The O(1/m 2 ) heavy quark-antiquark potential at nite temperature

Transcription

1 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 1 / 9 The O(1/m ) heavy quark-antiquark potential at nite temperature Carla Marchis University of Turin Department of Physics Supervisors: Prof. Michele Caselle Prof. Marco Panero September 3, 15

2 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 / 9 1 Static potential Wilson loop Eective sting theory Eective eld theories NRQCD and pnrqcd 3 The eective string theory formulation Mapping 4 Potential at zero temperature The 1/m potential 5 Potential at nite temperature Static potential

3 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 3 / 9

4 Static potential Wilson loop Wilson loop A gauge invariant quantity of particular interest for this study is the Wilson loop 1 : W (r, T ) = Tr [Pe ig ] r T dzµ A µ T Fixing the unitary gauge along the time direction, the expectation value of the Wilson loop is the analogue of the meson-meson correlator and so it is related to the interquark potential by the area-perimeter law: r V (r)t W (r, T ) e T (σrt +c(r+t )+k) = e 1 K. G. Wilson, Phys. Rev. D, 339 (1974) Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 4 / 9

5 Static potential Wilson loop Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 5 / 9 Static potential So for the static potential we have: 1 V (r) = lim ln W (r, T ) = σr + c T T where σ is the string tension which has the dimension of a squared mass. This potential is a conning potential. Connement is usually associated to the creation of a ux tube between the quark and antiquark and thus it is necessary to consider also the quantum uctuations of the ux tube. The theory which describes these quantum uctuations is the eective string theory (EST).

6 Static potential EST Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 6 / 9 Eective string theory In the EST framework: lim W (r, T ) = T Dξ l e S string where ξ l = ξ l (τ, z), l = 1,..., 4 are the components of the string. The simplest eective string action is the Nambu Goto action: T r S NG = σ dτ dz g where g is the determinant of the induced metric and τ and z are the longitudinal coordinates of the string. If we x the physical gauge and we consider only the rst two terms of the expansion of g, the string action may be written as: T r S NG = σ dτ dz (1 + 1 ) µξ l µ ξ l

7 Static potential EST Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 7 / 9 Static potential So we have: lim W (r, T ) = T Dξ l e σ T dτ r dz(1+ 1 µξl µ ξ l ) This leads to the following expression of the static potential: π(d ) 4r V () (r) = σr π 1r + µ is the rst quantum correction of the potential, known as Lüscher term.

8 EFT N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Rev. Mod. Phys. 77, 143 (5) Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 8 / 9 Eective eld theories The expression of the potential is given in terms of expectation values of Wilson loop with the insertion of suitable operators, within an expansion in 1/m, m beeing the mass of the heavy quarks. The basic idea behind EFTs is that to describe observables of a particular energy region, one can integrate out the degrees of freedom of the other regions. This produces an eective action involving only the degrees of freedom in the region we are interested in. Heavy quarkonia are characterized by three widely separated scales: the hard scale (the mass m of the heavy quarks); the soft scale (the relative momentum of the heavy quark-antiquark p p mv, v 1); the ultra soft scale (the typical kinetic energy E mv of the heavy quark and antiquark). Moreover, by denition of heavy quark, m Λ QCD.

9 EFT NRQCD and pnrqcd Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 NRQCD and pnrqcd In order to calculate the expression of the heavy quark-antiquark potential up to order 1/m it is necessary to integrate out in two steps the hard and soft scales. The two QCD eective eld theories that arise from integrating out the scales m and mv are called non relativistic QCD (NRQCD) and potential non relativistic QCD (pnrqcd). NRQCD 3 is obtained integrating out the degrees of freedom with energies much larger than mv; has an ultraviolet cut-o Λ which is Λ QCD Λ m; the Lagrangian is organized in powers of 1/m. pnrqcd 4 it is obtained integrating out the degrees of freedom with energies much larger than mv ; has two UV cut-os: mv Λ 1 mv and mv Λ m. 3 W. E. Caswell and G. P. Lepage, Phys. Lett. B 167, 437 (1986) 4 A. Pineda, J. Soto, Nucl. Phys. B 64, 48 (1998)

