Viscous Conformal Gauge Theories
VViscous Conformal Gauge Theories
Arianna Toniato, Francesco Sannino, and Dirk H. Rischke
2, 3 CP -Origins & the Danish Institute for Advanced Study, Danish IAS,University of Southern Denmark, Campusvej 55, DK–5230 Odense, Denmark Institute for Theoretical Physics, Goethe University,Max-von-Laue-Str. 1, D–60438 Frankfurt am Main, Germany Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China
We present the conformal behavior of the shear viscosity-to-entropy density ratio and the fermion-number diffusion coefficient within the perturbative regime of the conformal window for gauge-fermion theories.
Preprint: CP -Origins-2016-052 DNRF90 I. INTRODUCTION
Gauge theories constitute the backbone of the standard model of particle interactions. Gauge theories exist in severaldifferent phases that are naturally classified according to the force measured between static sources. Knowledge ofthe phase diagram proves crucial when investigating extensions of the standard model both for particle physics andcosmology. A special class of gauge theories are the ones that are fundamental according to Wilson [1, 2], meaningthat they possess a complete (in all couplings) ultraviolet (UV) fixed point either of non-interacting (asymptoticallyfree [3–7]) or of interacting nature (asymptotically safe [8]). Complete asymptotically safe quantum field theorieswere discovered only very recently [9, 10], widening the horizon of fundamental theories that can be used for novelphenomenological applications [11] beyond the traditional asymptotically free paradigm [3, 4]. The thermal propertiesof completely asymptotically safe field theories were elucidated in Ref. [12].Here we focus our attention on asymptotically free gauge theories featuring gauge and fermion degrees of freedomthat develop an infrared (IR) interacting fixed point. We henceforth push forward our program to systematicallyunderstand, in a rigorous manner, the dynamics of these theories at zero [13–18] and non-zero matter density [19, 20],by analysing their conformal viscous behavior as a function of the number of flavors . Because of the perturbativenature of the theories investigated here, along the full energy range, our investigation of their viscous properties isalso much better controlled than for QCD-like theories. This is so because at very high energies the theory is non-interacting and at very low energies the theory reaches an IR perturbative fixed point. Furthermore, the value of thegauge coupling at the IR fixed point can be made arbitrarily small by changing the number of flavors and colors ofthe theory. This allows us to consistently truncate the perturbative expansion within the range of convergence of thetheory.We henceforth determine the conformal behavior, as a function of the number of flavors, for the shear viscosity-to-entropy density ratio and the fermion-number diffusion coefficient. By adapting the results of Ref. [23] we learnthat, as we decrease the number of flavors below the loss of asymptotic freedom, their IR fixed point values decrease.Furthermore, for a given number of flavors within the perturbative conformal window both coefficients decrease withdecreasing temperature (once we multiply the diffusion coefficient by the temperature) from their infinite value inthe deep UV down to the value at the IR fixed point. We represent the results for three colors as a function of thenumber of flavors, but to the order investigated here the results are similar for any other fermion representation.We organise this paper as follows. In Sec. II we shortly review the theory, introduce the notation, and provide thesalient zero and non-zero temperature properties. This is followed by the determination of the transport coefficientsin Sec. III. Here we will comment on our findings and finally conclude in Sec. IV. A crucial property was unveiled in Ref. [9], i.e., the Yukawa interactions, mediated by the scalars, compensate for the loss of asymptoticfreedom due to the large number of gauged fermion flavors and therefore cure the subsequent growth of the gauge coupling. The furtherinterplay of the gauge, Yukawa, and scalar interactions ensures that all couplings reach a stable interacting UV fixed point, allowing fora complete asymptotic safety scenario in all couplings [9]. This is different from the complete asymptotic freedom scenario [5–7] whereall couplings vanish in the UV. Systematic analytic studies of the conformal window of non-supersymmetric field theories beyond perturbation theory re-started in Refs.[21, 22]. Here the reader will also find a complete list of earlier references. a r X i v : . [ h e p - ph ] J u l r T ( r ) C ( r ) d ( r ) N − N NG N N N − N +22 ( N − N +2) N N ( N +1)2 N −
22 ( N +1)( N − N N ( N − TABLE I. Relevant group factors for the representations used throughout this paper. However, a complete list of all the groupfactors for any representation and the way to compute them is available in Table II and the appendix of Ref. [22].
