black hole backgrounds

Yi-ze Cai School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China Zi-qiang Zhang [email protected] School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

Abstract

We perform the potential analysis for the holographic Schwinger effect in spinning Myers-Perry black holes. We compute the potential between the produced pair by evaluating the classical action of a string attaching on a probe D3-brane sitting at an intermediate position in the AdS bulk. It turns out that increasing the angular momentum reduces the potential barrier thus enhancing the Schwinger effect, consistent with previous findings obtained from the local Lorentz transformation. In particular, these effects are more visible for the particle pair lying in the transversal plane compared with that along the longitudinal orientation. In addition, we discuss how the Schwinger effect changes with the shear viscosity to entropy density ratio at strong coupling under the influence of angular momentum.

pacs:

11.25.Tq, 11.15.Tk, 11.25-w

I Introduction

In the vacuum of quantum electrodynamics (QED), virtual electron-position pairs could be materialized and become real particles in the presence of a strong electric-field. This phenomenon has been termed Schwinger effect. The production rate $\Gamma$ of electron-positron pairs for the case of weak-coupling and weak-field was first studied by Schwinger in 1951 JS

\Gamma\sim exp\Big{(}{\frac{-\pi m^{2}}{eE}}\Big{)},

(1)

where $m$ , $e$ , $E$ are an electron mass, an elementary electric charge and an external electric field, respectively. One can see that there is no critical field in (1). Later, the calculation of $\Gamma$ for the case of arbitrary-coupling and weak-field was considered in IK

\Gamma\sim exp\Big{(}{\frac{-\pi m^{2}}{eE}+\frac{e^{2}}{4}}\Big{)},

(2)

one can find that there is a critical electric field at $eE_{c}=(4\pi/e^{2})m^{2}\simeq 137m^{2}$ in (2), but this value doesn’t meet the weak-field condition $eE\ll m^{2}$ . Therefore, it seems that one could hardly get the critical field under weak-field condition.

In fact, Schwinger effect is not limited to QED but ubiquitous for quantum field theory (QFT) coupled to an U(1) gauge field. However, it is difficult to deal with this issue with the standard method in QFT. A possible way is to employ the AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ; MadalcenaReview . Using AdS/CFT, Semenoff and Zarembo pioneered the holographic Schwinger effect in 2011 GW . They pointed out that a supersymmetric Yang-Mills (SYM) coupled with an U(1) gauge field can be realized by breaking the gauge group from $SU(N+1)$ to $SU(N)\times U(1)$ via the Higgs mechanism. In doing so, Schwinger effect could be modelled in the higgsed $\mathcal{N}=4$ SYM. The production rate of the fundamental particles (W-boson supermultiplet or quarks) at large $N_{c}$ (color number) and large $\lambda$ (’t Hooft coupling) are evaluated as GW

\Gamma\sim exp\Big{[}-\frac{\sqrt{\lambda}}{2}\Big{(}\sqrt{\frac{E_{c}}{E}}-\sqrt{\frac{E}{E_{c}}}\Big{)}^{2}\Big{]},\qquad E_{c}=\frac{2\pi m^{2}}{\sqrt{\lambda}},

(3)

interestingly, the value of $E_{c}$ coincides with the DBI result GW . Subsequently, there are many works to develop and extend this idea. For instance, the universal aspects of holographic Schwinger effect for general backgrounds was discussed in YS2 . The holographic Schwinger effect with constant electric and magnetic fields was studied in SB ; YS3 . Furthermore, the potential barrier for holographic Schwinger effect was explored in various backgrounds YS ; YS1 ; KB ; MG ; ZQ ; ZQ1 ; LS ; ZR ; swe . Other related results can be found in DK ; SCH ; KHA ; KHA1 ; KG ; LS1 ; yz ; yd ; swe1 ; WF .

In this work, we shall explore the holographic Schwinger effect in spinning black hole backgrounds by means of the AdS/CFT correspondence. The motivations are as follows: First, Schwinger effect may have a connection with the heavy-ion collisions experiments, where strong electro-magnetic fields and color fields could be induced due to the collision of heavy ions. Second, it has been reported that the noncentral collisions tends to deposit high angular momentum in the quark gluon plasma (QGP) generated in heavy ion collisions and such angular momentum may give rise to significant observable effects in QGP nat ; zt ; fb ; xg ; lg . Already, various observables with respect to QGP have been studied under the influence of angular momentum from holography. Such as jet quenching parameter js ; bmc , drag force iy ; iy1 ; ana , energy loss kb ; mat ; df , confinement/deconfinement nr ; xc and running coupling constant xc1 . Other related results can be found in sb ; bm ; bm1 ; hb ; an ; sy ; aa . In previous literature, we have investigated the holographic Schwinger effect in a soft wall model yz by taking a local Lorentz transformation mb ; ce ; am to the static frame of a small segment of the rotating medium. But this approach has its limitations: its metric can only describe a small neighbourhood around $l=l_{0}$ and a domain less than $2\pi$ wide of the rotating medium jx , where $l_{0}$ represents the radius to the rotating axis. Are there any other ways besides the local Lorentz transformation to mimic the rotating QGP? One possible approach is to utilize the rotating black holes, e.g., Kerr-AdS₅ metric sw . Recently, the shear viscosity to entropy density ratio $\eta/s$ has been calculated in five-dimensional Myers-Perry black holes. These black holes are a form of spinning five-dimensional AdS black holes and have been found as vacuum solutions within Einstein gravity mg . In particular, for these solutions, the boundary is compact and the dual SYM lives on $S^{3}\times\mathbb{R}$ . So if one is interested in a dual to a spinning fluid, e.g., QGP, in flat space $\mathbb{R}^{3,1}$ , one could consider the large black holes. Furthermore, if one tends to think of a regime of large temperature in order to have a dual field theory on a non-compact spacetime (this case maybe more relevant for applications to heavy ion collisions), one could consider the planar limit black brane as a limit of the large black holes. For these reasons, we would like to reexamine the holographic Schwinger effect in five-dimensional Myers-Perry black holes. We want to see whether the results obtained from these spinning black hole backgrounds are in line with those from the local Lorentz transformation. Also, by comparing with the results of mg , we want to see how the Schwinger effect changes with $\eta/s$ at strong coupling under the influence of angular momentum.