10 EFT NRQCD and pnrqcd 5 A. Pineda, A. Vairo, Phys. Rev. D 63, 547 (1) [hep-ph/9145] Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 1 / 9 The potentials We are considering a heavy quark located at x 1 and a heavy antiquark located at x, both of mass m. Up to order 1/m the quark-antiquark potential can be written as the sum of three terms 5 where: V () (r) = σr π 1r V = V () + V (1/m) + V (1/m ) is the static potential; V (1/m) (r) = m V (1,) (r) is the 1/m potential; V (1/m) = 1 m { V (,) (r) + V (1,1) (r) } is the 1/m potential.

11 EFT NRQCD and pnrqcd Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 11 / 9 The 1/m potential The 1/m potential is given by: where: V (1,) (r) = 1 g is the coupling constant; E 1 (t) stands for E(t, x 1 ); W / W ; dt t g E 1 (t) g E 1 () c. O 1 (t 1 )O (t ) c = O 1 (t 1 )O (t ) O 1 (t 1 ) O (t ) is the connected correlator with O 1 (t 1 ), O (t ) operators inserted on the Wilson loop at times t 1 t.

12 EST formulation Mapping 6 G. Peréz-Nadal and J. Soto, Phys. Rev. D 79, 114 (9) [hep-ph/811.76] Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 1 / 9 Mapping In order to nd the EST representation of chromoelectric and chromomagnetic operator insertions, it is convenient to give a gauge invariant expression of all the fundamental blocks of the theory. 6 To do so we make use of two Grassmann elds, ψ and χ: ψ annihilates a static source in the fundamental representation at point r/ = (,, r/); χ creates a static source in the anti-fundamental representation at point - r/. The expectation value of the rectangular Wilson loop can be rewritten as: W (T, r) = O( T, r)o ( T, r) O(t, r) = χ (t)φ(t, r ; t, r )ψ(t) where φ(t, r; t, r ) is the straight Wilson line joining r and r at the time t.

13 EST formulation Mapping Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 13 / 9 Mapping The insertion of chromoelectric and chromomagnetic operators correspond to the insertions of the following gauge invariant operators: ψ (t)e i (t, r )ψ(t), χ (t)e i (t, r )χ(t) ψ (t)b i (t, r )ψ(t), χ (t)b i (t, r )χ(t) In order to level out the notation we will denote: ψ (t)f i (t, r )ψ(t) =... Fi 1(t)... χ (t)f i (t, r )χ(t) =... Fi (t)... where F stands for either the chromoelectric or the chromomagnetic eld.

14 EST formulation Mapping Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 14 / 9 Mapping It is now possible to nd a representation of chromoelectric and chromomagnetic operators in terms of string variables. To do so we make use of the global symmetry of the system, the D h group (the symmetries of a diatomic molecule) and time reversal. Rotations with respect to the z-axis E i (t, z) R ij E j (t, z) B i (t, z) R ij B j (t, z) ψ(t) ψ(t), χ(t) χ(t) Reection with respect to the xz-plane E i (t, z) ρ ij E j (t, z) B i (t, z) ρ ij B j (t, z) ψ(t) ψ(t), χ(t) χ(t) Rotations with respect to the z-axis ξ i (t, z) R ij ξ j (t, z) Reection with respect to the xz-plane ξ i (t, z) ρ ij ξ j (t, z)

15 EST formulation Mapping Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 14 / 9 Mapping It is now possible to nd a representation of chromoelectric and chromomagnetic operators in terms of string variables. To do so we make use of the global symmetry of the system, the D h group (the symmetries of a diatomic molecule) and time reversal. CP T E i (t, z) (E i ) T (t, z) B i (t, z) (B i ) T (t, z) ψ(t) χ (t), χ(t) ψ (t) E i (t, z) E i ( t, z) B i (t, z) (B i ) T (t, z) ψ(t) ψ( t), χ(t) χ( t) CP T ξ i (t, z) ξ i (t, z) ξ i (t, z) ξ i ( t, z)

