QCD quark condensate in external magnetic fields
We present a comprehensive analysis of the light condensates in QCD with 1+1+1 sea quark flavors (with mass-degenerate light quarks of different electric charges) at zero and nonzero temperatures of up to and external magnetic fields . We employ stout smeared staggered fermions with physical quark masses and extrapolate the results to the continuum limit. At low temperatures we confirm the magnetic catalysis scenario predicted by many model calculations while around the crossover the condensate develops a complex dependence on the external magnetic field, resulting in a decrease of the transition temperature.
Introduction.—Strong (electro)magnetic fields prominently feature in various physical systems. They play an essential role in cosmology, where magnetic fields of and may have been present Vachaspati (1991); Enqvist and Olesen (1993) during the strong and electroweak phase transitions of the universe, respectively. Magnetic fields with strengths up to ( GeV) are also generated in non-central heavy ion collisions Skokov et al. (2009); Voronyuk et al. (2011); Bzdak and Skokov (2012); Deng and Huang (2012) at the Relativistic Heavy Ion Collider (RHIC) or the Large Hadron Collider (LHC). Furthermore, for certain classes of neutron stars like magnetars, magnetic fields of the order of have been deduced Duncan and Thompson (1992). In addition to this phenomenological relevance, external (electro)magnetic fields can be used to probe the dynamics of strongly interacting matter, i.e. the vacuum structure of Quantum Chromodynamics (QCD).
One of the most important aspects of QCD is chiral symmetry breaking. At zero quark masses the chiral condensate is an order parameter. It vanishes at high temperatures where chiral symmetry is restored but develops a nonzero expectation value in the hadronic phase. In nature, quark masses are nonzero and the corresponding quark condensates, though only approximate order parameters, still exhibit this characteristic behavior around the transition temperature between the hadronic and the quark-gluon plasma phases. Lattice simulations revealed that for physical quark masses this transition is an analytic crossover Aoki et al. (2006a), leading to a transition temperature which may depend on the observable used for its definition.
In the response of QCD to external magnetic fields, ‘magnetic catalysis’ refers to an increase of the condensate with . This implies a -dependence of as well. Almost all low-energy models and approximations to QCD Gusynin et al. (1996); Shushpanov and Smilga (1997); Agasian and Shushpanov (2000); Agasian (2001); Cohen et al. (2007); Andersen (2012a, b); Klevansky and Lemmer (1989); Menezes et al. (2009); Gatto and Ruggieri (2011, 2010); Kashiwa (2011); Andersen and Khan (2012); Avancini et al. (2012); Fukushima and Pawlowski (2012); Mizher et al. (2010a); Gatto and Ruggieri (2011); Andersen and Tranberg (2012); Kanemura et al. (1998); Klimenko (1992); Alexandre et al. (2001); Scherer and Gies (2012); Johnson and Kundu (2008); Preis et al. (2011); Gusynin et al. (1996); Shushpanov and Smilga (1997); Agasian and Shushpanov (2000); Cohen et al. (2007); Agasian (2001); Mizher et al. (2010b); Gasser and Leutwyler (1987a, b); Gerber and Leutwyler (1989) as well as lattice simulations in quenched theories Buividovich et al. (2010); Braguta et al. (2010) and at larger than physical pion masses in QCD D’Elia et al. (2010); D’Elia and Negro (2011) and in the SU(2) theory Ilgenfritz, E.-M. and others (2012) found and to increase with . Exceptions in this respect with a decreasing function are the results obtained within two-flavor chiral perturbation theory Agasian and Fedorov (2008), in the linear sigma model without vacuum corrections Fraga and Mizher (2009) and in the bag model Fraga and Palhares (2012).
In contrast to the majority of the above results, our large-scale study of QCD in external magnetic fields with physical pion mass and results extrapolated to the continuum limit Bali et al. (2012) has revealed the transition temperature to decrease as a function of the external magnetic field. This applies to the ’s defined from the quark condensate, the strange quark number susceptibility and the chiral susceptibility. In particular, we found the condensate to depend on in a non-monotonous way in the crossover region.
In Ref. Bali et al. (2012) we have already pointed out two rationales why former lattice studies are at variance with these recent findings: coarser lattices and larger quark masses. Obviously, it is also very important to address the differences between our QCD results and many model and chiral perturbation theory (PT) predictions, especially since the latter methods can be used to investigate regions that are not easily accessible to lattice simulations, e.g., QCD at a non-vanishing baryon density.