The organization of the paper is as follows. In the next section, we briefly recall the spinning Myers-Perry black holes given in sw ; gw ; gw1 . In section 3, we perform the potential analysis for the holographic Schwinger effect in these backgrounds and analyze how angular momentum modifies the production rate. Finally, we give our conclusions and discussions in section 4.

II Setup

The more familiar metric of the five-dimensional spinning black holes is written by Hawking et al. sw

$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle-\frac{\Delta}{\rho^{2}}(dt_{H}-\frac{a\sin^{2}\theta_{H}}{\Xi_{a}}d\phi_{H}-\frac{b\cos^{2}\theta_{H}}{\Xi_{b}}d\psi_{H})^{2}$	(4)
	$\displaystyle+$	$\displaystyle\frac{\Delta_{\theta_{H}}\sin^{2}\theta_{H}}{\rho^{2}}(adt_{H}-\frac{r_{H}^{2}+a^{2}}{\Xi_{a}}d\phi_{H})^{2}+\frac{\Delta_{\theta_{H}}\cos^{2}\theta_{H}}{\rho^{2}}(bdt_{H}-\frac{r_{H}^{2}+b^{2}}{\Xi_{b}}d\psi_{H})^{2}+\frac{\rho^{2}}{\Delta}dr_{H}^{2}$
	$\displaystyle-$	$\displaystyle\frac{\rho^{2}}{\Delta_{\theta_{H}}}d\theta_{H}^{2}+\frac{1+\frac{r_{H}^{2}}{L^{2}}}{r_{H}^{2}\rho^{2}}(abdt_{H}-\frac{b(r^{2}+a^{2})\sin^{2}\theta_{H}}{\Xi_{a}}d\phi_{H}-\frac{a(r^{2}+b^{2})\cos^{2}\theta_{H}}{\Xi_{b}}d\psi_{H})^{2},$

with

$\displaystyle\Delta$	$\displaystyle=$	$\displaystyle\frac{1}{r_{H}^{2}}(r_{H}^{2}+a^{2})(r_{H}^{2}+b^{2})(1+\frac{r_{H}^{2}}{L^{2}})-2M,$
$\displaystyle\Delta_{\theta_{H}}$	$\displaystyle=$	$\displaystyle 1-\frac{a^{2}}{L^{2}}\cos^{2}\theta_{H}-\frac{b^{2}}{L^{2}}\sin^{2}\theta_{H},$
$\displaystyle\rho$	$\displaystyle=$	$\displaystyle r_{H}^{2}+a^{2}\cos^{2}\theta_{H}+b^{2}\sin^{2}\theta_{H},$
$\displaystyle\Xi_{a}$	$\displaystyle=$	$\displaystyle 1-\frac{a^{2}}{L^{2}},$
$\displaystyle\Xi_{b}$	$\displaystyle=$	$\displaystyle 1-\frac{b^{2}}{L^{2}},$	(5)

where $t_{H}$ is the time, $L$ is the AdS radius, $r_{H}$ represents the AdS radial coordinate, $(\phi_{H},\psi_{H},\theta_{H})$ are the angular Hopf coordinates. ${a,b}$ denote two independent angular momentum parameters which could generate all possible rotations. Here we will focus on the case of $a=b$ , which is referred as the simply spinning Myers-Perry black holes gw ; gw1 .

