yui guhguyg

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4.10. Lecture X 201

The Sudakov form factor for the photon-quark-antiquark ver-tex is defined as

Z1

2 p+, ,g ,m,md u(p) V

p+, p, ,g ,m,md

Z1

2

p, ,g ,m,md

v(p),

where md is any IR regulator. We find that

V V(p+, p, ,g ,m,md).

Now the form factor becomes,

u(p) v(p) (p+, p, ,g ,m,md),where

(p+, p, ,g ,m,mg ) = V(p+, p

, ,g ,m,md)Z12

p, ,g ,m,md

Z12 p+, ,g ,m,md .

Hard

Collinear

Soft

Collinear

The computation of involves loop integrals with loop momen-tum, say, k. We can split the domain of integration as follows:


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202 4. Perturbative Quantum Chromodynamics

1. the hard region, namely k > Q, for all ;

2. the soft region, namely k Q, 1 for all ;

3. the collinear (to p) region, namely k+ Q, k 2Q,k Q;

4. the collinear (to p) region, namely k Q, k+ 2Q,k Q.

Using the standard power counting techniques in the axial gaugeone can separate hard, soft and collinear regions of the loop inte-grals to all orders in perturbation theory. How small k is, deter-mines the degree of collinearity and how small k is, determinessoftness. Noting that

d|k||k|

dk

k( ) log2(Q),

we find that the typical form of the perturbation series looks like

=

n=0

g2n

a(n)2n log

2n

Q

+ a

(n)2n1 log

2n1

Q

+

,

where the first non-trivial term in the series is called the leadinglog contribution. The sum of the leading logs leads to

exp

Cg2 log2

Q

,

where the constant C is calculable perturbatively and is greaterthan zero. Hence, in the large Q limit, the sum of leading ordercontributions vanishes exponentially. This does not necessarilymean that = 0 in the large Q limit. To show that it hap-pens, one has to include all the non-leading terms in the sum aswell. Moreover, the leading order behaviour comes purely fromthe renormalisation constants Z and Z which are unphysical.


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4.11. Lecture XI 203

4.11 Lecture XI

In this lecture, we shall argue in favour of some exact results which

are valid beyond p erturbation theory. To this end, we shall workwith the functions log V, log Z and logZ. Let us define,p+, p, ,g ,m,mg

=

log V

logp+

p+, p, ,g ,m,mg

=

log V

logp.

One can prove the following:Theorem: The contributions to and come only from hardloops, hence they do not contain any infrared or collinear diver-gences.

(q)

qis(p)

qis(p)

g(k)

Consider the correction to the photon-quark-antiquark vertexdue to a gluon exchange. The amplitude for this process can bewritten as the integral (over k) of

u(p)iN

(/p /k + m) (/p /k + m)((p k)2 m2 + i) ((p + k)2 m2 + i)

v(p).

Let us study this amplitude in the limit k 0. Using the equa-tion of motion, we find that the numerator of the expression insidethe square bracket is

(2p) (2p) = 4p+p .


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The same result may be obtained by dropping m and noting that/p p+, /p p+ and using {, } = 0, {+, } = 4.Likewise, the denominator simplifies to

14(p k)(p k) =

1

4p+pk+k.

From the above, we find that the integrand goes as N+/k+kand since N+ 1/k2, the vertex function behaves as

d4k1

k2k+k.

This is logarithmically divergent. This proves that, to order g2,log V /logp+ and log V /logp are free of soft and collinear

divergences.Now recall that p+, p = Z12 (p+)Z12 (p)V(p+, p). Dif-

ferentiating with respect to p+ and p respectively,

log

logp+= A(p+) + (p+, p),

log

logp= B(p) + (p+, p),

where

A(p+) =log

Z

logp+, B(p) =

log

Z

logp .

Lorentz invariance demands that p+, p

is a function of the

product p+p, i.e., = p+p

. Therefore, loglogp+ =

log logp

and thus:

A(p+) B(p) = (p+, p) (p+, p)= (p+, p, , g)

= p+, p,, g()

=

p+/,p/,, g()

,


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where, in the last two steps, we have used the RG equations andscaling arguments respectively. Now, using the fact that the cou-pling constant evolves by the RG equation

dg()d

= (g())

with the boundary condition g() = g, we arrive at

logp++

logp (g)

g= 0.

This, in turn, implies that

A

logp+ (g) A

g=

B

logp (g) B

g.