16 EST formulation Mapping Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 15 / 9 Mapping From these symmetry considerations the following mapping is obtained:... E l 1(t)... =... Λ z ξ l (t, r/) E l (t)... =... Λ z ξ l (t, r/) B l 1(t)... =... Λ ɛ lm t z ξ m (t, r/) B l (t)... = Λ ɛ lm t z ξ m (t, r/) E 3 1(t)... =... Λ E 3 (t)... =... Λ B 3 1(t)... =... Λ ɛ lm t z ξ l (t, r/) z ξ m (t, r/) B 3 (t)... = Λ ɛ lm t z ξ l (t, r/) z ξ m (t, r/)... where l and m label the transverse coordinates, l, m = 1, ; ɛ lm is such that ɛ 1 = 1 and ɛ lm = ɛ ml ; Λ, Λ, Λ, Λ are unknown constants of mass dimension and of order Λ QCD. The heavy quark is located at x 1 = (,, r/) and the heavy antiquark at x = (,, r/). E i (t) and B i (t) stand for E(t, x i ) and B(t, x i ).

17 EST formulation Mapping Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 16 / 9 Field correlators It is then possible to write the expectation value of chromoelectric and chromomagnetic elds in terms of derivatives of two points eld correlators G lm (it, z; it, z ) = ξ l (it, z)ξ m (it, z ), which are given by: ( ξ l (it, z)ξ m (it, z ) = δlm 4πσ ln cosh [ π r (t t ) ] + cos [ π r (z + z ) ] ) cosh [ π r (t t ) ] cos [ π r (z z ) ].

18 Potential at zero temperature The 1/m potential Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 17 / 9 V (1,) (r) V (1,) (r) = 1 dt t g E 1 (t) g E 1 () c E l 1(it )E m 1 () c = Λ 4 z z ξ l (it, r/)ξ m (, r/) = ( lm Λ4 cosh[t = δ 4πσ π/r] + cos[(z + z z z ln cosh[t π/r] cos[(z z )π/r]) z=z = r which gives us: E 1 (it ) E 1 () c = πλ4 σr sinh ( ) πt r

19 Potential at zero temperature The 1/m potential Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 17 / 9 V (1,) (r) V (1,) (r) = 1 dt t g E 1 (t) g E 1 () c The above integral becomes: V (1,) (r) = πg Λ 4 4σr ( dt t sinh i πt ) r To compute this integral it is necessary to perform a Wick rotation to imaginary time t it. Moreover it suers from a UV divergence that can be regulated by introducing a cut-o ɛ for small times. V (1,) (r) = g Λ 4 πσ ln(σr ) + µ 1.

20 Potential at zero temperature The 1/m potential Results We are considering the following model: a quark-antiquark pair, both of mass m, bound by the potentials we have just computed. In the centre of mass frame, the Hamiltonian of the system is H = p /m + V. The potential V reads 7 : V (r) σr π 1r + 1 σ m π ln(σr ) 1 9ζ(3)σ r. m π 3 Each term of the heavy quark-antiquark potential up to order 1/m has been compared individually with lattice data, whenever they were available. There still does not exist any numerical simulations that was tted with the complete potential 8. 7 N. Brambilla, M. Groher, H. E. Martinez and A. Vairo, Phys. Rev. D 9, 1143 (14) [hep-ph/ v] 8 We refer to the master thesis of Simone Bacchio for a rst attempt in this direction Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 18 / 9

21 Potential at nite temperature Static potential Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 19 / 9 Static potential In order to insert the temperature in the system under study, let us consider the Euclidean space-time with compactied time direction. In this situation T = 1/L. The description of a heavy quark-antiquark pair is done through the correlator of two Polyakov loops, P(r/)P ( r/) e LVq q(r), which have opposite orientations. L r

