(2) PHYS. REV. D 98, 031501 (2018). ALEJANDRO AYALA et al. In this work, we study the thermomagnetic behavior of the strong coupling αs . We resort to using the renormalization group equation (RGE), applied to the one-loop gluon vacuum polarization, computed in the presence of a magnetic field at zero and finite temperature. To understand the thermomagnetic evolution of αs , we study the limiting regimes where the magnetic field is considered either in the weak or in the strong field case, both at zero and at high temperature. A renormalization group flow analysis for large ratios of the magnetic field to the temperature squared, applied to the relevant two fermion operator, has been recently carried out using the gauge-gravity correspondence in Ref. . Also, a RGE analysis of the strong coupling running with temperature at zero magnetic field, based on the use of temperature-dependent renormalization constants, has been carried out in Ref. . Let Pðq2 ; T; jeBj; αs Þ be the unrenormalized coefficient of a given tensor structure upon which the gluon polarization can be decomposed at finite temperature T and in the presence of a constant magnetic field jeBj. Since the thermomagnetic medium breaks Lorentz invariance, P can depend separately on the square of the components of the four-momentum qμ . However, for notational simplicity and latter convenience, we write q2 for the momentum dependence. We scale the energy variables appearing in P by the renormalization ultraviolet energy scale μ, writing jeBj ¼ μ2 ðjeBj=μ2 Þ;. T 2 ¼ μ2 ðT 2 =μ2 Þ;. q2 ¼ μ2 ðq2 =μ2 Þ:. ð1Þ. Therefore, we have Pðq2 ; T; jeBj; αs Þ ¼ μD Pðq2 =μ2 ; T 2 =μ2 ; jeBj=μ2 ; αs Þ; ð2Þ where D ¼ 2 is the energy dimension of P. Since μ is arbitrary, the statement that P should be independent of this scale is provided by the RGE , namely, ∂ ∂ μ þ αs βðαs Þ − γ Pðq2 ; T; eB; αs Þ ¼ 0; ð3Þ ∂μ ∂αs where βðαs Þ is the QCD beta function defined by ∂αs ; ∂μ. ð4Þ. ∂ ln Z−1 ; ∂μ. ð5Þ. αs βðαs Þ ¼ μ and γ¼μ. with Z being the gluon vacuum polarization wave function renormalization. The beta function encodes the dependence of the strong coupling on the renormalization scale and. thus the ultraviolet properties of the theory. It is well known that the QCD beta function is negative and that to one-loop level it is given by βðαs Þ ¼ −b1 αs ;. 1 ð11N c − 2N f Þ; 12π. b1 ¼. ð6Þ. with N c the number of colors and N f the number of active flavors. To set up the stage for the analysis, let us first recall how the usual evolution of the strong coupling with the momentum scale is established. Starting from Eq. (3) and considering that the only energy scale in the function P is q2 , we have ∂ ∂ μ þ αs βðαs Þ − γ Pðq2 ; αs Þ ¼ 0: ð7Þ ∂μ ∂αs We now introduce the variable t ¼ lnðQ2 =μ2 Þ;. ð8Þ. where Q2 is the momentum transferred in a given process. Notice that the reference scale μ2 is usually large enough, so as to make sure that the calculation is well within the perturbative domain, therefore Q2 < μ2 . After this change of variable, the RGE becomes ∂ ∂ − þ αs βðαs Þ − γ Pðq2 ; αs Þ ¼ 0: ð9Þ ∂t ∂αs Using the method of the characteristics , one obtains the relation between the coupling values evaluated at Q2 and the reference scale μ2 as Z. 1 dt ¼ − 2 2 b1 tðQ ¼μ Þ tðQ2 Þ. Z. αs ðQ2 Þ. dαs : α2s. ð10Þ. αs ðμ2 Þ ; 1 þ b1 αs ðμ2 Þ lnðQ2 =μ2 Þ. ð11Þ. αs. ðQ2 ¼μ2 Þ. Solving for αs ðQ2 Þ, we obtain αs ðQ2 Þ ¼. from where it is seen that as Q2 increases the coupling decreases. In the presence of a magnetized medium and/or a heat bath, there appear extra energy scales in the analysis. The question we set up to answer is how the coupling evolves as a function of the magnetic field and/or temperature scales when we take as the reference scale the momentum transferred in the given processes, Q2 and the smallest of either T 2 and jeBj. The strategy we follow is to use the explicit dependence of a certain P on the corresponding energy variables to study the relationship that changes on these scales bare with changes on the coupling constant, by. 031501-2.
