Applications of dimensional reduction to electroweak and QCD matter
aa r X i v : . [ h e p - ph ] S e p HU-P-D144
Applications of dimensional reductionto electroweak and QCD matter
Mikko Veps¨al¨ainen Theoretical Physics Division, Department of Physical Sciences andHelsinki Institute of PhysicsP.O. Box 64, FIN-00014 University of Helsinki, Finland
Abstract
This paper is a slightly modified version of the introductory part of a doctoral dis-sertation also containing the articles hep-ph/0311268, hep-ph/0510375, hep-ph/0512177and hep-ph/0701250. The thesis discusses effective field theory methods, in particu-lar dimensional reduction, in the context of finite temperature field theory. We firstbriefly review the formalism of thermal field theory and show how dimensional re-duction emerges as the high-temperature limit for static quantities. Then we applydimensional reduction to two distinct problems, the pressure of electroweak theoryand the screening masses of mesonic operators in hot QCD, and point out the simi-larities. We summarize the results and discuss their validity, while leaving all detailsto original research articles. E-mail: Mikko.T.Vepsalainen@helsinki.fi ontents i hapter 1 Introduction
The physics of interactions between elementary particles is described to an amazing ac-curacy by the standard model of particle physics. It ties three of the four fundamentalinteractions, namely the electromagnetic, weak and strong interactions, together underthe conceptual framework of relativistic quantum field theory. Scattering processes andbound states involving few particles are well described by the model, although many openquestions, mostly related to strongly interacting states, remain. In the energy region cur-rently accessible to experiments we have therefore full reason to believe that this theoryis correct.When matter is heated high above everyday temperatures, its neutral constituents aretorn apart into an interacting plasma of elementary particles. At temperatures of the sameorder or higher than the particle masses this necessitates combining quantum statisticalmechanics with relativistic field theory. The interactions between individual particles arestill governed by the standard model interactions, but the effects of hot medium changetheir long-distance behavior and give rise to many-particle collective modes.Experimentally such extreme conditions are accessible in relativistic heavy ion collisionscurrently produced at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven, and,starting this year, also at Large Hadron Collider (LHC) at CERN. In these experiments twoheavy nuclei collide against each other, forming a finite volume of extremely hot matter.The matter described by the theory of strong interactions, quantum chromodynamics(QCD), goes through a phase transition to a deconfined phase of color-charged particlesforming a quark-gluon plasma, which then rapidly cools as it expands. This kind oftemperatures were also present in the very early universe, whose expansion is sensitive tothe equation of state of both QCD and electroweak matter.The formalism for finite temperature quantum field theory arises naturally from the pathintegral quantization of field theories. The time coordinate is extended to complex valuesto account for varying the fields over statistical ensemble, and the functional integral is overall field configurations periodic or antiperiodic in the imaginary time. When temperatureis larger than any other scale in the process, the excitations in the imaginary time canbe integrated out and the physics of static quantities is described by a three-dimensionaleffective theory. This is known as dimensional reduction [5]. The effective theory can besystematically derived, and it exhibits the same infrared behavior as the full theory. Atfinite temperatures the main advantage in using a dimensionally reduced effective theoryin perturbative computations is the ability to systematically treat the various infrareddivergences, as well as the resummations needed to cure them, in a simpler setting.1imensional reduction has been successfully applied over the years to compute manybosonic quantities both perturbatively and in combination with lattice simulations. In theQCD sector, the three-dimensional formulation known as EQCD has made it possible toperturbatively compute the pressure up to the last perturbative order g ln g [6–13], andthe result has also been extended to nonzero chemical potentials [14]. Lattice implementa-tions of EQCD have been used to compute the static correlation lengths of various gluonicoperators [15–20]. There are also recent developments in formulating an effective theorypreserving the spontaneously broken Z (3) symmetry of the deconfined phase [21], whichis explicitly broken in EQCD [22]. Besides QCD, the electroweak symmetry breaking hasalso been solved in detail using lattice simulations in dimensionally reduced effective the-ory [23–28], motivated by the possibility of a first order electroweak phase transition beingthe origin of the observed baryon asymmetry in the universe.There are only few applications of dimensional reduction to fermionic observables, be-cause the fermion fields are integrated out from the three-dimensional effective theory. Thissimplifies the computation of bosonic quantities tremendously, but the accessible fermionicobservables are then limited to those that can be inferred from vacuum or bosonic ones,such as quark number susceptibilities χ ij = ∂ p/∂µ i ∂µ j [29]. Systematic application of di-mensional reduction to fermionic operators was developed in [30], inspired by the progressin heavy quark effective theories.The use of dimensional reduction is restricted to time-independent quantities. It shouldbe mentioned here that for real-time computations there exists another scheme of resum-ming the light particle self-energy corrections to regulate some of the infrared divergences,namely the hard thermal loop (HTL) approximation [31, 32]. Both schemes succesfullyresum the one-loop infrared divergences, but in general the HTL Green’s functions aremore complicated, since they carry the full analytic structure of the original theory. Itis also very hard to systematically improve the HTL approximation beyond the leadingorder.In this thesis we study two applications of dimensional reduction to the standard model,the perturbative evaluation of the electroweak pressure and the next-to-leading order cor-rection to screening masses of mesonic operators. The thesis is organized as follows. Inchapter 2 we first review the formalism of thermal quantum field theory, and then dis-cuss dimensional reduction in the context of general effective theories in section 2.2. Inchapter 3 we go through the computation of the electroweak theory pressure, with specialattention given to the behavior near the phase transition. We combine the result with thepreviously known QCD pressure in section 3.3 and study the convergence of the series andthe deviation from the ideal gas for physical values of parameters. Results for a simpler,weakly coupled SU(2) + Higgs theory are also shown for comparison.In chapter 4 we review our work on meson correlators. After a short motivation usinglinear response theory, we compute the leading order correlators at zero and finite density.Then we proceed to derive a dimensionally reduced effective theory for the lowest fermionicmodes and solve the O ( g ) corrections to screening masses. Finally, we compare withrecent lattice determinations of the masses and discuss the differences. Chapter 5 containsour conclusions. 2 hapter 2 Thermal field theory
In this chapter we will first review how the thermodynamical treatment of quantum fieldtheory can be formulated in terms of Euclidean path integrals. We then proceed to discussdimensional reduction, which is the underlying effective theory method used in all theresearch papers included in this thesis.
The statistical properties of relativistic quantum field theory are most naturally describedusing the grand canonical ensemble. Since particles can be spontaneously created andannihilated, the microcanonical or canonical ensembles with fixed particle numbers cannotbe built, but instead one would have to use the conserved quantities like electric charge.To avoid this kind of complicated constraints on field configurations, it is generally easierto fix the mean values of energy and conserved commuting number operators using theLagrange multipliers β = 1 /T and µ i , respectively. This is the grand canonical ensemble.The thermodynamical properties of the system are given by the partition function andits derivatives. In quantum mechanics the partition function is defined as the trace of thedensity matrix ρ , Z ( T, V, µ i ) ≡ Tr ρ = Tr e − β ( H − µ i N i ) , (2.1)where H and N i are the Hamiltonian and conserved number operators, respectively. Thethermal average of an operator is then defined as h A i = 1 Z Tr ρA , (2.2)and the usual thermodynamic quantities like pressure, entropy, energy and particle num-bers are given by the partial derivatives p = T ∂ ln Z∂V , S = ∂T ln Z∂TN i = T ∂ ln Z∂µ i , E = − pV + T S + µ i N i . (2.3)In quantum mechanics the evaluation of the trace in Eq. (2.1) is simple, one just takesany complete orthonormal basis {| n i} , preferably eigenstates of H − µN if these are known,and sums over h n | ρ | n i . The same procedure can in principle be applied to field theory,3here the sum over basis vectors is replaced by a functional integral in the space of fieldconfigurations.Field theories are usually defined in the Lagrangian formalism, and finding the Hamil-tonian function required for computation of the partition function in Eq. (2.1) can bequite involved, in particular in the context of gauge theories. One has to fix the gaugeand then carefully separate the canonical variables from auxiliary ones depending on thechosen gauge [33]. In addition to the usual canonical equations of motion, the fields areconstrained by the gauge condition and the field equation for the auxiliary field, whichcan be interpreted as the Gauss’ law.Once the Hamiltonian has been found, we can insert a complete set of eigenstates | φ ( x ); t i of the field operator ˆ φ ( x ) in the Heisenberg picture to compute the partitionfunction. This gives Z ( T, V, µ i ) = Z [d φ ] h φ ( x ); t | e − β ( H − µ i N i ) | φ ( x ); t i , (2.4)where the integration is over all canonical variables. From the time-dependence of thefield operator it follows thatˆ φ ( x , t ) = e iHt ˆ φ ( x , e − iHt ⇒ | φ ( x ); t i = e iHt | φ ( x ); 0 i . (2.5)Eq. (2.4) can then be viewed as the transition amplitude for the field to return to the samestate after an imaginary time − iβ , when the time-development is given by the Hamiltonian H − µ i N i , Z ( T, V, µ i ) = Z [d φ ] h φ ( x ); t − iβ | φ ( x ); t i . (2.6)Dividing the time interval into infinitesimally small pieces and inserting at every point acomplete set of position and momentum eigenstates this can be cast into a path integralform (for details see e.g. [34, 35]) Z ( T, V, µ i ) = Z D φ D π exp (cid:20) i Z t − iβt d t ′ Z d x ˙ φ ( x , t ′ ) π ( x , t ′ ) − H ( φ, π ) + µ i N i ( φ, π ) (cid:21) , (2.7)where H and N are the Hamiltonian and number densities, respectively, and ˙ φ ≡ ∂ t φ .When H − µ i N i is at most quadratic in canonical momenta, the momentum integrationcan be done. In gauge theory it is useful to first reintroduce the Gauss’ law by treating thetemporal gauge field component A a as an independent variable, which, when integratedover, would be replaced by the stationary value satisfying Gauss’ law.Performing the momentum integrations, we get back to the Lagrangian formulation Z ( T, V, µ i ) = Z D Φ exp (cid:20) i Z t − iβt d t ′ Z d x L ′ (Φ , ˙Φ) (cid:21) , (2.8)where the integration is now over both canonical and auxiliary fields. The Lagrangian L ′ usually differs from the one we started with. In particular, the momenta in Eq. (2.7) mustbe replaced with the values solved from˙ φ ( x , t ) = δδπ ( x , t ) ( H [ φ, π ] − µ i N i [ φ, π ]) , (2.9)4o that in the end we have L ′ = π ( φ, ˙ φ ) ˙ φ − H ( φ, π ( φ, ˙ φ )) + µ i N i ( φ, π ( φ, ˙ φ )) . (2.10)Moreover, in a gauge theory one usually includes an additional gauge fixing term intothe Lagrangian using Grassmannian ghost fields in order to have less constraints on theintegration variables.As can be seen in Eq. (2.6), the partition function is computed as an integral overamplitudes with the same field configuration at both end points, φ ( t − iβ, x ) = φ ( t, x ). Forfermionic variables it follows from the anticommutation properties of Grassmann variablesthat the trace has to be computed with antiperiodic condition ψ ( t − iβ, x ) = − ψ ( t, x )instead. Both boundary conditions can be verified by inspecting the two-point function,taking into account the correct time ordering of the fields [34].When extending the time coordinate to complex values, the integration path is no longerunique. It can be chosen to fit the problem in question, with some minor restrictions.The time arguments of the operators whose thermal averages we are computing shouldobviously lie on the integration path. Also, the imaginary part of t should be nonincreasingin order to have a well-defined propagator. There are two conventional choices for the path,leading to two different ways of computing at finite temperatures.First, one can choose to include the whole real axis by first integrating from − t to t ,then down to t − iσ , with 0 ≤ σ ≤ β , back to − t − iσ and finally down to − t − iβ ,in the end letting t → ∞ (see e.g. [36]). This approach leads to the so-called real-timeformalism, which has the advantage that one can directly compute real-time quantitieswithout having to analytically continue the final results to Minkowski space. However,in this formalism the number of degrees of freedom is doubled, with unphysical fieldsliving on the lower horizontal part of the integration path and mixing with the physicalones. This in turn requires the propagators to be extended to 2 × t ( τ ) = t − iτ , τ = 0 ..β , whichleads to the so-called imaginary time formalism. The choice of t does not affect theresults, so one can choose t = 0 and replace the time coordinate in Eq. (2.8) by τ = it : Z ( T, V, µ i ) = Z per . D Φ exp (cid:20)Z β d τ Z d x L ′ E (Φ , ˙Φ) (cid:21) . (2.11)The functional integral is over periodic or antiperiodic fields as described above, and theEuclidean Lagrangian L ′ E is the same as in Eq. (2.10), rotated to Euclidean space withthe replacements t = − iτ , γ E = γ M , A E = iA M ,∂ t = i∂ τ , γ Ei = − iγ iM , A Ei = A iM , (2.12)where ‘E’ and ‘M’ stand for Euclidean and Minkowski space quantities, respectively. Inthe following we will always work in Euclidean space unless otherwise mentioned, and dropthe ‘E’ superscripts. In the above equation, A µ represents any four-vector, in particularthe gauge field components. There is no doubling of degrees of freedom in this formal-ism, and for static quantities, such as the free energy or screening masses, it is usuallysimpler to compute in imaginary time. Other results have to be analytically continued toreal time arguments, and while in principle this can be done with some mild regularity5ssumptions [37], in practice some additional model assumptions are required to carry outthe continuation.Because the fields are required to be periodic, the imaginary time direction can beviewed as a closed circle with circumference β = 1 /T . The momentum component in acompact dimension is quantized, so the fields can be decomposed in the momentum spaceas Fourier series φ ( τ, x ) = T ∞ X n = −∞ φ n ( x ) e iω n τ , ω n = (cid:26) nπT (bosons)(2 n + 1) πT (fermions) , (2.13)where ω n are referred to as Matsubara frequencies [38]. From the gauge transformationrule for the gauge field components A µ → Ω A µ Ω − − ig ( ∂ µ Ω)Ω − , Ω( x ) = exp[ igT a α a ( x )] (2.14)it is easy to see that the gauge transformation functions α a have to be periodic as well, sothe ghost fields will have bosonic Matsubara frequencies despite of their anticommutingnature. The thermal environment changes the boundary conditions and the propagators from theirzero-temperature forms. Fortunately, this does not introduce any new ultraviolet diver-gences, but the usual renormalization procedure remains unchanged and the countertermshave precisely the same values as at T = 0 (depending on the scheme). Intuitively this iseasy to understand, since only the excitations with wavelengths & β can see the periodicityof the time direction, while the renormalization is only concerned with divergences relatedto the short distance behavior of Green’s functions. The divergence structure is then pre-cisely the same as in the zero-temperature theory and one can choose a T -independentrenormalization scheme such as the MS scheme.To see this in some more detail, we note that the free propagator at finite tempera-ture can be viewed as an explicitly periodic combination of zero-temperature Euclideanpropagators [36], S F ( τ, x ; T ) = ∞ X n = −∞ S F ( τ + nβ, x ; T = 0) , ≤ τ < β . (2.15)The zero-temperature ultraviolet divergences requiring renormalization arise from theshort-distance singularities at x = 0. The only term in the above sum where we canhave x = ( τ + nβ ) + x = 0 is the n = 0 term, which does not depend on temperature.The divergences of the thermal propagator are therefore correctly removed by the T = 0counterterms. At higher order diagrams these divergences are multiplied by T -dependentfinite parts of the diagram, so the general proof of renormalizability and T -independenceof counterterms is somewhat more involved, but it follows from a similar decompositionof propagator into a singular T = 0 part and an analytic T -dependent part [39].As the parameters of the theory are renormalized, they also run with the scale accordingto the renormalization group equations. The actual equations are again the same as in T = 0 theory, but the choice of renormalization point is complicated by the appearance6f new scales πT and µ in addition to the external scales present in the problem, as wellas the the scales gT , g T generated dynamically by interactions. If these scales are verydifferent, removing the large logarithms by a suitable choice of scale may prove difficult,and a careful analysis of the scale hierarchy is required to construct a good perturbativeexpansion.While the ultraviolet divergences are unaffected by the finite temperature, at the infraredend the situation is very different. The finite extent of the temporal direction causes thefield components with wavelengths ≫ /T to see the space effectively as three-dimensional,and this gives rise to many new infrared divergences. These will be treated in more detailin the following section. In this section we will review the rationale for dimensional reduction in the more generalcontext of low-energy effective field theories. We will also discuss the finite-temperatureinfrared divergences and the resummations needed to get rid of them.