In order to analyze conveniently, one can employ more convenient coordinates and reparameterize the mass following km

$\displaystyle t$	$\displaystyle=$	$\displaystyle t_{H},$
$\displaystyle r^{2}$	$\displaystyle=$	$\displaystyle\frac{a^{2}+r_{H}^{2}}{1-\frac{a^{2}}{L^{2}}},$
$\displaystyle\theta$	$\displaystyle=$	$\displaystyle 2\theta_{H},$
$\displaystyle\phi$	$\displaystyle=$	$\displaystyle\phi_{H}-\psi_{H},$
$\displaystyle\psi$	$\displaystyle=$	$\displaystyle-\frac{2at_{H}}{L^{2}}+\phi_{H}+\psi_{H},$
$\displaystyle b$	$\displaystyle=$	$\displaystyle a,$
$\displaystyle\mu$	$\displaystyle=$	$\displaystyle\frac{M}{(L^{2}-a^{2})^{3}}.$

Then the metric (4) can be simplified to

ds^{2}=-(1+\frac{r^{2}}{L^{2}})dt^{2}+\frac{dr^{2}}{G(r)}+\frac{r^{2}}{4}((\sigma^{1})^{2}+(\sigma^{2})^{2}+(\sigma^{3})^{2})+\frac{2\mu}{r^{2}}(dt+\frac{a}{2}\sigma^{3})^{2},

(7)

with

$\displaystyle G(r)$	$\displaystyle=$	$\displaystyle 1+\frac{r^{2}}{L^{2}}-\frac{2\mu(1-\frac{a^{2}}{L^{2}})}{r^{2}}+\frac{2\mu a^{2}}{r^{4}},$
$\displaystyle\mu$	$\displaystyle=$	$\displaystyle\frac{r_{h}^{4}(L^{2}+r_{h}^{2})}{2L^{2}r_{h}^{2}-2a^{2}(L^{2}+r_{h}^{2})},$
$\displaystyle\sigma^{1}$	$\displaystyle=$	$\displaystyle-\sin\psi d\theta+\cos\psi\sin\theta d\phi,$
$\displaystyle\sigma^{2}$	$\displaystyle=$	$\displaystyle\cos\psi d\theta+\sin\psi\sin\theta d\phi,$
$\displaystyle\sigma^{3}$	$\displaystyle=$	$\displaystyle d\psi+\cos\theta d\phi.$	(8)

where the range of the coordinates is

-\infty<t<\infty,\ r_{h}<r<\infty,\ 0\leq\theta\leq\pi,\ 0\leq\phi\leq 2\pi,\ 0\leq\psi<4\pi.

(9)

here $r_{h}$ is the outer horizon, defined by $G(r_{h})=0$ . It should be noted that the Myers-Perry black holes defined by (7) have two instabilities km . First, a superradient instability has been found, which occurs at large angular velocities $|\Omega L|>1$ . In order to avoid this, we consider $|\Omega L|<1$ here. The second instability (Gregory-Laflamme instability) was found at small horizon radius $r_{h}\sim L$ . This instability is not within the range of parameters which we consider $r_{h}\gg L$ .

As prophesied above, in this work we are most interested in rotating QGP, so we tend to consider the large black hole limit, since this limit would be more relevant for applications to heavy ion collisions mg . For this purpose, one adopts the following coordinate transformation

$\displaystyle t$	$\displaystyle=$	$\displaystyle\tau,$
$\displaystyle\frac{L}{2}(\phi-\pi)$	$\displaystyle=$	$\displaystyle x,$
$\displaystyle\frac{L}{2}\tan(\theta-\frac{\pi}{2})$	$\displaystyle=$	$\displaystyle y,$
$\displaystyle\frac{L}{2}(\psi-2\pi)$	$\displaystyle=$	$\displaystyle z,$
$\displaystyle r$	$\displaystyle=$	$\displaystyle\widetilde{r},$	(10)

then the coordinates in the new $(\tau,\widetilde{r},x,y,z)$ coordinates become

$\displaystyle\tau$	$\displaystyle\rightarrow$	$\displaystyle\beta^{-1}\tau,$
$\displaystyle x$	$\displaystyle\rightarrow$	$\displaystyle\beta^{-1}x,$
$\displaystyle y$	$\displaystyle\rightarrow$	$\displaystyle\beta^{-1}y,$
$\displaystyle z$	$\displaystyle\rightarrow$	$\displaystyle\beta^{-1}z,$
$\displaystyle\widetilde{r}$	$\displaystyle\rightarrow$	$\displaystyle\beta\widetilde{r},$
$\displaystyle\widetilde{r_{h}}$	$\displaystyle\rightarrow$	$\displaystyle\beta\widetilde{r_{h}},\ (\beta\rightarrow\infty)$	(11)

where $\beta$ is an appropriate power of a scaling factor.

As a result, one obtains a Schwarzschild black brane metric that has been boosted about the $\tau$ - $z$ plane

ds^{2}=\frac{r^{2}}{L^{2}}(-d\tau^{2}+dx^{2}+dy^{2}+dz^{2}+\frac{r_{h}^{4}}{r^{4}(1-\frac{a^{2}}{L^{2}})}(d\tau+\frac{a}{L}dz)^{2})+\frac{L^{2}r^{2}}{r^{4}-r_{h}^{4}}dr^{2},

(12)

note that for $a=0$ in (12), the Schwarzschild black brane is reproduced.

The temperature of this boosted black brane reads

T=\frac{r_{h}\sqrt{L^{2}-a^{2}}}{\pi L^{3}}.