(3) THERMOMAGNETIC EVOLUTION OF THE QCD STRONG … means of the RGE. We now proceed to the analysis for the case when the corresponding function P is computed in the four limiting cases: the weak and strong magnetic fields both at zero and high temperatures. In the weak field limit, namely, jeBj < Q2 and at T ¼ 0, the gluon vacuum polarization (omitting the diagonal color structure) can be expressed as [17,18] (see also Ref. ) k. μν μν μν ⊥ 0 Πμν weak ¼ Pweak Πk þ Pweak Π⊥ þ Pweak Π0 ; μν with Πμν k ¼ gk − qμ qν q2. qμk qνk q2k. μν , Πμν ⊥ ¼ g⊥ þ. qμ⊥ qν⊥ , q2⊥. ð12Þ. μν and Πμν 0 ¼g −. μν − Πμν k − Π⊥ , where, when taking the magnetic field μν along the ẑ axis, we have gμν k ¼ ð1; 0; 0; −1Þ, g⊥ ¼ μν μ μν μ μν ð0; −1; −1; 0Þ, gμν ¼ gμν k þ g⊥ , qk ¼ gk qν , q⊥ ¼ g⊥ qν , q2k ¼ðq0 Þ2 −ðq3 Þ2 , q2⊥ ¼ðq1 Þ2 þðq2 Þ2 , and q2 ¼ q2k − q2⊥ . k. Let us look at the coefficient Pweak . In Ref. , it has been shown that with N f ¼ 3 active flavors its explicit expression is k. Pweak. dλB ; λB. ð16Þ. λB ¼ CB e−t ;. ð17Þ. dt ¼ −. the solutions of which are λq ¼ Cq e−t ;. where Cq and CB are integration constants to be determined from the initial condition for the evolution. Upon combining Eq. (17), we can write λq þ λB ¼ ðCq þ CB Þe−t ¼ e−t ;. ð18Þ. where we have chosen that at t ¼ 0, for the initial condition of the evolution, λq ¼ 1 and λB ¼ 0. Therefore, Cq þ CB ¼ 1. Also, for the subsequent evolution, we take Q2 fixed, and thus we refer the evolution of jeBj to the reference scale Q2 ; namely, we take λB ¼ jeBj=Q2 . Therefore, we can write ð19Þ. . ∂ ∂ k þ αs βðαs Þ − γ̃ Pweak ¼ 0; ∂t ∂αs. ð13Þ. Z. tðλB ¼jeBj=Q2 Þ. tðλB ¼0Þ. where γ̃ ¼ γ − D. Using the method of the characteristics, we can write. dt ¼ −. 1 b1. Z. αs ðQ2 þjeBjÞ. αs ðQ2 Þ. dαs : α2s. ð21Þ. Solving for αs ðjeBjÞ ≡ αs ðQ2 þ jeBjÞ, we obtain. ð14Þ. ð15Þ. ð20Þ. from where the relation between the coupling values evaluated at jeBj and the reference scale Q2 can be expressed as. αs ðjeBjÞ ¼. Using the RGE, Eq. (7), we get ∂ ∂ ∂ k − λB þ αs βðαs Þ − γ̃ Pweak ¼ 0; −λq ∂λq ∂λB ∂αs. dλq ; λq. Notice that in this case, as opposed to Eq. (8), the evolution energy scale appears in the denominator of the logarithmic function in Eq. (19). Therefore, Eq. (15) becomes. where in the second line we have scaled the momentum components and the square root of the magnetic field intensity by μ and in the third line we have introduced the dimensionless scale factors λB and λq , one for each of pﬃﬃﬃﬃﬃﬃﬃﬃﬃ the magnetic jeBj=μ and momentum q=μ powers of the dimensionless