Notice that the LHS of the above equation is function ofp+ alone,

while the RHS depends only on p. This is possible if they areindependent of both p+ and p. In other words,

A

logp+ (g) A

g= C(, g ,m,mg),

B

logp (g) B

g= C(, g ,m,mg),

where C is a constant independent of p+ and p.The solution to the first order differential equations above can

be obtained as follows. First, we use scaling arguments to write

Ap+

, , g(), m , mg = C(g(), m , mg ) .Next, integrating with respect to the scale parameter ,

dA

p+

, , g(), m , mg

=

p+

1

d

C(, g(), m , mg) ,

we find:

Ap+, , g(), m , mg

= A

,, g(p+), m , mg

+

p+

dx

xC(, g(x), m , mg) .


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It follows that

Z1

2 (p+) = Z1

2 () exp

p+

dy

y y

dx

xC(, g(x), m , mg )

+

p+

dx

xA (,, g(x), m , mg )

.

Moreover, in case g is a constant,

Z1

2 (p+) = Z1

2 () exp

1

2C(,g,m,mg) log

2

p+

+A(,,g,m,mg )log

p+

, (4.105)

which vanishes as p+

(C < 0). We can find a similar result

for Z(p). It can be shown that the scale dependent couplingconstant does not affect our conclusion as they can only generatenon-leading terms such as log log(p+).

Finally, we can find V using

log V

log Q=

+

p+=p=Q.

Therefore,

log V

logp++

log V

logp=

+

(Q,,g)

= + (Q,, g())=

+

(Q/, g())

=

+

(,, g(Q)) ,

where, again, we have used RG and scaling arguments as before.Integrating, we find

V(Q) = V() exp

Q

dx

x

+

(,, g(x))

.


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If the coupling constant does not depend on the scale,

V(Q) exp

+

(,, g) log

Q

. (4.106)

Once again, the scale dependent coupling produces only termswhich have less leading logarithms.

Therefore the Eqs.(4.106), (4.105) and the analogue of the lat-ter for Z are exact leading order results. Corrections to theseinvolve less dominant terms such as log logp+, etc.


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Bibliography

[1] T.P. Cheng and L.F. Li, Gauge theory of elementary particle

physics, Clarendon Press, Oxford (1984).[2] T. Muta, Foundations of quantum chromodynamics: an in-

troduction to perturbative methods in gauge theories, WorldSci. Lect. Notes Phys. 5, 1 (1987).

[3] M.E. Peskin and D.V. Schroeder, An introduction to quantumfield theory, Addison-Wesley (1995).

[4] P. Ramond, Field theory: a modern primer, Front. Phys. 74,1, Addison-Wesley (1989).

[5] G. Sterman, An introduction to quantum field theory, Cam-

bridge University Press (1993).

[6] S. Weinberg, The quantum theory of fields, Vol. 1: Foun-dations, Vol. 2: Modern applications, Cambridge UniversityPress (1995).

The above are some text books on quantum field theory thattreats perturbative QCD. For more details on the subject ofthe lectures X and XI, see:

[7] J.C. Collins, Algorithm to compute corrections to the Sudakovform-factor, Phys. Rev. D22 (1980) 1478.

[8] A. Sen, Asymptotic behavior of the Sudakov form-factor inQCD, Phys. Rev. D24 (1981) 3281.


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210 5. Quark-Gluon Plasma

5.1.1 Confinement & why quark-gluon plasma is im-

portant

Quantum chromodynamics (QCD) is now the widely accepted the-ory of strong interactions which describes the theory of interac-tions of quarks and gluons, the basic building blocks of protons andneutrons. Unlike the electroweak theory, the coupling of QCD, scan have large values in many interesting physical applications,although QCD has been tested extensively and successfully in ex-periments for small s. The property of asymptotic freedom ofQCD states that for sufficiently large Q2 (or small distances) therunning coupling s(Q

2) is small. I refer you to Ashoke Sens lec-tures in this volume for further details on perturbative QCD andits successes1.

However, QCD also has to explain how the entire spectrum ofobserved hadrons arises and what their masses, decay constantsetc. are. Even for certain electroweak decays, which naively shouldbe computable using perturbative techniques, one needs hadronicmatrix elements where the corresponding s is at low Q

2 andtherefore large. The phenomenon of confinement of quarks orgluons is a qualitative prediction expected from QCD: Indeed, itis otherwise a mystery as to why no free quarks or gluons have beenobserved in any experiments in spite of vigourous searches. Usualperturbation theory fails for all these situations. New tools ortechniques are needed if the beloved Standard Model is to accountfor the observed physics in its entirety.

