Holographic Schwinger effect in confining D3-brane background with chemical potential Zi-qiang Zhang,1, ∗ De-fu Hou,2, † Yan Wu,1, ‡ and Gang Chen1, § 1
School of mathematical and physics, China University of Geosciences, Wuhan 430074, China 2 Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan 430079,China
arXiv:1604.00095v1 [hep-ph] 1 Apr 2016
Using the AdS/CFT correspondence, we invetigate the Schwinger effect in confining D3-brane background with chemical potential. We calculate the potential between a quark-antiquark pair in an external electric field. It is shown that the presence of chemical potential tends to suppress the pair production effect. PACS numbers: 11.25.Tq, 11.15.Tk, 11.25-w
I.
INTRODUCTION
Schwinger effect is known as the pair production in an external electric field in QED [18]. The virtual electronposition pairs can become real particles when a strong electric-field is applied. The production rate Γ has been calculated in the weak-coupling and weak-field approximation long time ago. Later, it is generalized to arbitrarycoupling and weak-field case in [2] Γ∝e
−πm2 eE
2
+ e4
(1)
where m and e are the mass and charge of the created particles respectively, and E is the external electric field. This non-perturbative effect can be explained as a tunneling process, and is not restricted to QED but usual for QFTs coupled to a U(1) gauge field. AdS/CFT, namely the duality between the type IIB superstring theory formulated on AdS5 × S 5 and N = 4 SYM in four dimensions, can realize a system that coupled with a U(1) gauge field[3–5]. Therefore, it is very interesting to consider the Schwinger effect in a holographic setup. After [6, 7] where the creation rate of the quark pair in N = 4 SYM theory was obtained firstly, there are many attempts to addressing Schwinger effect in this direction. For instance, the universal aspects of this effect in the general backgrounds are investigated in [8]. The pair production in confining background is studied in [9, 10]. The potential barrier for the pair creation is analyzed in [11]. The pair production in the conductivity of a system of flavor and color branes is studied in [12]. The Schwinger effect with constant electric and magnetic fields is investigated in [13, 14]. The non-relativistic Schwinger effect is discussed in [15]. The Gauss-Bonnet corrections to this effect is investigated in [16]. This effect is also studied with some AdS/QCD models [17, 18]. For a recent review on this topic, see Ref.[19]. The Schwinger effect has been discussed in confining D3-brane background which is related to the finite temperature [9]. Actually, the QGP is usually assumed to carry a finite, albeit small, baryon number density. So it is necessary to take into account the influence of the chemical potential on Schwinger effect. In this paper, we will add the chemical potential to the confining D3-brane background. We would like to see how chemical potential affects the pair production in this case. It is the motivation of the present work. The organization of this paper is as follows. In the next section, the background with chemical potential is introduced. In section 3, we perform the potential analysis for confining D3-brane with chemical potential. The last part is devoted to conclusion and discussion.
∗ Electronic
address: address: ‡ Electronic address: § Electronic address: † Electronic
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2 II.
BACKGROUND WITH CHEMICAL POTENTIAL
In the framework of the AdS/CFT duality, N = 4 SYM theory with a chemical potential is obtained by making the black hole in the holographic dimension charged. The metric can be written as [20] ds2 = gµν dxµ dxν r2 L2 = 2 [−(dx0 )2 + (dx1 )2 + (dx2 )2 + f (r)(dx3 )2 ] + 2 f (r)−1 dr2 + L2 dΩ25 , L r
(2)
with f (r) = 1 − (1 + Q2 )(
rt 4 rt ) + Q2 ( )6 , r r
(3)
where r denotes the radial coordinate of the black brane geometry, r = rt √ is the horizon, and the AdS space radius L 1 L2 λ. relates the string tension 2πα ′ to the ’t Hooft coupling constant by α′ = The temperature of the boundary theory is given by 1 − 21 Q2 rt , πL2 √ where the charge Q of the black hole is in the range 0 ≤ Q ≤ 2. The chemical potential µ is related to Q as following: √ 3Qrt µ= . L2 T =
(4)
(5)
Note that, When Q = 0 (zero chemical potential), the usual confining D3-brane background is reproduced. Actually, the chemical potential implemented in this way is not the quark (or baryon) chemical potential of QCD but a chemical potential corresponding to the R-charge of N = 4 SYM theory. However, in this context it can serve as a simple way of introducing finite density effects into the system. The same metric has been used to investigate the ”heavy quarks and mesons” in [20]. III.
POTENTIAL ANALYSIS
To proceed, the world sheet in the bulk can be parameterized by τ and σ, then the Nambu-Goto action for the world-sheet becomes Z Z p 1 1 S= (6) dτ dσL = dτ dσ detGab . ′ ′ 2πα 2πα By using the static gauge
x0 = τ,
x1 = σ,
(7)
and suppose the radical for the radial direction r = r(σ),
x2 = const,
x3 = const.