ratios, respectively. Notice that Eq. (13) satisfies ∂ ∂ ∂ k μ þ λq þ λB − D Pweak ¼ 0: ∂μ ∂λq ∂λB. dt ¼ −. Q2 t ¼ ln 2 : Q þ jeBj. 2 qk 2αs 2 2 2 ¼− ð2qk þ q⊥ Þ ; jeBj 9π ðq2 Þ3 2α ¼ −μ2 s ðjeBj2 =μ4 Þ 9π 2 2 qk =μ 2 2 2 2 × ð2qk =μ þ q⊥ =μ Þ ; ðq2 =μ2 Þ3 λ4 2α ¼ −μ2 B2 s ðjeBj2 =μ4 Þ λq 9π 2 2 qk =μ 2 =μ2 þ q2 =μ2 Þ ; ð2q × ⊥ k ðq2 =μ2 Þ3. PHYS. REV. D 98, 031501 (2018). αs ðQ2 Þ 2. Q Þ 1 þ b1 αs ðQ2 Þ lnðQ2 þjeBj. :. ð22Þ. From Eq. (22), we see that as the magnetic field intensity increases the coupling increases with respect to its corresponding value at the reference scale Q2 . Next, let us consider the strong field limit jeBj > Q2 still for T ¼ 0. Working in the lowest Landau level (LLL) approximation, the only nonvanishing coefficient is that of the structure Πμν k , which for three active flavors and a vanishing quark mass (m) is given by [18,20,21]. 031501-3.
(4) PHYS. REV. D 98, 031501 (2018). ALEJANDRO AYALA et al. k. 4 αs 2 jeBje−q⊥ =2jeBj 3π λ2q 2 −ð 2 Þðq⊥ =2jeBjÞ 2 2 4 αs jeBj e λB ¼ −μ λB ; 2 3π μ. Pstrong ¼ −. ð23Þ. where once again, in the second line, we have scaled the momentum components and the square root of the magnetic field intensity by μ and have introduced the dimensionless scale factors λB and λq , one for each of the pﬃﬃﬃﬃﬃﬃﬃﬃﬃ magnetic jeBj=μ and momentum q=μ powers of the dimensionless ratios, respectively. It is easy to check that Eq. (23) satisfies Eq. (14), and thus, upon using the RGE, Eq. (7), we obtain ∂ ∂ ∂ k −λq − λB þ αs βðαs Þ − γ̃ Pstrong ¼ 0: ð24Þ ∂λq ∂λB ∂αs. the temperature by μ and have introduced the dimensionless scale factors λB , λq , λm , and λT , one for each of the pﬃﬃﬃﬃﬃﬃﬃﬃﬃ magnetic jeBj=μ, momentum q=μ, mass m=μ, and temperature T=μ powers of the dimensionless ratios, respectively. It is easy to check that Eq. (26) satisfies X ∂ ∂ k μ þ λ − D PT-B ¼ 0 ð27Þ ∂μ i¼q;B;m;T i ∂λi and that upon using the RGE Eq. (3) one obtains X ∂ ∂ k λi þ αs βðαs Þ − γ̃ PT-B ¼ 0: − ∂λ ∂α i s i¼q;B;m;T As before, we can write. Using the same arguments as for the weak field case, which implies starting the evolution from the fixed scale Q2 < jeBj, we once again obtain for the relation between the coupling evaluated at jeBj and the reference scale Q2 αs ðjeBjÞ ¼. αs ðQ2 Þ 2 : Q 1 þ b1 αs ðQ2 Þ ln Q2 þjeBj. ð25Þ. The results in Eqs. (22) and (25) show that for T ¼ 0, αs is an increasing function of jeBj, when referred to the scale Q2 . This result, although not necessarily evident, is in retrospect almost trivial: Since the reference scale is taken as the square