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7.2 Midpoints of the Current Correlators at Finite

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07

16 24 24(II) 32 48 cont 16 32 40

1283xNτ 1.5Tc 3.0Tc1.5Tc1.2Tc G1(τT=0.5)/G2(τT=0.5) (V/AV)TI

(V/AV)NP -(Vii/Aii)NP

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14

16 24 24(II) 32 48 cont 16 32 40

1283xNτ 1.5Tc 3.0Tc1.5Tc1.2Tc G1(τT=0.5)/G2(τT=0.5) (PS/S)TI

(V/AV)NP

Figure 7.6:Left: The ratio of the midpoints of the vector and axial vector channel renor-malized with non-perturbative and tadpole improved renormalization con-stants. Right: A comparison of the non-perturbative vector to axial vector ratio and the tadpole improved pseudo scalar to scalar.

To study these effects we show the ratio of the midpoints of the vector to axial vec-tor and the pseudo scalar to scalar correlation functions for the available lattices with Nσ = 128 in Fig.7.6. In addition the data atT ≃1.45Tc is used to extrapolate the ratio to the continuum. Whereby only the results with approximately the same quark mass were used for the extrapolation, i.e. Nτ = 24, 32 and 48. Note that in order to check for finite mass effects we show the results for both available quark masses on theNτ = 24 lattice.

On the left of Fig.7.6 we focus on the vector to axial vector ratio and show the full vector to full axial vector ratio once using the non-perturbative and once the tadpole improved renormalization constants given in Tab.2.4.1. Additionally we give the non-perturbatively renormalized ratio of the spatial components of the vector and axial vector channels. Clearly the results are very much compatible and we observe a systematic de-creasing trend with inde-creasing cut-off. However, all our results can be approximately contained in a region deviating in around 1% or 3% across our calculations and the continuum extrapolation.

On the right of Fig.7.6we compare the non-perturbative full vector to axial vector ratio with the tadpole improved pseudo scalar to scalar. We observe a strong cut-off depen-dence and a deviation from unity of something between 10% and 6% in the pseudo scalar by scalar case.

Comparing the two quark masses atNτ = 24 in both figures finite mass effects are seen to be small as they deviate from one another only on the level of 0.5%. As such they are not strong enough to explain the deviation from unity of the results in the vector/axial vector and the pseudo scalar/scalar ratios.

With finite mass effects small and a visible deviation from unity also in the continuum

limit we identify two sources for the apparent mismatch. As such one possibility is that the observed deviation indeed arises from the errors of the renormalization constants.

The other possibility is that the corresponding symmetries are not yet restored in the available temperature range.

Fortunately we can turn to a number of studies to get a feeling which of the two scenarios might be applicable, see e.g. [115,116] and references therein. As such using an improved staggered fermion formulation on dynamicalNf = 2 + 1 configurations the recent calcu-lation of [116] could show that in the intermediate temperature region 1.2Tc .T .1.5Tc both chiral symmetry and the anomalousUA(1) exhibit signs of restoration.

If we assume this to be the case in our study the deviation from unity would be mostly due to the uncertainties of the renormalization constants.

If this is indeed so we can estimate the error induced by these uncertainties especially in the pseudo scalar/scalar case. As here it is not possible to take renormalization in-dependent ratios, as was done in the vector channel, when examining the correlation functions. In the pseudo scalar channel additional error bands on the correlator data, in e.g. Fig.7.2, of the order of 5−10% due to renormalization could explain the offset of the individual lattice results. Subsequently the discrepancies would not be due to lattice effects.

In this thesis we computed meson correlation functions in the deconfined phase of quenched QCD. For the calculation we invoked the framework ofO(a2)-improved Wilson-Clover fermions at light quark masses. We computed the vector correlation function on lattices of sizeNσ3×Nτ, with 32≤Nσ ≤128 andNτ = 16, 24, 32 and 48. ForNτ = 16 we calculated GH(τ T) on lattices with spatial extent Nσ = 32, 64, 96 and 128, in or-der to quantify finite volume effects at fixed values of the lattice cut-off. For Nτ = 24 we checked that the quark masses used in our calculations are indeed small enough on the scale of the temperature to be ignored in the analysis of our correlation functions.

On the largest spatial lattice, Nσ = 128, we performed calculations for four different values of the lattice cut-off by choosing Nτ = 16, 24, 32 and 48 and at the same time changing the value of the gauge coupling β such that the temperature is kept constant, T ≃ 1.45Tc. Finally we varied the temperature at Nσ = 128 keeping the cut-off scale fixed and varying the temporal extentNτ = 16, 32 and 40, corresponding to the respec-tive temperatures T ≃3.0Tc, 1.45Tc and 1.2Tc.

With these datasets we were able to undertake physics analysis at an unprecedented precision and our main results are summarized in the following.