II. REVIEW OF THE HOT CONFORMAL FREE ENERGY DENSITY @ O ( g ) AND ENTROPYDENSITY
Our starting point is a generic asymptotically free gauge theory with N f Dirac flavors transforming according tothe representation r of the underlying gauge group.The relevant group-normalization factors are:Tr[ T ar T br ] = T [ r ] δ ab , T ar T ar = C [ r ] , (1)where T ar is the a -th group generator in the representation r and a = 1 , . . . , d [ G ]. We denote with d [ r ] the dimensionof the representation, and with G the adjoint representation. The quantities T [ r ] and C [ r ] are related via the identity C [ r ] d [ r ] = T [ r ] d [ G ]. We summarise useful group theory factors in Table I.The β function up to four-loop order, β ( g ) = − β (4 π ) g − β (4 π ) g − β (4 π ) g − β (4 π ) g + O ( g ) , (2)was computed in Ref. [24]. As is the case for the free energy, the four-loop β function is also computed in the MSscheme, thus no ambiguities in the scheme dependence of the expressions arise. Only β and β are scheme-independentand read: β = 113 C [ G ] − T [ r ] N f , (3) β = 343 C [ G ] − (cid:18) C [ G ] + 4 C [ r ] (cid:19) T [ r ] N f . (4)Asymptotic freedom is lost when the lowest-order coefficient, β , changes sign. This occurs for N AF f = 114 C [ G ] T [ r ] . (5)For a given fermion representation, the second coefficient, β , is negative below and near this critical number of flavorsand an IR-stable fixed point develops, which is known as the Banks-Zaks fixed point [25]. Such a theory displayslarge-distance conformality. The value of the coupling at the IR fixed point, g ∗ , is such that β ( g ∗ ) = 0, and it is givenat next-to-leading order by: g ∗ = − (4 π ) β β . (6)The IR fixed point disappears, at two-loop level, when β changes sign. This occurs for: N Lost f = 17 C [ G ]10 C [ G ] + 6 C [ r ] C [ G ] T [ r ] . (7)The free energy density is known up to the order g log(1 /g ) [26] but for this exploratory study it is sufficient tostop at order g , where it reads: fπ T = − d [ G ]9 (cid:20)
15 + 720 d [ r ] d [ G ] N f − (cid:18) C [ G ] + 52 T [ r ] N f (cid:19) g ( T )(4 π ) (cid:21) , (8)where T is the temperature of the theory and we traded the renormalization scale by T . In the deep UV, i.e., attemperatures sufficiently high that the physics is dominated by the asymptotically free fixed point, the couplingvanishes logarithmically and the UV free energy density is the one of a free gas of gluons and fermions: f UV π T = − d [ G ]9 (cid:20)
15 + 720 d [ r ] d [ G ] N f (cid:21) . (9)This is the trivial conformal limit while the interacting conformal free energy density in the deep IR is obtained byreplacing the coupling constant with the Banks-Zaks fixed point value g ∗ [20]: f IR π T = f ∗ π T = − d [ G ]9 (cid:34)
15 + 720 d [ r ] d [ G ] N f + (cid:0) C [ G ] + T [ r ] N f (cid:1) (11 C [ G ] − T [ r ] N f )34 C [ G ] − (20 C [ G ] + 12 C [ r ]) T [ r ] N f (cid:35) , We observe immediately that due to the conformal large-distance nature of our theories the dependence of the freeenergy density on the energy scale is only via the temperature, which factors out leaving behind a numerical factorcontaining information on the specific theory studied.The entropy density s can be determined via its relation with the free energy density: s π T = − π T dfdT = ˆ f + β ( g )4 ∂ ˆ f∂g , (10)with f = − ˆ f ( g ( T )) π T . At fixed points, where the β function vanishes, s F P π T = − f F P π T . (11)Having at our disposal the precise expressions of both the entropy and free energy density we can now move to thetransport coefficients that encode further important dynamical properties of the theory. III. FLAVOR AND TEMPERATURE DEPENDENCE OF THE CONFORMAL SHEAR VISCOSITYAND FERMION-NUMBER DIFFUSION COEFFICIENTS
We are now ready to unveil the dependence on the number of flavors for relevant transport coefficients such asthe shear viscosity and fermion-number diffusion coefficient for several gauge theories at perturbatively trustableinteracting fixed points. We will also analyse the temperature dependence of the mentioned transport coefficients,once the number of flavors and colors are fixed to some value in the perturbative conformal window.In order to determine the transport coefficients, the authors of Refs. [23, 27] used kinetic theory in which coupledBoltzmann equations describe the evolution of the phase-space density of distinct particle species. The transportcoefficients can be read off from the stress-energy tensor of the theory, which in turn is determined once the phase-space densities of all the particle species are known. In Refs. [23, 27] analytic expressions for the transport coefficientsare given, which approximately reproduce the numerical results. The result for the shear viscosity, in the next-to-leading-log approximation, is: η (cid:39) d [ G ] ζ (5) (cid:18) π (cid:19) ( v T c − v ) T g ( T ) ln( A T /m D ) , (12)where c = ( d [ G ] C [ G ] + N f d [ r ] C [ r ]) (cid:18) d [ G ] C [ G ] 00 N f d [ r ] C [ r ] (cid:19) + 9 π N f d [ r ] C [ r ] d [ G ] (cid:18) − − (cid:19) ,v = (cid:18) d [ G ] N f d [ r ] (cid:19) ,m D = 13 (cid:18) C [ G ] + N f C [ r ] d [ r ] d [ G ] (cid:19) g T , (13) N f A B .