Gauge theories formulated on discrete space-time lattice havethe potential to handle all the questions above. Over the past twodecades or so, tremendous progress has been made in establishingthe answers to the questions above. QCD on a discrete space-timelattice postdicts confinement, hadron masses and various weakdecay matrix elements. A true test of a theory and/or a newtechnique is in confronting its predictions with the experimentssuccessfully. Lattice techniques have yielded a non-perturbative

1We shall attempt to provide references in the text. A brief list of usefulreferences and guide to further reading is given at the end.


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5.1. Introduction 211

predictions of QCD : A phase transition to a new state of stronglyinteracting matter, called Quark-Gluon Plasma (QGP). It turnsout that such a state can be, and may have been, produced in

heavy ion collisions in CERN and very recently in RelativisticHeavy Ion collider (RHIC) at the Brookhaven National Lobora-tory (BNL), New York. If so, this would constitute the observationa deconfined state of quarks and gluons, and thus the first exper-imental proof of confinement at lower temperatures and densitiesby implication. The mystical nature of quarks and gluons, whichare felt but not seen in most experiments makes the confine-ment hypothesis quite a bit unconvincing/unreal. Lattice QCDnot only demonstrates it to be an in-built feature of the theory, italso predicts that confinement can be overcome if extreme condi-tions are created in the laboratory or the early universe.

These lectures are devoted to explaining

how QCD provides the basic, fundamental information ofQGP from first principle, and

how this new state QGP could be produced and detected inthe laboratory.

As you will see the first part is theoretically well developed andone can ask innovative questions to obtain detailed information on

QGP. As I will outline, however, there are quite a few conceptualquestions as well. Some are technical but some are still stum-bling blocks. No satisfactory treatment of QCD at finite density(or equivalently at finite baryonic chemical potential) has so faremerged, depriving us of a first principles approach to investigateexciting speculations as colour superconductivity or the strangequark stars. The second part is rewarding for the connection itmakes with the real world but is theoretically less developed. Infact, there are many challenges here to even develop the rightframework or at least, to establish the existence of QGP experi-mentally in the face of somewhat weaker theoretical setting.


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5.1.2 A simple model for quark-gluon plasma: the

MIT bag model

As already outlined above, one needs new non-perturbative tech-niques to extract information on the hadron spectrum and, con-sequently, to deal with hadronic world under extreme conditions.Before we turn to lattice QCD, let us obtain some physical insightby considering simple model of hadrons which exploit the basicfeatures of QCD under these conditions. MIT bag model is onesuch model which incorporates both confinement of quarks andgluons and the property of asymptotic freedom of QCD. It doesso by treating the ordinary vacuum as a medium in which hadronexists as a bag (or a bubble). As shown in Fig. 5.1, a colour neu-tral baryon in this model has (almost) free coloured quarks insideit which cannot escape out due to the bag shown as a boundary.

Thus the bag incorporates the physics of confinement while thefree quarks inside account for asymptotic freedom.

Figure 5.1: A baryon in the bag model.

Assuming the vacuum to have an energy density B, the energyof such a hadron is given by,

EH =4

3R3B +

C

R, (5.1)

where C/R is the contribution coming from the kinetic energyestimated by using the uncertainty principle and the first term is


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the energy needed to create the bag. A stable hadron H will resultwhen

EH

R = 4R2

B C

R2 = 0,

yielding RH =

C

4B

14

.

This, in turn, implies that the mass of the hadron is given by

MH =4

3R3HB +

4BR4HRH

=16

3BR3H. (5.2)

Note that1

4R2EHR

is the radial pressure on the bag surface,

being the force per unit area of its surface. So for a static bag the

pressure P = B + C/4R4H = 0. Thus, the vacuum pressureB balances the kinetic energy pressure C/4R4H to create a sta-ble hadron. A more detailed calculation, taking the appropriatefull kinetic energy term, enables to fix the bag constant B fromthe experimental data on hadron masses.

Imagine now a box of volume V having some hadrons. Increaseits density of hadrons or increase the temperature T of the box.As the bags (hadrons) begin to overlap, one can envisage a stateof matter in the box where the mobility of the quarks will extendbeyond the size of an individual bag. This is the state we will callquark-gluon plasma in which colour can flow over distance scalesmuch larger than typical hadron size.