The Light Quark Vector Spectral Function at Vanishing Momentum in the Continuum Limit of Quenched QCD

At the fixed value of temperature T ≃1.45Tc we performed a systematic and detailed analysis of the vector correlation functions at vanishing momentum. Analyzing different lattice cut-off values combined with an analysis of finite volume and quark mass effects allowed us to extrapolate the vector correlation function to the continuum limit for the first time in a renormalization independent fashion for a large interval of Euclidean times, spanning from the midpointτ T = 0,5 toτ T ≃0.2. In this interval we determined the correlation function to an unprecedented accuracy at the 1% level. Additionally we computed the first two thermal moments of the correlation functions, the second being equivalent to the curvature of the correlator at the midpoint of the finite temperature Euclidean time interval.

Subsequently we analyzed the continuum extrapolated correlation functions invoking several fit Ans¨atze that differ in their low frequency structure. As a result we find the vector correlator to be best fitted by a simple Ansatz of the free spectral function times a perturbatively motivated correction factor k(T) plus a phenomenologically inspired Breit-Wigner term centered at ω= 0.

This Ansatz already gives small χ2/d.o.f. and describes the data well. Systematically

changing the low frequency structure of the Ansatz spectral function, we are able to estimate the systematic uncertainty of the low energy structure of the spectral function.

Some features of the spectral functions are robust, as such the spectral function for fre-quenciesω/T & (2−4) is close to its free form. In this regime it would be interesting to replace the component proportional to the free continuum spectral function in our Ansatz by the hard thermal loop spectral function. This is easily possible and will be done in the near future.

For energies in the regionω/T .(1−2) the spectral function is significantly larger than the free result, but smaller than the HTL spectral function, which diverges at small energies. As a result the the resulting thermal dilepton rate is an order of magnitude larger than the leading order Born rate at energiesω/T ≃1.

Finally we accurately and systematically determined the electrical conductivity from our resulting spectral functions, the thus obtained value isσ/T = (1/3−1)·Cem. To our knowledge this is the first time a fully non-perturbative estimate of this transport coefficient is given including also systematic uncertainties.

To quantitatively analyze in what fashion our results and the enhancement in the low frequency region can account for the experimentally observed dilepton rates in this energy region [38,117], we need to extend our analysis to incorporate the temperature and momentum dependence of our results. Then a complete analysis of dilepton rates that takes into account the hydrodynamic expansion of dense matter created in heavy ion collision will become possible [108].

First steps in this direction have been undertaken in this thesis and we quickly summarize the corresponding results in the following.

Results on the Temperature Dependence at Vanishing Momentum

Fixing the cut-off to β= 7.457 and thus a−1 = 12.864GeV we performed an analysis of the vanishing momentum vector correlation function by varying Nτ = 16, 32 and 40.

With these values we obtain the temperaturesT ≃3.0, 1.45 and 1.2Tc, respectively.

We carefully analyzed the temperature dependence of the vector correlation function, the quark number susceptibility and the thermal moments of the correlator. The latter were seen to be almost constant inT /Tc, implying only little change in the underlying spectral functions.

Additionally we fitted the data using the simple free continuum plus Breit-Wigner Ansatz motivated atT ≃1.45Tc. Here our systematic analysis is limited to determining the dependence on the Euclidean time window of the fit, as a continuum extrapolation that eliminates the lattice effects was not yet possible within the available computing resources.

The resulting parameters are then analyzed for the temperature dependence of the width and height of the Breit-Wigner peak. For all these quantities the parameters are seen to be almost constant in units of temperature, implying a linear dependence on the temperature. As such the width increases linearly while the height drops linearly with

A similar behavior is observed for the temperature dependence of the electrical con-ductivity, whereby the corresponding results are within 7% of each other over the tem-perature range. As a result we could establish also a linear dependence in temtem-perature for the electrical conductivity, in accordance with perturbation theory [111].

Finally we checked of a sum rule stating the area under the Breit-Wigner peak to be constant and temperature independent [112]. Here the resulting values are seen to match within errors. However the large uncertainties encountered make it difficult to firmly establish the validity of the sum rule, even though our results lean towards this direction.

For the temperature evolution additional lattice calculations, preferably with a subse-quent continuum extrapolations are highly desirable.

Exploratory Study of the Vector Correlation Function at Finite Momentum In the case of finite momentum we restricted our analysis to the lattice sized 1283×48 and the lattice momenta ~k = (0,0, pz) where pz = 0,1,2,3, whereby the physical momenta are given by |p|/T = 2π· |~k| ·Nτ/Nσ. We could show that the longitudinal component of the vector correlation function exhibits only very little momentum dependence, both in the correlator and its thermal moments.

This is markedly different in the transverse case. Here a large momentum dependence was seen in the correlator and a somewhat weaker but nevertheless visible dependence in the second thermal moment.

All results are seen to lie on top of each other atτ T ≤0.15, implying the correction factor k(T) to be momentum independent.

Additionally the ratios of the data and the corresponding free correlators exhibit a decreasing trend with increasing Euclidean time, taking into account the correction factor this means the low frequency part of the interacting spectral function is in fact smaller than that of the free case.

