The effects of deformation parameter on thermal width of moving quarkonia in plasma
aa r X i v : . [ h e p - t h ] M a y Prepared for submission to JHEP
The effects of deformation parameter on thermalwidth of moving quarkonia in plasma
J. Sadeghi, a, S. Tahery, a, a Sciences Faculty, Department of Physics, University of Mazandaran, Iran
E-mail: [email protected] , [email protected] Abstract:
In general we can say that the thermal width of quarkonia corresponds toimaginary part of it’s potential. Gravity dual of theories give explicit form of potentialas V Q ¯ Q . Since there is an explicit formula for ImV Q ¯ Q one can consider different gravityduals and study the results of contribution of various parameters. Variable gravity dualsof moving pair in plasma have different results for potential. Our paper shows that defor-mation parameter c in warp factor leads to new results that we present them for arbitraryangles of the pair with respect to it’s velocity. We compare our results with the case thatno deformation parameter is in metric background. We will see that the thermal widthof the pair increases with increasing deformation parameter. Also, for nonzero values ofdeformation parameter the pair feels moving plasma in all distances. In addition our resultsindicate that contribution of deformation parameter leads to larger dissociation length forthe moving pair reverse to the effect of the pair’s velocity in the plasma. ontents Q ¯ Q in an deformed AdS, perpendicular case 23 Q ¯ Q in an deformed AdS at arbitrary angles 94 Concolusion 15 When we want to study Q ¯ Q interaction we should consider the effect of the medium inmotion of Q ¯ Q , because this pair is not produced at rest in QGP. So, the velocity of thepair through the plasma has some effects on its interactions that should be taken into ac-count. The interaction energy has a finite imaginary part at finite temperature that can beused to estimate the thermal width of the quarkonia [1, 2]. Calculations of Im V Q ¯ Q relevantto QCD and heavy ion collisions were performed for static Q ¯ Q pairs using pQCD [3] andlattice QCD [4–6] before AdS/CFT.The AdS/CFT is a correspondence [7–10] between a string theory in AdS space and a con-formal field theory in physical space-time. It leads to an analytic semi-classical model forstrongly coupled QCD. It has scale invariance, dimensional counting at short distances andcolor confinement at large distances. This theory describes the phenomenology of hadronicproperties and demonstrate their ability to incorporate such essential properties of QCD asconfinement and chiral symmetry breaking. In the AdS/CFT point of view the AdS playsimportant role in describing QCD phenomena. So in order to describe a confining theory,the conformal invariance of AdS must be broken somehow. Two strategies AdS/QCDbackground have been suggested in the literatures hard-wall model [11–16] and soft-wallmodel [17–35]. In hard-wall model to impose confinement and discrete normalizable modesthat is to truncate the regime where string modes can propagate by introducing an IRcutoff in the fifth dimension at a finite value z ∼ QCD . Thus, the hard-wall at z breaksconformal invariance and allows the introduction of the QCD scale and a spectrum of parti-cle states, they have phenomenological problems, since the obtained spectra does not haveRegge behavior. To remedy this it is necessary to introduce a soft cut off, using a dilatonfield or using a warp factor in the metric [11, 22]. These models are called soft wall models.The soft-wall and hard-wall approach has been successfully applied to the description ofthe mass spectrum of mesons and baryons, the pion leptonic constant, the electromagneticform factors of pion and nucleons, etc. On the other hand the study of the moving heavyquarkonia in space-time with AdS/QCD approach plays important role in interaction en-ergy [36–39]. By using different metric backgrounds we see different effects on interaction– 1 –nergy.Evaluation of Im V Q ¯ Q will yield to determine the suppression of Q ¯ Q in heavy ion collision[40].The main idea is using boosted frame to have Re V Q ¯ Q and Im V Q ¯ Q [41] for Q ¯ Q in a plasma.From viewpoint of holography, the AdS/CFT correspondence can describe a “brocken con-formal symmetry”, when one adds a proper deformed warp factor in front of the AdS metricstructure [42–62]. So, e cz is a positive quadratic