22 Potential at nite temperature Static potential Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 / 9 Static potential In the r L limit the expectation value of two Polyakov loops is computable by means of a modular transformation: ln P(r/)P ( r/) LV () (r). The static potential at nite temperature can be obtained using the eective string, and is given by: V () (r) = σ(t )r ( σ(t ) = σ 1 π ) T 3σ T C = 3σ π

23 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 1 / 9 In the L r limit the logarithm of the expectation value of two Polyakov loops is assumed to be: ln P( r )P ( r ( ) L V () (r) + m V (1,) (r) + 1 { V r (,) (r) + V r (r)} ) (1,1). m In the r L limit we will assume that the modular transformation acts on the logarithm of the expectation value of the two Polyakov loops as follow: ln P( r )P ( r ) r ( V () (L) + m V (1,) (L) + 1 m {V (,) (L) + V (1,1) (L)} ).

24 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 / 9 Collecting a common factor L where ln P( r )P ( r ) Lσ(T )r ( ) = L σ () (T ) + σ (1/m) (T ) + σ (1/m) (T ) r, σ () is the string tension in the static case, σ (1/m) = m σ(1,) is the 1/m correction to the string tension, σ (1/m) = 1 m { σ (,) + σ (1,1)} is the 1/m correction to the string tension.

25 Potential at nite temperature 9 A. Allais and M. Caselle, JHEP 91 (9) 73 [hep-lat/81.84v] Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 3 / 9 L r limit In this regime the two string eld correlator can be written as 9 : G lm (w; w ) = δlm πσ log θ 1 [π(w w )/r] θ [π(w + w )/r] where w embed the time and spatial direction of the two Polyakov loops (w = z + iτ), and θ 1 (w) and θ (w) are the Jacobi theta functions dened as: θ 1 (w) = q 1/4 θ (w) = q 1/4 n= ( 1) n q n(n+1) sin[(n + 1)w] n= q n(n+1) cos[(n + 1)w] with: q = e iπτ ; τ = il/r.

26 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 4 / 9 L r limit Taking the L limit, i.e. q we obtain: G lm (τ, z; τ, z ) = δlm sin[ π((z z )+i(τ τ )) r ] ln πσ cos[ π((z+z )+i(τ τ )) r ] Using the properties of goniometric and hyperbolic functions one obtains: [ ] G lm (τ, z; τ, z ) = δlm cosh[ π(τ τ ) r ] + cos[ π(z+z ) r ] ln 4πσ cosh[ π(τ τ ) r ] cos[ π(z z ) r ] which is exactly the two string eld correlator that we used to compute the potential at zero temperature, so all the results that we have already exposed follow from it.

27 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 5 / 9 r L limit In this regime it is convenient to perform a modular transformation τ 1/τ of the expression of the correlator we previously used. In this way one obtains: G lm (w; w ) = δlm ln θ 1 [iπ(w w )/L] πσ θ 4 [iπ(w + w )/L] + δlm Re(w)Re(w ) σ rl where θ 1 (w) and θ 4 (w) are the Jacobi theta functions and w = z + iτ. In the r limit, the correlator becomes: [ G lm (τ, z; τ, z ) = (ln(q δlm 1/4 ) + 1 πσ ln cosh[ π(z z ) L ] cos[ π(τ τ ) L ] This is the expression of the two string elds correlator which will be used to obtain the results in the r L limit. ]).

28 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 6 / 9 The 1/m correction σ (1,) = 1 L L Performing a Wick rotation one obtains σ (1,) = 1 L L dt t g E 1 (t) g E 1 () c. dτ τ g E 1 ( iτ) g E 1 () c.

29 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 6 / 9 The 1/m correction σ (1,) = 1 L L Performing a Wick rotation one obtains σ (1,) = 1 L L dt t g E 1 (t) g E 1 () c. dτ τ g E 1 ( iτ) g E 1 () c.