5.1.3 -T phase diagram

The above qualitative physical picture can be made more quanti-tative. Recall that the partition function of a system,

Z = Tr

exp

H N

T

gives the entire thermodynamics. Here, H is the hamiltonian ofthe system and N is its some conserved number, for us here the


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baryon number. is the corresponding chemical potential. Allphysical observables can be obtained as appropriate derivatives ofthe partition function. E.g., the energy density of the system is

given by

EV

=1

V ZTr

Hexp

H N

T

=T2

V

ln Z

T+ n,

where,

n NV

=1

V ZTr

Nexp

H N

T

=

T

V

ln Z

is the (baryon) number density.The phase of our box above at low T (or low density) can

be approximated by a gas of noninteracting hadrons (ideal gas ofhadrons), whereas the high T phase (or the high density phase)can be assumed to be an ideal gas of quarks and gluons, but withthe vacuum pressure. For noninteracting fermions or bosons, onecan easily evaluate the (log of the) partition function to be

ln Z =gV

62T

0

dp p4

p2 + m2

1

e

ET

+

+1

e

E+T

+

, (5.3)

where E =

p2 + m2, = 1 for fermions and bosons respec-tively and g denotes the degeneracy due to spin polarizations. Form = 0, the integrals can be evaluated easily:

T(ln Z)f =gfV

12

7

302T4 + 2T2 +

4

24

,

T(ln Z)B =gBV

902T4 (5.4)

Problem 1: Derive Eq.(5.4) from Eq.(5.3).


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Using the relation P = T ln Z/V, one can write down thepressure in the two phases. Since we expect these two phases tobe the relevant ones at the two ends ofT (or ), one may invoke the

Gibbs principle to select the favoured phase of maximum pressure.Let us consider some special cases to illustrate this and obtain the-T phase diagram for our simple model of hadrons.

Case I T = 0 , = 0.The hadronic phase has the usual pions, kaons, protons, -

mesons, etc. with masses given by m = 140 MeV, mK = 495and mp = 940 MeV, m = 770 MeV. If the temperatures of in-terest are sufficiently low, as we will shortly see is the case, thenthe hadronic phase can be approximated by three pions since theheavier particles will not contribute due to a typical suppressionfactor exp(E/T) exp(M/T). We can simplify further byassuming the pions to be massless bosons so that one can usethe analytic solutions (5.4); this assumption can be trivially re-moved by doing the integrals (5.3) numerically. The pressure ofthe hadronic phase is then given by

PH =2

30T4.

For the quark-gluon phase, we only need to get the correct de-grees of freedom, since gluons are massless and only light quarkswill be relevant. Note that quarks with masses mq < QCD are

light quarks whereas mq > QCD are heavy, QCD is knownfrom experiments to be 200250 MeV. One mostly deals withlight quarks in QGP as the Boltzmann suppression factor ofexp(mq/T) is substantial for the charm, bottom or top quark.Indeed, the strange quark with a mass of100-150 MeV is alreadya borderline case, as shown below. Assuming therefore masslessquarks of three colours and two flavours, i.e., Nc = 3, Nf = 2,one gets for quarks, gf = 2 3 2 = 12, and for gluons,gB = 2 (N2c 1) = 16, where the extra factor of 2 in eachcase is due to the possible spin polarizations. Using these valueswith (5.4), one obtains the pressure of the quark-gluon phase as :


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Pq+g =37

902T4 B. (5.5)

Note the presence of the bag pressure term above which reducesthe pressure of the quarks and gluons and keeps them from flyingapart.

The thermodynamically favoured state of minimum thermo-dynamic potential or maximum pressure for low temperatures isthe hadronic phase since PH > 0, Pq+g < 0 for it. On the otherhand, when T is large, Pq+g > PH and the quark-gluon phase ispreferred by the Gibbs criterion. The phase transition will oc-cur at a temperature Tc found by equating PH(Tc) = Pq+g(Tc).

One finds Tc =

45B/172 14 = 0.72 B

1

4 . Since the nucleonmass Mp =

163 R

3B = 1 GeV, and its radius Rp = 1 f m

(= 200MeV)1

, one can estimate B1/4

31/4

200 MeV 150MeV. This, in turn, implies a Tc 100 140 MeV where thehigher value is obtained if one uses a higher average value Mpto take care of the presence of various higher resonances. Notethat mK/Tc, m/Tc, mN/Tc >> 1 and therefore can be a posteri-ori ignored in obtaining the pressure of hadronic phase and thusTc( = 0). It is easy to compute the energy densities EH(Tc) andEq+g(Tc) of the two phases at the transition point Tc. Since thefermionic and bosonic energy densities are give by

Ef(T, = 0) = gf12

7

102T4, (5.6)

EB(T, = 0) = gB30

2T4. (5.7)

The energy densities of the two phase at Tc are found to be

Ep+q(Tc) = 710

2T2c +8

152T4c + B,

EH(Tc) = 2T4c10

T4c . (5.8)

The pressure is continuous at Tc by construction, but as isevident above, the energy density jumps discontinuously. The


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latent heat at this first order phase transition is

L =

Ep+q(Tc)

EH(Tc) =

682

45

T4c (5.9)

Tc

SB

T

1

0

Figure 5.2: The energy density vs. T in bag model for = 0.