One of the fundamental properties of physics is that phenomena at some specific distancescale can be effectively described by a theory which does not depend on the physics at muchshorter scales. This is fortunate, for otherwise we would not even be able to describe thetrajectory of a thrown ball without knowledge of beyond the standard model physics. Thesame behavior, known as decoupling, is also present in quantum field theory, where it is byno means obvious that the heavy particles inevitably occurring as internal legs in Feynmandiagrams can be neglected. The proof that the high-energy modes only contribute to long-distance phenomena by renormalization of the parameters and by corrections suppressedby inverse powers of the heavy masses is contained in the celebrated theorem of Appelquistand Carazzone [40]. From this point of view, every physical theory can be viewed as aneffective theory, equivalent to the underlying more fundamental theory in some finiteenergy range.Formally, if the underlying theory is known, the effective theory for light modes φ l canbe written as a path integral over the heavy modes Φ h , e iS eff [ φ l ] = Z D Φ h exp iS [ φ l , Φ h ] , (2.16)where the effective action S eff [ φ l ] is in general a non-local functional of the light fields.Analytically the path integral can only be computed in the Gaussian approximation aroundsome given field configuration ¯Φ h , S [ φ l , Φ h ] ≈ S [ φ l , ¯Φ h ] + Z d d x δSδ Φ h ( x ) (cid:12)(cid:12)(cid:12) Φ h =¯Φ h (cid:0) Φ h ( x ) − ¯Φ h ( x ) (cid:1) + 12 Z d d x d d y δ Sδ Φ h ( x ) δ Φ h ( y ) (cid:12)(cid:12)(cid:12) Φ h =¯Φ h (cid:0) Φ h ( x ) − ¯Φ h ( x ) (cid:1) (cid:0) Φ h ( y ) − ¯Φ h ( y ) (cid:1) . (2.17)Choosing ¯Φ h to be a saddle point of the action, δS [ φ l , Φ h ] /δ Φ h = 0, the integration overΦ h gives the effective action (for bosonic Φ h ) as S eff [ φ l ] = S [ φ l , ¯Φ h ] + i δ Sδ Φ h ( x ) δ Φ h ( y ) (cid:12)(cid:12)(cid:12) Φ h =¯Φ h , (2.18)7here the last term depends on φ l both directly through S and through the saddle pointcondition which makes ¯Φ h a functional of φ l . The Gaussian approximation corresponds tothe one-loop level in heavy-loop expansion; if we want to go beyond that the path integralcan no longer be computed analytically, but we have to resort to perturbation theory orsome other approximation.While the heavy fields can be integrated out as shown above, the resulting effectiveaction is generally a complicated nonlocal functional of the light modes and cannot becast in the form of an effective local Lagrangian density without some additional approx-imations. An often used method is the derivative expansion, where the non-local termsare expanded in the light field momenta p over the heavy field mass M , leading to S eff = Z d d x L eff + X n O n (cid:16) pM (cid:17) n , (2.19)where O n represent operators suppressed by powers of the heavy mass. In terms of Feyn-man diagrams this means that the effective action is computed with only heavy fields onthe internal lines, since the action is made local in the light fields. The form of Eq. (2.19) isprecisely what should be expected based on the decoupling theorem: parameter renormal-izations and heavy mass suppressed operators. There is a twist, however, since the lightparticle momenta need not be small when the non-local operator is embedded in a multi-loop graph and interacts with heavy fields, and the derivative expansion may then fail.For example, in the large-mass expansion at zero temperature [39,41] it is well known thatone needs to take into account also the diagrams with light internal lines in order to getthe correct low-energy effective Lagrangian. This will also be the case in the dimensionallyreduced effective theory at high temperatures, as we will show later on.As an illuminating example, consider a theory with two scalar fields [42], L = 12 ∂ µ φ∂ µ φ − m φ − V ( φ ) + 12 ∂ µ Φ ∂ µ Φ − M Φ + 12 λφ Φ , (2.20)in the limit m ≪ M . This is similar to the situation at finite temperature where φ can bethought as the static ( n = 0) Matsubara mode, while the heavy field mass is of the order2 πT . In this model the dependence on the heavy field is quadratic, so we can exactlyintegrate out Φ, giving S eff = S [ φ ] + i − ∂ − M + λφ ) = S [ φ ] − i ∞ X k =1 λ k k Tr [( ∂ + M ) − φ ] k , (2.21)where in the last step we have dropped a φ -independent term and expanded in the smallcoupling λ . The first term in the sum ( k = 1) is represented by Fig. 2.1(a) and contributesby a local term to the mass renormalization, − iλ ∂ + M ) − φ = iλ Z d d x φ ( x ) Z q q − M + iǫ = − λM π ) (cid:18) ǫ + 1 − ln M µ (cid:19) Z d d x φ ( x ) , (2.22)where we have used dimensional regularization to control the ultraviolet divergence in themomentum integration, with the conventions Z q ≡ (cid:18) e γ µ π (cid:19) ǫ Z d d q (2 π ) d , d = 4 − ǫ . (2.23)8a) (b) (c) Figure 2.1:
Diagrams in the effective action for a theory with two scalars. Solid lines representthe heavy field, dashed lines the light one.
Here µ is the (arbitrary) dimensional regularization scale, modified to include the constantstypical of the MS scheme.The λ -term, however, already shows where the derivative expansion causes problems.A straightforward computation of the diagram in Fig. 2.1(b) gives − iλ ∂ + M ) − φ ] = λ π ) Z d d x d d y Z k φ ( x ) φ ( y ) e − ik · ( x − y ) ×× (cid:18) ǫ − ln M µ − Z d t ln (cid:20) − t (1 − t ) k M (cid:21)(cid:19) . (2.24)The first two k -independent terms contribute to the renormalization of the 4-point vertex.The remaining logarithm is a non-local operator connecting two φ products at differentpoints. For small k the integrand can be expanded in k /M , leading to a series of localfour-point derivative couplings of the form φ ( ∂ /M ) n φ . However, when this operator ispart of a larger diagram there is no guarantee that k is small.For example, the diagram in Fig. 2.1(c) with one light and two heavy internal linesis not produced by the effective theory expanded this way. All loop momenta can belarge, and therefore the expansion in k /M is not reliable. Computing this diagram israther nontrivial [43], but one can show that if the ultraviolet divergences are removed inthe MS scheme, the diagram does not vanish in the limit M → ∞ . To have an explicitdecoupling where all graphs containing heavy internal lines are suppressed one should usea renormalization scheme where the counterterm is the negative of the graph expanded inthe light masses and momenta [39]. At finite temperatures this may be difficult becauseof the additional infrared divergences. Moreover, we would prefer to use the MS schemewhere the counterterms are already known to high order and have a simple structure.Because of the difficulties in integrating out the heavy fields as described above, at higherorders it is usually safer to construct the effective Lagrangian explicitly by matching theGreen’s functions. The decoupling theorem states that in a renormalizable theory theparameters in the effective theory can be chosen in such way that the Green’s functionsof light fields differ from those computed in the full theory by terms suppressed by powersof the heavy mass, G N ( p , . . . , p N ; g, G, m, M, µ ) = h | T φ ( p ) . . . φ ( p N ) | i full = z − N/ G ∗ N ( p , . . . , p N ; g ∗ , m ∗ , µ ) [1 + O (1 /M a )]= z − N/ h | T φ ∗ ( p ) . . . φ ∗ ( p N ) | i eff [1 + O (1 /M a )] , (2.25)where M and G are the masses and couplings in terms involving heavy fields, whilethose for terms with only light fields are labeled m, g . The corresponding effective theoryparameters are m ∗ , g ∗ and φ ∗ = z / φ . We can use this information directly and write9own the most general light mode Lagrangian which respects the symmetries of the originaltheory, and then compute a number of N -point functions (usually N = 2 , , Field theories at finite temperature contain many new mass scales in addition to thosegiven by the parameters of the zero-temperature Lagrangian. Besides the temperatureitself there are dynamically generated scales related to collective modes and screeningphenomena, and the particle masses are modified by thermal effects as well. Renormal-izing the theory in the minimal subtraction scheme gives rise to logarithms of the typeln( m /µ ), where m can be any of the different scales in the theory. In particular, largescales do not decouple but instead give contributions that grow logarithmically with thescale. This seems to make perturbation theory useless in theories with vastly differentmass scales, since we cannot choose a renormalization scale that simultaneously makes allthe logarithms small. As a result, terms in the perturbative expansion contain powers oflarge logarithms in addition to small coupling and need not decrease at higher orders.To be more specific, in gauge theory the electric and magnetic screening scales are oforder gT and g T , respectively, and thus there is a clear hierarchy of scales in the smallcoupling region where we would want to use perturbation theory. The solution is, asdiscussed above, either to use a more complicated renormalization scheme or to formulatean effective theory and continue using the MS scheme [44]. As it turns out, it is simplerto carry out the computations using the effective theory. We will mostly concentrate ongauge theories in what follows, in particular on QCD and electroweak theory.In the imaginary time formalism we can write the four-dimensional theory in terms ofthe Matsubara modes of Eq. (2.13). For generic bosonic and fermionic fields the free partof the action (without any chemical potentials, although they could easily be included) is S = Z β d τ Z d x φ † [ − ∂ + m b ] φ + ¯ ψ ( /∂ + m f ) ψ = T ∞ X n = −∞ Z d x φ † n (cid:0) − ∂ i + [(2 πnT ) + m b ] (cid:1) φ n + ¯ ψ n [ i (2 n + 1) πT γ + γ i ∂ i + m f ] ψ n , (2.26)which can be viewed as a three-dimensional Euclidean theory of an infinite set of fieldswith masses M n = ω n + m . If the temperature is much higher than the particle masses,we can use the arguments of the previous section and try to formulate an effective theoryfor the light modes with M n ≪ T , or the bosonic zero-modes since they are the onlymodes with ω n = 0. This theory loses all dependence on the (imaginary) time coordinate,so we have effectively reduced the number of dimensions to three. From the point of viewof modes with wavelengths much larger than 1 /T the finite temporal direction of length β has shrunk to a point.While the dimensionally reduced theory cannot give any information about the timedependence of the theory, for static Green’s functions the effective theory gives correctresults up to corrections of order m / ( πT ) , where m is any of the light masses. Notein particular that at high enough temperatures the highest unintegrated mass scales arethe dynamically generated scales ∼ gT , so the corrections to the effective theory are10omparable with the higher orders of perturbation theory and both have to be taken intoaccount to get a consistent perturbative expansion. To gain control over which operatorsto include, power counting rules have to be established for given momentum region. Athigher orders it will be necessary to include nonrenormalizable operators into the effectivetheory, especially if one wishes to have a theory that produces all static Green’s functionsto given order. In many cases, like when computing the free energy, it is sufficient to useonly the couplings present already in the original theory, in which case the effective theoryis super-renormalizable because of the lower dimensionality.The main advantage in using an effective theory at high temperatures is in the infraredphysics. In general, if the theory contains massless bosonic fields one expects more severeinfrared singularities when going to finite temperature, since the Bose–Einstein factor inreal-time propagators behaves as n B ( E ) = 1 e βE − e βk − → βk as k → . (2.27)This can be also understood in the imaginary time formalism, where the zero Matsub-ara mode behaves like a massless particle in three dimensions, and lower dimensionalitygenerally makes the infrared behavior worse. It is well known that in Yang–Mills theoriesperturbation theory at finite temperatures suffers from many infrared problems, becomingfinally completely non-perturbative at O ( g ) [45, 46]. These problems are related to mass-less particles, in particular to the gauge fields, whose screening by medium effects is notcorrectly reproduced by the na¨ıve perturbation theory. By definition, the dimensionallyreduced theory has the same infrared limit as the original theory, while being computa-tionally simpler. The leading order contribution coming from scales of order T can beincluded in the parameters of the effective theory via the matching procedure, which isinfrared safe, and the infrared peculiarities can then be studied in a simpler setting. Inparticular, the dimensionally reduced effective theory does not contain any fermionic fields,which makes it easier to study non-perturbatively using lattice simulations.The electric screening effects can be included by reorganizing the perturbative expan-sion. Computing the one-loop self-energy of a zero-mode gauge field component A µ , wefind that in the limit of vanishing momentum it behaves asΠ µν ( ω n = 0 , k → ∝ g T δ µ δ ν . (2.28)The temporal component develops a thermal mass of order gT , while the other componentsremain massless. In the soft limit where k . gT it is not consistent to treat this self-energyas perturbation, but it should be included in the propagator instead. This means that weshould sum all diagrams with an arbitrary number of self-energy insertions on the temporalgluon line to get consistent O ( g ) results, which is often referred to as resummation. In fourdimensions one has to be careful not count any diagram twice because of this summation;usually this is done by adding and subtracting a term containing the self-energy in theLagrangian, L = L + L I = ( L + δ L ) + ( L I − δ L ) (2.29)and treating the subtracted term as an interaction. In the dimensionally reduced theorythe resummation is simpler, since the thermal mass for A comes out naturally from thematching procedure. Moreover, there is no risk of double counting diagrams, since thethermal mass is only created by n = 0 and fermionic modes (the mass can be computed11n the k = 0 limit, and the dimensionless graphs vanish in dimensional regularization),which are not present in the effective theory. Note that the electric mass does not breakthe remaining gauge invariance, since when restricting to bosonic zero modes only we arealso forced to only consider τ -independent gauge transformations. The transformationrule in Eq. (2.14) then boils down to A a = 2 Tr A T a → A Ω − T a = exp[ igα c τ cab ] A b , (2.30)so in the three-dimensional theory A becomes a massive scalar transforming in the adjointrepresentation of the gauge group. The remaining gauge invariance in three dimensionsprevents the spatial gauge field components from developing a mass term.In the magnetic sector there are infinitely many diagrams that all contribute at order g , and, unlike for the electric mass, they appear with so different and complex topologiesthat they cannot be resummed in a simple way to tame the infrared singularities. In fact,there is no gauge-invariant magnetic mass term that could be included in the Lagrangianfor perturbatively computing beyond O ( g ), but instead the magnetic screening has tobe treated non-perturbatively. In the very low momentum region the fields with thermalmasses ∼ gT can be integrated out as well, leaving a three-dimensional pure gauge theorywith coupling ˜ g = g T , which is the only dimensionful parameter in the Lagrangian. Inthis theory there is no small dimensionless parameter to do perturbation