25 2.867 3.177TABLE II. Values of the coefficients A and B [28] appearing in the next-to-leading-log expressions of the shear viscosity andthe fermion-number diffusion coefficient, for N = 3 and different values of N f . with m D the Debye mass and A a numerical coefficient that has a mild dependence on the number of flavors andcolors. The numerical values of A relevant for the cases studied in this paper are reported in Table II.Because of the overall T dependence of the shear viscosity it is convenient to normalise it to the entropy density.The so constructed ratio reads at a generic fixed point η F P s F P = A ( N f , N ) g ∗ ln[ B ( N f , N ) g − ∗ ] , (14)with A ( N f , N ) and B ( N f , N ) calculable definite positive and smooth functions of the number of colors and flavors,with g ∗ = g ∗ ( N f , N ) the value of the coupling at the fixed point.As expected at non-interacting fixed points, such as the UV fixed point, the ratio diverges. On the other hand atthe interacting IR fixed point the ratio approaches a finite value controlled by a small non-vanishing δ = N AFf − N f .In the left panel of Fig. 1 we plot ( η/s ) IR as function of the number of flavors, for fermions in the fundamentalrepresentation with N = 3. When decreasing the number of flavors below the asymptotically free boundary, wherethe shear viscosity diverges, we observe a dramatic decrease while still remaining much above the bound η/s ≥ / (4 π )conjectured by AdS/CFT [29]. It is natural to expect that, as we further decrease the number of flavors, the IRratio further decreases to reach a minimum value at the lower boundary of the conformal window. Below this criticalnumber of flavors we expect the onset of chiral symmetry breaking and the theory in the deep IR becomes a theoryof non-interacting pions with again a divergent value of this quantity.In the right panel of Fig. 1 we present the temperature dependence of the shear viscosity over the entropy densityfor several values of N f . The quantity η/s depends on the temperature over a reference scale Λ via the gauge coupling.The reference energy scale is chosen to be the one for which the β function displays a minimum occurring betweenthe trivial UV and interacting IR fixed points. The energy scale Λ is therefore defined by: g ( T = Λ) = 35 g ∗ . (15)For N f = 6, for which the theory does not display an IR perturbative fixed point, Λ is taken to be the scale atwhich the one-loop gauge coupling diverges as function of the temperature. The ratio η/s decreases as we decreasethe temperature for different values of the number of flavors within the conformal window. However for N f = 15 weobserve that a minimum develops around T = Λ. This happens because for this value of N f there is a temperaturefor which 4 ln (cid:16) ATm D (cid:17) = 1, which corresponds to a mininum for the g − ln (cid:16) ATm D (cid:17) − function.We now move our attention to another relevant transport quantity, the fermion-number diffusion coefficient. Thediffusion coefficient for the net number density of the fermion flavor a is given in Ref. [23] and reads, at the next-to-leading-log level: D a = 6 ζ (3) π C [ r a ] (cid:20) f ¯ fh (cid:88) b T [ r b ] λ b + 3 π C [ r a ] (cid:21) − T − g ln( B T /m D ) , (16)where the sum extends over all particle species b that the fermion species a can scatter with in the process ab → ab ,mediated by a gauge boson. Particles and antiparticles are counted separately, and the same goes for the helicitystates: this means that we have to count a factor of four for every Dirac fermion, and a factor of two for gauge bosons.Furthermore, λ b = 1 if the particle b is a fermion, and λ b = 2 if it is a boson. B is a numerical coefficient, whosevalues relevant for the cases studied in this paper are reported in Table II.We can specialize Eq. (16) to our theory with SU ( N ) gauge symmetry and N f fermions, all in the same represen-tation r . We obtain: D = 6 ζ (3) π C [ r ] (cid:20) N f T [ r ] + 4 N + 3 π C [ r ] (cid:21) − T − g ln( B T /m D ) . (17) ���� ���� ���� ���� ���� ������������������� � � � ( η / � ) � � N f = N f = N f = N f = N f = - ���� - ��� � ��� ������������������� � �� � / Λ η / � FIG. 1. Left Panel: η/s evaluated at the IR fixed point, as a function of the number of flavors, for fermions in the fundamentalrepresentation with N = 3 colors. Right Panel: η/s as function of the temperature over the RG scale Λ for different values of N f in the conformal window and one outside corresponding to N f = 6, for N = 3 colors. Although N f = 14 still displays apotential IR fixed point the IR dynamics of η/s cannot be accessed perturbatively. The horizontal line at the bottom is theconjectured AdS/CFT bound. At very low energies, where the coupling is frozen at the fixed-point value g ∗ , the dimensionless quantity ( T D ) IR can be plotted as a function of the number of flavors. This is represented in the left panel of Fig. 2, for the caseof fermions in the fundamental representation and N = 3. As for the case of the shear viscosity-to-entropy density ���� ���� ���� ���� ���� ���� ���� �������������� � � ( � � ) � � N f = N f = N f = N f = N f = - ���� - ��� � ��� ���������������� � �� � / Λ � � FIG. 2. Left Panel: (