(8)
We have L=
s
r4 1 2 r˙ + 4 , f L
(9)
with r˙ = dr/dσ. Now that L does not depend on σ explicitly, we have the conserved quantity, L−
∂L r. ˙ ∂ r˙
(10)
3 It is clear that the boundary condition at σ = 0, implying r˙ = 0 at r = rc with rt < rc < r0 , then the conserved quantity can be written as r4
then a differential equation is derived,
q
1 2 f r˙
+
r4 L4
= const = rc2 L2 ,
(11)
p r r2 r4 − rc4 dr rt rt r˙ = 1 − (1 + Q2 )( )4 + Q2 ( )6 . = dσ rc2 L2 r r By integrating (12) the separation length x of quark-antiquark pair on the probe brane becomes Z dy 2L2 1/a q , x= r0 a 1 2 4 y (y − 1)[1 − (1 + Q2 )( b )4 + Q2 ( b )6 ] ay
(12)
(13)
ay
where the following dimensionless parameters have been introduced, r rc rt y≡ , a≡ , b≡ . rc r0 r0
(14)
Plugging (12) into (9), one can obtain the classical action. So the sum of Coulomb potential(CP) and static energy(SE) of string can be written as Z r0 a 1/a y 2 dy q VP E+SE = . (15) ′ πα 1 b 6 b 4 ) + Q2 ( ay ) ] (y 4 − 1)[1 − (1 + Q2 )( ay Therefore, the total potential Vtot can be obtained by adding the potential energy of quark pair in electric field as the following: Vtot = VP E+SE − Ex Z r0 a 1/a y 2 dy q = ′ b 6 b 4 πα 1 ) + Q2 ( ay ) ] (y 4 − 1)[1 − (1 + Q2 )( ay Z 1/a r0 α dy q , − b 6 b 4 πα′ a 1 ) + Q2 ( ay ) ] y 2 (y 4 − 1)[1 − (1 + Q2 )( ay
(16)
where the following quantities have been introduced, α≡
E , Ec
Ec ≡
1 r02 , 2πα′ L2
(17)
where Ec corresponds to the critical electric field obtained from the DBI action. We have checked that the total potential Vtot in confining D3-brane background [9] can be derived from (16) if we neglect the effect of chemical potential by plugging Q = 0 in (16). Before discussing the results, let us recall the potential analysis in general backgrounds in Ref [8]. There are two 1 critical values of the electric field, Es = TzF2 and Ec = TzF2 with TF = 2πα ′ . When E < Es , the pair is confined and no s 0 Schwinger effect can occur. When Es < E < Ec , the potential barrier is present and the pair production is described as a tunneling process, and the production rate is exponentially suppressed. As E increases, the potential barrier decreases gradually. At last, it vanishes at E = Ec . When E > Ec , no tunneling occurs and the production rate is not exponentially suppressed any more, the pair production is catastrophic and the vacuum becomes totally unstable. We now discuss the results. The total potential (16) can be plotted versus distance x numerically. Here FIG.1 is plotted for a fixed value of b (fixed temperature) and Q (related to chemical potential) with different values of α. The left is plotted for a small value of chemical potential (Q = 0.2) while the right is for a larger value of chemical potential (Q = 1.4). In all of the plots from top to down α = 0.1, 0.25, 0.6, 1.0, 1.3. From the figure, we can indeed see that there exist two critical values of the electric field: one is related to α = 1(E = Ec ), the other is near α = 0.25 (E = Es = 0.25Ec ). This result is consistent with that in [9]. But as the chemical potential increases, the shape of barrier changes significantly i.e. the height and the width of barrier increase. Then one finds that larger chemical potential causes a higher potential barrier. As we know, higher potential barrier makes the produced pair harder to escape. Therefore, we conclude that the chemical potential tends to suppress the pair production effect.
4 V (x)
V (x)
tot
0.8
tot
0.8
Q=0.2
0.6
0.6
0.4
0.4
0.2
Q=1.4
0.2
x 0.0
x
0.0 0.0
0.5
1.0
1.5
2.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
0.5
1.0
1.5
2.0
FIG. 1: Plots of total potential versus distance x with two different values of Q (related to chemical potential). Here b=0.5 and 2L2 /r0 = r0 /(πα′). Left: Q = 0.2; Right: Q = 1.4. In all of the plots from top to down α = 0.1, 0.25, 0.6, 1.0, 1.3
IV.
CONCLUSION AND DISCUSSION
In this paper, we have investigated the influence of chemical potential on the pair production effect in an external field. The effect was considered in gravity dual of quark-gluon plasma with chemical potential background. The quark-antiquark potential was obtained by calculating the Nambu-Goto action of string attaching the rectangular Wilson loop at the probe brane. From the results, we can indeed see that there exist two critical values of the electric field. This is consistent with that in [9]. In addition, it is shown that the chemical potential causes a higher potential barrier. So we conclude that the chemical potential tends to suppress the pair production effect. Interestingly, the presence of instantons density (order parameter q) also suppresses the pair production effect [21]. However, since the holographic setup of the Schwinger effect is closely related to the string theory, it is necessary to take into account various stringy corrections. For instance, R2 corrections, R4 corrections and so on, we hope to report our progress in this regard in future. V.
ACKNOWLEDGMENTS
This research is partly supported by the Ministry of 502 Science and Technology of China (MSTC) under the 973 503 Project No. 2015CB856904(4). Zi-qiang Zhang is supported by the NSFC under Grant Nos.11547204. Gang Chen is supported by the NSFC under Grant Nos 11475149. De-fu Hou is partly supported by NSFC under Grant Nos. 11375070 and 11221504.
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