correction with z, the fifth dimension.One natural question is about the connection between the warp factor and the potential V Q ¯ Q . In this work, the procedure of [40] is followed to evaluate the imaginary part ofpotential for an AdS metric background with deformation parameter in warp factor. It isinteresting to see “ what will happen if meson be in a deformed AdS?”It is a trend to see the effects of deformation parameter on Re V Q ¯ Q and Im V Q ¯ Q which areevidences for “usual” or “unusual" behavior of meson in compare with the c = 0 case. Asexpected in the limit of c → , all results are equal to the results of AdS case. All aboveinformations give us motivation to work on effect of the deformation parameter in AdS metric background on real and imaginary parts of potential. So, we organized the paperas follows. In section 2, we discuss the case where the pair is moving perpendicularly tothe joining axis of the dipole in deformed AdS, we assume this metric background for Q ¯ Q and find some relations for real and imaginary parts of potential. This example will be pre-sented with some numerical results for different values of deformation parameter. Then weconsider general orientation of Q ¯ Q in section 3 and follow the procedure as before. Section4 would be our conclusion and some suggestions for future work. Q ¯ Q in an deformed AdS, perpendicular case In this section we consider soft-wall metric background with deformation parameter in warpfactor at finite temperature case. So, we present general relations for real and imaginaryparts of potential when the dipole is moving with velocity η perpendicularly to the wind[40].In our case we apply the general result for deformed AdS, the dual gravity metric will beas: ds = e A ( z ) [ − f ( z ) dt + Σ i =3 i =1 dx + 1 f ( z ) dz ] , (2.1)Where A ( z ) = − ln zR + cz and f ( z ) = 1 − ( zz h ) . As mentioned before c is deformationparameter and R is the AdS curvature radius, also ≤ z ≤ z h , z h = πT and T is boundaryfield theory’s temperature. We have a dynamic dilaton in action for the background andwe write our calculations in string frame. If dilaton is such that it enters directly in theworldsheet action in the form φR . May be our concern is about the effect of a nontrivialdilaton profile to the string action. But somewhen people neglect it at the first step [63] andleave it for future study. Then one can check that the integral on the action correspond toworldsheet with higher genus. This means that we are doing string interactions and goingto higher order in string perturbation theory. But now, for leading order calculations ingenus, we need not to bother with this term even if the geometry has a dynamical dilaton.– 2 –n the other hand one trace of dynamical dilaton can appear via temperature if we wantto calculate it with [64] approach. So, the exact temperature will be in hand. But we referthe reader to [62] for the reasons that in deformed AdS model with quadratic correctionin warp factor the “temperature” takes the form of AdS-SW BH temperature. So, we havea deformed AdS which in the limit c → becomes AdS . This comparing results helpus to underestand the effects of deformation parameter on the physical quantities suchas interaction energy. Our calculations in the cases of LT , ReV Q ¯ Q and ImV Q ¯ Q give usmotivation to compare results between different values of deformation parameter.From metric background (2.1) one can obtain: G = R z [1 − ( zz h ) ] e cz (2.2) G xx = R z e cz (2.3) G zz = R z [1 − ( zz h ) ] − e cz , (2.4)with these definitions, ˜ M ( z ) ≡ M ( z ) cosh η − N ( z ) sinh η (2.5) ˜ V ( z ) ≡ V ( z ) cosh η − P ( z ) sinh η (2.6) M ( z ) ≡ G G zz (2.7) V ( z ) ≡ G G xx (2.8) P ( z ) ≡ G xx (2.9) N ( z ) ≡ G xx G zz , (2.10)we continue with hamiltonian, H ( z ) ≡ s ˜ V ( z )˜ V ∗ ˜ V ( z ) − ˜ V ∗ ˜ M ( z ) , (2.11)where ˜ V ∗ means ˜ V ( z ∗ ) and ∗ is the deepest position of the string in the bulk.The equation of motion and the boundary conditions of the string relates L (length of theline joining both quarks) with z ∗ as follows, L Z Λ r ∗ drH ( r ) . (2.12)So, for the corresponding case we have, L − Z z ∗ dzH ( z ) . (2.13)– 3 –n order to relation between S str and z ∗ we find the regularized integral [41] as, S regstr = Tπα ′ Z ∞ r ∗ dr q ˜ M ( r ) s ˜ V ( r )˜ V ( r ∗ ) ˜ V ( r )˜ V ( r ∗ ) − ! − / − p M ( r ) − Tπα ′ Z r ∗ r h dr p M ( r ) , (2.14)and we