(13)

Incidentally, the $\eta/s$ in spinning Myers-Perry black holes is given by mg

	$\displaystyle\frac{\eta_{\perp}}{s}$	$\displaystyle=$	$\displaystyle\frac{1}{4\pi},$
	$\displaystyle\frac{\eta_{\parallel}}{s}$	$\displaystyle=$	$\displaystyle\frac{1}{4\pi}(1-a^{2}),$		(14)

one can see that $\eta/s$ depends on the angle between the spatial direction of the measurement and the angular momentum. For more details about the spinning Myers-Perry black holes, we refer to sw ; gw ; gw1 ; km .

III Potential analysis in holographic Schwinger effect

In this section we investigate the behavior of the Schwinger effect for the background (12) following YS . The Nambu-Goto action is

S=T_{F}\int d\xi d\eta\mathcal{L}=T_{F}\int d\xi d\eta\sqrt{g},

(15)

where $T_{F}=\frac{1}{2\pi\alpha^{\prime}}$ is the fundamental string tension. $\alpha^{\prime}$ is related to $\lambda$ via $\frac{L^{2}}{\alpha^{\prime}}=\sqrt{\lambda}$ . $g$ denotes the determinant of the induced metric

g_{\alpha\beta}=g_{\mu\nu}\frac{\partial X^{\mu}}{\partial\sigma^{\alpha}}\frac{\partial X^{\nu}}{\partial\sigma^{\beta}},

(16)

with $g_{\mu\nu}$ and $X^{\mu}$ being the metric and target space coordinate, respectively.

It can be seen from (12) that the boost exists in the $\tau$ - $z$ plane, implying the angular momentum can distinguish the different orientations of the particle pair e.g., $(Q\bar{Q})$ axis with respect to the direction of rotation (defined here to be $z$ axis). Two extreme cases are worthy of note: transverse case (the pair’s axis is on the $x-y$ plane) and parallel case (the pair’s axis is on the $z$ axis). Next, we will examine the two cases in turn.

III.1 Transverse to rotation direction

First we consider the transverse case. Without loss of generality, one could assume that the pair’s axis is along the $x$ direction,

\tau=\xi,\qquad x=\eta,\qquad y=0,\qquad z=0,\qquad r=r(\eta).

(17)

Given that, the induced metric can be written as

g_{00}=-\frac{r^{2}}{L^{2}}+\frac{r_{h}^{4}}{r^{2}(L^{2}-a^{2})},\qquad g_{01}=g_{10}=0,\qquad g_{11}=\frac{r^{2}}{L^{2}}+\frac{L^{2}r^{2}}{r^{4}-r_{h}^{4}}\dot{r}^{2},

(18)

where $\dot{r}=\frac{dr}{d\eta}$ .

The Lagrangian density reads

\mathcal{L}=\sqrt{A(r)+B(r)\dot{r}^{2}},

(19)

with

A(r)=\frac{r^{4}}{L^{4}}-\frac{r_{h}^{4}}{L^{2}(L^{2}-a^{2})},\qquad B(r)=-\frac{L^{2}r_{h}^{4}}{(L^{2}-a^{2})(r^{4}-r_{h}^{4})}+\frac{r^{4}}{r^{4}-r_{h}^{4}}.

(20)

One can see that $\mathcal{L}$ does not depend on $\eta$ explicitly, so the Hamiltonian is conserved,

\mathcal{L}-\frac{\partial\mathcal{L}}{\partial\dot{r}}\dot{r}=Constant.

(21)

Imposing the boundary condition at $\eta=0$

\frac{dr}{d\eta}=0,\qquad r=r_{c}\qquad(r_{h}<r_{c}),

(22)

one gets

\frac{dr}{d\eta}=\sqrt{\frac{A^{2}(r)-A(r)A(r_{c})}{A(r_{c})B(r)}},

(23)

where $A(r_{c})=A(r)|_{r=r_{c}}$ .

Integrating (23), the inter-distance between the particle pair is obtained

x^{\perp}=2\int_{r_{c}}^{r_{0}}dr\sqrt{\frac{A(r_{c})B(r)}{A^{2}(r)-A(r)A(r_{c})}},

(24)

where we have placed the probe D3-brane at an intermediate position $r=r_{0}$ rather than close to the boundary. Such operations could yield a finite mass which then makes sense of the production rate GW .

Substituting (19), (23) into (15), the sum of Coulomb potential and static energy is obtained

V_{CP+E}=2T_{F}\int_{r_{c}}^{r_{0}}dr\sqrt{\frac{A(r)B(r)}{A(r)-A(r_{c})}}.

(25)

To proceed, we calculate the critical field. The DBI action is

S_{DBI}=-T_{D3}\int d^{4}x\sqrt{-\det(G_{\mu\nu}+\mathcal{F}_{\mu\nu})},

(26)

with

T_{D3}=\frac{1}{g_{s}(2\pi)^{3}\alpha^{\prime^{2}}},\qquad\mathcal{F}_{\mu\nu}=2\pi\alpha^{\prime}F_{\mu\nu},

(27)

where $T_{D3}$ refers to the D3-brane tension.