theory with, butthe infrared dynamics of nonabelian gauge theory is inherently nonperturbative.To see how the matching of parameters in the dimensionally reduced theory goes inpractice, we will take a closer look at the mass parameters, following to some extent [11,23].The masses can be found by comparing the static two-point functions computed in boththeories. For simplicity, we will use a scalar particle with a small zero-temperature mass m . gT as an example and work to order g , which is sufficient for many computations,in particular for determining the free energy to order g as in [2, 3].In the full theory the inverse propagator can be written as k + m + Π( k ) = k + m + Π( k ) + Π ( k ) , (2.31)where Π( k ) includes the diagrams with at least one heavy internal line, while Π ( k ) isthe contribution of n = 0 modes only. In the effective theory the same function reads k + m + Π ( k ) . (2.32)The contribution coming from the non-static modes, Π( k ), is of order g T , and thematching has to carried out in the region where the effective theory is valid, k . gT . Sinceintegration over massive modes is infrared safe, the renormalized self-energy Π( k ) has noinfrared divergence and can be expanded in k /T ,Π( k ) = Π(0) + k dd k Π(0) + O (cid:18) g k T (cid:19) , (2.33)where the terms left out are of order g T . Further expanding each term in loop expansionwith coupling g , Π( k ) = ∞ X n =1 Π ( n ) ( k ) , where Π ( n ) ( k ) ∼ O ( g n ) , (2.34)12he inverse propagator in Eq. (2.31) reads, including terms up to O ( g ), k (cid:18) k Π (1) (0) (cid:19) + m + Π (1) (0) + Π (2) (0) + Π ( k ) . (2.35)The massive modes correspond to poles in the propagator, or the zeros of the inversepropagator, so we set the expressions in Eqs. (2.32),(2.35) equal to zero and solve for k .Equating the pole locations in both theories, we find the matching condition m + Π ( k ) = (cid:18) − dd k Π (1) (0) (cid:19) h m + Π (1) (0) + Π (2) (0) + Π ( k ) i . (2.36)By construction, the infrared behavior contained in the soft self-energies Π and Π is thesame in both theories, so this relation is infrared safe. The difference is of order g ,Π ( k ) = Π ( k ) (cid:2) O ( k /T ) (cid:3) , Π ( k ) ∼ g m ≈ g T , (2.37)so, working at order g , we can drop all terms containing Π , Π from the matchingcondition. We are then left with an equation for the three-dimensional mass parameter m = m + Π (1) (0) + Π (2) (0) − (cid:16) m + Π (1) (0) (cid:17) dd k Π (1) (0) . (2.38)As a by-product we also found the field normalization factor to order g , since from lookingat the coefficients of k in both propagators we can write φ d = 1 T (cid:20) k Π (1) (0) (cid:21) φ d . (2.39)The factor 1 /T here stems from the overall factor T in Eq. (2.26), which is conventionallyabsorbed into the fields and couplings of the 3d theory.It should be noted that Eq. (2.38) only contains contributions from the heavy scale T , whereas the infrared sensitive parts Π and Π drop out. The mass parameter m regulates the infrared behavior of the dimensionally reduced theory, but it is a completelyperturbative quantity and should not be confused with the actual screening lengths thatare sensitive to infrared physics. In particular, the thermal mass of the adjoint scalar A in the dimensionally reduced theory agrees with the electric screening mass m el only atorder g , beyond which m el becomes sensitive to the magnetic screening [47], while m onits part is given to O ( g ) by the completely perturbative expression in Eq. (2.38).Apart from the gauge fields, the only other elementary boson in the standard modelis the Higgs field, which has a negative mass parameter − ν in the phase of unbroken SU (2) × U (1) symmetry. Near the electroweak phase transition the Higgs field mass isa special case in the power counting, since the T = 0 mass parameter and the thermalcorrections almost cancel each other, giving m ∼ − ν + g T ∼ g T or smaller, depending on how close to the phase transition we choose to work. To havea better separation of scales, it is necessary to integrate out the fields with masses ∼ gT when computing close to the electroweak phase transition, as we did in [3]. This leads toa theory containing only the Higgs field and spatial gluons. The thermal mass m ( T ) isthe leading term in the Higgs field effective potential, which drives the phase transition.13he above matching computation gives another example of how the expansion in loopsand momenta can be identified when the the correct momentum region is known. At hightemperatures, the mass parameters can be estimated as gT and the momenta at most ofthe same magnitude, in the region where dimensional reduction is valid. The requiredlevel of matching is determined by the problem in question and the accuracy goal onewants to reach. For example, for computing the free energy to order g we needed thecouplings only at tree-level, but the mass parameters to two-loop ( g ) order.A more general analysis given in [23] states that in order to have a theory which givesthe same light field Green’s functions as the full theory up to corrections of order O ( g ),we need to match the parameters at least to this order. To be more precise, the couplingconstants are required to one-loop level g = T ( g + g ) and adjoint scalar (temporalgauge field component) masses to two-loop accuracy m = T ( g + g ). If the theorycontains a light scalar field such as the Higgs field, its thermal mass should be computedto three-loop level m = − ν + T ( g + g + g ), since the first terms cancel each other,and the mass is of order g T close to the phase transition. The same analysis shows thatbeyond O ( g ) it is necessary to include non-renormalizable 6-dimensional operators intothe effective theory.Apart from the simple power counting, the importance of the higher order operatorsinevitably resulting from the reduction step is difficult to estimate. In [23, 48] it is notedthat in both abelian and SU(2) Higgs models these operators are further suppressed bysmall numerical coefficients in addition to powers of the coupling constant, and thus giveonly very small contributions. The operators following from the second reduction step,where the scales ∼ gT are integrated out to give a pure gauge theory, can be consistentlytreated as perturbations with respect to the tree-level Lagrangian, as discussed in [49].For matching purposes we still need to compute some Green’s functions in the full theory,but using the effective theory this only has to be done once, after which the computationscan be carried out in the simpler effective theory. For both QCD [9, 11] and electroweaktheory [23] the matching has been carried out explicitly to order g , and for a generictheory containing scalars, fermions and gauge fields the rules given in [23] can be used tofind the parameters of the effective theory. The QCD coupling has even been matched totwo-loop [ g = T ( g + g + g )] level in [50].While the effective theory approach saves us from computing multiple complicated sum-integrals, at finite temperatures the main advantages of dimensional reduction lie in theeasy way to organize the resummations and separating the contributions of different scales.Eventually non-perturbative methods such as lattice simulations are needed to handle theinfrared limit, but the dimensional reduction methods allow us to work out the parame-ters with completely perturbative methods, and then apply the computationally intensivemethods to the simpler three-dimensional theory. Lattice simulations in the dimensionallyreduced theory are easier because there is one spatial dimension less, no fermions and theshortest scales . /T have been integrated out.14 hapter 3 Pressure of the standard model
At high temperatures the local SU (2) L × U (1) Y gauge symmetry of electroweak theoryis restored. The phase transition is driven by the Higgs field, whose effective potentialis modified by thermal corrections in such way that the vacuum expectation value of thefield vanishes when the temperature is raised. Because of the possibility of the phase tran-sition being strongly first order and contributing to the baryon number asymmetry, theeffective potential has been extensively studied both by 1-loop [51–53] and 2-loop [54–56]perturbative calculations and by dimensional reduction [23] combined with lattice simu-lations [24–28]. In those works it was shown that in the standard model the electroweakphase transition is a crossover for realistic Higgs masses.Apart from the effective potential computations, the thermodynamics of electroweaktheory has not been studied in detail. In [2, 3] we computed the most fundamental ther-modynamic quantity, the free energy, for electroweak theory at high temperatures. Thiscomputation is very similar to the evaluation of the free energy in QCD, with the maindifferences coming from the presence of a light scalar field driving the phase transitionand the multitude of scales and couplings leading to a very complicated general struc-ture. Together with the QCD result and the few terms mixing the strong and electroweakcouplings, this computation gives us the free energy of the full standard model.Partial derivatives of the free energy give the basic thermodynamical quantities as inEq. (2.3). It should be noted here that we are computing in the grand canonical ensemble,whose partition function gives the grand potential Ω = − T ln Z , but at zero chemicalpotentials this can be identified with the free energy F = Ω+ µ i N i . In the thermodynamicallimit V → ∞ the free energy density equals the pressure, F = − pV , so for simplicity wewill we talking about pressure from now on.The energy density and pressure are particularly interesting, since they control the ex-pansion of the universe at its very early stages. Temperatures higher than the electroweakcrossover cannot be reached experimentally, but they were present in the early universe.The relic densities of particles decoupling from the ordinary matter are sensitive to theevolution of the universe, which in turn is governed by the equation of state. Recentmeasurements of the cosmic microwave background suggest a sizeable amount of colddark matter, which could be explained by weakly interacting massive particles (WIMPs)(see [57] for a review). Given a theory describing WIMPs, we need to know the evolutionof the universe at the time of their decoupling as well as at later times to make predictionsof the present situation. In [58] it is estimated that a 10% change in the equation of stateleads to 1% difference in relic densities, which is visible in future microwave observations.15 .1 Perturbative evaluation of the pressure The electroweak sector of the standard model is given by the Euclidean Lagrangian L = 14 G aµν G aµν + 14 F µν F µν + D µ Φ † D µ Φ − ν Φ † Φ + λ (Φ † Φ) + ¯ l L /Dl L + ¯ e R /De R + ¯ q L /Dq L + ¯ u R /Du R + ¯ d R /Dd R + ig Y (cid:16) ¯ q L τ Φ ∗ t R − ¯ t R (Φ ∗ ) † τ q L (cid:17) , (3.1)where G aµν = ∂ µ A aν − ∂ ν A aµ + gǫ abc A bµ A cν and F µν = ∂ µ B ν − ∂ ν B µ are the field strengths ofthe weak and hypercharge interactions, Φ is the Higgs field and the covariant derivativesact on the chiral fermion fields and the Higgs field as usual (for details, see Eq. (2.3)of [2]). We only include the Yukawa coupling for the top quark, since for other particlesthe Yukawa couplings (which are proportional to particle masses in the broken symmetryphase) are orders of magnitude smaller.When the Euclidean action is given, the pressure can be computed as the logarithm ofthe partition function, p ( T ) = lim V →∞ TV ln Z D A D ψ D ¯ ψ D Φ exp (cid:20) − Z β d τ Z d x L ( A, ¯ ψ, ψ, Φ) (cid:21) , (3.2)where the path integral is over all fields in the Lagrangian. As described in the previouschapter, a straightforward perturbative evaluation of the path integral Eq. (3.2) failsbecause of infrared divergences. The solution is to resum a class of diagrams by means ofan effective theory, using dimensional reduction.In the first level of dimensional reduction all non-static modes, in particular all fermions,are integrated out. This leads to an effective theory S E , whose parameters are matchedby perturbative computations in the full theory with no resummations, p ( T ) ≡ p E ( T ) + lim V →∞ TV ln Z D A k D A D Φ exp ( − S E ) . (3.3)Note in particular the appearance of parameter p E ( T ), which is the contribution of thenon-static modes, or scales ∼ πT , to the pressure. This parameter can be also viewed asthe matching coefficient of the unit operator by looking at the (unnormalized) expectationvalue of unit operator in both theories, h i full = Tr 1 · ρ full = Z full = e − F/T = e − F E /T +ln Z E = e − F E /T h i E , (3.4)where F = − pV . Since the matching is infrared safe, all parameters of S E and also p E are series in g . In addition to curing some of the infrared problems, this approachmakes full use of the scale hierarchy T ≫ gT ≫ g T by separating the contribution fromeach scale into successive effective theories, whose contributions enter at different levels ofperturbation theory. For example, it is easy to see that the dimensionally reduced theory S E in Eq. (3.3) starts to contribute at level T m ∼ g T .The effective theory S E still contains two different scales gT and g T , the latter of whichis related to non-perturbative magnetic screening effects. If one wishes to go further usingperturbation theory, it is useful to integrate out the electric scales gT as well, giving p ( T ) ≡ p E ( T ) + p M ( T ) + TV ln Z D A k exp ( − S M ) , (3.5)16here S M only contains the spatial gauge fields. Close to the phase transition this step ismore complicated because the scalar mass is very light, and deserves a separate discussionin section 3.2. The only dimensional parameter in S M is the gauge coupling ˜ g ≈ g T ,so this theory begins to contribute at order T ˜ g ∼ g T , this term being completely non-perturbative. We have only kept terms of order g in our calculation of the pressure, sothis non-perturbative contribution can be dropped. The purpose of this second reductionstep is that the computation of p M in the first effective theory S E can be considered asa matching computation, without having to worry about the resummations needed forspatial gauge fields.The electroweak theory contains many dimensionless coupling constants, and we need toestablish a power counting between them in order to determine which terms to include inthe perturbative expansion. We have decided to use the weak gauge coupling as reference,and, denoting the strong and hypercharge couplings by g s and g ′ , respectively, make thesimple choice λ ∼ g ′ ∼ g s ∼ g Y ∼ g , ν . g T , (3.6)which corresponds to three-loop expansion in all couplings. Numerically this is not the bestchoice, since the strong and Yukawa couplings are large compared to electroweak couplings,and we underestimate their importance. However, trying to incorporate higher orders of g Y would require four-loop sum-integrals, which we do not know how to perform. Thestrong coupling is even harder, since the g s order suffers from the same infrared problemsas any nonabelian gauge theory. In [23] the rule g ′ ∼ g is used, but there is no dangerand practically no extra work in keeping terms of order g ′ as well. It should be also keptin mind that the one-loop renormalization group running of the couplings is such that g ′ ( T ) grows with temperature, whereas g ( T ) decreases.After all the preparations are done, it remains to actually compute the pressure. Westart by evaluating p E to three-loop order. At the 4d full theory level the Higgs massparameter ν is treated as a perturbation, so we expand the propagators in ν . This ispossible since the matching procedure is infrared safe. The resulting massless sum-integralscan be evaluated using the methods developed in [9] and can be conveniently read fromthe Appendix A of [11]. The largest work lies in writing down all the required diagramswith correct symmetry and group theory factors, reducing them to integrals given in [11]and summing everything together. Note that the diagrams with only static modes do nothave to explicitly subtracted, since they vanish in the dimensional regularization due tolack of dimensionful parameter.Schematically, the generic form of p E is p E ( T ) = T α E + X i g i α Ei + 1(4 π ) X ij g i g j α Eij + ν T " α Eν + 1(4 π ) X i g i α Eiν + ν (4 π ) α Eνν + T · O ( g ) , (3.7)where the summation is over g i = g , g ′ , g s , g Y , λ and the values of all nonzero coefficients(all combinations except α Eλs , α