30 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 6 / 9 The 1/m correction σ (1,) = 1 L L Performing a Wick rotation one obtains σ (1,) = 1 L L dt t g E 1 (t) g E 1 () c. dτ τ g E 1 ( iτ) g E 1 () c. E 1 (τ ) E 1 () c = πλ4 σ L 1 sin ( πτ L )

31 Potential at nite temperature The 1/m correction σ (1,) = 1 L L Performing a Wick rotation one obtains σ (1,) = 1 L The above integral becomes: L σ (1,) (L) = πg Λ 4 σ L 3 dt t g E 1 (t) g E 1 () c. dτ τ g E 1 ( iτ) g E 1 () c. L dτ τ sinh ( πτ L ). This time integration suers from a UV divergence which may be regulated by introducing a cut-o ɛ for small times: ( T σ (1,) (T ) = σ T 4π ln σ ) + µ 1, Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 6 / 9

32 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 7 / 9 Results Putting together the above results, the expression of the heavy quark-antiquark potential up to O(1/m ) at nite temperature is V = σ(t )r with ( σ(t ) σ 1 π T + 1 T 3σ m π ln ( T σ ) 1 ) Aσ. m π 3 It is now interesting to study how the corrections we computed aect the deconnement temperature. We called and numerically solved X = T σ 1 π 3 X + σ m X π ln (X ) σ A m π =. 3

33 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 7 / 9 Results Putting together the above results, the expression of the heavy quark-antiquark potential up to O(1/m ) at nite temperature is V = σ(t )r with ( σ(t ) σ 1 π T + 1 T 3σ m π ln ( T σ ) 1 ) Aσ. m π 3 It is now interesting to study how the corrections we computed aect the deconnement temperature. We called and numerically solved X = T σ 1 π 3 X + σ m X π ln (X ) σ A m π =. 3

34 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 8 / 9 Deconnement temperature

35 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Conclusions The corrections to the string tension that we computed lead to a decrease of the deconnement temperature when the mass of the quarks decreases. This general behaviour is in qualitatively agreement with the one obtained through numerical lattice simulations. The next step will be to quantitatively compare this theoretical prediction with lattice data.

36 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Matching condition and Wilson loop E (x 1, x, p 1, p ) = h s (x 1, x, p 1, p ). Consider a heavy quark of mass m 1 located at x 1 and a heavy antiquark of mass m located at x. The Hamiltonian of NRQCD is expressed as a power expansion in 1/m H = H () + H(1,) m 1 + H(,1) m + H(,) m 1 + H(,) m + H(1,1) m 1 m. Splitting the Hamiltonian as H = H () + H I, the eigenvalues E n (x 1, x ; p 1, p ) of H can be obtained from the static solution by working out formulas analogous to the one used in standard quantum mechanics perturbation theory. Hence, it is possible to write also E n within an expansion in 1/m: E n = E n () + E n (1,) + E n (,1) + E n (,) m 1 m m 1 + E n (,) m + E n (1,1) m 1 m

37 Potential at nite temperature Matching condition and Wilson loop The the Hamiltonian of pnrqcd is: V is the quark-antiquark potential a where: V () (r) = σr π 1r V (1/m) (r) = V (1,) (r) m 1 V (1/m) = V (,) (r) m 1 E (x 1, x, p 1, p ) = h s (x 1, x, p 1, p ). h s (x 1, x, p 1, p ) = p 1 + p + V. m 1 m V = V () + V (1/m) + V (1/m ) is the static potential; + V (,1) (r) m + V (,) (r) m is the 1/m potential; + V (1,1) (r) m 1 m is the 1/m potential. a Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9

38 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 The potentials It is useful to separate in the 1/m potential a spin-dependent (SD) from a spin-independent (SI) part: where V (,) = V (,) SD + V (,) SI, V (,) = V (,) SD + V (,) SI, V (1,1) = V (1,1) SD + V (1,1) SI, V (1,1) SI = 1 { V (,) SI = 1 { V (,) SI = 1 { p 1 p, V (1,1) p } (r) + V (,) L (r) r L 1 + V r (,) (r), } (r) + V (,) L (r) r L + V r (,) (r), } (r) V (1,1) L (r) r (L 1 L + L L 1 ) + V r (1,1) (r). p 1, V (,) p p, V (,) p