Assuming the electric field is turned on along the $x$ direction YS , one gets

G_{\mu\nu}+\mathcal{F}_{\mu\nu}=\left(\begin{array}[]{cccc}-\frac{r^{2}}{L^{2}}+\frac{r_{h}^{4}}{r^{2}(L^{2}-a^{2})}&2\pi\alpha^{\prime}E&0&\frac{ar_{h}^{4}}{r^{2}L(L^{2}-a^{2})}\\ -2\pi\alpha^{\prime}E&\frac{r^{2}}{L^{2}}&0&0\\ 0&0&\frac{r^{2}}{L^{2}}&0\\ \frac{ar_{h}^{4}}{r^{2}L(L^{2}-a^{2})}&0&0&\frac{r^{2}}{L^{2}}+\frac{r_{h}^{4}a^{2}}{r^{2}L^{2}(L^{2}-a^{2})}\end{array}\right),

(28)

yielding

\det(G_{\mu\nu}+\mathcal{F}_{\mu\nu})=\frac{r^{2}}{L^{2}}[(2\pi{\alpha}^{\prime}E)^{2}(\frac{r^{2}}{L^{2}}+\frac{r_{h}^{4}a^{2}}{L^{2}r^{2}(L^{2}-a^{2})})+\frac{r^{2}r_{h}^{4}}{L^{4}(L^{2}-a^{2})}-\frac{r^{2}r_{h}^{4}a^{2}}{L^{6}(L^{2}-a^{2})}-\frac{r^{6}}{L^{6}}].

(29)

Plugging (29) into (26) and making the probe D3-brane located at $r=r_{0}$ , one gets

S_{DBI}=-T_{D3}\frac{r_{0}}{L}\int d^{4}x\sqrt{\frac{r_{0}^{2}r_{h}^{4}a^{2}}{L^{6}(L^{2}-a^{2})}+\frac{r_{0}^{6}}{L^{6}}-\frac{r_{0}^{2}r_{h}^{4}}{L^{4}(L^{2}-a^{2})}-(2\pi{\alpha}^{\prime}E)^{2}(\frac{r_{0}^{2}}{L^{2}}+\frac{r_{h}^{4}a^{2}}{L^{2}r_{0}^{2}(L^{2}-a^{2})})}.

(30)

To avoid the action (30) being ill-defined, one needs

\frac{r_{0}^{2}r_{h}^{4}a^{2}}{L^{6}(L^{2}-a^{2})}+\frac{r_{0}^{6}}{L^{6}}-\frac{r_{0}^{2}r_{h}^{4}}{L^{4}(L^{2}-a^{2})}-(2\pi{\alpha}^{\prime}E)^{2}(\frac{r_{0}^{2}}{L^{2}}+\frac{r_{h}^{4}a^{2}}{L^{2}r_{0}^{2}(L^{2}-a^{2})})\geq 0,

(31)

which leads to

E\leq T_{F}\sqrt{\frac{\frac{r_{0}^{6}}{L^{4}}+\frac{r_{0}^{2}r_{h}^{4}a^{2}}{L^{4}(L^{2}-a^{2})}-\frac{r_{0}^{2}r_{h}^{4}}{L^{2}(L^{2}-a^{2})}}{r_{0}^{2}+\frac{r_{h}^{4}a^{2}}{r_{0}^{2}(L^{2}-a^{2})}}}.

(32)

As a result, the critical field is

E_{c}^{\perp}=T_{F}\sqrt{\frac{\frac{r_{0}^{6}}{L^{4}}+\frac{r_{0}^{2}r_{h}^{4}a^{2}}{L^{4}(L^{2}-a^{2})}-\frac{r_{0}^{2}r_{h}^{4}}{L^{2}(L^{2}-a^{2})}}{r_{0}^{2}+\frac{r_{h}^{4}a^{2}}{r_{0}^{2}(L^{2}-a^{2})}}},

(33)

one can see that $E_{c}^{\perp}$ depends on $T$ , $r_{0}$ and $a$ .

Finally, the total potential for the transverse case can be written as

$\displaystyle V_{tot}^{\perp}(x)$	$\displaystyle=$	$\displaystyle V_{CP+E}-Ex^{\perp}$	(34)
	$\displaystyle=$	$\displaystyle 2pr_{0}T_{F}\int_{1}^{1/p}dy\sqrt{\frac{A(y)B(y)}{A(y)-A(y_{c})}}$
	$\displaystyle-$	$\displaystyle 2pr_{0}T_{F}\alpha\sqrt{\frac{\frac{r_{0}^{6}}{L^{4}}+\frac{r_{0}^{2}(qr_{0})^{4}a^{2}}{L^{4}(L^{2}-a^{2})}-\frac{r_{0}^{2}(qr_{0})^{4}}{L^{2}(L^{2}-a^{2})}}{r_{0}^{2}+\frac{(qr_{0})^{4}a^{2}}{r_{0}^{2}(L^{2}-a^{2})}}}\int_{1}^{1/p}dy\sqrt{\frac{A(y_{c})B(y)}{A^{2}(y)-A(y)A(y_{c})}},$

where

\alpha\equiv\frac{E}{E_{c}^{\perp}},\qquad y\equiv\frac{r}{r_{c}},\qquad p\equiv\frac{r_{c}}{r_{0}},\qquad q\equiv\frac{r_{h}}{r_{0}},