Esν , and α Es , α Ess since we exclude the pure QCD terms)can be found in the Appendix A of [2]. To keep track of different contributions they aregiven in terms of the group theory constants, which for SU(2) read T F = 1 / C F = 3 / d F = 2, C A = 2 and d A = 3. 17e have normalized the pressure so that (the real part of) the pressure at the symmetricphase vanishes at zero temperature, p ( T = 0) = 0, in order to exclude the large vacuumenergy contribution and the related divergences. This normalization is already taken intoaccount in Eq. (3.7), where we have subtracted a term proportional to ν computed atzero temperature. The T = 0 computation differs from the high-temperature expansionin Eq. (3.7), and the difference is contained in the remaining coefficient α Eνν .The renormalization of the 4d parameters does not remove all divergences, but 1 /ǫ terms can be found in most of the coefficients of O ( g ) terms, corresponding to infrareddivergences that cancel against similar terms in p M . Having stated above that the matchingcomputation is infrared safe, we should elaborate on the nature of these divergences andtheir cancellation a bit more. Dimensional regularization simultaneously handles both theinfrared and ultraviolet limit, and it is not easy to tell the divergences apart.The electroweak theory is known to be renormalizable. The computation of p E cannotthus contain any ultraviolet divergences, since they are removed by the counterterms.However, there are diagrams that are both ultraviolet and infrared divergent and vanishin dimensional regularization. We can see how they behave through the following simpleexample. Consider the logarithmically divergent integral Z k k ≡ (cid:18) e γ µ π (cid:19) ǫ Z d − ǫ k (2 π ) − ǫ k , (3.8)which vanishes in dimensional regularization. This can be written as a sum of two integrals,one divergent at the ultraviolet and the other at the infrared momenta, Z k k ( k + m ) + m k ( k + m ) = Z k Z d x k + xm ] + 2 m (1 − x )[ k + xm ] = 116 π (cid:18) e γ µ m (cid:19) ǫ Z d x Γ( ǫ ) x − ǫ + Γ(1 + ǫ )(1 − x ) x − − ǫ = 116 π (cid:18) ǫ UV − ǫ IR (cid:19) . (3.9)The renormalization counterterms remove the ultraviolet divergence 1 /ǫ UV here, leavingthe infrared divergent part − /ǫ IR . The vanishing diagram therefore contributes with aninfrared divergence when renormalized.The scaleless diagrams at finite temperature are precisely those with only static modes,and no summations over the Matsubara frequencies. The dimensionally reduced effectivetheory contains the same diagrams, but with self-energy corrections resummed to givemasses on some propagators. These masses do not change the ultraviolet behavior of thediagram, but regularize the infrared limit, so the divergence structure is just 1 /ǫ UV , whichprecisely cancels against − /ǫ IR from the full theory computation.The contribution of the scales gT can be calculated from the path integral in Eq. (3.3)once the dimensionally reduced theory is known. Before matching, we need to considerthe most general renormalizable (in 3d) Lagrangian respecting the symmetries of the fulltheory, S E = Z d x G aij G aij + 14 F ij F ij + ( D i Φ) † ( D i Φ) + m Φ † Φ + λ (Φ † Φ) + 12 ( D i A a ) + 12 m A a A a + 14 λ A ( A a A a ) + 12 ( ∂ i B ) + 12 m ′ B B + 14 λ B B + h Φ † Φ A a A a + h ′ Φ † Φ B B − g g ′ B Φ † A a τ a Φ , (3.10)18here we have included masses and quartic self-interactions for the scalar fields A , B ,the former temporal components of the gauge field. The field A transforms in the adjointrepresentation of SU(2), wheres B does not interact with the gauge fields due to theabelian nature of U(1).In order to relate the pressure p M to the full theory we need to know the parameters ofthe effective theory S E in terms of the full theory couplings. For the couplings the leadingorder results are sufficient, since the two-loop diagrams in the effective theory are alreadyof order g . The matching then boils down to absorbing the factor of T / to couplings togive them the correct dimensions, g = g T , g ′ = g ′ T ,λ = λT , λ A,B = O ( g ) ,h = g T , h ′ = g ′ T . (3.11)The quartic couplings for the adjoint scalars are of higher order than we need in ourcomputation.The matching of the mass parameters is more complicated. Since the leading order(one-loop) diagrams are of the order
T m ∼ g T , we need order g terms in the expres-sions for the masses to get the pressure up to O ( g ). Moreover, the two-loop diagramscontain ultraviolet divergences, so we need also the O ( ǫ ) terms for the masses when usingdimensional regularization. The mass parameters in electroweak theory have already beencomputed in [23] apart from the g ǫ terms, which we have evaluated in [2]. The massesare found by matching the two-point functions at vanishing external momentum as inEq. (2.38).The general form of the adjoint scalar masses is m = T (cid:20) g (cid:0) β E + β E ǫ + O ( ǫ ) (cid:1) + g (4 π ) ( β E + O ( ǫ )) + O ( g )+ g (4 π ) (cid:18) β Eλ λ + β Es g s + β EY g Y + β E ′ g ′ + β Eν − ν T (cid:19)(cid:21) , (3.12)and similarly for m ′ D . The coefficients β Ex can be found in the Appendix B.1 of [2]. Itshould be noted that there are no divergences in these coefficients, but the renormalizationof the 4d theory is enough to make the adjoint scalar masses finite. In the 3d theory theseparameters are renormalization group invariant to this order, and only start running atorder g with terms proportional to λ A and g λ A [55].The fundamental scalar mass has 1 /ǫ divergences that are not removed by the renor-malization of the full theory. These are again related to the infrared limit of the staticmodes and are removed by the counterterms in the effective theory. Looking from thedimensionally reduced theory, the matching procedure produces the bare mass which wecan either split into the renormalized mass and counterterms, or continue using the massparameter with 1 /ǫ terms included as we did in [2]. Either way, the divergences will cancelin the final result for the pressure, and m itself is not a physical parameter we would beinterested to study in detail.Using the MS scheme in the effective theory to renormalize the Higgs mass, we get thefinite result m (Λ) = − ν + T (cid:18) C F g + 116 g ′ + 16 ( d F + 1) λ + 112 N c g Y (cid:19) ǫ T (cid:0) g β A + g ′ β B + λβ λ + g Y β Y (cid:1) + − ν (4 π ) (cid:0) g β νA + g ′ β νB + λβ νλ + g Y β νY (cid:1) + T (cid:20) g (4 π ) β AA + g ′ (4 π ) β BB + g g ′ (4 π ) β AB + λg (4 π ) β Aλ + λg ′ (4 π ) β Bλ + λ (4 π ) β λλ + g g Y (4 π ) β AY + g ′ g Y (4 π ) β BY + g s g Y (4 π ) β sY + λg Y (4 π ) β λY + g Y (4 π ) β Y Y (cid:21) , (3.13)which depends on the MS renormalization scale Λ replacing µ in Eq. (2.23). The param-eters are linear combinations of ζ ′ ( − γ E and ln(Λ / πT ), and they are given explicitlyin the Appendix B.2 of [2]. The mass counterterm can be read from the matching com-putations, δm = T (4 π ) ǫ (cid:18) − g + 764 g ′ + 1532 g g ′ − λg − λg ′ + 3 λ (cid:19) . (3.14)This coincides with the counterterm computed directly from the effective theory S E ,1(4 π ) ǫ (cid:18) − g + 564 g ′ + 1532 g g ′ − λ g − λ g ′ + 3 λ + 32 h − h g + 2 h ′ (cid:19) , (3.15)when the relations (3.11) between couplings are taken into account. Since the 3d theoryis super-renormalizable and has only a finite number of divergent graphs, the countertermin Eq. (3.15) is actually an exact result, without any higher order corrections [55].Since we are computing only vacuum diagrams in the effective theory, matching thefields is not required, apart from what was included in the mass parameter computations.All the required parameters are then known, and the pressure p M can be computed byevaluating all one-particle irreducible vacuum diagrams up to three-loop level in terms ofthese parameters. Apart from a gauge boson loop with both Higgs and A self-energycorrections, all the required 3d integrals are computed in [11]. Because of the massivepropagators the general structure of the result is much more complicated than for p E : p M ( T ) T = 14 π d F (cid:0) m + δm (cid:1) / (cid:20)
23 + ǫ (cid:18)
169 + 43 ln µ m (cid:19)(cid:21) + 14 π (cid:18) d A m + 13 m ′ (cid:19) + 1(4 π ) (cid:20) − d F ( d F + 1) λ m − d F d A h m m D − d F h ′ m m ′ D − (cid:18) C F g + 14 g ′ (cid:19) d F m (cid:18) ǫ + 32 + 2 ln µ m (cid:19) − C A d A g m (cid:18) ǫ + 34 + ln µ m D (cid:19)(cid:21) + 1(4 π ) h g m B AAf + g ′ m B BBf + g g ′ m B ABf + g m D B AAa + g λ m B Aλf + g ′ λ m B Bλf + λ m B λλf + h m B hhf + h m D B hha + h ′ m B ′ hhf + h ′ m ′ D B ′ hhb + g g ′ m b ( m ) + g g ′ m D b ( m D ) + g g ′ m ′ D b ( m ′ D ) + d F m ( d A h m D + h ′ m ′ D ) + d m (cid:18) d A h m D + h ′ m ′ D (cid:19) + g C A C F d F (cid:18) m m D ln m D + m m + m m ln m D + m m D (cid:19) + d F ( d F + 1) λ ( d A h m D + h ′ m ′ D ) + g h m D B Aha + g ′ h ′ m ′ D B ′ Bhb + g h ′ m ′ D B ′ Ahb + g ′ h m D B Bha + g h m B Ahf (cid:21) . (3.16)20he coefficients B xyz and the coefficient function b ( x ) are linear combinations of 1 /ǫ , ln 2, π and ln( µ /M ), where µ is the 3d dimensional regularization scale and M can be anycombination of the different mass parameters m D , m ′ D , m . The detailed expressions canbe found in the Appendix C of [2].The part of the pressure coming from the electric scales in Eq. (3.16) has many newfeatures that are not present in the corresponding computation for QCD. In particular,the only dimensional parameters in dimensionally reduced QCD (known as EQCD) are g and m D , so the possible terms are, for dimensional reasons, m , g m and g m D , withdivergent coefficients containing ln( µ /m D ). This is in sharp contrast with the abundanceof different terms in Eq. (3.16); not only are there many combinations of couplings andmasses, but also completely new kinds of expressions like m i /m j and ln( m i /m j ).All the coefficients B xyz and b ( x ) of the O ( g ) terms have ultraviolet divergences, butthey cancel against the 1 /ǫ terms in the mass counterterm δm . The renormalized mass inEq. (3.13) depends on the renormalization scale Λ through logarithms ln(Λ / πT ), whichcome with the divergences as usual. They cancel against the corresponding logarithmsln( µ /M ) in Eq. (3.16), leaving terms like g m D ln( m D /T ) ∼ g ln g . This kind of termsare not present in EQCD, where all the mass parameters are finite at O ( g ). If we choose µ = Λ, the scale dependence in p M vanishes completely at O ( g ) and O ( g ) when therunning of the 4d couplings is taken into account.The remaining 1 /ǫ terms shown explicitly at two-loop level in Eq. (3.16) cancel againstthe infrared divergences in p E , Eq. (3.7). Also there the cancellation between terms comingfrom scales πT and gT results in large logarithms of order g ln g . These terms are alsopresent in the QCD pressure, where they were originally derived from the requirementthat the pressure should not depend on the scale at O ( g ) [8].The presence of terms proportional to ln( g ) shows that we cannot choose the scalein such way that the large logarithms would completely vanish. The use of an effectivetheory to separate the contributions from different scales is often advocated by the absenceof large logarithms, but as we see, the infrared divergences mix the different scales, andlogarithms of ln( gT /T ) are left in the final result. A stronger argument for formulating theproblem in terms of effective theories is the proper handling of resummations and isolatingthe non-perturbative infrared behavior into a simpler theory, as discussed in section 2.2.2. One of the most interesting properties of the electroweak theory is the crossover phasetransition. When the temperature is lowered, the effective potential of the Higgs fielddevelops a new minimum at some finite value h Φ i 6 = 0. This phase transition gives massesto all quarks and leptons (except neutrinos) as well as to those gauge bosons mediating theweak interactions that correspond to the broken part of the eletroweak SU(2) L × U(1) Y symmetry.In the deconfinement phase transition of QCD the strong coupling grows very largeand perturbation theory cannot be used to study the phase transition. This is not thecase in the electroweak symmetry breaking, since the confinement radius of weak interac-tions is tremendously large ( g has a Landau pole at 1 / Λ EW ≈ m) compared to therelevant distance scales at the electroweak transition temperatures (1 /T c ∼ − m for T c ∼
200 GeV). We can then try to use perturbation theory in studying the electroweakphenomena close to the phase transition, in particular to extend our previous computation21f the pressure down to transition temperatures. Besides the pressure playing a centralrole in describing thermodynamics near the transition, this also allows us to test the va-lidity of dimensional reduction at the phase transition, something that cannot be done inQCD.Note that perturbation theory is not able to describe the transition itself correctly withphysical Higgs masses. Both two-loop effective potential calculations [55], although limitedto m H . m W , and the ǫ -expansion analysis extrapolated to ǫ = 1 [59] suggest the presenceof a first order phase transition for large Higgs masses, while in reality the transition isof crossover type for m H &
72 GeV [25, 27, 28]. In the unbroken phase the Higgs fieldhas a non-vanishing vacuum expectation value, which we would have to incorporate inour computations to build the perturbative expansion around the true, physical vacuum.Neverthless, because the coupling constant stays small in the transition we are able toperturbatively compute physical quantities while approching the transition from above,as long as we stay in the symmetric vacuum.The pressure computed in Eqs. (3.7),(3.16) cannot be directly continued down to thephase transition, because it becomes singular as we approach the critical temperature(which exists in perturbation theory). The reason for this is that the thermally correctedHiggs mass m in Eq. (3.13) becomes very small at the phase transition, and finally turnsnegative a little below the transition. At leading order it is this negative mass parameterthat makes the symmetric phase unstable and drives the phase transition. The singulareffects of small thermal Higgs mass can be seen in p M , Eq. (3.16), which contains termslike m /m and m D ln( µ /m ).The origin of this problem is in our power counting, which assumed m ∼ m D ∼ gT .Close to T c this assumption fails and we run into the same infrared problems as in theoriginal theory, since the resummation of non-static modes no longer gives a finite mass tothe Higgs field. In Fig. 3.1 we have plotted the ratio of the renormalized Higgs mass to theSU(2) adjoint scalar mass in both the full standard model (including strong interactions)and the SU(2) + Higgs theory which we have used as a weakly coupled toy model. Thenumerical values of the parameters are given in section 3.3, and the renormalization scalefor m is chosen as Λ = 2 πT . As the figure shows, the mass ratio drops steeply as thetemperature approaches the phase transition, and at T − T c .