39 Potential at nite temperature The potentials It is useful to separate in the 1/m potential a spin-dependent (SD) from a spin-independent (SI) part: where V (,) = V (,) SD + V (,) SI, V (,) = V (,) SD + V (,) SI, V (1,1) = V (1,1) SD + V (1,1) SI, V (,) SD = V (,) LS (r)l 1 S 1, V (,) SD = V (,) LS (r)l S, V (1,1) SD = V (1,1) L (r)l 1S 1 S V (1,1) L (r)l S 1 S 1 + V (1,1) S (r)s 1 S + V (1,1) (r)s 1 (ˆr), and L i = r p i is the orbital momentum of the particle (i = 1) and the antiparticle (i = ), S i = σ i / is the spin operator, with i = 1, and S 1 (ˆr) 3ˆr σ 1ˆr σ σ 1 σ. Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 S 1

40 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Matching condition and Wilson loop At order 1/m: E n (1,) = + 1 () k g E 1 n () k n E n () E () k. Using the matching condition between NRQCD and pnrqcd one obtains: V (1,) = + 1 () k g E 1 () k E () E (). k The next step consists in expressing the above equation in terms of Wilson loop with eld strength component insertions: V (1,) (r) = 1 dt t g E 1 (t) g E 1 () c.

41 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Matching condition and Wilson loop The way to prove the latter equivalence is the following. Consider the connected correlator of two chromoelectric elds. Analogously to what was done for the static potential, a complete set of intermediate states k k () () k is inserted into the Wilson loop average, giving: g E 1 (t) g E 1 () c = k = k () g E 1 k () () k g E 1 () () i(e e k E () )t () g E 1 k () e i(e () k E () )t. Now consider the following integral: 1 () i(e dt t e k E () )t 1. (E () k E () )

42 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 The 1/m potentials V (,) p (r) = i ˆriˆr j dt t g E i 1(t)g E j () c, 1 V (,) L (r) = i 4 (δij 3ˆr iˆr j ) V (,) LS V (1,1) p (r) = c(1) F r i r dt t g E i 1(t)g E j 1 () c, (r) =iˆr iˆr j dt t g E i 1(t)g E j () c, V (1,1) L (r) = i (δij 3ˆr iˆr j ) V (1,1) LS1 V (1,1) S V (1,1) S1 c(1) F (r) = r i r (r) = c(1) F 3 c(1) (r) = F 4 i c() F c() F dt t g B 1(t) g E 1() + c(1) S r r ( r V () ), iˆr iˆr j dt dt t g E i 1(t)g E j () c, dt t g B 1(t) g E (), dt g B 1(t) g B () 4(d sv + d vv C f )δ 3 (r), [ g i1(t)g ] j δij B B () g 3 B 1(t) g B (),

43 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 The potentials V (,) r (r) = πc f α sc (1) D δ 3 (r) ic(1) F 4 V (1,1) dt g B 1(t) g B 1() c + 1 ( ) r V (,) p i t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E 1(t 3) g E 1() c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i 1(t 1)g E 1(t ) g E 1() d (1) f 3 abc d 3 x lim g G µν(x)g a µα(x)g b να(x), c T r (r) = 1 ( ) r V (1,1) p + (d ss + d vsc f )δ 3 (r) t1 t i dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E (t 3) g E () c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i 1(t 1)g E (t ) g E () c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i (t 1)g E 1(t ) g E 1() c.

44 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Field correlators E i 1 (it)ej 1 () c is easily calculated as follows: E l 1(it)E m 1 () c = Λ 4 z z ξ l (it, r/)ξ m (, r/) = ( lm Λ4 cosh[(t t = δ 4πσ )π/r] + cos[(z + z z z ln cosh[(t t )π/r] cos[(z z )π/r]) z=z = r t = which gives us: E i 1(it)E j () 1 c = δ ij πλ 4 ( πt ) 4σr sinh r where δ ij = for i or j = 3 and δ ij = δ ij for i, j = 1,. This is due to the fact that the third component of the chromoelectric eld is mapped into a constant, and so the quantity E 3 1 (it)e3 1 () c is equal to zero.