(35)

$\displaystyle A(y)$	$\displaystyle=$	$\displaystyle\frac{(pr_{0}y)^{4}}{L^{4}}-\frac{(qr_{0})^{4}}{L^{2}(L^{2}-a^{2})},$
$\displaystyle B(y)$	$\displaystyle=$	$\displaystyle-\frac{L^{2}(qr_{0})^{4}}{(L^{2}-a^{2})((pr_{0}y)^{4}-(qr_{0})^{4})}+\frac{(pr_{0}y)^{4}}{(pr_{0}y)^{4}-(qr_{0})^{4}},$
$\displaystyle A(y_{c})$	$\displaystyle=$	$\displaystyle\frac{(pr_{0})^{4}}{L^{4}}-\frac{(qr_{0})^{4}}{L^{2}(L^{2}-a^{2})}.$	(36)

The analysis of (34) will be provided together with the parallel case later.

III.2 Parallel to rotation direction

Now let’s move on to the parallel case. Assuming the the particle pair’s axis is aligned in the $z$ direction,

\tau=\xi,\qquad x=0,\qquad y=0,\qquad z=\eta,\qquad r=r(\eta).

(37)

Through similar calculations, the inter-distance, the critical electric field and the total potential are obtained as

x^{\parallel}=2pr_{0}\int_{1}^{\frac{1}{p}}dy\sqrt{\frac{A_{1}(y_{c})B_{1}(y)}{A_{1}^{2}(y)-A_{1}(y)A_{1}(y_{c})}},

(38)

E_{c}^{\parallel}=T_{F}\sqrt{\frac{r_{0}^{4}}{L^{4}}+\frac{(qr_{0})^{4}a^{2}}{L^{4}(L^{2}-a^{2})}-\frac{(qr_{0})^{4}}{L^{2}(L^{2}-a^{2})}},

(39)

	$\displaystyle V_{tot}^{\parallel}(x)$	$\displaystyle=$	$\displaystyle 2pr_{0}T_{F}\int_{1}^{1/p}dy\sqrt{\frac{A_{1}(y)B_{1}(y)}{A_{1}(y)-A_{1}(y_{c})}}$		(40)
		$\displaystyle-$	$\displaystyle 2pr_{0}T_{F}\alpha\sqrt{\frac{r_{0}^{4}}{L^{4}}+\frac{(qr_{0})^{4}a^{2}}{L^{4}(L^{2}-a^{2})}-\frac{(qr_{0})^{4}}{L^{2}(L^{2}-a^{2})}}\int_{1}^{1/p}dy\sqrt{\frac{A_{1}(y_{c})B_{1}(y)}{A_{1}^{2}(y)-A_{1}(y)A_{1}(y_{c})}},$		(40)

with

$\displaystyle A_{1}(y)$	$\displaystyle=$	$\displaystyle\frac{(pr_{0}y)^{4}}{L^{4}}-\frac{(qr_{0})^{4}}{L^{2}(L^{2}-a^{2})}+\frac{a^{2}(qr_{0})^{4}}{L^{4}(L^{2}-a^{2})},$
$\displaystyle B_{1}(y)$	$\displaystyle=$	$\displaystyle\frac{(pr_{0}y)^{4}}{(pr_{0}y)^{4}-(qr_{0})^{4}}-\frac{L^{2}(qr_{0})^{4}}{(L^{2}-a^{2})((pr_{0}y)^{4}-(qr_{0})^{4})},$
$\displaystyle A_{1}(y_{c})$	$\displaystyle=$	$\displaystyle\frac{(pr_{0})^{4}}{L^{4}}-\frac{(qr_{0})^{4}}{L^{2}(L^{2}-a^{2})}+\frac{a^{2}(qr_{0})^{4}}{L^{4}(L^{2}-a^{2})},$

where $y,p,q$ are the same as in (35) and $\alpha\equiv E/E_{c}^{\parallel}$ . We have checked that by plugging $a=0$ in (34) or (LABEL:V1), the results of SYM (without rotation) YS can be reproduced.

Refer to caption — Figure 1: $V_{tot}(x)$ versus $x$ with $a=0.2$ . Left: Transverse case. Right: Parallel case. In both panels from top to bottom $\alpha=0.8,0.9,1,1.1$ , respectively.

Before going on, we determine the values of some parameters. First, we take $T_{F}=L=1$ , similar to YS . Moreover, we choose a large (fixed) temperature of $T=100/\pi$ , performing a planar limit on the geometry yielding a black brane, as follows from mg . In addition, it has been suggested mg that the spinning black brane (12) would be unstable at sufficiently large angular momentum $a\approx 0.75L$ . To alleviate this, we consider $a<0.75L$ in calculations.

Let’s discuss results. In fig.1, we plot $V_{tot}(x)$ as a function of $x$ for different values of $\alpha$ with fixed $a=0.2$ (other cases with different $a$ have similar picture), where the left panel is for the transverse case while the right is for the parallel case. From both panels, one can see that for $\alpha<1$ (or $E<E_{c}$ ), the potential barrier is present and the Schwinger effect can occur as tunneling process. With the increase of $E$ , the potential barrier decreases and finally vanishes at $\alpha=1$ (or $E=E_{c}$ ). For $\alpha>1$ ( $E>E_{c}$ ), the vacuum becomes unstable catastrophically. These results fall in line with YS .