50 GeV the assertion m ∼ m D clearly fails.The solution is to resum yet another class of diagrams, the adjoint scalar self-energycorrections on the fundamental scalar line. We then have one more level of dimensionalreduction, and instead of Eq. (3.5) the pressure is given by p ( T ) ≡ p E ( T ) + p M1 ( T ) + p M2 ( T ) + TV ln Z D A k D Φ exp ( − S M ) , (3.17)where p E is the same as before, Eq. (3.7), and p M1 is computed from the effective theory S E in Eq. (3.10), treating now the light scalar mass m as a perturbation and expandingthe integrals in m /m . The contribution from scales m . g T is contained in the con-tribution p M2 , which is computed from an effective theory containing only the fundamentalscalar field and the spatial components of the gauge bosons, S E2 = Z d x G aij G aij + 14 F ij F ij + ( D i Φ) † ( D i Φ) + e m Φ † Φ + ˜ λ (Φ † Φ) , (3.18)where the gauge couplings and the scalar self-coupling do not get any matching correctionsat this level, (˜ g , ˜ g ′ , ˜ λ ) = ( g , g ′ , λ ). The mass parameter, however, now also resums22
200 400 600 800 1000T [GeV]00.10.20.30.40.5 m / m D Standard modelSU(2) + Higgs
Figure 3.1:
The ratio of the fundamental scalar thermal mass to the SU(2) adjoint scalar mass.The regularization scale is chosen as Λ = 2 πT . the adjoint scalar loops, which we have to compute up to one-loop level ( g in our powercounting), e m = m − π (cid:18) d A h m D + 14 g ′ m ′ D (cid:19) − π (cid:20) d A h m D (cid:18) µ m D (cid:19) + 14 g ′ m ′ D (cid:18) µ m ′ D (cid:19)(cid:21) ǫ + O ( g ) . (3.19)Apart from the O ( ǫ ) terms, this expression has been previously computed in [23]. Thecorrection is of the same order g as the leading term m , showing that resummation isnecessary to get consistent results.For simplicity, we have neglected the order g corrections to the scalar mass. This can bejustified by power counting arguments, since for m ∼ g T the two-loop corrections wouldcontribute parametrically at order m g ∼ g / T , which is strictly speaking higher than O ( g ) we are considering here. Dropping terms suppressed by √ g may not be numericallyjustified, but we expect their effect to be small, in particular because this only concernsfour out of more than a hundred degrees of freedom in the standard model. The practicalreason is that we want to avoid computing all three-loop diagrams in the effective theory,as well as the renormalization and scale dependence of the Higgs mass at two-loop level.Evaluating all the three-loop vacuum diagrams of theory S E1 and two-loop diagrams of S E2 (there are only three of them), and setting the number of fermion families to n F = 3,we get p M2 T = d F π e m − e m (4 π ) (cid:20) d F ( d F + 1)˜ λ + 12 d F (cid:18) C F ˜ g + 14 ˜ g ′ (cid:19) (cid:18) ǫ + 3 + 4 ln ˜ µ e m (cid:19)(cid:21) , (3.20)23 M1 T = 14 π (cid:18) d A m + 13 m ′ (cid:19) + 1(4 π ) C A d A g m (cid:18) − ǫ − − ln µ m D (cid:19) − π ) ǫ (cid:20) C A C F g m D + d A h m D + h ′ m ′ D + 14 C F g g ′ ( m D + m ′ D ) (cid:21) d F
2+ 1(4 π ) (cid:26) g m D (cid:20) C d A (cid:18) − − π (cid:19) + C A C F d F (cid:18) − −
34 ln µ m D (cid:19)(cid:21) + g g ′ C F d F (cid:20) ( m D + m ′ D ) (cid:18) − − µ m D + m ′ D (cid:19) − m D ln µ m D − m ′ D ln µ m ′ D (cid:21) + h m D d A d F (cid:18) − − µ m D (cid:19) + h ′ m ′ D d F (cid:18) − − µ m ′ D (cid:19)(cid:27) . (3.21)The O ( g ) two-loop divergences in p M1 cancel against p E as before. In addition, thereare 1 /ǫ terms left in p E whose coefficients by themselves are of order g but combine toa term proportional to m , which cancels against the divergence in p M2 . Unlike in thehigh-temperature case, there are also O ( g ) divergences in p M1 , but these go away whenthe one-loop corrections in e m multiplying the “sunset” diagram in p M2 are included. Thescale dependence cancels along with divergences, if we take into account the running of thecouplings in the full theory, and set all regularization scales to be the same, Λ = µ = ˜ µ .The phase transition can now be safely approached, since the result (3.20) for p M2 isperfectly well-behaved as e m goes to zero. To see how the multitude of terms we have computed affects the physical pressure, wehave to supply some numbers for the parameters of the 4d Lagrangian, Eq. (3.1). Apartfrom the yet undiscovered Higgs particle mass, the standard model parameters have beenmeasured to great precision in collider experiments, in particular LEP. The values ofcouplings can be determined from their tree-level relations to various mass parameters, ν ( m Z ) = 12 m H , λ ( m Z ) = 1 √ G µ m H ,g ( m Z ) = 4 √ G µ m W , g ′ ( m Z ) = 4 √ G µ (cid:0) m Z − m W (cid:1) ,g Y ( m Z ) = 2 √ G µ m t , α s ( m Z ) = 0 . , (3.22)where m W = 80 .
40 GeV, m Z = 91 .
19 GeV and m t = 174 GeV are the masses of the Wand Z bosons and the top quark, respectively, and G µ = 1 . · − GeV − is the Fermicoupling constant [60]. The cited values are what we have used in [2, 3], and they remainpractically unchanged in the more recent Review of Particle Physics [61], with only thestrong coupling being slightly smaller, α s ( m Z ) = 0 . m H &
114 GeV but leave it otherwise unknown. We have usedthe value m H = 130 GeV in all our analysis. The pressure of the full standard model isvery insensitive to the Higgs mass when we are not close to the phase transition, although T c itself depends on m H . In [2] we have shown that increasing m H to 200 GeV causes arelative change of 10 − in the pressure.In addition to the standard model, we have also studied the simpler SU(2) + Higgstheory, for which the corresponding results can be found from those computed above bysetting g ′ = g s = g Y = n F = 0. Besides simpler analytic expressions, this model has some24
00 1000 10000T [GeV]0.850.90.951p /p g g + g g + g + g g + g + g + g Figure 3.2:
Pressure of the standard model at different orders of perturbation theory. advantages over the standard model when we want to study the behavior of dimensionalreduction near the critical temperature. The Higgs field drives the phase transition, butit only represents four of the 106.75 effective degrees of freedom in the standard model,so its effects on the pressure are hard to see. In SU(2) + Higgs model the correspondingnumber is only 10. Also, this toy model is weakly coupled, with λ ≈ .
20 and g ≈ . g s and g Y . More details on computations in this model can be foundin [62]. Thermodynamics of SU(2) + fundamental Higgs theory have also been studied onlattice [63].In Fig. 3.2 we have plotted the pressure of the full standard model up to O ( g ), com-bining the results in Eqs. (3.7),(3.16) and the pure QCD pressure taken from [9–12] (withone-loop quark diagrams subtracted to avoid double counting), p ( T ) = p E ( T ) + p M ( T ) + p QCD ( T ) + T · O ( g ) . (3.23)The radiative electroweak corrections also need to be taken into account in the two-loopelectric gluon mass m which we insert in p QCD ( T ) above. They have been computedin [2], and used in the leading order term 2 m / π .Although the physical pressure is independent of the renormalization scale, the per-turbative expansion depends on the scale through the renormalization of parameters atorders higher than those included in the computation, and we need to fix the scale to definethe couplings. As the remaining scale dependence is cancelled by higher order terms, themagnitude of the unknown corrections can be estimated by varying the scale. We havechosen to use Λ = 2 πT , having shown that the scale dependence is indeed weak.The pressure in all our plots is normalized to the Stefan–Boltzmann result of non-25nteracting gas of relativistic particles, p = π T (cid:18) d A + 2( N c −
1) + 2 d F + 2 78 n F [ d F + 1 + N c ( d F + 2)] (cid:19) = π T (cid:26) . T isactually α E of Eq. (3.7) + the gluon contribution.Fig. 3.2 shows that the perturbative expansion does not converge very well at moderatetemperatures, but instead the O ( g ) correction is even larger than any of the preceedingterms. This behavior is known in QCD, and the strong coupling constant is still largeat electroweak temperatures, g s ( m Z ) ≈ .
48. In the full standard model the stronglyinteracting degrees of freedom sum up to 79, or 74% of the number in Eq. (3.24), so theterms with gluon exchange clearly dominate the higher order corrections, together withthe few terms containing Yukawa interactions. The pressure lies 5-10% below the idealgas result, and begins to converge very slowly at temperatures in TeV range.Close to the phase transition we have to use the resummed result of Eq. (3.17) for thepressure. In Fig. 3.3 we have plotted both the high-temperature result p HT ( T ) and lightscalar mass resummed result p PT ( T ). As the figure shows, the high-temperature result isvery sharply peaked at T c , whereas the corrected computation goes through the transitionsmoothly. Of course, below T c the system goes to the nonsymmetric ground state whosepressure is larger than the symmetric phase pressure plotted in Fig. 3.3, since a thermalsystem always tries to minimize the free energy, or maximize the pressure. When e m becomes negative slightly below T c , the symmetric phase pressure develops an imaginarypart, which can be interpreted as the decay rate of the unstable symmetric phase [64]. Inthis region we have plotted the real part of the pressure in our figures.The two curves in Fig. 3.3 differ by a term roughly proportional to T , but the differenceis only about 0.3%. This is because in p PT we have resummed another class of diagramsthat are not present in p HT . We have also left out all three-loop diagrams in S E2 , whichwould contribute at order e m g , and the O ( g ) corrections to m . In particular the termswith strong and Yukawa couplings might be important even at this order.Corresponding plots for the simpler SU(2) + fundamental Higgs theory are shown inFig. 3.4 and Fig. 3.5, with parameters taken from Eq. (3.22) using m W = 80 GeV and m H = 130 GeV. In both figures we have also plotted an approximation of the broken phasepressure below T c to indicate the phase transition at T c ≈
220 GeV. It can be derived fromthe two-loop (order g ) computations of the effective potential [54] and the pressure inthe symmetric phase by using p BP ( T, φ ) = p SP ( T ) − V eff ( T, φ ). Fig. 3.4 should not betrusted near the transition, but away from T c it shows that in the weakly coupled modelthe perturbative expansion converges nicely and settles to a level which is about 2% belowthe ideal gas pressure. Note that in the one-loop renormalization λ (2 πT ) grows with T ,so there is no reason to expect free theory behavior even at very high temperatures.Close to the phase transition Fig. 3.5 shows a similar peak in p HT as in the standardmodel, while p PT does not see the transition at all until e m becomes negative at about15 GeV below T c . The constant (times T ) difference between the two pressures is largerthan in the standard model, about 2%. The couplings are all small here, so we expect thatthe three-loop diagrams in in S E2 and the O ( g ) corrections to the fundamental scalar massare not important. However, in this model larger fraction of the overall pressure comesfrom the Higgs sector, so the result is more sensitive to different resummations on thescalar propagator. 26
00 150 200 250 300T [GeV]0.850.860.870.880.89p / p p HT p PT
140 1600.860.865
Figure 3.3:
The pressure of the standard model near the phase transition. Here p HT stands forthe high-temperature calculation, p PT is resummed for the light scalar mass.
100 1000 10000T [GeV]0.960.9811.021.04p / p g (symmetric phase)g (broken symmetry phase)g (symmetric phase)g (broken symmetry phase)g (symmetric phase)g (symmetric phase) Figure 3.4:
The pressure of SU(2) + Higgs theory.
00 220 240 260 280 300T [GeV]0.960.9811.02p / p broken phasep HT p PT up to g p PT up to g Figure 3.5:
The pressure of SU(2) + Higgs theory near the phase transition. Here p HT stands forthe high-temperature calculation, while p PT has Higgs mass corrections resummed. Also shown isthe O ( g ) pressure in the broken symmetry phase. We have computed the pressure of the full standard model to order g both near thephase transition and at high temperatures. In principle, it is possible to go one step furtherand compute the coefficient of the last perturbatively accessible term of order g ln g . Thiscomputation has been carried out in QCD [12], but because the QCD pressure dominatesthe standard model pressure and g s is large, we expect advances in understanding the QCDpressure to be more important than four-loop diagrams in electroweak theory. Anothercomputable term would be the neglected three-loop diagrams and O ( g ) mass correctionsin S E2 close to the phase transition, contributing parametrically at order g . , but theseaffect only the Higgs sector and are probably too small to have any physical implications.28 hapter 4 Mesonic correlation lengths
Dimensional reduction is only useful in cases where the physical quantity of interest is time-independent. We can then average over any time coordinates, or, in terms of momentumspace Green’s functions, take the limit p ≡ ω → m ≈ e T /
3. It should be noted that in general the massesof real-time bound states can be very different from the corresponding screening masses.Static correlators of bosonic operators have been succesfully studied using dimensionallyreduced effective theories. For example, various gluonic correlation lengths have beenmeasured by implementing the three-dimensional theory of static gluons (EQCD) on lattice[15–20]. In these works the fermionic modes are integrated out as in sections 2.2 and 3.1,leaving a theory of soft gluonic excitations around the perturbative vacuum. However, thisis not the only option when deriving an effective theory, but we can as well expand aroundany other saddle point of the action, corresponding to a choice ¯Φ h = 0 in Eq. (2.17). Inparticular, not all fermionic modes need to be integrated out, but the effective theory canconstructed around some specific fermionic state. This opens a possibility to study alsofermionic correlators using dimensional reduction.Of particular interest are operators consisting of a light quark-antiquark pair propa-gating in the hot medium. In [65] it was suggested that at scales comparable to themagnetic scale 1 /g T the spectrum of quark-gluon plasma consists of color-singlet modesonly, while the colored excitations are dynamically confined. The lowest lying excitationsat these scales would then be the various glueball modes and the mesonic and baryonicstates consisting of two and three quarks, respectively. In order to better understand thelong-distance behavior of the plasma, the properties of these states have been measuredin detail on lattice. Most of these studies have been devoted to Euclidean correlators,or the screening properties of these operators, due to the inherently Euclidean nature oflattice simulations. In particular the spectrum of the hadronic screening masses has been29arefully measured, the first simulations dating back 20 years [66, 67].While combining perturbative calculations with lattice simulations using dimensionalreduction has been very useful when measuring the glueball spectrum, the hadronic screen-ing masses have been measured using expensive 4d simulations. However, the need foranalytical tools is even greater in the fermionic sector, where the lattice simulations havedifficulties in treating the light dynamical quarks correctly. The situation is yet worsewhen we allow for quark chemical potentials, which make the fermion determinant com-plex and ruin the conventional importance sampling. On the other hand, operators builtout of quark fields are usually less infrared sensitive, so perturbation theory should moreapplicable in computing their properties.The first attempts to determine the screening masses of mesonic states at high tempera-tures using dimensional reduction were more of a qualitative nature, since they knowinglyleft out corrections of the same order as the leading term [68, 69]. In particular, the scaleinside the logarithm in the two-dimensional Coulomb potential ∼ ln r was not fully iden-tified. A more systematic approach was developed by Huang and Lissia in [30], where itwas shown that the dimensionally reduced theory for fermionic modes can be formulatedin terms of massive non-relativistic quarks in 2+1 dimensions. They also discussed thecorrect power counting of different operators, and computed one-loop corrections to thequark self-energy and the quark-gluon interaction vertex. However, although the effectivetheory was derived in order to calculate screening quantities, the authors did not proceedto compute any masses in that work.Following [30], we used similar methods in [1] to derive an effective three-dimensionaltheory for the lowest fermionic modes ± πT , which dominate the mesonic correlator atlarge distances. This theory takes the form of non-relativistic quarks coupled to EQCD,or the spatial gluons and an adjoint scalar field. Because the fermionic sector of thetheory is very similar in form and power counting to the effective theory for heavy quarksin four dimensions known as “non-relativistic QCD”, we have named the reduced theoryNRQCD . Using this theory, we were able to compute the next-to-leading order correctionto mesonic screening masses in perturbation theory, and this computation was extendedto finite quark chemical potentials in [4]. In this chapter we will review these results. The correlation functions usually computed in theoretical calculations are related to phys-ically measurable quantities through linear response theory. Our short presentation herefollows mostly [36], and is somewhat biased towards screening physics.Consider perturbing the system in equilibrium with some external probe, described byan interaction Hamiltonian V ( t ) which vanishes for t <
0. In the Schr¨odinger picture thetime-development of an unperturbed state is given by the time-independent Hamiltonian H , while the effect of V ( t ) can be written in terms of a time-development operator U ( t ), | ψ S ( t ) i = e − iHt | ψ S (0) i ≡ e − iHt | ψ H i , | ψ ′ S ( t ) i = e − iHt U ( t ) | ψ S (0) i , where U ( t ) satisfies i∂ t | ψ ′ S ( t ) i = H | ψ ′ S ( t ) i + e − iHt i∂ t U ( t ) | ψ S (0) i = ( H + V ) | ψ ′ S ( t ) i⇒ i∂ t U ( t ) = e iHt V ( t ) e − iHt U ( t ) , U ( t ) = 1 for t < . (4.1)30n the last equation V H ( t ), the potential in the unperturbed Heisenberg picture is seento emerge. If V is small, U ( t ) can be solved recursively as a series in V by integratingEq. (4.1), U ( t ) = 1 − Z t d t V H ( t ) − Z t d t Z t d t V H ( t ) V H ( t ) + O ( V ) . (4.2)The change in the expectation value of an arbitrary operator ˆ O ( t ) in the Schr¨odingerpicture is then δ h ˆ O ( t ) i ≡ h ψ ′ S ( t ) | ˆ O ( t ) | ψ ′ S ( t ) i − h ψ S ( t ) | ˆ O ( t ) | ψ S ( t ) i = − i Z ∞ d t ′ h ψ H | θ ( t − t ′ )[ ˆ O H ( t ) , V H ( t ′ )] | ψ H i , (4.3)where the operators and the state vectors are now all in the Heisenberg picture with theunperturbed Hamiltonian H . In particular, Eq. (4.3) applies to the eigenstates of H ,so we can sum over all states in the ensemble with appropriate weights, and replace theexpectation value in a specific state by thermal average.Typically, the external interaction can be written as a time-dependent c-number source v ( t, x ) coupled to the system through some current J built of field operators, V ( t ) = Z d x J ( ˆ φ ( t, x )) v ( t, x ) , (4.4)and the response of the system is measured through the same current. Eq. (4.3) can thenbe written in terms of the retarded correlation function D R , δ h J ( ˆ φ ( t, x )) i = − i Z ∞ d t ′ Z d x ′ v ( t ′ , x ′ ) h θ ( t − t ′ )[ J ( ˆ φ H ( t, x )) , J ( ˆ φ H ( t ′ , x ′ ))] i≡ − Z d x ′ D R ( x − x ′ ) v ( x ′ ) , (4.5)where the lower limit of the time integration can be extended to −∞ because v ( t, x ) = 0for t <
0. In this expression the response of the system is clearly separated into theretarded propagator, which is specific to the thermal system in question, convoluted witha factor v ( x ) depending on the details of the perturbation.The excitations of the system manifest themselves as large responses to an externalperturbation. Going into the momentum space, the Fourier transform of Eq. (4.5) reads δ h J i ( ω, k ) = − iD R ( ω, k ) v ( ω, k ) , (4.6)so this is equivalent to having a pole in the propagator for some values of frequency ω andmomentum k . In general there are some real-time excitations, whose frequencies dependon the momentum through the dispersion relation ω = ω ( k ). In addition, at the staticlimit ω → ω and k , since the thermal environment breaks the Lorentz invariance. The screening states31nd the real-time excitations describe completely different physics, and the correspondingpoles need not be related at finite temperature. One should also be careful when takinglimits ω, k →
0, since different orders of limits may give different results, and the correctprocedure depends on the physical situation. A good comparison of screening and real-time quantities in the context of solvable 2+1-dimensional Gross–Neveu model can befound in [70].