obtain the following results LT = 2 π y h q − y h cosh η Z ∞ dy r ( y − y h )[ e cy hπ T ( y − ( y − y h cosh η ) − (1 − y h cosh η )] (2.15)where y = z ∗ z and y h = z ∗ z h S regstr = T √ λy h { Z ∞ dy [ e cy hπ T ( y − ) ( y − y h cosh η ) r ( y − y h )[ e cy hπ T ( y − ( y − y h cosh η ) − (1 − y h cosh η )] − e cy hπ T y ] − Z dy e cy hπ T y } , (2.16)Where λ = R α ′ is and α ′ is the ’t Hooft coupling of the gauge theory. Finally, we find thereal part of potential as ReV Q ¯ Q = S regstr T .Now we present a derivation of relation for imaginary part of potential from [41]. Thereader can see more details in that reference. From there we can say one should considerthe effect of worldsheet fluctuations around the classical configuration z c ( x ) , z ( x ) = z c ( x ) → z ( x ) = z c ( x ) + δz ( x ) . (2.17)And then the fluctuations should be taken into account in partition function so one arrivesat, Z str ∼ Z D δz ( x ) e iS NG ( z c ( x )+ δz ( x )) . (2.18)Then there is an imaginary part of potential in action so , by dividing the interval region ofx into N points where N −→ ∞ that should be taken into account at the end of calculationwe arrive at, Z str ∼ lim N →∞ Z d [ δz ( x − N )] . . . d [ δz ( x N )] exp i T ∆ x πα ′ X j q M ( z j )( z ′ j ) + V ( z j ) . (2.19)Notice that we should expand z c ( x j ) around x = 0 and keep only terms up to second orderof it because thermal fluctuations are important around z ∗ which means x = 0 , z c ( x j ) ≈ z ∗ + x j z ′′ c (0) , (2.20)– 4 –ith considering small fluctuations finally we will have, V ( z j ) ≈ V ∗ + δzV ′∗ + z ′′ c (0) V ′∗ x j δz V ′′∗ , (2.21)where V ∗ ≡ V ( z ∗ ) and V ′∗ ≡ V ′ ( z ∗ ) . With (2.20), (2.21) and (2.19) one can derive (2.22),(2.23) and (2.24), S NGj = T ∆ x πα ′ q C x j + C (2.22) C = z ′′ c (0)2 (cid:2) M ∗ z ′′ c (0) + V ′∗ (cid:3) (2.23) C = V ∗ + δzV ′∗ + δz V ′′∗ . (2.24)For having ImV Q ¯ Q = 0 the function in the square root of (2.22) should be negative.then, we consider j-th contribution to Z str as, I j ≡ δz jmax Z δz jmin d ( δz j ) exp (cid:20) i T ∆ x πα ′ q C x j + C (cid:21) , (2.25)For every δz between minimum and maximum of it’s values which are the roots of C x j + C in δz , one leads to C x j + C < . The extermal value of the function D ( δz j ) ≡ C x j + C ( δz j ) (2.26)is, δz = − V ′∗ V ′′∗ . (2.27)So, D ( δz j ) < −→ − x c < x j < x c leads us to have an imaginary part in square root,where, x c = s C (cid:20) V ′ ∗ V ′′∗ − V ∗ (cid:21) . (2.28)If the square root in (2.28) is not real we should take x c = 0 . After all these conditionswe can approximate D ( δz ) by D ( − V ′∗ V ” ∗ ) in I j , I j ∼ exp " i T ∆ x πα ′ s C x j + V ∗ − V ′ ∗ V ′′∗ . (2.29)The total contribution to the imaginary part, will be in hand with continuum limit. So, Im V Q ¯ Q = − πα ′ Z | x |
1) and (3.24)– 11 – dχd ˜ σ (cid:19) = p q (cid:18) y − cosh ηy − (cid:19) , (3.25)where we defined the dimensionless variables y ≡ z h /z , χ ≡ X d /z h and ˜ σ ≡ σ/z h aswell as the dimensionless integration constants q ≡ Q z h /R and p = K z h /R also theboundary conditions become, y (cid:18) ± π LT θ (cid:19) = 0 χ (cid:18) ± π LT θ (cid:19) = ± π LT θ. (3.26)So, (3.11), (3.12) and (3.10) lead to (3.27), (3.28) and (3.29), LT π sin θ = q Z ˜Λ y ∗ dy q (( e cπ T y )( y − − p )( y − cosh η ) − q ( y −
1) and (3.27) LT π cos θ = p Z ˜Λ y ∗ dy y − cosh ηy − q (( e cπ T y )( y − − p )( y − cosh η ) − q ( y − . (3.28) (( e cπ T y ∗ )( y ∗ − − p )( y ∗ − cosh η ) − q ( y ∗ −
1) = 0 . (3.29)Therefore the real part of potential is, Re V Q ¯ Q T √ λ = Z ∞ y ∗ dy e cπ T y ( y − cosh η ) q ( y − cosh η )( e cπ T y ( y − − p ) − q ( y − − e c π T y − Z y ∗ dye c π T y . (3.30)And from (3.23) we arrive at imaginary part of potential as, Im V Q ¯ Q T √ λ = − π e c π T y ∗ × [ cy ∗ πT − πT y ∗ − cπTy ∗ (cos ˜ θ +cosh η sin ˜ θ )] y ∗ π T − c − c π T y ∗ − ( c π T y ∗ + cy ∗ )(cos ˜ θ +cosh η sin ˜ θ )] − ( y ∗ − (cos ˜ θ + cosh η sin ˜ θ )) r y ′′ (0) ( y ∗ − cosh ηy ∗ − ) + [ cy ∗ π T − y ∗ − cy ∗ π T (cos ˜ θ +cosh η sin ˜ θ )]2 sin ˜ θ y ′′ (0) + b tan ˜ θ ( y ∗ − . (3.31)We proceed by solving (3.29) numerically to have y ∗ as a function of q and p, then (3.27),(3.28), (3.30), (3.31) will be functions of p and q. On the other hand for finding p as afunction of q, one can solve (3.27) and (3.28) for fixed θ , after doing all these, LT as afunction of q is in hand. Before we start to