Invoking a set of toy models we then were capable to identify, what kind of behavior is favored by both channels in the interacting case. To this extent we employed first a time-independent contribution plus the free continuum and second a time-dependent contribution modeled by the corresponding free behavior at low frequency plus the free continuum. We could show that the longitudinal vector correlation function is very well describable by a model that incorporates none or very little time dependence both in the correlator and its thermal moments. While the transverse was seen to hold a middle ground between the constant and modified free behavior. As such the corresponding model parameters deviate less from the free case in the longitudinal channel than in the transverse.

The results obtained are encouraging and suggest a determination of the spectral

func-tion from our data is indeed possible.

Additionally we studied the connection between the time-time and longitudinal vector correlation functions at finite momentum. We could derive two exact relations between the longitudinal and the time-time correlation functions. The first approximately con-nects the time-time correlator with a sum of thermal moments of the longitudinal case, while the second links the longitudinal correlator with the second derivative of the time-time case. These two relations are subsequently checked and an excellent agreement is observed in our data. As a consequence we could estimate the effect of a contribution linear in frequency in the longitudinal spectral function [114] on the time-time correla-tor. We found this contribution to be divergent. Such a divergence is unphysical and we therefore exclude the possibility of such a non-zero intercept in the longitudinal case.

Remarks on the Pseudo Scalar and Renormalization Issues

Finally we turned to evaluating the pseudo scalar correlation function and the midpoints of the correlators of all channels, both at vanishing momentum.

As in the case of the vector correlator, we could combine a study of lattice, quark mass and cut-off effects for the pseudo scalar. However here the possibility to take renor-malization independent ratios does not arise and a continuum extrapolation contains the additional uncertainties due to the only perturbatively known renormalization con-stants. Nevertheless the extrapolation was done both for the correlator and its thermal moments, revealing large non-perturbative effects atT ≃1.45Tc.

These effects were seen to persist also at T ≃ 1.2Tc and T ≃ 3.0Tc, even though in a somewhat reduced sense in the latter case.

AtT ≃1.45Tc we then used a MEM analysis to obtain an estimate of the pseudo scalar spectral function. Using a number of default models which include Breit-Wigner con-tributions in the low frequency regime, it turned out that MEM strongly disfavors the presence of such a transport peak. Instead a dominant (non-perturbative) peak struc-ture arises atω/T ≃(5−20), suggesting the survival of a resonance in the pseudo scalar channel.

To further understand the effects in the pseudo scalar channel it is important to quan-tify more rigorously the errors of the renormalization constants. Additionally a better theoretical understanding of the effects expected would be desirable.

Touching on the topic of the uncertainties of the renormalization constants, we com-pare the midpoints of the pseudo scalar and scalar as well as the vector and axial vector correlation function. Assuming to be in the symmetry restored phase atT ≃1.45Tc, we observe the results of the vector/axial vector ratio to be within 2% of unity while the pseudo scalar/scalar results differ by about 10%.

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“All the world’s a stage,

And all the men and women merely players;

They have their exits and their entrances;

And one man in his time plays many parts;

His acts being seven ages...“

- William Shakespeare

For me one of these ages is the age of education in Bielefeld.

And as this wonderful act of my life draws to a close, I pause to thank all those who have played their role on my stage and who will hopefully continue to do so in the future.

Even if I cannot name all of you, for that would fill many thick volumes, I hope each of you knows in his heart that they belong on this list.

Foremost I wish to thank Olaf Kaczmarek, Frithjof Karsch and Edwin Laermann for their supervision and constant support throughout this work.

I deeply acknowledge the valuable lessons they tried to teach me, and I am sure I will benefit from them for all times.

Thank you for giving me the privilege to learn from the best.

Indispensable for the success of this work in an entirely different arena I wish to thank Gudrun Eickmeyer and Susi von Reder.

Thank you for always being there and having an open ear for our problems.

You have made things possible that are nearly impossible anywhere else.

In my time here in Bielefeld many have passed through our work group and it is them that I wish to thank for their help and discussions,

just to name a few these are Wolfgang Unger, Sabine B¨onig, Jack Liddle, Nils H¨uske, Rossella Falcone, Jens Langelage, Yannis Burnier, Jan M¨oller, Marina Seikel, Wolfgang S¨oldner and many more.

At this point I want to especially thank my comrade-in-arms Heng Tong Ding.

I would never want to miss the discussions and arguments we had about MEM and spectral functions.

And I very cherish the memories of the times in Paris and Brookhaven.

Looking back at my time in Bielefeld,

I realize I never could have done it without my friends:

Stefan Fr¨ohlich, Simon Hennig, Andre Lampe, Marc L¨ollmann, Maik Stuke, Borello, Pulki and recently Fabian and Idir.

My heart is heavy that I will not be there for our lunch date and I wish you guys all the best.

Kisses go out to Susanne,

you have helped me in more ways than you will ever know.

And I love you for every one.

Finally I thank my family and my parents

for providing me the environment and possibilities to follow my dreams.

14th September, 2011

Anthony S. Francis

In loving memory of Jason S. Francis.