In the present work we introduce a holographic prescription to independently describe sea and valence quarks in the context of the gaugegravity correspondence. We use such prescription to perform an initial calculation that permits us to compare our results with those obtained through lattice techniques when studding magnetic catalysis and its inverse. We find in agreement with previous studies that the elaborated behavior of the condensate is mostly attributable to the sea quarks rather than the valence which show a quite featureless participation.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']
Identifying sea and valence quarks in a magnetically driven catalysis
{'Identifying sea and valence quarks in a magnetically driven catalysis.': 'Daniel Kosoi 1, ∗ and Leonardo Patiño 1, ‡  \n1 Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal 70-542, CP 04510, Ciudad de México, México.  \nIn the present work we introduce a holographic prescription to independently describe sea and valence quarks in the context of the gauge/gravity correspondence. We use such prescription to perform an initial calculation that permits us to compare our results with those obtained through lattice techniques when studding magnetic catalysis and its inverse. We find, in agreement with previous studies, that the elaborated behavior of the condensate is mostly attributable to the sea quarks, rather than the valence which show a quite featureless participation.', 'I. INTRODUCTION AND MAIN RESULTS': 'Since its first form, speculated in the 1970s, the phase diagram of Quantum Chromodynamics (QCD) has become increasing more complex, considering dependencies on parameters other than the temperature and the baryon chemical potential, like the masses of the quarks or separated chemical potential for each flavor. These and other elements have been incorporated [1-3] into the description that is conjectured to be provided by the gauge/gravity correspondence [4]. The motivation to use such tool is that there are regions of the aforementioned phase diagram that escape the usual perturbative treatment of quantum field theory, making the so called holographic calculations an appealing alternative. Another agent that has proven to have a relevant influence on the shape of the phase diagram is an intense external magnetic field, expected to be generated in non-central high energy collision experiments where the state of matter known as quark gluon plasma (QGP) is produced. Even the most conservative estimations predict an intensity of 10 -1 m π 2 for this magnetic field, while the more extreme go as high as 15 m π 2 [5]. Regardless of the specific value, any intensity in this range is guarantied to have consequences not only on the phase diagram, but also importantly, on the interpretation of experimental results, so a lot of effort has been placed to understand the related phenomenology.  \nA five dimensional background was constructed in [6] to model a strongly coupled plasma subjected to an external magnetic field, and it was latter uplifted to ten dimensions [7] so that fields in the fundamental representation could be incorporated. In this latter work we showed that the quasinormal modes of a fundamental scalar operator accommodated themselves in Landau like levels in the reference frame of the plasma. In the present letter we step away from the rest frame of the plasma so that a dispersion relation can be extracted, exhibiting that these modes indeed behave like quasi-particles, and letting us evaluate the impact of the magnetic field over their condensate at either constant kinetic momentum  \nor magnetic field. The benefit of independently controlling these two parameters is that sea quarks and valence quarks react differently to the presence of an intense magnetic field [8]. The field theory expressions in this latter article allow us to identify the gravitational dual to the condensate of both kinds of quarks, and use this to show that the valence condensate in particular depends solely on the combination of canonical momentum and magnetic field specifically given by the kinetic momentum.  \nThe final result turns out to be holographically quite intuitive, indicating that the effect of the magnetic field over the sea quarks is codified in the back reaction of the geometry to its presence, while that over the valence quarks is reflected on the impact over the perturbations of the flavor branes.  \nSince our analysis is done using the behavior of the quark condensate as characterizing element, we seize the opportunity to exhibit the existence of inverse magnetic catalysis and magnetic catalysis for different ranges of the magnetic field, consistently with the results reported in [9].  \nAlong the way we supplement the study of the quasiparticles displaying their width as a function solely of either the magnetic field or the kinetic momentum, showing that the former is always destabilizing, while the latter has the opposite effect.', 'II. GRAVITATIONAL MODEL AND PREVIOUS RESULTS': "To provide a gravitational dual of a strongly coupled plasma with degrees of freedom in the fundamental representation subject to a constant magnetic field, in [7] we constructed the ten dimensional uplift of the five dimensional background introduced in [6], and embedded a D7-brane on it. This was done in such a manner that the fundamental degrees of freedom were massless, and left us proceeded to study the perturbations of the brane that are dual to scalar excitations of said fundamental fields in the gauge theory.  \nOur results in [7] were obtained using either gauge, Landau A = Bxdy or symmetric A = B/ 2( xdy -y dx ) , to introduce the magnetic field B = d A , but for the sake of concreteness, we will employ the former in what follows. Once Ladau gauge has been adopted, the ten  \ndimensional metric, consistent with the symmetries introduced in [6], reads  \nds 2 10 = -U ( r ) dt 2 + 1 U ( r ) dr 2 + V ( r ) ( dx 2 + dy 2 ) + W ( r ) dz 2 + [ dθ 2 +sin 2 θd ˜ ϕ 2 1 +cos 2 θ ( dϑ 2 +sin 2 ϑd ˜ ϕ 2 2 +cos 2 ϑd ˜ ϕ 2 3 )] , (1)  \nwith d ˜ ϕ i = dϕ i + 2 √ 3 Bxdy . The directions t, x, y and z are dual to those of the spacetime where the field theory lies, while r is the radial holographic coordinate, on which the functions U ( r ) , V ( r ) , and W ( r ) , depend solely, and the compact directions are described by the second line in (1).  \nFor stability reasons [7], the brane is embedded to extend in the r, t, x, y, z directions and wrap the 3-cycle at the end of (1) given by ϑ, ϕ 2 , and ϕ 3 . In this manner, the embedding can be described placing the brane at constant ϕ 1 and writing θ as a function of r . From (1) we notice that as usual in this descriptions, it is convenient to use χ ( r ) = sin[ θ ( r )] to study the profile of the brane. The object of interest to our work are the perturbations δχ to the aforementioned embedding, of which in [7] we took the particular case χ ( r ) = 0 so that the resulting equation for the brane perturbations, when expressed as the product δχ ( r, t, x, y, z ) = χ t ( t ) χ x ( x ) χ y ( y ) χ z ( z ) χ r ( r ) /χ r ( r ∞ ) , reduced to  \n[ 3 UVW ' χ ' r +6 W ( V U ' χ ' r + UV ' χ ' r + UVχ '' r ) (2)  \n+6 V W ( 3 -∂ 2 t χ t Uχ t + ∂ 2 y χ y V χ y + ∂ 2 z χ z Wχ z ) χ r ] χ x + Wχ r ( 6 ∂ 2 x χ x -8 B 2 x 2 χ x ) = 0 ,  \nwith a general solution that includes the factor χ t ( t ) χ y ( y ) χ z ( z ) = e -i ( ωt -k y y -k z z ) , and which can be further separated as  \n( 3 + ω 2 U -k 2 y V -k 2 z W ) V Wχ r + 1 2 UVW ' χ ' r + W ( V U ' χ ' r + UV ' χ ' r + UVχ '' r ) = 2 E Wχ r , (3)  \nand  \n1 2 [ -∂ 2 x χ x + e 2 B 2 x 2 χ x ] = E χ x , (4)  \nIn the expressions above e = 2 √ 3 , and we have taken into account that U ( r ) , V ( r ) , and W ( r ) , are only functions of the radial coordinate r , denoting the differentiation with respect to the latter by a prime.  \nThe details of the ten dimensional background, and the embedding of the brane, can be found in [7], but for completeness we should mention that there is a 5form that plays a relevant role in the construction of the uplift, but has no effect on the embedding nor on our current calculations, and that the 1-forms d ˜ ϕ i = dϕ i +  \n2 √ 3 Bxdy in (1) show that, as the U (1) field is encoded in these compact directions, the quantity e = 2 √ 3 correctly represents the charge with respect to such fields.  \nWe began our previous study of the equations above noticing that for the solutions to describe acceptable embeddings, the separation constant E had to take values on the discrete spectrum  \nE n = ( n + 1 2 ) ω c , (5)  \nwhere ω c = e B was identified as the cyclotron frequency and the integer n with the Landau level number.  \nWe continued by focusing on the k y = k z = 0 case, and looked for the complex values of ω for which the solutions to (3) were ingoing and had a normalizable profile in the radial direction, since these are dual to quasinormal modes of the scalar excitations.  \nIt was the energy obtained from the latter frequencies that we proved to closely follow the dispersion relation characteristic of Landau levels, presenting a small deviation as expected from modes that are not fully stable. Furthermore, when we studied the ratio of the with of the states over their energy as a function of the magnetic field, we observed a behavior reminiscent of magnetic catalysis for large intensities of such field, and inverse magnetic catalysis for small ones, all with respect to an identifiable critical intensity.", 'III. ROLL OF THE SEA AND VALENCE QUARKS IN (INVERSE) MAGNETIC CATALYSIS': 'The difference between the rolls that sea and valence quarks play for (inverse) magnetic catalysis has been pointed out in investigations concerning the origin of such phenomenon. The aim of the present letter is to exploit this difference to identify the elements of our gravitational construction that are dual to each of these types of quarks, or at least, that codify the two different effects.  \nIsolating the effect of the magnetic field was not possible in our previous work, where the calculations were made in the rest frame of the plasma and as the intensity of said field became larger, its direct impact was certainly augmented, but also unavoidably increased the kinetic momentum associated to each Landau level.  \nIn what follows we will consider more general frames by allowing nonzero values of k z in (3). Since the square of the kinetic momentum of the n th Landau level in this scenario is given by k K 2 = 2 e b ( n +1 / 2)+ k 2 z , introducing a non-vanishing k z permits the exploration of the properties of the quasinormal modes as functions of the intensity of the magnetic field in a range 0 ≤ b ≤ k K / 2 e ( n +1 / 2) , while the kinetic momentum is kept constant by varying k z from k K to 0.  \nSince b explicitly appears in (3), it is clear that changing its value has a direct impact on this equation, but  \njust as important is the implicit modification due to the fact that the form that U ( r ) , V ( r ) , and W ( r ) have as functions of r depends on the intensity of the magnetic field. Since this metric functions are only known to us numerically, we need to resource to such methods to solve Eq. (3).', 'IV. FINDING THE QUASINORMAL FREQUENCIES': "As can be consulted in [7], close to the horizon the metric functions are given by the expansions  \nU ( r ) = 6 r h ( r -r h ) + ∞ ∑ i =2 U i ( r -r h ) i , V ( r ) = ∞ ∑ i =0 V i ( r -r h ) i , W ( r ) = 3 r h 2 ( r -r h ) 0 + ∞ ∑ i =1 W i ( r -r h ) i , (6)  \nwhich through Eq. (3) show that the radial profile of the embedding behaves like ( r -r h ) iα , with α = ± ω 4 πT = ± ω 6 r h , in the r → r h limit. The ingoing wave requirement is imposed by choosing the negative sign for α and approximating χ ( ω,n ) r ( r ) near the horizon through the resulting series  \nχ ( ω,n ) r ( r ) ≃ ( r -r h ) -i ω 6 r h χ (0) r [ 1 + C (1) ( r,ω,n ) ( r -r h ) + C (2) ( r,ω,n ) ( r -r h ) 2 + O ( r -r h ) 3 ] , (7)  \nwhere χ (0) r is a free global factor due to the linear character of (3), while  \nC (1) ( r,ω,n ) = 1 108 r h 2 V ( r h ) 2 (3 r h -iω ) { b 2 ω (5 ω -3 ir h ) + 18 k 2 z V ( r h ) 2 -6 V ( r h ) [ -18( n +1 / 2) e br h 2 + V ( r h ) ( 27 r h 2 -15 ir h ω + ω 2 )] } C (2) ( r,ω,n ) = 1 23328 r h 4 V ( r h ) 4 (18 r h 2 -9 ir h ω -ω 2 ) { b 4 ω ( -252 r h 2 ω -468 ir h 3 +35 ir h ω 2 +25 ω 3 ) +324 k 4 z V ( r h ) 4 +12 b 2 V ( r h ) [ 18( n +1 / 2) e br h 2 ( 36 r h 2 -12 ir h ω +5 ω 2 ) + V ( r h ) ( -459 r h 2 ω 2 -288 ir h 3 ω +972 r h 4 +97 ir h ω 3 -5 ω 4 ) ] + 36 V ( r h ) 2 [ 324( n +1 / 2) 2 e 2 b 2 r h 4 -36( n +1 / 2) e br h 2 V ( r h ) ( 99 r h 2 -24 ir h ω + ω 2 ) + V ( r h ) 2 ( -126 r h 2 ω 2 -2268 ir h 3 ω +2673 r h 4 -45 ir h ω 3 + ω 4 ) ] -36 k 2 z V ( r h ) 2 [ B 2 (72 r 2 h -24 ir h ω -5 ω 2 ) +6 V ( r h )( -18( n + 1 2 ) ebr 2 h + V ( r h )(99 r 2 h -24 ir h ω + ω 2 ))] } . (8)  \nIn the asymptotic region r →∞ the metric functions  \nare described by  \nU ( r ) = r 2 + U 1 r + U 2 1 4 + 1 r 2 ( U -2 -2 3 b 2 log r ) + U 1 1 r 3 ( -U -2 -1 3 b 2 + 2 3 b 2 log r ) + O ( 1 r 4 ) , V ( r ) = r 2 + U 1 r + U 2 1 4 + 1 r 2 ( V -2 + 1 3 b 2 log r ) + U 1 1 r 3 ( -V -2 + 1 6 b 2 -1 3 b 2 log r ) + O ( 1 r 4 ) , W ( r ) = r 2 + U 1 r + U 2 1 4 + 1 r 2 ( -2 V -2 -2 3 b 2 log r ) + U 1 1 r 3 ( 2 V -2 -1 3 b 2 + 2 3 b 2 log r ) + O ( 1 r 4 ) , (9)  \nand the radial profile therefore by  \nχ ( ω,n ) r ( r ) ≃ χ r ( -1) [ 1 r -U 1 2 r 2 + ( ω 2 -k 2 z -2( n +1 / 2) e b ) log r 2 r 3 -3 U 1 ( ω 2 -k 2 z -2( n +1 / 2) e b ) log r 4 r 4 + U 1 ( ω 2 -k 2 z -2( n +1 / 2) e b + U 1 2 ) 1 4 r 4 ] + χ r ( -3) [ 1 r 3 -6 U 1 1 4 r 4 ] + O ( 1 r 5 ) , (10)  \nas can be verified by substituting (9) in (3).  \nThe expansion coefficients χ ( -1) r and χ ( -3) r are respectively related to the source and the vacuum expectation value of the excitation dual to χ . The quasinormal modes of such excitation are given by the normalizable solutions, identified as those for which χ ( -1) r vanishes, since as can be seen in (10), this is the coefficient that multiplies the no-normalizable part of the radial profile.  \nThe quest now is to obtain the quasinormal frequencies as functions of the dimensionless parameters b/T 2 and k K 2 /T 2 . To this end, we notice that the specific behavior of χ ( r ) over any member of our family of backgrounds, at a given temperature T and intensity b of the magnetic field, is parametrized by ω, k y , k z , and n . We fix our attention on the lowest Landau level set by n = 0 in (5), and chose k y = 0 in (3) while employing k z as described above to select a value for k K 2 = e b + k z 2 . Numerically solving the latter equation under these circumstances and near horizon conditions given by (7) for a particular value of ω , permits us to use the asymptotic behavior of the solution χ ( r ) to extract the coefficients χ ( -1) r ( ω ) and χ ( -3) r ( ω ) corresponding to this frequency. Performing such integration for values of ω that explore the complex plane, starting at the origin and searching for the nearest locus Re[ χ ( -1) r ( ω ) ]=0 and Im[ χ ( -1) r ( ω ) ]=0, leads us to extract the frequency of the quasinormal mode as the value at which these lines intersect. Repeating the procedure above for several k K 's  \nover a series of backgrounds with a range of temperatures and intensities for the magnetic field renders the desire function ω ( b/T 2 , k K 2 /T 2 ) .", 'V. THE COMPLEX CONDENSATE': "Once the quasinormal frequencies have been found, they can be used to determine the associated condensate dual to the coefficient χ ( -3) r . The calculation results in a complex function of b/T 2 and k K 2 /T 2 .  \nTo understand the meaning of both parts, real and imaginary, in the dual gauge theory, we first note that even if we have kept our focus on δχ ( r, x, y, z ) = sin [ δθ ( r, x, y, z )] , the embedding of the D7-brane also accepts perturbations δϕ ( r, x, y, z ) over its position in ϕ 1 that we fixed as part of our construction. The holographic dictionary developed in Appendix A of [10] describes how δχ (equal there to -δχ ) and δϕ are respectively related to scalar and pseudoscalar excitations in the gauge theory. To exhibit the correspondence stated above, the authors in [10] use the near boundary expansion (10) and the fact that the lowest component of the massless modes Φ 7 , 7 of the open string sector stretching from the D7-brane to itself, is a complex scalar Φ 0 7 , 7 = 1 √ 2 ( X 8 + iX 9 2 πℓ 2 s ) that describes the fluctuations X 8 and X 9 of such brane in the 2-dimensional space perpendicular to it. If the fiducial embedding is taken at ϕ 1 = 0 , δϕ corresponds precisely to the phase of the scalar field Φ 0 7 , 7 , while if δχ remains real it can be roughly thought as the modulus.  \nWe have indeed set ϕ 1 = 0 because, as described in [7], a consistent solution to the equations of motion can be found by studying the δχ mode with the δϕ mode turned off. Even though this certainly restricts the leading order of the perturbation near the boundary to only capture information of the dual scalar excitation, the complex nature of the radial equation leads to solutions χ ( ω,n ) r ( r ) that become complex as soon as they enter the bulk. This is why the coefficient χ ( -3) r is a complex number, which imaginary part is dual to the condensate of the pseudoscalar excitation in the gauge theory, because it constitutes the right subleading perturbation to the phase of the scalar field Φ 0 7 , 7 = 1 √ 2 ( X 8 + iX 9 2 πℓ 2 s ) that δϕ would directly produce. The reading of the condensate of the dual scalar excitation as the real part of -√ λN f N c T 3 χ ( -3) r / 8 , where λ is 't Hooft coupling, N f the number of D7branes, N C the number of black D3-branes, and T the temperature, is unchanged from the one in [10], while the one we have just provided about the imaginary component of the same expression is obtained from the variation, consisting of a phase rotation, of its respective Lagrangian.", 'VI. VALENCE AND SEA CONTRIBUTIONS TO THE CONDENSATE': "As we mentioned previously, there have been several works studying the valence and sea quarks effects in (inverse) magnetic catalysis, and particularly in [8] a way to separate and explore each effect by itself is presented. Unlike our case, in said work the field under consideration is fermionic, and its condensate is determined to be given by  \nψψ ( b ) = 1 Z ( b ) ∫ D Ue -S g det ( / D ( b ) + m ) Tr ( / D ( b ) + m ) -1 ,  \nwhere Z ( b ) = ∫ D Ue -S g det ( / D ( b ) + m ) . (11)  \nFrom this expression the valence and sea condensates are defined as  \nψψ val ( b ) = 1 Z (0) ∫ D Ue -S g det ( / D (0) + m ) Tr ( / D ( b ) + m ) -1 , ψψ sea ( b ) = 1 Z ( b ) ∫ D Ue -S g det ( / D ( b ) + m ) Tr ( / D (0) + m ) -1 , (12)  \nreflecting that in the presence of a magnetic field, the valence effect is codified in a shift of the lower mode of the Dirac operator, while the sea quarks affect the condensate through the fermion determinant. It is also shown, using lattice QCD techniques, that the valence quarks only contribute to magnetic catalysis by enhancing the condensate, but the sea contribution suppresses it, promoting quiral symmetry restoration and presenting inverse magnetic catalysis. This last phenomenon being more noticeable when near the critical temperature and independent of the central value around which the gauge field is explored.  \nMotivated by the distinction above to create an entry of the holographic dictionary, we will employ our construction to identify the shift in spectrum as the effect on the perturbation δχ caused by the explicit appearance of b in (3), and the change in the integration measure as dual to the backreaction of the bulk geometry (1) to such magnetic field.  \n̸  \nIn the scalar case that we are working on this can be concretely implemented by holographically computing the valence condensate < O m > val ( b ) using the coefficient χ ( -3) ( b ) extracted from the solution to (3) at b = 0 , but with U 0 , V 0 , and W 0 the metric functions of a black D3-brane at b = 0 .  \nConversely, for the sea condensate < O m > sea ( b ) the coefficient χ ( -3) ( b ) that should be used in our holographic calculation is the one extracted from the solution  \nto (3) at b = 0 , but with U, V, and W the metric functions that have fully backreacted to the presence of the magnetic field.  \nGiven that the valence condensate is inherently easier to calculate, we follow the same strategy as in [8] and study it along with the complete condensate. The procedure described above to compute the valence condensate simplifies (3) to  \n[3 r 3 ( r 4 -1) + r 5 ω 2 ] χ r +(1 -6 r 4 +5 r 8 ) χ ' r + r ( r 4 -1) 2 χ '' r = r ( r 4 -1)(2 E + k z 2 ) χ r , (13)  \nwhere the metric functions of the black D3-brane have been used and k y has been set to zero as in previous sections.  \nFrom (5) we see that the quantity 2 E + k z 2 in parenthesis on the right hands side of (13) is the kinetic momentum k K 2 = eb + k z 2 for n = 0 , showing that the valence condensate depends on k z and b only through k K , and not on their independent values.", 'VII. RELATION TO THE QUASI-PARTICLE MODELS': 'A quasi-particle model of the quark-gluon plasma [11] has been successfully used to describe certain aspects of it [12], including scenarios where it is magnetized [13].  \nBefore we study the behavior of the condensate, it is interesting to see that the quasinormal modes in our work follow a dispersion relation consistent with said model. To this end, we would like to display the energy of the mode as a function of the kinetic momentum at constant magnetic field. Since the kinetic momentum k K is defined by the relation k K 2 = eb + k z 2 , isolating the effect that it has on the quasinormal modes is a simple task, as it suffice to explore the result of changing k z while keeping b constant. To graphically compare the results for different values of b/T 2 , it turns out to be convenient to compensate for the shift that the field introduces, so the results will be plotted as functions of ( k K 2 -eb ) /T 2 .  \nIn figure 1 we show the energy of the mode, given by the real part of the quasinormal frequency squared, as a function of the kinetic momentum. We notice that it is monotonically increasing and becomes linear for large values of k K 2 /T 2 . One can observe that the dispersion relation for a massless transversal field in a thermal theory [14], characterized by the asymptotic behaviors  \nRe[ ω 2 ] ≃ ω p 2 + αk K 2 for k K ≪ eT, and Re[ ω 2 ] ≃ m T 2 + k K 2 for k K ≫ eT, (14)  \nis recovered. This behavior is a strong indication that the quasinormal modes are indeed acting like quasi-particles in the plasma.  \nIn figure 2 we present the width m 0 Γ = 2 Re [ ω ] Im [ ω ] of the mode over its energy E = √ Re [ ω 2 ] . The curves  \nFIG. 1: Real part of the squared quasinormal frequency as a function of the kinetic momentum k K 2 at fixed magnetic field intensity. The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to b/T 2 = { 0 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively.  \n<!-- image -->  \nFIG. 2: Width of the unstable states dual to the quasinormal modes as a function of the kinetic momentum ( k K 2 -eb ) /T 2 at fixed magnetic field intensity. The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to b/T 2 = { 0 , 4 π 2 / 9 , 16 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively.  \n<!-- image -->  \ndecrease with increasing kinetic momentum, consistent with the expected stabilizing effect. We also observe that the plots accommodate from bottom to top for increasing values of the magnetic field, suggesting a destabilizing effect that shall become clear when inspecting the behavior as a function of b .  \nAs mentioned before, for any given value k K of the ki-  \nnetic momentum and not to exceed it, the magnetic field intensity can only be in the interval 0 ≤ b ≤ k K /e . To be able to make comparisons for plots at different values of fixed kinetic momentum, we normalize the horizontal axis by the maximum value that b can take.  \nIn figure 3 we show the dependence of Γ /E on b/b max at fixed k K 2 /T 2 . The curves are monotonically increasing for every value of the momentum, reinforcing that the magnetic field has a destabilizing effect on the quasiparticles. It is of note that as k K 2 grows, the graphs accommodate form top to bottom, consistently with the notion that the kinetic momentum has a stabilizing effect.  \nFIG. 3: Width of the unstable states dual to the quasinormal modes as a function of the magnetic field b/T 2 at fixed kinetic momentum. The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to k K 2 /T 2 = { 2 π 2 / 45 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively.  \n<!-- image -->  \nPart of our motivation to work at fixed kinetic momentum becomes apparent on the light of the conclusions above, since had we increased b without using k z to compensate the change in k K , we would have seen the result of competing destabilizing and stabilizing effects, given that b also augments the kinetic momentum.', 'VIII. SCALAR CONDENSATE': 'We now present the results for the scalar condensate. In figure 4 we display the dimensionless [10] quantity -8 ⟨ O m ⟩ √ λN f N c T 3 at fixed b/T 2 as a function of ( k K 2 -eb ) /T 2 , where the choice of variables is for the same reasons as above. The curves are all monotonically increasing, showing that the kinetic momentum promotes spontaneous symmetry breaking for the scalar excitation when both, valence and sea contributions are considered.  \nIn figure 5 the full scalar condensate is shown as a function of the magnetic field at constant kinetic momentum.  \nWe can see that for certain values of k K 2 /T 2 , there is an inflection point as b/T 2 gets closer to its maximal value, and the condensate begins to decrease. The fact that this phenomenon appears only for the plots at larger k K 2 /T 2 indicates that such inverse magnetic catalysis is associated to large magnetic fields, that as we explained, are only achievable at large kinetic momentum.  \nTo separate now the valence condensate we follow the ideas developed in section VI, where we proposed for the valence contribution to be calculated by turning off the magnetic field in the background, while leaving it intact in the equations for the D7-brane perturbation. As we saw, this made the quasinormal modes to depend only on the kinetic momentum. To verify that we had indeed set to zero the field in every place of our code where it was necessary, we numerically confirmed that the same results were obtained for any given k K 2 /T 2 , regardless of the value of b leading to it and imputed in (13) through the substitution of E = 1 2 eb .  \nThe observation in the previous paragraph means that the behavior of the valence condensate of the scalar excitation as a function of the kinetic momentum at any constant magnetic field, is fully captured by the b/T 2 = 0 case in figure 4. This plot recovers the result in [8] that determines the effect of increasing the kinetic momentum to be an enhancement of the valence condensate, promoting spontaneous symmetry breaking.  \nIt is important to acknowledge that in contrast to the results reported in [8], where the valence condensate is a convex function of the kinetic momentum, our curve is concave instead. We believe this difference to arise from the sensibility of the calculations to the value of the quark mass, since while in the aforementioned paper a small mass approximation is used, we work on the quiral limit where m q = 0 .  \nFIG. 4: Scalar condensate as a function of the momentum ( k K 2 -eb ) /T 2 at fixed magnetic field intensity b/T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to  \n<!-- image -->  \nb/T 2 = { 0 , 4 π 2 / 9 , 16 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively.  \nFigure 7 shows the pseudoscalar condensate as a function of the magnetic field at fixed kinetic momentum. For the plots at lower fixed values of k K 2 /T 2 the only effect that we observe is enhancement, in the sense that the magnitude of the condensate grows when increasing the intensity of the field, preventing restoration of the quiral symmetry. Nonetheless, for larger values of k K 2 /T 2 there is a point after which increasing the magnetic field reduces the magnitude of the condensate, getting closer to an spontaneous restoration of quiral symmetry, and thus presenting inverse magnetic catalysis.  \n<!-- image -->  \nFIG. 5: Scalar condensate as a function of the magnetic field intensity b/T 2 at fixed kinetic momentum k K 2 /T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to k K 2 /T 2 = { 2 π 2 / 45 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively.  \n<!-- image -->', 'IX. PSEUDOSCALAR CONDENSATE': 'FIG. 6: Pseudoscalar condensate as a function of the momentum ( k K 2 -eb ) /T 2 at fixed magnetic field intensity b/T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to b/T 2 = { 0 , 8 π 2 / 9 , 16 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively.  \n<!-- image -->  \nThe last result we present before the discussion of our findings is what we concluded in section V to be the pseudoscalar condensate, graphed below in figure 6 as a function of ( k K 2 -eb ) /T 2 for fixed b/T 2 . We see that the value of this condensate starts negative, signaling an instability in the brane embedding [15], at some point it starts growing with ( k K 2 -eb ) /T 2 until it becomes positive, and then keeps rising. This again shows that the kinetic momentum has a stabilizing effect over the quasiparticles. As shown in figure 6, for the lower and higher values of b/T 2 that we explored, b/T 2 = { 0 , 4 π 2 / 9 , 40 π 2 / 9 } , this  \nFIG. 7: Pseudoscalar condensate as a function of the magnetic field intensity b/T 2 at fixed kinetic momentum k K 2 /T 2 . The dotted, dot dashed, dashed (medium), dashed (large) and continuous curves correspond to k K 2 /T 2 = { 2 π 2 / 45 , 4 π 2 / 9 , 8 π 2 / 3 , 40 π 2 / 9 , 64 π 2 / 9 } respectively  \npseudoscalar condensate has an inflection point now close to ( k K 2 -eb ) /T 2 = 0 , that is, it decreases at first with ( k K 2 -eb ) /T 2 , and after a certain value of this variable it begins to increase monotonically. For the intermediate magnitudes, b/T 2 = { 16 π 2 / 9 , 8 π 2 / 3 } , the value of such condensate always increases with the kinetic momentum.  \nAs in the scalar case, the valence condensate depends solely on the kinetic momentum, therefore changing b while keeping k K fixed will have no effect, and the behavior with the latter as a variable is fully capture by the plot at b = 0 in figure 6.', 'X. DISCUSSION AND FUTURE WORK': 'The main objective of the preset work was to take advantage of a background with a fully backreacted magnetic field, and use the relevance that the presence of such field has in high energy collisions, to introduce a holographic prescription that distinguishes valence from sea quarks in the QGP, and perform a first test of such prescription.  \nAs part of the construction we verified in section VII that the modes used in our study followed the dispersion relation expected for a massless transversal field in a  \nthermal theory, allowing them to be interpreted as quasiparticle states. In passing we showed that for these quasiparticles, momentum has an stabilizing effect while the magnetic field destabilizes them.  \nIntended to use it as an analyzing tool, we stopped and noticed that the condensate of both the scalar and pseudoscalar excitations have a nonzero value. In section V we pointed out that from the perspective of the gravity side, this happens because of the complex nature of the perturbation equations of an embedded brane that crosses the horizon of the background. It is interesting now to interpret the above in the field theory model of pseudo-particle as a consequence of their finite width, causing that even if the excitation is exclusively done on the scalar, it induces a nonzero value for both scalar and pseudoscalar condensates.  \nIt is at this point that the importance of using our gravitational configuration becomes evident. It comes from our holographic prescription which assumes that the physics of the sea quarks is captured by the effect of the background on the embedding, while that of the valence quarks is encoded explicitly in the perturbation equations, provoking the necessity to create a scenario where such entities can be modified independently. That is exactly what a brane embedded in a background that includes a fully backreacted magnetic field provides, since it enabled us to change the background to independently probe its effect on the perturbations of the embedding, or modify the value of the intensity of the field directly in the embedding equation to see what results of this action.  \nThe procedure above lead us to the particular result of an equation to describe the valence excitations where the canonical momentum and the magnetic field exclusively enter through their combination given by the kinematic momentum k K 2 = eb + k z 2 of the lowest level n = 0 . This is one of the main reasons why among our goals was to have control over k K and generate data keeping it constant while varying b .  \nWith all of the above in hand, the numerical results for the scalar condensate showed that when in a constant magnetic field, the effect of the momentum was to drag the system away from quiral symmetry restoration. Differently, when working at fixed k K 2 /T 2 , and for high enough values of such constant, the curves exhibited both, enhancement of the condensate at low values of b in comparison to its maximal value b max = k K 2 /e , and suppression, with respect to the maximal point, as the intensity of the field approaches such upper limit. This increasing and decreasing behavior when augmenting the intensity of the field is precisely what is respectively termed magnetic catalysis and inverse magnetic catalysis, but before we say more about it, let us revisit another result.  \nAs mentioned earlier, the fact that the valence condensate depends only on k K means that the numerical results for its scalar part are described by the curve at b = 0 in figure (4). What is also true is that therefore this curve can be seen equally well as depicting the valence con-  \nFIG. 8: Total (dotted) and valence (points) scalar condensates as a function of the magnetic field intensity b/T 2 in the rest frame of the plasma. The thick line corresponds to the best fitting second order polynomial for the valence condensate data  \n<!-- image -->  \n6 . 05 × 10 -2 +1 . 86 × 10 -4 ( b/T 2 ) -4 . 86 × 10 -7 ( b/T 2 ) 2  \ndensate calculated in the rest frame as a function of the magnetic field, making it actually usable for comparisons with measurements performed in such frame. Furthermore, since the curve so obtained only presents magnetic catalysis, we conclude that any inverse catalysis of this type for a scalar condensate should be attributed to the sea quarks, consistently with the conclusion in [8] and providing a first corroborating test for our holographic prescription. The comparison of the full scalar condensate and the valence contribution to it, both in the rest frame is presented in figure 8.  \nConcerning the pseudoscalar condensate, our numerical results show that it is negative over a range of values for the momentum and magnetic field. As a function of k K alone, it starts negative close to the minimum value that the constant magnetic field permits, and evolves in such a manner that becomes positive as k K 2 grows. Since a negative condensate indicates an instability of the brane embedding in the gravity side, this result is another confirmation that the momentum has a stabilizing effect on the system.  \nWe already notice that when plotted as a function of the intensity of the magnetic field at any constant kinetic momentum, the magnitude of the pseudoscalar condensate increases with b when such quantity is far bellow the maximum value it can take. We also pointed out in passing that for curves traced at larger values of k K , this tendency is reversed and the magnitude of this condensate begins to decrease as the field approaches its maximum intensity for such kinetic momentum. We would like to add two further observations, one being that even if the way we present our results is suggestive about the transition to inverse magnetic catalysis mentioned above happening because of the large kinetic momentum at which certain curves are traced, it would probably be more ac-  \nFIG. 9: Total (dotted) and valence (dots) pseudoscalar condensate as a function of the magnetic field intensity b/T 2 in the rest frame of the plasma. The thick line corresponds to the best fitting second order polynomial for the valence condensate data  \n<!-- image -->  \n-8 . 23 × 10 -3 +2 . 08 × 10 -5 ( b/T 2 ) + 2 . 30 × 10 -7 ( b/T 2 ) 2  \ncurate to think of it as occurring for very large intensities of the magnetic field. The need to clarify this comes from using b/b max as a variable so that the full range of b would be covered from 0 to 1, and the plots could be compared regardless of how largely different their maximum intensities were. A side effect of this is not making it visually clear how large the direct intensity is at which the magnitude of the condensate begins to decrease. The correctness of the latter point of view is supported by the plot of the pseudoscalar condensate as a function of b in the rest frame of the plasma shown in figure 9, where the field can take arbitrarily high intensities and we indeed see how inverse magnetic catalysis appears as that happens. For comparison, we have included in the same figure 9 the plot of the valence contribution to the pseudoscalar condensate, as obtained from the trace at b = 0 in figure 6 by the considerations mentioned above. The other observation is that from the plots included in figure 7, it would be imaginable that for other graphs, done at larger values of k K , the change in tendencies for b close to b max would be intense enough to make the condensate positive. Even if a full plot done at such high values of k K is too computationally demanding just to confirm this perception, we have actually verified that at b/b max = 0 . 91 for k K 2 /T 2 = 152 π 2 / 9 , the pseudoscalar condensate is indeed positive.  \nNow that figures 8 and 9 have been presented, it is time to notice that all the distinctive features of the behavior of both condensates, including magnetic catalysis and its inverse, seem to be attributable to the sea quarks, since either of the valence contributions is well approximated by a simple function of second order in b . We do not think much should be made out of the particular fittings, but it is interesting to see how featureless the valence plots turn out to be in comparison with those of the full condensate.  \nEven if the analysis above exhausts the scope of the current work, it also indicates some elements that require further investigation.  \nOne example is that the curves traced in figure 6 at the lower and higher values of b present an inflection point close to k K 2 -eb = 0 , while those at intermediate fixed intensities do not. We currently do not posses a clear understanding of the reason behind this, and believe it deserves further clarification.  \nAlso, equations (11) are presented in [8] as an approximation, in the sense that the addition of both contributions is not expected to match the value of the condensate computed using all the elements. Our holographic construction presents the opportunity to verify how accurate this approximation is, but obtaining the sea condensate demands so much computational time that it has to be addressed on its own. As a first glance of the interest that such an investigation would bear, we present the plots in figure 10, where the subtraction of total minus valence, that should approximate the sea contribution, is plotted for both condensates. It stands out that the scalar and pseudoscalar are almost reflections of each other with respect to the intermediate line we have drawn in between them.  \nFIG. 10: Difference between total and valence condensates for the scalar (dashed) and pseudoscalar (dot dashed) case as a function of the magnetic field intensity b/T 2 in the rest frame of the plasma.  \n<!-- image -->  \nOne last topic that can be addressed using our construction is the effect of a magnetic field over the thermal mass [16], in our case, of the quasi-particles dual to the modes we studied. This could be done performing a very precise calculation to extract the value of m T that better fits the asymptotic behavior in (14) for different intensities of the magnetic field.', 'XI. ACKNOWLEDGMENTS': "We are very thankful to Alejandro Ayala for meaningful discussions about the thermal dispersion relations.  \nThe work of D.K. is supported by CONAHCyT MSc.  \n- [1] Andreas Karch and Emanuel Katz. 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