T D ) IR as a function of the number of flavors, for the case of fermions in the fundamental representationand N = 3. Right Panel: T D as a function of the temperature, for different values of N f and N = 3. ratio, we observe that ( T D ) IR diverges as g ∗ approaches the origin when increasing the number of flavors towardsthe asymptotic freedom boundary. As for the shear viscosity-to-entropy density ratio, in the right panel of Fig. 2 wealso plot T D as a function of temperature for different values of the number of flavors in the conformal window andfor N f = 6.One last comment has to be made about the applicability of the next-to-leading-log approximation for the transportcoefficients in the conformal window. The presence of a perturbative IR fixed point allowed us to apply the next-to-leading-log results in the whole energy range, from the UV, where the theory is asymptotically free, down to theIR. However, particular care has to be taken to decide whether the values obtained in the deep IR can be trusted.We chose to illustrate the results for the case of three colors and for different values of the number of flavors withinthe perturbative conformal window. N f = 15 is the last value at which we could observe the expected behavior ofthe transport coefficients as a function of the temperature, i.e., to run from a divergent value in the UV down toa constant finite value in the IR. For N f = 14 the next-to-leading-log expression for the transport coefficients doesnot stabilise at a finite value in the IR, but instead diverges at low energies, showing that the next-to-leading-logapproximation cannot be trusted any longer. In fact, following Ref. [27] one can argue that the next-to-leading-logresult is very close to the full leading-order result (and therefore trustable) as long as m D /T ≤
1. This requirement issatisfied in our analysis provided N f is larger than 16 .
25, de facto further limiting the window of applicability of theperturbative analysis. The values of m D /T at the IR fixed point for N = 3 and the values of N f within the conformalwindow that have been considered in this paper are reported in Table III. N f ( m D /T ) IR
14 3 . . . .
25 0 . m D /T evaluated at the IR fixed point for N = 3 and different values of N f in the conformalwindow. It can be observed that the constraint m D /T ≤ N f = 16 . IV. CONCLUSIONS
We determined the shear viscosity-to-entropy density ratio and the fermion-number diffusion coefficient within theperturbative regime of the conformal window for gauge-fermion theories. Our formalism is valid for any fermionicmatter representation, while the physical results, which are expected to hold generically, were elucidated via a three-color gauge theory as functions of the number of flavors in the fundamental representation. We observed that whenthe number of flavors decreases from the value at the loss of asymptotic freedom both the shear viscosity-to-entropydensity ratio and the fermion-number diffusion coefficient measured at the IR fixed point dramatically decrease.Furthermore, for a given number of flavors within the perturbative conformal window both coefficients decrease(albeit not monotonically for N f = 15) with the temperature from their divergent value in the UV down to the valueat the IR fixed point. More specifically we discovered that down to 15 flavors the next-to-leading-log results exhibitthe expected behavior of stabilising at a constant finite value in the IR. For N f = 14 the next-to-leading-log resultsdiverge at low energy, showing that the next-to-leading-log approximation cannot be trusted even qualitatively. Infact, following Refs. [23, 27] one can consider a more restrictive constraint for the next-to-leading-log approximationto be quantitatively accurate. The latter requires m D /T ≤ N f larger than 16 . η/s at the IR fixed point drops significantly when going from 16.25 to 15 flavors showing that a modestchange in the number of flavors dramatically affects the dynamics of the theory encoded in the transport coefficients.Higher-order corrections are needed to reach lower values of N f within the conformal window for the transportcoefficients. In contrast, at zero temperature one observes that perturbation theory allows to go quite low in thenumber of flavors within the conformal window [14–17]. Although unproven it is reasonable to expect that theminimum of η/s as function of temperature in QCD lies below the lowest value of η/s obtained at the bottom of theconformal window, and therefore lower than the one obtained near 15 flavors.To conclude, the salient results of our analysis are: • We provided theoretically relevant examples in which the perturbative estimate of the transport coefficients canbe used along the entire RG flow from the UV to the IR without loosing their validity. • We determined the range of applicability of those results within the conformal window of QCD and QCD-liketheories.Our computations delineate and extend the range of applicability of the perturbative transport coefficients to therelevant subject of the conformal window in QCD and QCD-like theories. The work serves as stepping stone forfuture studies in this direction.
ACKNOWLEDGMENTS
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