45 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Field correlators For the other correlators one proceeds in a similar way and obtains: E i 1(it)E j () c = δ ij πλ 4 4σr B l 1(it)B l Λ 1() c = π3 4σr 4 l=1 ( πt ), cosh r sinh 4 ( πt r ( πt ), sinh 6 r ) [ + cosh ( πt )], r Λ B 3 1(it)B 3 1() c = π4 16σ r 6 [ Λ ( E 1(it 1) E 1(it )E 1(it 3) E 1() 8 c = π sinh πt ) ( ) sinh π(t1 t 3) 8σ r 4 r r ( + sinh πt1 ) ( sinh π(t t 3) r r [ Λ ( E 1(it 1) E 1(it )E (it 3) E () 8 c = π cosh πt ) ( ) cosh π(t1 t 3) 8σ r 4 r r ( + cosh πt1 ) ( cosh π(t t 3) r r )], )].

46 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V (,) Potential at nite temperature (r) and V (1,1) (r) p p V (,) p (r) = i ˆriˆr j dt t g E i 1(t)g E j () 1 c The correlator of two chromoelectric elds is contracted with r = (,, r) and hence vanishes because r i δ ij =. So one obtains: V (,) p (r) =. For the same aforementioned reason one gets: V (1,1) p (r) =.

47 V (,) Potential at nite temperature (r) and V (1,1) (r) L L V (,) L (r) = i 4 δij dt t g E i 1(t)g E j () 1 c. In order to compute this integral one has to proceed in the same way followed for the 1/m potential. V (,) L (r) = rg Λ 4 π σ dx X sinh X. Since the integration over X is one of the representations of the Riemann zeta-function of argument 1, nally one obtains: In a similar way one gets: V (,) L (r) = g Λ 4 6σ r. V (1,1) L (r) = g Λ 4 6σ r. 1 The representation to which we are referring is ζ(s) = s+1 1 t s 4 Γ(s+1) dt, which can be sinh t found in [?]. Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9

48 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (,) (r) Potential at nite temperature r (r) = πc f α sc (1) D δ 3 (r) ic(1) F 4 V (,) dt g B 1(t) g B 1() c + 1 ( ) r V (,) p i t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E 1(t 3) g E 1() c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i 1(t 1)g E 1(t ) g E 1() d (1) f 3 abc d 3 x lim g G µν(x)g a µα(x)g b να(x), c T

49 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (,) (r) Potential at nite temperature r (r) = πc f α sc (1) D δ 3 (r) ic(1) F 4 V (,) dt g B 1(t) g B 1() c + 1 ( ) r V (,) p i t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E 1(t 3) g E 1() c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i 1(t 1)g E 1(t ) g E 1() d (1) f 3 abc d 3 x lim g G µν(x)g a µα(x)g b να(x), c T

50 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (,) (r) Potential at nite temperature r (r) = πc f α sc (1) D δ 3 (r) ic(1) F 4 V (,) i f 3 abc d (1) dt g B 1(t) g B 1() c t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E 1(t 3) g E 1() c d 3 x lim g G µν(x)g a µα(x)g b να(x). c T

51 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (,) (r) Potential at nite temperature r (r) = πc f α sc (1) D δ 3 (r) ic(1) F 4 V (,) i f 3 abc d (1) dt g B 1(t) g B 1() c t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E 1(t 3) g E 1() c d 3 x lim g G µν(x)g a µα(x)g b να(x). c T µ 3 + µ 4 r + µ 5 r + π3 c (1) F g Λ 4 6σ r 5

52 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (,) (r) Potential at nite temperature r (r) = πc f α sc (1) D δ 3 (r) ic(1) F 4 V (,) i f 3 abc d (1) dt g B 1(t) g B 1() c t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E 1(t 3) g E 1() c d 3 x lim g G µν(x)g a µα(x)g b να(x). c T ζ(3)g 4 Λ 8 r π 3 σ