of the transferred momentum in a given process, the magnetic field scale jeBj does not compete with any other energy scale and is thus the only one that drives the evolution. The reference scale Q2 , although large, is always smaller than either the combined scale Q2 þ jeBj or jeBj, which in turn determines the sign of the logarithmic function appearing either in Eq. (22) or (25). We now turn to study the finite temperature case. Given that there is no need to assume a given hierarchy between T 2 and jeBj, the calculation is more straightforwardly performed when working in the LLL approximation. This is a suitable approximation when one or the other scale is considered as being the largest one. As is shown in Ref. , in this approximation, the coefficient for the parallel unrenormalized gluon polarization tensor structure is given by k PT-B. λ2 4 αs jeBj −ðλ2q Þðq2⊥ =2jeBjÞ e B ¼ 3 π μ2 2 2 2 2 2 λ m =μ ðq3 =μ Þ þ ðq20 =μ2 Þ × ln 2m 2 2 2 ; ðq2k =μ2 Þ π λT T =μ μ2 λ2B. ð26Þ. where we have again scaled the momentum components, the square root of the magnetic field, the quark mass, and. ð28Þ. dt ¼ −. dλq ; λq. dt ¼ −. dλB ; λB. dt ¼ −. dλm ; λm. dt ¼ −. dλT ; λT. ð29Þ. the solutions of which are λq ¼ Cq e−t ;. λB ¼ CB e−t ;. λm ¼ Cm e−t ;. λT ¼ CT e−t ;. ð30Þ. where Cq , CB , Cm , and CT are integration constants to be determined from the initial condition for the evolution. We can again combine Eq. (30) and write λq þ λB þ λm þ λT ¼ ðCq þ CB þ Cm þ CT Þe−t :. ð31Þ. Let us first consider the case where jeBj is the largest of the energy (squared) scales. The lesson we learned from the analysis at T ¼ 0 is that one should choose the reference scale as the sum of the rest of the energies (squared) other than the one that is evolving. Since for the case of three active quark flavors the quark mass is small with respect to the rest of the energy scales involved, we neglect the quark mass and use as the reference scale the combination Q2 þ T 2 . Therefore, in a fashion similar to the analysis of the T ¼ 0 case, we obtain αs ðjeBjÞ ¼. αs ðQ2 þ T̃ 2 Þ 2 2 ; Q þT 1 þ b1 αs ðQ2 þ T̃ 2 Þ ln Q2 þT 2 þjeBj. ð32Þ. where, since the analysis is only valid at leading order, we have taken T̃ as a representative fixed value of T in the energy domain of interest to compute the value of the coupling at the reference scale. From Eq. (32), we see that the coupling grows with the field intensity. We now consider the case where T 2 is the largest of the energy (squared) scales. In this case, the analysis yields. 031501-4.