In this Letter, we present a detailed analysis of the dependence of the light quark condensates on and on the temperature , based on the simulations described in Bali et al. (2012) and new simulations at . The data — all continuum extrapolated — are presented in ways that will enable to refine model assumptions and parameters. We also perform a first comparison to PT and to Polyakov-Nambu-Jona-Lasinio (PNJL) model predictions. We aim at a better understanding of the physical mechanisms behind the differences. This in turn should be of phenomenological relevance.
Below we introduce our notations and the simulation setup. We then present our results and compare them to PT and PNJL predictions, both at zero and nonzero temperatures.
Condensate on the lattice at nonzero .—We study QCD coupled to a constant external magnetic field , pointing in the positive direction. Such a field can be implemented by multiplying the links of the lattice by complex phases. The specific choice of these phases and our setup are detailed in Ref. Bali et al. (2012). The external field couples only to the quark electric charges with labelling the different flavors. Thus, the magnetic field only appears in combinations .
where the smallest quark charge (that of the down quark), enters, with being the elementary charge. Here is the number of lattice sites in a spatial direction (our lattices are symmetric in space). Similarly, counts the lattice points in the temporal direction. The spatial volume of the system is given by and the temperature is related to the inverse temporal extent of the lattice as .
The quark condensate can be derived from the partition function, which in the staggered formulation of QCD with three flavors () is given by the functional integral,
where is the inverse gauge coupling, the gauge action and the fermion matrix. For we use the tree-level improved Symanzik action, while in the fermionic sector we employ a stout smeared staggered Dirac operator . The details of the lattice action can be found in Refs. Aoki et al. (2006b); Bali et al. (2012). The lattice sizes range from to for the zero temperature simulations, while at non-vanishing we investigate , and lattices. We set the quark masses to their physical values, with mass-degenerate light quarks: . The electric charges of the quarks are , therefore we need to treat each flavor separately. The line of constant physics (LCP) was determined by fixing the ratios and to the experimental values. The lattice spacing is defined by keeping fixed, for details see Ref. Borsányi et al. (2010). At the continuum limit corresponds to . At nonzero temperature, it is convenient to define the continuum limit as , keeping fixed.
The quark condensate is defined as the derivative of with respect to the lattice mass parameter
To carry out the continuum limit, the lattice condensate needs to be renormalized since it contains additive (for ) and multiplicative divergences. These cancel Bali et al. (2012) in the following combination,
where, to obtain a dimensionless quantity, we divided by the combination which contains the zero-field pion mass and (the chiral limit of the) pion decay constant Colangelo and Dürr (2004). This specific combination enters the Gell-Mann-Oakes-Renner relation,
Note that the normalization in definition (4) can easily be converted into the slightly different one employed in former studies by the Budapest-Wuppertal collaboration (e.g. Refs. Aoki et al. (2006a); Borsányi et al. (2010)) and in Ref. Bali et al. (2012). We define the change of the condensate due to the magnetic field as
Note that the term cancels from this difference. In our normalization, Eq. (6) defines the change of the condensate caused by a nonzero , in units of the chiral condensate at and . This normalization will be advantageous when comparing the lattice results to and model predictions, which are usually given in units of . The is included in Eq. (4) so that the chiral limit of the condensate is fixed to 1 at , and approaches 0 as . At nonzero quark mass will still start from 1 at . At very high temperatures, however, it is well known from the free case Dolan and Jackiw (1974); Weinberg (1974) that the condensate receives a contribution . This term is negligible for the temperatures under study and it cancels exactly from .
Results.—In Fig. 1 we display the renormalized difference as a function of at , for five different lattice spacings. We carry out the continuum limit by fitting the results to a lattice spacing-dependent spline function (for a similar fit in two dimensions see Endrődi (2011)). This function is defined on a set of points and is parameterized by two values at each such node, in the form , to reflect the -scaling of our action. The parameters and are obtained by minimizing the corresponding . The systematic error of the limit is determined by varying the node positions. We find that lattice discretization errors become large at high magnetic fields due to saturation of the lattice magnetic flux Bali et al. (2012), therefore we only include points with . In Fig. 1 we also show the continuum limit of the difference .