53 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (,) (r) Potential at nite temperature r (r) = πc f α sc (1) D δ 3 (r) ic(1) F 4 V (,) i f 3 abc d (1) dt g B 1(t) g B 1() c t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E 1(t 3) g E 1() c d 3 x lim g G µν(x)g a µα(x)g b να(x). c T V r (,) (r) = ζ(3)g 4 Λ 8 r + µ π 3 σ 3 + µ 4 r + µ 5 r 4 + π3 c (1) F g Λ + πc f α s c (1) D δ 3 (r) 6σ r 5 d (1) 3 f abc d 3 x lim g G µν(x)g a µα(x)g b να(x) c T

54 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (1,1) (r) Potential at nite temperature r (r) = 1 ( ) r V (1,1) p t1 t i dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E (t 3) g E () c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i 1(t 1)g E (t ) g E () c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i (t 1)g E 1(t ) g E 1() c V (1,1) + (d ss + d vsc f )δ 3 (r).

55 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (1,1) (r) Potential at nite temperature r (r) = 1 ( ) r V (1,1) p t1 t i dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E (t 3) g E () c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i 1(t 1)g E (t ) g E () c ( + 1 t1 ) i r dt 1 dt (t 1 t ) g E i (t 1)g E 1(t ) g E 1() c V (1,1) + (d ss + d vsc f )δ 3 (r).

56 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (1,1) (r) Potential at nite temperature V r (1,1) (r) = i t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E (t 3) g E () c + (d ss + d vsc f )δ 3 (r).

57 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (1,1) (r) Potential at nite temperature V r (1,1) (r) = i t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E (t 3) g E () c + (d ss + d vsc f )δ 3 (r). ζ(3)g 4 Λ 8 r π 3 σ

58 Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 V r (1,1) (r) Potential at nite temperature V r (1,1) (r) = i t1 t dt 1 dt dt 3(t t 3) g E 1(t 1) g E 1(t )g E (t 3) g E () c + (d ss + d vsc f )δ 3 (r). V r (1,1) (r) = ζ(3)g 4 Λ 8 r + (d π 3 σ ss + d vs C f )δ 3 (r).

59 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 The 1/m potentials V (,) p (r) =, V (1,1) p (r) =, V r (,) (r) = ζ(3)g 4 Λ 8 r + µ π 3 σ 3 + µ 4 r + µ 5 + πc f α s c (1) D δ 3 (r) d (1) 3 f abc r + π3 c (1) F g Λ 4 6σ r 5 d 3 x V r (1,1) (r) = ζ(3)g 4 Λ 8 r + (d π 3 σ ss + d vs C f )δ 3 (r). lim g G µν(x)g a µα(x)g b να(x), c T

60 Potential at nite temperature 11 N. Brambilla, D. Gromes and A. Vairo, Phys. Rev. D 64, 399 (1) [hep-ph/1468] Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Fixing the coecients Poincaré invariance xes some of the eld normalization constants Λ, Λ, Λ, Λ, because it requires some equations to be exactly fullled by the potentials. For our purposes it is sucient to know that there exists an equation that relates the momentum-dependent potentials with the static potential 11 : r dv () + V (,) L dr V (1,1) L =. This equation is fullled in the EST only if gλ = σ.

61 Potential at nite temperature Carla Marchis (UniTo) O(1/m ) heavy Q Q potential at nite T 9/3/15 9 / 9 Field correlators E 1 (τ) E 1 () c = πλ4 σ L 1 sin ( πτ L ), E 1 (τ 1 ) E 1 (τ )E 1 (τ 3 ) E 1 () c = π Λ 8 E 1 (τ 1 ) E 1 (τ )E (τ 3 ) E () c =. σ L4 [ ( sin πτ ) ( ) sin π(τ1 τ 3 ) L L ( ) ( )] πτ + sin 1 sin π(τ τ 3 ), L L