To understand how angular momentum modifies the Schwinger effect, we plot $V_{tot}(x)$ against $x$ for different values of $a$ with fixed $\alpha=0.9$ in fig.2, where the left panel is for the transverse case while the right is for the parallel case. In both panels from top to bottom $a=0,0.3,0.7$ , respectively. From these figures it is clear to see that as $a$ increases the height and width of the potential barrier both decrease. As we know, the higher (or the wider) the potential barrier, the harder the produced pairs escape to infinity. One can thus conclude that the inclusion of angular momentum decreases the potential barrier thus enhancing the Schwinger effect. In other words, the presence of angular momentum enhances the production rate. These results are consistent with previous findings obtained from a soft wall model yz . Moreover, by comparing the two panels, one finds angular momentum has important effect for the transverse case comparing with the parallel case.

Also, one can examine how angular momentum affects the critical electric field. To this end, we plot $E_{c}/E_{c0}$ versus $a$ in fig.3, where the left panel is for the transverse case while the right is for the parallel case, and $E_{c0}$ represents the critical electric field at $a=0$ . One can see that $E_{c}/E_{c0}$ decreases as $a$ increases. In particular, when $a=0.7$ , the ratio decreased by about 8 percent for the transverse case and 4 percent for the parallel case. It is known that the smaller the critical electric field, the easier the tunneling process. This is in agreement with the previous potential analysis.

IV Conclusion and discussion

In this paper, we investigated the effect of angular momentum on holographic Schwinger effect in spinning Myers-Perry black holes. Along with the prescription in YS , we calculated the potential between the produced pair by evaluating the classical action of a string attaching on a probe D3-brane sitting at an intermediate position in the AdS bulk. It is shown that the inclusion of angular momentum reduces the potential barrier thus enhancing the Schwinger effect. Namely, producing particle pairs would be easier in rotating medium, in accordance with previous findings obtained from the local Lorentz transformation yz . Also, the results show that angular momentum has important effect for the particle pair lying in the transversal plane compared with that along the longitudinal orientation.

Moreover, the results may provide an estimate of how the Schwinger effect changes with $\eta/s$ at strong coupling. From (14) one sees that $\eta_{\perp}/s$ is not affected by $a$ but $\eta_{\parallel}/s$ decreases as $a$ increases. Here we will not make much comment on why only one of the shear viscosities saturates the bound, while the other may violate the bound (a similar situation appeared in some anisotropic backgrounds je ; rc ; ar ). We talk about $\eta_{\parallel}/s$ . From the above analysis one finds that increasing $a$ leads to decreasing $\eta_{\parallel}/s$ thus making the fluid becomes more ”perfect”. On the other hand, increasing $a$ leads to enhancing the Schwinger effect. Taken together, one may conclude that at strong coupling as $\eta/s$ decreases the Schwinger effect is enhanced.

However, there are some problems worthy for further studies. First, here we just considered spinning Myers-Perry black holes ( $a=b$ ), what will happen for general situation ( $a\neq b$ )? Moreover, the potential analysis for Schwinger effect considered here is basically within the Coulomb branch associated with the leading exponent corresponding to the on-shell action of the instanton. One can research the full decay rate if possible.

V Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) with No. G1323523064.