calculate ImV Q ¯ Q we should obtain ˜ θ , y ′′ (0) and ˜ b . The ˜ θ is obtained from (3.25) at ˜ σ = 0 and y = y ∗ . We should solve (3.24) and (3.25)with use of boundary conditions (3.26) , to evaluate y ′′ (0) and ˜ b . After doing all these– 12 – −1 q L T c/T =0c/T =25c/T =50 Figure 8 . LT as a function of q at η = 1 and θ = π . different values of deformation parameter arecontributed. The solid green curve corresponds to cT = 0 , the dotted red curve to cT = 25 andthe dashed blue curve to cT = 50 . −2.5−2−1.5−1−0.5 LT R e V QQ / λ . T c/T =0 η =1 Figure 9 . ReV Q ¯ Q as a function of LT at η = 1 and θ = π and deformation parameter is zero calculations numerically, with y ′′ (0) and ˜ b known, we can calculate ImV Q ¯ Q as a functionof q. So, we will survey LT ( q ) , also real and imaginary parts of potential as a function of LT .The result of cases with a fixed η and different choices of θ besides a fixed θ and differentchoices of η have been studeid in [40], so we proceed by fixing both of them and choosingdifferent values of deformation parameter.In Fig. 8 we show LT as a function of q for a fixed orientation of the dipole , fixed η and dif-ferent values of deformation parameter. we know LT max depends strongly on the rapidity η and it decreases with increasing η [40]. In our plots, we can see that LT max which indicates– 13 – LT R e V QQ / λ . T c/T =25 η =1 Figure 10 . ReV Q ¯ Q as a function of LT at η = 1 and θ = π and scaled deformation parameter is cT = 25 LT R e V QQ / λ . T c/T =50 η =1 Figure 11 . ReV Q ¯ Q as a function of LT at η = 1 and θ = π and scaled deformation parameter is cT = 50 the limit of validity of classical gravity calculation, increases with increasing deformationparameter.In Figs. 9,10 and 11 we present ReV Q ¯ Q as a function of LT. We can see for small valuesof LT which means short distances or small temperatures, there is a difference between c = 0 and c = 0 cases. As we expected, when deformation parameter contributes to thecalculation, the interaction of the pair is relevant with plasma, it is similar to the result ofperpendicular case. The other point is that real part of potential has no intense alterationwith varying angle for any value of deformation parameter.In Fig. 12 we can see ImV Q ¯ Q as a function of LT. It shows that for angle θ < π with decreas-– 14 – I m V QQ / λ . T c/T =0c/T =25c/T =50 Figure 12 . Imaginary part of potential as a function of LT , at a fixed velocity η = 1 and θ = π .different values of scaled deformation parameter are contributed. The solid green curve correspondsto cT = 0 , the dotted red curve to cT = 25 and the dashed blue curve to cT = 50 . ing angle, the imaginary part of potential becomes smaller for any values of deformationparameter. In this article, we have used the method of [40] to investigate the real and imaginary parts ofpotential for moving heavy quarkonia in plasma with a gravity dual which has deformationparameter in warp factor. At the first step we considered Q ¯ Q pair oriented perpendicularlyto the hot wind and after that we extended all calculations to arbitrary angles. We saw thatfor both perpendicular and arbitrary angle cases, the limit of classical gravity calculationincreases with increasing deformation parameter. Also for nonzero values of c the pair feelsmoving plasma even in short distances, but for c = 0 case the pair does not feel movingplasma at some small values of LT as we expected. We indicated when nonzero values ofdeformation parameter contribute to the imaginary part of potential, the thermal width ofquarkonia increases with increasing deformation parameter. Results of perpendicular casein compare with arbitrary angle θ < π showed that with decreasing angle, the imaginarypart of potential becomes smaller for any values of deformation parameter, but real part ofpotential has no intense alteration with varying angle for any value of deformation param-eter.Another interesting problem is instead of using the soft wall model we use hyperscalingviolation metric background and discuss the moving mesons and investigate real and imagi-nary parts of potential. This problem with corresponding metric background for the movingmeson in plasma media is in hand. – 15 – cknowledgement The authors are grateful very much to S. M. Rezaei for support and valuable activity innumerical calculations.
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