(5) THERMOMAGNETIC EVOLUTION OF THE QCD STRONG … g αs ðQ2 þ jeBjÞ ; αs ðjeBjÞ ¼ g ln 2Q2 þjeBj 2 1 þ b1 αs ðQ2 þ jeBjÞ Q þjeBjþT. Pðαs Þ ¼. ð33Þ. where we have also used a representative fixed value of g to evaluate the coupling at the the field strength jeBj reference scale, to account for the fact that the analysis is only valid at leading order. From Eq. (33), we see that the coupling decreases with the field intensity. We note that, contrary to the T ¼ 0 case, now the magnetic field has an extra scale to compete with. When the field intensity is the largest of these scales, the result is as for the T ¼ 0 case, where jeBj is the only relevant reference scale and this in turn implies that the coupling increases. Conversely, when the temperature is the largest scale, the field strength appears both in the numerator and in the denominator of the argument of the logarithmic function, which in turn makes the coupling to decrease with the field strength. Figure 1 shows the dependence of the strong coupling as a function of the magnetic field strength. We have chosen a fixed large value of Q2 ¼ 1 GeV2 . The black solid and red dash-dot lines show the cases for T ¼ 0 and jeBj > T 2 , with T ¼ 1.1 GeV, respectively, corresponding to jeBj being the largest of the energy scales. For both cases, we observe that the coupling increases with the field strength. The blue dash line shows the case for jeBj < T 2 with T ¼ 1.5 GeV. We observe that in this case, when the temperature is the largest of the energy scales, the coupling decreases as a function of the field strength. We also find the solution of the evolution equation for Pk in the four studied regimes. This is obtained in general as . FIG. 1. Strong coupling running as a function of the magnetic field strength. The curves are computed using a fixed transferred momentum Q2 ¼ 1 GeV2 . The black solid line shows the case for T ¼ 0. The blue dash line shows the case when jeBj < T 2 computed with T ¼ T̃ ¼ 1.5 GeV. The red dash-dot line is the case for jeBj > T 2 computed with T ¼ ˜ ¼ 1.2 GeV2 . 1.1 GeV and jeBj. PHYS. REV. D 98, 031501 (2018) Pðα0s Þ exp. Z −. αs. α0s. γ̃ðα0 Þ dα ; βðα0 Þ 0. ð34Þ. where, as before, γ̃ ¼ γ − D, with γ the one-loop gluon field anomalous dimension , given explicitly by 13 8 α CA þ T F N f s ≡ Aαs : γ¼ a− 3 3 4π. ð35Þ. We consider N f ¼ 3 active flavors, work in the Landau gauge a ¼ 0, and remember that T F ¼ 1=2 and CA ¼ N c , with N c ¼ 3 the number of colors. From Eq. (34), we obtain that in each of the studied regimes the parallel component of the gluon polarization tensor is given by k. 2 A b1 Q þ jeBj 2 αs ðQ2 Þ ¼ ; 2 2 Q αs ðQ þ jeBjÞ ðα0s Þ. Pweak;strong ðαs Þ k. Pweak;strong k. 2 Q þ T 2 þ jeBj 2 ¼ f2 ðα0s Þ Q2 þ T. PT<jeBj ðαs Þ k. PT<jeBj. . f2 Þ bA αs ðQ2 þ T 1 × ; 2 2 αs ðQ þ T þ jeBjÞ 2 k PT>jeBj ðαs Þ Q þ T 2 þ jeBj 2 ¼ k g PT>jeBj ðα0s Þ Q2 þ jeBj A g b1 αs ðQ2 þ jeBjÞ × : αs ðQ2 þ T 2 þ jeBjÞ. ð36Þ. We notice that the Pk ’s grow with the field strength. The growth is tamed by the temperature such that for the case with T 2 > jeBj the growth is less pronounced. In conclusion, from an analysis of the RGE for the gluon polarization tensor in the presence of a magnetic field and at finite temperature, we have shown that when the magnetic field is the largest of the energy scales the strong coupling increases with the field strength. Conversely, when the temperature is the largest of the scales, there is a turnover behavior, and the strong coupling decreases as a function of the field strength. We emphasize that the renormalization program is performed integrating out the degrees of freedom above the ultraviolet scale μ. Therefore, the running of the strong coupling accounts for the competition of magnetic and thermal energy scales much below μ. This is in contrast to other schemes adopted for instance in Refs. [14,24], where an attempt is made to integrate out degrees of freedom down to the thermal scale. Since the strength of the quark condensate is a measure of the quark-antiquark binding and this in turn depends on the strength of the QCD coupling, these results can help one understand the IMC phenomenon observed by LQCD.. 031501-5.
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