References

(1) J. S. Schwinger, Phys. Rev. 82 (1951) 664.
(2) I. K. Affleck, O. Alvarez and N. S. Manton, Nucl. Phys. B 197 (1982) 509.
(3) J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).
(4) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998).
(5) O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000).
(6) G. W. Semenoff and K. Zarembo, Phys. Rev. Lett. 107 (2011) 171601.
(7) Y. Sato and K. Yoshida, JHEP 1312 (2013) 051.
(8) S. Bolognesi, F. Kiefer, E. Rabinovici, JHEP 1301, 174 (2013).
(9) Y. Sato, K. Yoshida, JHEP 1304, 111 (2013).
(10) Y. Sato and K. Yoshida, JHEP 1308 (2013) 002.
(11) Y. Sato and K. Yoshida, JHEP 1309 (2013) 134.
(12) K. B. Fadafan and F. Saiedi, Eur.Phys.J. C (2015) 75:612.
(13) M. Ghodrati, Phys. Rev. D 92, 065015 (2015).
(14) Z. q. Zhang, Nucl. Phys. B 935 (2018) 377.
(15) Z. q. Zhang, X. r. Zhu and D.f. Hou, Phys. Rev. D 101, 026017 (2020).
(16) L. Shahkarami, M. Dehghani, P. Dehghani, Phys. Rev. D 97, 046013 (2018).
(17) Z. R. Zhu, D. f. Hou and X. Chen, Eur.Phys.J.C 80 (2020) 6, 550.
(18) S.w. Li, S.-k. Luo, H.-q. Li, JHEP 08 (2022) 206.
(19) D. Kawai, Y. Sato, and K. Yoshida, Int. J. Mod. Phys. A 30 (2015) 1530026.
(20) S. Chakrabortty and B.Sathiapalan, Nucl.Phys. B 890 (2014) 241.
(21) K. Hashimoto and T. Oka, JHEP 1310 (2013) 116.
(22) K. Hashimoto, T. Oka, and A. Sonoda, JHEP 1406 (2014) 085.
(23) K. Ghoroku, M. Ishihara, JHEP 09 (2016) 011.
(24) L. Shahkarami, F. Charmchi, Eur. Phys. J. C 79 (2019) 4, 343.
(25) Y. z. Cai, R. p. Jing, Z. q. Zhang, Chin.Phys.C 46 (2022) 10, 104107.
(26) Y.Ding, Z. q. Zhang, Chin.Phys.C 45 (2021) 1, 013111.
(27) S.w. Li, Eur.Phys.J.C 81 (2021) 9, 797.
(28) W. Fischler, P.H. Nguyen, J.F. Pedraza, W. Tangarife, Phys. Rev. D 91, 086015 (2015).
(29) The STAR Collaboration, Nature 548 (2017) 62-65.
(30) L. G. Pang, H. Petersen, Q. Wang, and X. N. Wang, Phys. Rev. Lett. 117, 192301 (2016).
(31) X. G. Huang, P. Huovinen, and X. N. Wang, Phys. Rev. C 84, 054910 (2011).
(32) F. Becattini, F. Piccinini, J. Rizzo, Phys. Rev. C 77, 024906 (2008).
(33) Z. T. Liang and X. N. Wang, Phys. Rev. Lett. 94, 102301 (2005); 96, 039901(E) (2006).
(34) J. Sadeghi and B. Pourhassan, Int. J. Theor. Phys. 50, 2305 (2011).
(35) B. McInnes, arXiv:1710.07442 [hep-ph].
(36) I. Ya. Arefeva, A.A. Golubtsova and E. Gourgoulhon, JHEP 04 (2021) 169.
(37) I. Ya. Arefeva, A. Bagrov and A. Koshelev, JHEP 07 (2013)170.
(38) A. N. Atmaja and K. Schalm, JHEP 1104 (2011) 070.
(39) K. B. Fadafan, H. Liu, K. Rajagopal, Eur. Phys. J. C 61 (2009) 553.
(40) M. Atashi, K. B. Fadafan, Phys. Lett. B 800 (2020) 135090.
(41) D. f. Hou, M. Atashi, K. B. Fadafan, Z.q. Zhang, Phys. Lett. B 817 (2021) 136279.
(42) N. R. F. Braga, L. F. Faulhaber and O. C. Junqueira, Phys. Rev. D 105, 106003 (2022).
(43) X. Chen, L. Zhang, D. Li, D. Hou and M. Huang, JHEP 07 (2021) 132.
(44) J. Zhou, S.w. Zhang, J. Chen, L. Zhang and X. Chen, Phys. Lett. B 844 (2023) 138116.
(45) S. Bhattacharyya, S. Lahiri, R. Loganayagam and S. Minwalla, JHEP 0809 (2008) 054.
(46) B. McInnes, Int. J. Mod. Phys. A 34, no. 24 (2019) 1950138.
(47) B. McInnes, Nucl. Phys. B 911 (2016) 173.
(48) H. Bantilan, T. Ishii and P. Romatschke, Phys. Lett. B 785, 201 (2018).
(49) A. A. Golubtsov, E. Gourgoulhon and M. K. Usova, Nucl.Phys.B 979 (2022) 115786.
(50) S.Y. Yang, R.D. Dong, D.f. Hou and H.c. Ren, Phys.Rev.D 107 (2023) 7, 076020.
(51) A. A. Golubtsova and N. S. Tsegelnik, Phys. Rev. D 107 no. 10, (2023) 106017.
(52) M. Bravo Gaete, L. Guajardo, and M. Hassaine, JHEP 04 (2017) 092.
(53) C. Erices and C. Martinez, Phys. Rev. D 97 no. 2, (2018) 024034.
(54) A. M. Avad, Class. Quant. Grav.20 (2003) 2827-2834.
(55) J. x. Chen, D.f. Hou and H.c. Ren, arXiv:2308.08126.
(56) S.W. Hawking, C.J. Hunter and M. Taylor, Phys. Rev. D 59 (1999) 064005.
(57) M. Garbiso and M. Kaminski, JHEP 12 (2020), 112.
(58) G.W. Gibbons, M.J. Perry and C.N. Pope, Class. Quant. Grav. 22 (2005) 1503.
(59) G.W. Gibbons, H. Lu, D.N. Page and C.N. Pope, Phys. Rev. Lett. 93 (2004) 171102.
(60) K. Murata, Prog. Theor. Phys. 121 (2009) 1099.
(61) J. Erdmenger, Phys. Lett. B 699 (2011) 301.
(62) R. Critelli, S.I. Finazzo, M. Zaniboni and J. Noronha, Phys. Rev. D 90 (2014) 066006.
(63) A. Rebhan and D. Steineder, Phys. Rev. Lett. 108 (2012) 021601.