Next, we address the condensate at nonzero temperature, carrying out a similar continuum extrapolation for as at , using three lattice spacings with , and . The increase of the difference is qualitatively similar for zero and nonzero temperatures in PT and in the PNJL model (see below). In QCD, however, the situation is quite different: in Fig. 2 we plot the continuum extrapolated lattice results for as functions of for several temperatures, ranging from up to . Note that the transition temperature varies from down to Bali et al. (2012). The increasing behavior of at low temperatures () corresponding to magnetic catalysis continuously transforms into a hump-like structure in the crossover region () and then on to a monotonously decreasing dependence (). We remark that — although in the high temperature limit the condensate and its dependence on are suppressed — at again starts to increase. Furthermore, we note that the strange condensate (with a definition similar to that in Eq. (4)) does not exhibit this complex dependence on and but simply increases with growing for all temperatures. This shows that the partly decreasing behavior near the crossover region only appears for quark masses below a certain threshold , inbetween the physical light and strange quark masses, .
Comparison to effective theories/models.—In Fig. 3 we compare our zero temperature QCD result for as a function of to the PT prediction Cohen et al. (2007); Andersen (2012a, b, ) and to that of the PNJL model Gatto and Ruggieri (2011); Ruggieri (), both at physical pion mass. We see that the PT prediction describes the lattice results well up to , while the PNJL model works quantitatively well up to . Note that, since the Polyakov loop at zero temperature vanishes, in the limit the PNJL model becomes indistinguishable from the NJL model with the same couplings.
In Fig. 4, the condensate Eq. (4) as a function of is compared to PT and to the PNJL model for different magnetic fields. At we use the continuum extrapolation for the condensate presented in Ref. Borsányi et al. (2010) (where lattices up to were employed), and complement this with the differences shown in Fig. 2. In addition to the continuum extrapolated lattice data we plot the PT curves for Gerber and Leutwyler (1989) and for Andersen (2012a, b, ), together with the PNJL model predictions Gatto and Ruggieri (2011); Ruggieri (). The results indicate that PT is reliable for small temperatures and small magnetic fields, , . (We remark that the inclusion of the hadron resonance gas contribution to the condensate in PT Gerber and Leutwyler (1989) improves the agreement with lattice results, as was shown at in Ref. Borsányi et al. (2010). One would expect a similar improvement at .) Since the PNJL model condensate is calculated using a Polyakov loop effective potential that was obtained from lattice results Gatto and Ruggieri (2011), differences between the model and our results at are expected to be large, as both the transition temperature and the transition strength (the slope of the condensate at ) strongly depend on the number of flavors. To enable a comparison, we linearly rescaled the temperature axis (only for the PNJL curves) to match our lattice inflection point at . Nevertheless, the -dependence of the condensate for the PNJL model also reveals qualitative differences in comparison to the QCD results.
Finally, in Fig. 5 we plot as a function of the temperature for several magnetic field strengths. At zero magnetic field isospin symmetry is exact since we employed mass-degenerate light quarks. As increases, due to the difference between the electric charges, develops a temperature-dependence similar to that of , see Fig. 4. The results for are also listed in Table 1.
Summary.—We determined the QCD light quark condensates at nonzero external magnetic field strengths for physical quark masses in the continuum limit. Our results are in quantitative agreement with chiral perturbation theory and PNJL model predictions for small magnetic fields and at small temperatures. Note that the constants within these parameterizations have not been adjusted to our data but were taken from the literature where they have been obtained at vanishing magnetic field. Unsurprisingly, PT fails in regions where pions cease to be the essential low energy degrees of freedom. While in the hadronic phase low energy models qualitatively reproduce the -dependence of the lattice data, they miss an important feature which becomes dominant for light quark masses and for temperatures around , see Fig. 2. Clearly, the coupling between the magnetic field and the gauge background is enhanced near the chiral limit: the smaller the quark mass, the more the fluctuations of the gauge field influence the quark determinant. Thus, for light quarks the indirect interaction between the gluonic degrees of freedom and the external field becomes more important. A possibility to separate this indirect effect would be to consider the sea and valence contributions to the condensate, as was performed in Ref. D’Elia and Negro (2011), which we plan to discuss in a forthcoming study.
Acknowledgments.— We thank J.O. Andersen, T. Cohen, E. Fraga, T. Kovács, M. Ruggieri and K. Szabó for useful discussions. Computations were carried out on the GPU cluster Egri et al. (2007) at Eötvös University Budapest and the BlueGene/P of FZ Jülich. This work was supported by the DFG (SFB/TR 55 and BR 2872/4-2) and the EU (ITN STRONGnet 238353 and ERC Grant 208740).
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