The Euclidean Lagrangian governing the behavior of quarks and gluons at finite temper-ature is L E = 14 F aµν F aµν + ¯ ψ ( γ µ D µ + M ) ψ , (4.7)where the covariant derivative is defined as D µ ψ ≡ ∂ µ ψ − igA aµ T a ψ and the gluon fieldstrength F µν = i/g [ D µ , D ν ], T a being the generators in the fundamental representationof SU( N c ). We consider N F flavors of degenerate quarks, so the quark field ψ is an N F -component vector in flavor space, and the mass matrix M is proportional to unit matrix, M = m · N F . For simplicity we will set m = 0 in most of what follows.Finite quark densities are included through chemical potentials µ f multiplying the quarknumber density operators N f = ¯ ψ f γ ψ f . This is precisely the same structure as in themomentum time-component, so the actual effect when doing computations in perturbationtheory is a shift in p , which in absence of chemical potential woud be one of the fermionicMatsubara frequencies, i [(2 n + 1) πT ] → i [(2 n + 1) πT − iµ f ]. In pure QCD all quarknumbers are conserved separately, so the chemical potential for each flavor can be chosenindependently. Weak interactions, on the other hand, mix different quark flavors, mostlyinside SU(2) doublets but also between families, because the weak interaction eigenstatesdiffer from the mass eigenstates (for a brief review on this mixing, see [61]). Because ofthis, only certain combinations of the baryon number and the different lepton numbersare conserved in the full standard model. We assume that the time scales of chemicalequilibration through weak interactions are much larger than the characteristic time scalesof QCD processes we are studying, even when discussing static correlators, and continueto use independent chemical potentials for ech flavor. In the numerical studies we will usetwo distinct cases, isoscalar ( µ u = µ d ≡ µ S ) and isovector ( µ u = − µ d ≡ µ V ) chemicalpotentials for illustration, but the analytical results are applicable to general µ f .The quark fields ¯ ψ , ψ can be used to define mesonic operators of different spin and flavorstructures. These operators can be thought of either as the currents coupling to externalperturbations as in Eq. (4.4) or as interpolating operators for physical particle states,although in the latter case it should be remembered that the real-time mesonic boundstates do not survive at very high temperatures, and that their properties may be verydifferent from the corresponding screening states. We denote O a = ¯ ψF a Γ ψ, (4.8)where Γ is one of { , γ , γ µ , γ µ γ } for scalar, pseudoscalar, vector and axial vector objects O a = S a , P a , V aµ , A aµ , respectively. The flavor strucure is written in terms of the identitymatrix F s and the traceless matrices F a , which satisfy F s ≡ N F , Tr [ F a F b ] = 12 δ ab , a, b = 1 , . . . , N F − . (4.9)32a) (b) (c) (d) Figure 4.1:
Classes of diagrams contributing to the meson correlator. Diagrams of type (d) onlyapply to flavor singlets.
We wish to compute the static correlators of the above operators. The particular cor-relators we are interested in are defined as C q [ O a , O b ] ≡ Z /T d τ Z d x e i q · x h O a ( τ, x ) O b (0 , ) i . (4.10)The quantity in this equation seems different from the retarded propagator defined inEq. (4.5). The number of relevant propagator-like functions at finite temperature is largebecause of the possibilities of having either real or imaginary time coordinates as wellas different orderings of the operators, but they are all related to the spectral function ρ ( ω, q ), which in turn is given by the analytic continuation of the Euclidean correlator inEq. (4.10). All these relations can be found in standard textbooks (see e.g. [36]), and agood summary is given in [71]. The relevant relation for our computations is D R ( q = 0 , q ) = Z ∞−∞ d ωπ ρ ( ω, q ) ω − iǫ = lim q → Z ∞−∞ d ωπ ρ ( ω, q ) ω − iq = lim q → C E ( q , q ) , (4.11)where C E ( q , q ) is the Euclidean correlator in Eq. (4.10) with an arbitrary (Euclidean)momentum zero-component q .We can use the rotational invariance of the system to choose the vector x , the directionin which we are measuring the correlations, to point in the x direction, and furtheraverage over the transverse x x -plane. The fundamental quantity we are studying hereis then the z -dependent correlator C z [ O a , O b ] = Z d x ⊥ C ( x ⊥ ,z ) [ O a , O b ] = Z /T d τ Z d x ⊥ h O a ( τ, x ⊥ , z ) O b (0 , , i . (4.12)At very high temperature the QCD coupling is small due to asymptotic freedom, andthe correlator C z can be computed using perturbation theory. The leading order resultis given by the free theory diagram Fig. 4.1(a) consisting of two noninteracting quarkspropagating in the hot medium. When the chemical potentials all vanish, this diagramcan be computed in the momentum space, and apart from constants corresponding toterms ∼ δ ( x ) in the coordinate space, the result is proportional to the function B (2 ω n ) ≡ Z d p (2 π ) ω n + p ][ ω n + ( p + q ) ] = i πq ln 2 ω n − iq ω n + iq , (4.13)summed over all fermionic momenta ω n with coefficients that depend on the spin structureof the operator in question. At large distances, or low momenta, the behavior of thecorrelator is dominated by the lowest singularities of this expression, located at q = ± i πT .If we for a moment imagine rotating the three-dimensional theory of the lowest Matsubaramodes to 2+1 dimensions and treating the direction of q , or x , as a time coordinate,this singularity becomes a branch cut on the real q -axis, and can be understood as the33hreshold of producing two free quarks of mass πT . When the interactions are turned on,we expect this singularity to convert into a pole corresponding to a bound state in the2+1-dimensional theory.It should be noted that the temporal components of vector and axial vector correlatorsdo not have the singularity associated with B (2 ω n ), since carrying out the Dirac algebragives these terms a prefactor q + 4 ω n , which regularizes the singularity at q = ± iω n .Computing in the configuration space, it is easy to see that these correlators are suppressedby powers of distance and decay even faster, since the contribution from the lowest sin-gularity is removed. The longitudinal components V and A of those correlators vanishcompletely, apart from contact terms, because of the current conservation ∂ µ V µ = 0.When the chemical potentials are turned on, even the free correlator becomes very hardto compute. This is due to the shift in the temporal momentum components, which causesthe correlator to mix different Matsubara modes even after integrating over τ -direction.If the two quarks have identical chemical potentials, which is the case for flavor singletsor isoscalar chemical potential, the correlator C z in Eq. (4.12) can be computed explicitly.For scalar operator the result is C z [ S a , S b ] = δ ab N c T πz sinh 2 πT z (cid:18) πT coth 2 πT z cos 2 µz + 2 µ sin 2 µz + 1 z cos 2 µz (cid:19) (4.14)= δ ab N c π z (cid:20) − (cid:18) µ π T + 112 µ π T (cid:19) (2 πT z ) + O ( z ) (cid:21) = δ ab N c T z e − πT z (cid:20)(cid:18) πT z (cid:19) cos 2 µz + µπT sin 2 µz (cid:21) + O ( e − πT z ) , where we have also indicated the limiting behavior at small and large distances, respec-tively. For µ = 0 this result agrees with the previous zero density computation in [72]. Theeffect of a small chemical potential on the correlator is seen as oscillations with wavelength l µ = π/µ inside the zero density envelope, while for a large µ ≫ πT the interaction withthe particle bath is so strong that correlator oscillates wildly, averaging to zero.The case of arbitrary chemical potentials is much harder because of the summationsmixing different modes, but a good approximation can be found by only taking into accountthe lowest modes ω n = ± πT , which dominate the correlator at large distances. The scalarcorrelator can then be written in terms of exponential integral function Ei( z ), which weapproximate at large z to leading order in 1 /z , resulting in C z [ S a , S b ] ≈ X ij F a ij F b ji N c T π (cid:16) − e − ∆ µ ij /T (cid:17) z e − πT z " µ ij − ∆ µ ij (2 πT ) + ¯ µ ! ¯ µ ij sin ¯ µ ij z + µ ij + ∆ µ ij (2 πT ) + ¯ µ ! πT cos ¯ µ ij z , (4.15)where ∆ µ ij ≡ µ i − µ j and ¯ µ ij ≡ µ i + µ j . Details of the computation and a more completeresult for the general case can be found in the Appendix A of [4]. In this correlatorthe scale of oscillations is ¯ µ ij , a direct generalization of the isoscalar case. In particular,for isovector chemical potentials the oscillation vanishes (although it reappears at termssuppressed by 1 /z as sin ∆ µ ij z , see [4]) and the correlator has the same functional formas in the µ = 0 case.Together all these results show that the free correlator falls off as exp( − πT z ), regardlessof chemical potentials. The finite density only shifts the singularity in q by an imaginary34 π Tz-0.500.51 ( π / N c ) z C z µ/π T=0 µ/π
T=0.3 µ/π
T=0.5 µ/π
T=0.7 µ/π
T=1
Figure 4.2:
The free scalar meson correlator for different values of isoscalar chemical potential. part, which shows as oscillations in C z . We define the screening mass as the coefficient ofthis exponential fall-off, or the real part of the pole location, because that is what deter-mines the asymptotic behavior of the correlator. In Fig. (4.2) we have plotted the isoscalarcorrelator Eq. (4.14) for different values of chemical potential. The figure shows that whenthe wavelength of the oscillations l µ = π/µ is large, µ . . πT , the oscillatory behaviorbecomes apparent only at distances where the correlator is exponentially suppressed, andit is hard to discern the cosine term from an increased exponential fall-off. For larger µ this difference is obvious, and even more so when studying the asymptotic behavior ofthe correlator, so in the leading order of perturbation theory m = 2 πT + O ( g ) is theconsistent definition also at finite density. Going beyond the leading order, the location of the singularity in the static meson corre-lator is modified by interactions with gluons. We would like to compute in perturbationtheory the next-to-leading order corrections to the screening masses, which requires com-puting the diagrams of types Fig. 4.1(b,c). Note that the flavor singlet correlations arealso mediated by purely gluonic states, some of which have masses lower than 2 πT . Thesestates are completely non-perturbative, so we only concentrate in this work on flavor non-singlets. The couplings of different quark operators to glueballs have been worked out in [1]and the masses of these gluonic operators for both zero and finite chemical potentials havebeen measured in [19].Computing the one-gluon diagrams is not enough, however, since one runs here intothe same infrared problems as always when computing with light particles at finite tem-35erature. In particular, for soft momenta (which implicitly requires bosonic zero mode ω n = 0) the one-loop correction to the gluonic zero mode propagator is of the same orderas the leading term and needs to be resummed into the propagator to get rid of infraredsingularities arising from soft gluons. As discussed in previous chapters, at order g thisresummation gives the static temporal gluon component an electric mass m E ∼ gT , whichwe have to include in the soft gluon propagators of diagrams Fig. 4.1(b,c).Another class of diagrams requiring resummations consists of the graphs with soft gluonexchanges, as shown in Fig. 4.1(b). The integration over q when going to coordinate spacegets the largest contribution from the poles at q ≈ ± iπT , and if the gluon momentum issmall, the additional quark and gluon propagators in the diagram are also nearly on-shell,1 //p ∼ O (1 /g T ), compensating for the factors of g from the vertices. To get a consistentnext-to-leading order correction we therefore have to sum over all diagrams with an arbi-trary number of soft gluon exchanges. This is a common requirement for having a boundstate in theory, since summing only a finite number of diagrams with free propagatorscannot give rise to new singularities, and it applies as well to the screening states underdiscussion. We wish to avoid the exceedingly complicated formalism of relativistic boundstates, and instead make use of the hierarchy between the Matsubara modes πT and themomentum scales where the resummations become necessary, p ∼ gT . This suggests usingan effective theory to organize the resummations of the soft modes, while simply includingthe effects of the large momenta in the parameters.The leading order computation shows that at large distances the screening correlatoris dominated by the lowest Matsubara modes ± πT , while the contribution coming fromthe other modes is exponentially suppressed. We concentrate on the lowest modes andintegrate over all excitations of momenta ∼ πT around these modes, which gives aneffective theory only containing the static ( ω n = 0) gluons and the fermionic modes with ω n = ± πT . In particular, all the other fermionic states are off-shell by 2 πT ≫ gT , so thistheory does not contain any creation or annihilation of quarks, but the only quark lines arethose entering and leaving the diagrams as external legs. Because we are computing nearthe screening pole, these external quarks are almost on-shell, p = 0, and the interactionswith soft gluons do not change that very much. The relevant expansion parameter isthen the “off-shellness” p + p ⊥ + p ∼ g T , which is of the same order as the momentaof the soft gluons. If we rotate the 3-dimensional action of the quark modes into the2+1-dimensional Minkowski space, the Matsubara mode ± πT can be viewed as a heavyquark mass, and the restriction to momenta much lower than this and the separation ofquarks from antiquarks effectively makes the quarks non-relativistic in 2+1 dimensions.Note that there is nothing special in the lowest Matsubara modes, but we could equallywell derive a similar theory for modes ω n = 3 πT , for example. The reason we have singledout the lowest modes is because they dominate the screening correlator, which we aim tocompute.The difference between the purely gluonic EQCD and the effective theory consideredhere is the state around which the expansion takes place, and is dictated by the physicalapplication we have in mind. EQCD is directed at computing either vacuum diagramsor gluonic correlators, which do not have any quarks on external legs. Any fermionicexcitation is then off-shell at least by 2 πT , and only contributes at the high-momentumintegration. The mesonic operators, on the other hand, are built out of quark fields, whichcannot be integrated out if we intend to compute with them. Still, given an external quarkstate, any additional quarks would again be very much off-shell, so expanding around that36tate in low momenta we can neglect all other fermionic excitations. To compare withQED, the relation between the two theories roughly corresponds to the Euler–Heisenbergeffective Lagrangian for photon-photon interactions and the non-relativistic quantum me-chanics used to compute hydrogen (or positronium) binding energies. In the former theelectrons have been integrated out completely, whereas in the latter we can mostly ignorethe contribution of the states with e.g. two electrons and a positron, not because the masswould be much higher (3 m e vs. m e ), but because they are off-shell by ∼ m e , which islarge compared to typical momenta ∼ αm e .The bosonic sector of the dimensionally reduced QCD is well-known [5], and in finitedensity it reads [19] L beff = 12 Tr F ij + Tr [ D i , A ] + m Tr A + ig π X f µ f Tr A + λ (1)E (cid:0) Tr A (cid:1) + λ (2)E Tr A , (4.16)where we have also included the cubic A self-coupling proportional to quark chemicalpotentials. The parameters are found by matching gluonic 2- and 3-point functions. Weonly need the adjoint scalar mass to one-loop level and other couplings at tree level, so m = g T N c N F π X f µ T ! , g = g T, λ (1 , = O ( g T ) . (4.17)The cubic and quartic A self-interactions can be ignored, since they contribute to mesoncorrelators only at order g and higher. The bosonic theory in Eq. (4.16) describes a3-dimensional gauge theory with a massive adjoint scalar A a , and has been extensivelyused to compute gluonic quantities at high temperatures, as discussed in the beginning ofthis chapter.On the fermionic sector the tree-level Lagrangian is just a sum over the parts of the fulltheory Lagrangian containing modes ω n = ± πT . For a single mode ω n this term reads L q = ¯ ψ [ iγ ω n + γ µ − igγ A + γ k D k + γ D ] ψ, (4.18)where k = 1 , A is the gluonic zero mode. We have separated the x -directionfrom the other spatial components, anticipating the choice to measure correlations in thatdirection. Note that the interaction with static gluons does not mix different fermionMatsubara modes, so we have a separate term like Eq. (4.18) for each mode we wish tocompute with.Using a non-standard representation for Dirac matrices (see [1, 4] for details) and de-composing the four-component spinor as ψ = (cid:18) χφ (cid:19) , (4.19)the Lagrangian can be written in a form where the fields χ and φ are light and heavy closeto the pole p = ip = i ( ω n − iµ ), respectively, while the roles are reversed at the otherpole p = − ip . Solving the equation of motion for the heavy component and expandingthe resulting non-local operators in 1 /p , we get the non-relativistic Lagrangian L q ≈ iχ † (cid:20) p − gA + D − p (cid:16) D ⊥ + g i [ σ i , σ j ] F ij (cid:17)(cid:21) χ + iφ † (cid:20) p − gA − D − p (cid:16) D ⊥ + g i [ σ i , σ j ] F ij (cid:17)(cid:21) φ + O (cid:18) p (cid:19) . (4.20)37 p − qq Figure 4.3:
One-loop correction to the quark self-energy.
All dependence on the chemical potential at this level is contained in the shift of thetemporal momentum component, p = ω n − iµ . The Lagrangian is easier to understand ifwe again imagine rotating to 2+1 dimensions and setting z = it . The zero-point energyis then given by p , while the other free terms combine to − ( i∂ t + ∇ / p ), the standardnonrelativistic kinetic term with mass p . If we forget the relativistic origin of these terms,there is no reason for the zero-point energy and the mass parameter in the kinetic termto be the same, and as we will see shortly, the loop corrections will give these parametersdifferent values. It should be noted that for ω n > φ has negative mass, andshould be interpreted as the antiparticle of χ .Already at the tree-level the expansion in 1 /p gives rise to an infinite number of terms,and beyond this we will have to take into account all possible terms allowed by symme-tries. To limit the possibilities, a power counting has to be established. Requiring all thetransverse momenta to be at most of the order of the electric mass, p ⊥ . gT , and thatthe terms in the action to be of order unity, we get χ ∼ / | x ⊥ | ∼ gT , A ∼ ( x / x ⊥ ) / ∼ g / T / . (4.21)The off-shellness ∆ p ≡ p ± ip corresponds to the kinetic energy in the 2+1 dimensionaltheory, as can be verified from the poles in the quark propagators, Eqs. (4.25),(4.26). Fornearly on-shell quarks we can then estimate the derivative ∂ by∆ p ∼ p ⊥ /p ∼ g T ⇒ ∂ ∼ g T acting on quarks . (4.22)Using this power counting, and keeping only terms required to give the screening mass toorder g , the final form of the fermionic Lagrangian is L feff = iχ † (cid:18) M − g E A + D − ∇ ⊥ p (cid:19) χ + iφ † (cid:18) M − g E A − D − ∇ ⊥ p (cid:19) φ , (4.23)where only the zero-point energy, which we will henceforth denote by M , needs to bematched beyond tree-level. This matching is carried out by comparing the poles in theone-loop corrected quark propagator depicted in Fig. 4.3. The details of the computationare given in the original papers, and the result is simply M = p + g C F T p (cid:18) µ π T (cid:19) = ω n − iµ + g C F T ω n − iµ ) (cid:18) µ π T (cid:19) . (4.24)It should be noted that for ω n = ± πT , which are the modes of interest here, the real partof M does not depend on µ .The free quark propagators following from Eq. (4.23) are h χ u ( p ) χ ∗ v ( q ) i = δ uv (2 π ) δ ( p − q ) − iM + ip + p ⊥ / p (4.25) h φ u ( p ) φ ∗ v ( q ) i = δ uv (2 π ) δ ( p − q ) − iM − ip + p ⊥ / p (4.26)38r in the configuration space h χ u ( x ) χ ∗ v ( y ) i = − iδ uv θ ( ω n ( x − y )) p π ( x − y ) e − M ( x − y ) − p x ⊥− y ⊥ )2( x − y (4.27) h φ u ( x ) φ ∗ v ( y ) i = − iδ uv θ ( ω n ( y − x )) p π ( y − x ) e − M ( y − x ) − p x ⊥− y ⊥ )2( y − x . (4.28)In these equations it is obvious that for ω n > χ propagates forward and φ backward in x , the time coordinate of the 2+1-dimensional theory, confirming ourinterpretation of φ as the antiparticle of χ . For negative modes these roles are reversed,with φ propagating forward in x . In [1] we expanded these propagators in p ⊥ /p insideloop integrals to make sure that the transverse momenta are parametrically smaller thanthe heavy scale ∼ T . However, as discussed in [73], while this works well for (single)heavy quark effective theory, in NRQCD there are difficulties at two-loop level with thisapproach, and it is preferable to keep the kinetic terms summed into the propagators.For our modest purposes there is no real difference, but in [4] we chose not expand intransverse momenta. For consistency, this requires that we expand the gluon field inmultipole expansion, which at this level boils down to disallowing transverse momentumtransfer from gluons to quarks. The masses and the quark-antiquark potential turn outto not depend on the way we treat the propagators, whereas in order to compute the fullcorrelators it is necessary to keep the kinetic terms resummed. As a check, we have shownthat NRQCD is able to reproduce the leading order scalar correlator in Eq. (4.14) exactly.In terms of the new fields χ and φ the operators whose correlators we intend to computecan be written as S : ¯ ψψ = χ † φ + φ † χ ,P : ¯ ψγ ψ = χ † σ φ − φ † σ χ ,V : ¯ ψγ ψ = χ † χ + φ † φ ,V k : ¯ ψγ k ψ = − ǫ kl ( χ † σ l φ − φ † σ l χ ) ,V : ¯ ψγ ψ = i ( χ † χ − φ † φ ) ,A : ¯ ψγ γ ψ = φ † σ φ − χ † σ χ ,A k : ¯ ψγ k γ ψ = − i ( χ † σ k φ + φ † σ k χ ) ,A : ¯ ψγ γ ψ = − i ( χ † σ χ + φ † σ φ ) . (4.29)It should be noted that the temporal and longitudinal components of vector and axialvector currents consist of terms like χ † χ and φ † φ , so their correlators are proportional to θ ( z ) θ ( − z ) and vanish at nonzero distances. For V and A this follows from current conser-vation, whereas the correlators for charges V and A are power-suppressed, as discussedin the previous section. Apart from those operators, the correlator in the effective theoryis independent of the spin structure, up to a multiplicative constant, since ( σ i ) . Theflavor structure on the other hand is significant if we allow for finite chemical potentials. Having derived the Lagrangian for NRQCD , which is just the sum of the bosonic part inEq. (4.16) and the fermionic part in Eq. (4.23), it remains to find the masses of mesonicoperators. These correspond to bound states in the 2+1-dimensional theory, followingfrom the summation of diagrams of type Fig. 4.1(b,c) with an arbitrary number of softgluon exchanges. In a nonrelativistic theory the resummation can be carried out by findingthe static potential for a χ ∗ φ pair, and then solving the resulting Schr¨odinger equation39ith this potential. Here “static” should be understood from the 2+1-dimensional pointof view, which means that the potential will be valid for large x . All these computationscan be performed using the effective theory just derived, since it is only the ω n = 0 gluonswith low momenta that need to be summed beyond the leading order.The static potential is written as an expansion in g r , V ( r ) ∼ g ln r + g r + O ( g r ) . (4.30)Using either the power counting 1 /r ∼ p ⊥ . gT or the leading order Schr¨odinger equa-tion we see that g r ∼ g . The leading Coulomb-type term g ln r in the potential isthen sufficient for computing O ( g ) corrections to screening masses, and can be evaluatedperturbatively by computing all one-gluon diagrams in the effective theory. The leadinglogarithmic term already gives a confining potential, so there is no qualitative differencein dropping the linear term, which is parametrically of order g .The static potential of a φ ∗ χ pair is computed by inserting a point-splitting in thecorrelator to give the quarks a small spatial separation in the transverse direction, andfinding the Schr¨odinger-type equation satisfied by this correlator at z → ∞ limit. Thedetails of this computation can be found Appendix B of [4], and the result is V ( r ) = g C F π (cid:16) ln m E r γ E − K ( m E r ) (cid:17) , (4.31)where K is a modified Bessel function. The result is both ultraviolet and infrared finiteonce we have resummed the gluon self-energy corrections to an electric mass m E , whileletting m E → m E , as the leading order is onlysensitive to the propagation of gluons in the hot medium while the quarks simply act asstatic color charges.The Schr¨odinger equation satisfied by the correlator was already found as an interme-diate result when computing the potential, and it reads (cid:20) ± ( M i + M j ) − ± p ∇ r + V ( r ) (cid:21) Ψ = m full Ψ , (4.32)where we have separated the variables as C ( r , z ) = Ψ ( r ) e − m full z , (4.33)and the ± signs apply for ω n = ± πT , respectively. The flavors of the quarks formingthe meson are labelled with indices i, j , and in general the flavor symmetry betweendifferent mesons is broken by the different chemical potentials. The parameters M and p are generally complex when computing with finite chemical potentials, but for oppositemodes they are related by M − = − M ∗ + and p − = − p ∗ , so the screening masses satisfyby m full , − = m ∗ full , + . Thus we only need to compute the masses for ω n = πT , andin addition these relations guarantee that the full correlator, which is the sum over allseparate Matsubara modes, behaves as C z [ O a , O b ] ∝ X ij F a ij F b ji m full , ij ) z − α ij ] exp[ − Re( m full , ij ) z ] , (4.34)40here α ij is the overall phase of the φ ∗ i χ j correlator. This is of the same form as theleading order term, with the real part of the mass parameter giving an exponential decaywhile the imaginary part contributes to a cosine-like oscillation term. In particular, thiscorrelator is real-valued even though the term coming from any single mode is in generalcomplex.The Schr¨odinger equation with potential Eq. (4.31) cannot be solved analytically, sowe have to find the eigenvalues m full using numerical computations. For this, we cast theequation into a dimensionless form, which depends on the values of the physical parametersonly through the dimensionless combinations ρ and ˆ E defined as ρ ≡ g C F πm (cid:18) ω n − iµ i + 1 ω n − iµ j (cid:19) − , g C F π ˆ E ≡ m full − M i − M j . (4.35)One should note in ρ the appearance of the reduced mass, typical of two-body problems.The physical screening mass is given by ˆ E which we solve numerically,Re( m full ) = Re( M i + M j ) + g C F π Re( ˆ E ) = 2 πT + g T C F π (cid:18)
12 + Re( ˆ E ) (cid:19) , (4.36)where in the last equality we have used the fact that for the lowest modes the real part of M is independent of µ .The numerical solution is found by assuming that the ground state is cylindricallysymmetric, solving the behavior of Ψ ( r ) around the origin and then integrating out tolarger r and requiring square integrability. In zero density this is easy since both thewave function and ˆ E are real, and trying different values of ˆ E on the real axis with thecondition that Ψ ( r ) vanishes at large distances is very fast. We find, with somewhatsuperfluous precision, ˆ E = . , ρ = 2 / , ( N F = 0)0 . , ρ = 1 / , ( N F = 2)0 . , ρ = 4 / , ( N F = 3) (4.37)for different numbers of dynamical quarks N F . The number of flavors only enters in m E , soeven the case N F = 0 makes sense if the creation and annihilation of quarks is suppressed,as in the so-called quenched simulations on lattice. When the chemical potentials areturned on, the numerical computation becomes more demanding, as all parameters andthe wave function have complex values, and the solution has to be searched for in thecomplex plane instead of limiting to the real axis. Moreover, the dependence on µ/πT cannot be solved analytically, but we have to repeat the process for each chemical potentialseparately.In Fig. 4.4 we have plotted the eigenvalues ˆ E for isoscalar chemical potential withdifferent numbers of dynamical quarks. We have also studied the case where N F = 3,but only the two flavors in the measured operator have nonzero chemical potentials. Thisshould more closely correspond to the situation in heavy ion collisions, where the twocolliding nuclei have finite up and down quark densities but vanishing net strangeness.The figure shows that the poles move along quadratic curves off the real axis, and the realpart of ˆ E , which is the contribution to the screening mass, grows with µ S for dynamicalquarks. In the quenched case the real part decreases and becomes negative at µ S ≈ . πT .At large values of the chemical potential the oscillations due to the imaginary part of41 /π T = 0.2 µ/π
T = 0.4 µ/π
T = 0.6 µ/π
T = 0.8 µ/π
T = 1 µ/π
T = 0 µ/π
T = 0.2 µ/π
T = 0.4 µ/π
T = 0.6 µ/π
T = 0.8 µ/π
T = 0 µ/π
T = 1 I m Ê N F =0N F =2N F =3N F =3, µ s =0 Figure 4.4:
The eigenvalue ˆ E with the lowest real part for isoscalar µ S = 0 . . . ˆ E become strong and the numerical integration is unstable, so it is hard to go beyond µ ∼ πT numerically. On the other hand, in deriving the effective theory using dimensionalreduction we assume that T is larger than any other mass scale, so our results cannot betrusted for µ & πT .The chemical potential enters the dimensionless Schr¨odinger equation only through theparameter ρ . For the specific cases of isoscalar and isovector chemical potential that westudy numerically, the dependence is ρ ∝ − i ˆ µ S N F + 3 N F ˆ µ S (isoscalar) , ρ ∝ µ V N F + 3 N F ˆ µ V (isovector) , ˆ µ ≡ µ/πT . (4.38)From Fig. 4.4 we can see the twofold influence of the chemical potential. When N F = 0,increasing the chemical potential just shifts ρ into more imaginary values, while its real partstays constant. The following increase in the absolute value of ρ decreases Re( ˆ E ) like inthe µ = 0 case, Eq. (4.37). For dynamical fermions this effect is more than compensatedby the increase in m E , which raises the potential and accordingly increases the energyeigenvalues.In Fig. 4.5 we plot the real part of ˆ E , which apart from scaling and an additive µ -independent constant is the same as the screening mass, see Eq. (4.36). The isoscalar datais the same as in Fig. 4.4 discussed above, while for isovector chemical potential we see thatthe mass decreases with µ V for N F <
3. This is in agreement with the previous discussionon the relation between | ρ | and ˆ E , and Eq. (4.38), where ρ for isovector chemical potentialincreases with µ for small number of dynamical fermions.Numerically the correction we have computed is small, even near the phase transition42 µ S / π T R e Ê µ V / π T-0.4-0.200.20.4 R e Ê N F =0N F =2N F =3N F =3, µ s =0 Figure 4.5:
The real part of the eigenvalue ˆ E for isoscalar (left) and isovector (right) chemicalpotentials. where the coupling constant itself is large. For N F = 2 the range of screening masses fordifferent values of chemical potentials fits in the intervalRe( m full ) ≈ πT + g × . , µ S /πT = 1 . . , µ/πT = 0 . . , µ V /πT = 1 . g E is estimated to be g /T ≈ . T ∼ T c [50], giving thenext-to-leading order corrections to the screening mass of about 6–8%. Nevertheless, aslong as the coupling is large there is no reason to expect that the next correction wouldbe smaller by a factor of the same magnitude.While the Schr¨odinger equation Eq. (4.32) cannot be solved exactly, we were able tofind a simple approximate dependence on the parameters while writing this introductorypart. Realizing that the modified Bessel function K ( m E r ) in the potential interpolatesbetween − ln( m E r/
2) and 0, we can try to estimate the potential by V ( r ) ≈ g C F π (cid:16) C ln m E r C (cid:17) , (4.40)where we expect 1 ≤ C ≤
2. For real values of the dimensionless parameter ρ defined inEq. (4.35) its effects can be scaled into the dimensionless variables, giving for the potentialin Eq. (4.40) ˆ E ( ρ ) = ˆ E (1) − C ρ . (4.41)In Fig. 4.6 we have plotted ˆ E vs. ln ρ for the numerical data we have computed, assumingthat the expression in Eq. (4.41) can be extended for complex values of ρ as well, if thebranch cut is introduced on the negative real axis. As the figure shows, for isovector µ with real ρ the behavior is extremely well described by Eq. (4.41), and the agreement isalso good for the complex values of ρ . Fitting a line to both real and imaginary parts43 ρ |-0.4-0.200.20.40.60.8 R e Ê µ V µ S ρ I m Ê Figure 4.6:
The numerical data in Figs. 4.4 and 4.5 parametrized by the logarithm of the dimen-sionless parameter ρ . separately, we get slopes -0.78 and -0.74, respectively. To desired accuracy, the screeningmasses from our computations can then be summarized by the two-parameter fit from theleft plot in Fig. 4.6, Re( m full ) = 2 πT + g T C F π (0 . − .
78 ln | ρ | ) , (4.42)where ρ can be read from Eq. (4.35). The static mesonic correlators considered here have also been measured in the latticesimulations. In zero density there is a long tradition of measuring the screening masses,for recent results see [74–77]. Early measurements gave large differences between themasses of different spin structure operators, but in recent works the general picture is thatalready at ∼ T c the masses come close to the ideal gas result, with ρ (vector) mesonslightly larger than π (pseudoscalar) and both lying 5–10% below the ideal gas result.The correction we have computed above is of the same magnitude, but with different sign.For all N F the µ = 0 results we have computed lie above the free theory result 2 πT ,approaching the ideal gas slowly as α s ( T ) gets smaller due to asymptotic freedom.The systematic errors in lattice simulations are related to dynamical quarks, light quarkmasses, and the difficulties in going to the infinite volume limit. In [76] the analysisof the infinite volume extrapolation is carried out, resulting in slightly higher massesthan those measured earlier, but even then the screening masses are clearly below theideal gas result. It should be noted that our perturbative result is above 2 πT also for N F = 0, so the potential difficulties with dynamical quarks cannot completely explain thedisagreement with lattice measurements. On the other hand, because the strong coupling44s large near T c , the higher order perturbative corrections can be as large as the O ( g ) termwe have computed, at least at temperatures within the reach of lattice measurements. Atasymptotically high temperatures, however, the perturbative calculation should be valid,so we expect the screening masses to cross above the free theory result at high enoughtemperature.Recently the screening masses have also been computed by evaluating the meson spectralfunction in the HTL resummation scheme and determining the fall-off of the correlatorthrough the spectral function [78]. This method is very orthogonal to our computations,but the weak coupling limit of those results is very similar, and can be summarized inour terms by setting ˆ E = 0 in Eq. (4.36). The difference can be explained by the softgluon contributions that were knowingly left out in [78]. At temperatures close to thephase transition the meson properties have been studied analytically in the Nambu–Jona-Lasinio model, which gives screening masses well below the free theory result [79]. Itshould be noted that this method is valid at very different temperature region, and doesnot necessarily contradict our high-temperature results.At finite density the lattice measurements of hadronic screening masses have been per-formed only recently [80]. Simulations at nonzero chemical potentials are difficult becauseof the complex fermion determinants, so these computations were carried out by expand-ing the masses as Taylor series in µ around µ = 0 and measuring the derivatives up tosecond order response. All measurements in [80] were carried out using N F = 2 flavorsof staggered fermions. The leading order term is just the µ = 0 mass which behaves asdescribed above, and the first derivative vanishes for both isovector and isoscalar chemicalpotentials. For the latter this follows from symmetry properties, whereas the responsewith isovector µ is explicitly measured to be zero. In our perturbative calculation thefirst derivative at µ = 0 vanishes as well for both chemical potentials, as required by thesymmetry ˆ E ( − µ ) = ˆ E ∗ ( µ ).The second derivatives give the µ -dependence of the masses as measured on lattice. Forisoscalar µ the measured second order response rises steeply at T c and settles somewhatbelow 2 /T at higher temperatures, for both π and ρ . In the isovector case the response issmall and negative, and approaches zero as temperature is raised. At small µ these resultsare in qualitative agreement with our perturbative calculation, but for isoscalar the actualnumbers differ by orders of magnitude. For comparison, we have fitted quadratic curvesto our data to extract the second order response, and the result isd Re( ˆ E )dˆ µ S = − . , ( N F = 0)0 . , ( N F = 2)0 . , ( N F = 3)0 . , ( N F = 3 , µ s = 0) d Re( ˆ E )dˆ µ V = − . , ( N F = 0) − . , ( N F = 2)0 . , ( N F = 3) − . , ( N F = 3 , µ s = 0) (4.43)for isoscalar and isovector, respectively. In terms of physical parameters T d m d µ = g C F π T d Re( ˆ E )dˆ µ , (4.44)which gives second derivatives of order T d m/ d µ ∼ ± .
02, two orders of magnitudesmaller than the value measured on lattice. While the masses in both perturbative andlattice computations are consistent with the free theory result when all possible errorsources are taken into account, the difference in the second derivatives with respect toisoscalar chemical potential is striking. One has to be careful, however, to define the massthe same way in both computations before making any comparison.45 .5.1 Definition of the screening mass at nonzero density
The correlator C z in Eq. (4.12) is a complicated function of z already at leading order,as the exact free theory result in Eq. (4.14) shows. The screening masses appear as polesin the momentum space Green’s functions, but in practice the correlator is measured inconfiguration space, and only exhibits simple exponential behavior at asymptotically largedistances, the coefficient of this exponential decay being the screening mass. An effective z -dependent screening mass can be defined as m ( z ) ≡ − C z ∂C z ∂z , (4.45)which in free theory approaches the screening mass roughly as 1 /z . In lattice simulationsthis complication is often removed by measuring the one-dimensional correlations betweenplanar sources. On a finite lattice with periodic boundary conditions the correlator thenbehaves as C ( z ) ∼ cosh( − mz ), which is used in fitting to extract m from data.When chemical potentials are turned on, the situation becomes more complicated. Thedefinition of the effective mass in Eq. (4.45) is useless because of the oscillations in thecorrelator which cause C z to periodically go through zero and to negative values, prevent-ing us from taking the z → ∞ limit. At short distances the fall-off is faster than at µ = 0because of the cosine term cos(2 µz ) ≈ − µ z , but this term does not contribute to thedecay at longer distances. The asymptotic behavior of the correlator is dominated by thecoefficient of the exponential fall-off, which corresponds to the real part of the momentumspace pole location. At large distances this coefficient can be extracted from data, butusually one cannot do measurements at arbitrarily large separations because of the finitelattice size and the exponentially small value of the correlator.When determining the mass from short-distance data the potential oscillatory termshave to be included in the fit in order to get reliable estimates. For example, the freeplane-plane correlator at finite chemical potential behaves as ∼ cos(¯ µz ) exp( − M z ) with¯ µ ≡ µ i + µ j and M = 2 πT . Simply fitting an exponential of the form C exp( − mz )overestimates the fall-off because of the oscillations, giving( m − M )( m + M ) = ¯ µ (3 m + M ) ⇒ m ≈ M (cid:18) µ M − ¯ µ M (cid:19) . (4.46)The derivatives with respect to µ are also affected by the unfortunate choice of the fittingfunction. For isoscalar chemical potential ¯ µ = 2 µ S , and even for the µ S -independentscreening mass 2 πT of the free theory Eq. (4.46) would give T d m d µ S (cid:12)(cid:12)(cid:12) µ =0 = 8 TM = 4 π , (4.47)which incidentally is of the same magnitude as the derivatives measured in [80]. In thecase of isovector chemical potentials ¯ µ = 0 and the oscillations vanish at leading order in1 /z , so this effect disappears for µ V .As discussed above, the actual lattice simulations are carried out by measuring thederivatives of correlators at µ = 0 instead of the full correlator at µ = 0. The expressionfor the second derivative of C ( z ) = A exp( − M z ), given that the first derivative vanishesat µ = 0, is [81] 1 C ( z ) d C ( z )d µ = 1 A d A d µ − z d M d µ = − z − z d M d µ , (4.48)46here in the last equality we have substituted A = cos(2 µz ) and the whole expression iswritten in the limit of infinitely long lattice. The oscillations should then show as quadratic z -dependence of the second derivative, but at least the data in [81] does not support theexistence of this kind of term.Given that the second order response does not show signs of oscillation in lattice sim-ulations, it is not clear if the cosine terms clearly visible at the leading order and inone-loop terms survive at the non-perturbative level. However, understanding the po-tentially complicated behavior of the correlator at short distances is necessary to extractreliable information from data measured at finite distances. In particular, the oscillatoryterms suggested by the perturbative calculations should probably be taken into accountwhen fitting the correlator at finite chemical potentials.47 hapter 5 Review and outlook
Dimensional reduction is an approximate method which is perfectly suited for computingstatic observables in weakly coupled high temperature field theory. It is based on theobservation that when the temperature is large and the coupling small, there is a clear scalehierarchy between the thermal, electric and magnetic scales πT , gT and g T , respectively,and makes full use of that hierarchy by applying effective field theory techniques. Inperturbation theory reorganizing the perturbative expansion by resummation of classesof diagrams is necessary in order to regulate the infrared divergences, and this can becarried out systematically using dimensional reduction, without worrying about doublecounting diagrams. For non-perturbative lattice simulations the dimensionally reducedeffective theory is appealing because of the analytic treatment of fermions, lower space-time dimension and the integration out of scales ∼ πT , allowing for larger lattice spacing.For these reasons, dimensional reduction has been extensively used to compute propertiesof both QCD and electroweak theory at high temperatures. However, it is not of use nearthe QCD phase transition, where α s is large and the weak-coupling hierarchy of scalesdisappears. It should be also noted that dimensional reduction explicitly breaks the Z (3)symmetry of degenerate QCD vacua above T c , and the fluctuations between these statesbecome important close to the phase transition.In this thesis, we have studied two applications of dimensional reduction to the standardmodel physics. The pressure of the electroweak theory, which was previously undeterminedbeyond the first terms, was computed up to three-loop level, or O ( g ), in [2, 3]. Combinedwith the previously known QCD pressure, this enables us to study the pressure of the fullstandard model above the electroweak phase transition. We have shown that the pressureof the symmetric phase goes smoothly through the transition point once the small valueof the Higgs field mass is taken into account in the effective theory. The pressure ofthe whole standard model to this order lies 10–15% below the ideal gas value, but theperturbative expansion does not converge well because of the large values of QCD and topquark Yukawa couplings. As a special case we have studied the weakly coupled SU(2) +fundamental scalar theory, where the convergence is good as expected.The correlation lengths of mesonic operators at high temperatures have been computedin [1] and extended to finite chemical potentials in [4]. We have computed the leadingorder (free) correlators in full QCD, and then formulated a dimensionally reduced effectivetheory which includes the lowest fermionic modes in order to compute the first pertur-bative correction of order g to masses. The fermionic theory is seen to correspond to anonrelativistic theory for heavy quarks in 2+1 dimensions, and draws inspiration from the48eavy quark effective theories used to compute the properties of quarkonia. The screeningmass correction we find is small and positive, and it depends in a complicated way on thechemical potentials and the number of dynamical quarks. The finite density result is onlyin qualitative agreement with lattice results, and we discuss the potential differences inmeasuring the mass at finite distances.Dimensional reduction has proved to be an efficient tool for computing bosonic ob-servables at high temperatures both in perturbation theory and combined with non-perturbative methods. 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