In summary, the features of the quark-hadron transition in thiscase are

1. a first order phase transition;

2. a huge latent heat:

L

T4c 15 or L = 4B, i.e. a strongdiscontinuity in the energy density;

3. as shown in Fig.5.2, E ESB from above in the T limit, where ESB is the ideal gas or Stefan-Boltzmann limit.

Case II T = 0 , = 0.Let us now consider the limiting case of T = 0 with nonzero

. As increases, the net baryon number increases. A sensibleapproximation to the hadronic phase is to assume dominance of


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protons and neutrons. Using again the ideal gas expressions inthis T 0 limit,

PH = M4

62

M 2

M22

M2 1 5

2

+3

2ln

M +

2

M2 1

,

nB =2

32

2 M2

32

and E = p + sT + n = nB pB , (5.10)where, nB is the baryon density and s is the entropy density.

Problem 2: Derive Eq.(5.10) from Eq.(5.3).

Using the expressions (5.4) for the ideal quark gas, one has

Pq =4q

22 B, nB = nq

3=

23q32

, and Eq =34q22

+ B. (5.11)

Since a baryon is made of three quarks, the baryon density isone third of the quark density. Accordingly, the baryon chemicalpotential is related to the quark chemical potential by = 3q.The same method as in case I above can be used to determinethe thermodynamically favoured phase at a given (hadronicfor low and quark for high ) and the chemical potential at

the transition. Thus at the phase transition Pq = PH whereasthe energy density may in general be discontinuous at c. Sincelatent heat L = Eq(c) EH(c) 0, one obtains the rela-tion

34q /2

2

+ B nB + PH 0. Using the continuity ofpressure PH(c) = Pq(c) =

4qc22 B, this inequality becomes

24qc/2

cnB 0. Finally, substituting for nB from (5.10)and using qc = c/3, it can be converted as an inequality for c:

c

c3

3 (2c M2)

0,


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which implies

M c 3M2

2. (5.12)

Substituting these limits in (5.10) and (5.11), one finds that forthe lower limit, c = M, PH = 0 = Pq, leading to B =

122

M3

4.

For the upper limit, c =3

2

2M,

Pq =1

22M4

64 B, and PB = M

4

62

33

64+

3

4ln 2

. (5.13)

From the equality of these two, one has B = M4

82

34 ln 2

. Thus

a solution for c exists provided,

1

22 M

3 4

B

3

4 ln 2 M

4

82, (5.14)

which is a very narrow range indeed. However, it does include thestandard B1/4 value we usually have: 0.158M B 14 0.164M.

In summary, the features of the quark-hadron transition in thiscase are the following.

1. A first order phase transition, depending on value ofB. Forthe highest value of B, the latent heat L = 0, leading to asecond order transition;

2. For the lowest value of B, one has the strongest first order

transition with L = 4B again;3. As , E ESB from above.Note that nB = 0.15fm

3 is the cold nuclear matter density.This corresponds to 0.78 GeV ( or q 0.26 GeV). Thebag model, on the other hand, suggests the highest possible c 1.06M = 1 GeV, which is a bit too low.

Case III T = 0 , = 0.This general case is not analytically tractable even for the

simplifying approximations we made above. However, the same


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method works in this case as well. The only key difference isthat the equation Pq+g(Tc, qc) = pH(Tc, c) = Pmeson(Tc) +Pnucleon(Tc, c) for a given pair (Tc, c) has to be solved numeri-

cally. This the leads to the popular T- phase diagram of stronglyinteracting matter, shown in Fig.5.3.

GeV

Hadrons

quarkgluonplasma

~1

T(MeV)

~150

Figure 5.3: The T phase diagram for bag model.

Simple though it may be for illustrative purposes, the bagmodel phase diagram is clearly not very satisfactory, althoughunfortunately it is heavily used in phenomenological analyses ofheavy ion data and the cosmology of the early universe. Its majordrawback is that the phase transition line appears by construction.Since one assumes the hadronic phase and the ideal quark-gluonphase to be that of the same matter, one is able to apply the Gibbs

criterion. There is no strong theoretical basis for this assumptionother than the naive picture we presented earlier. On a somewhattechnical level, it is too simplistic to assume an ideal gas pictureor a lack of interaction (apart from B). This can, of course, beimproved by incorporating more hadrons and resonances as well.Similarly, the ideal quark-gluon gas can be improved upon byincluding weak perturbative QCD interactions.

One can thus take recourse to better models or insist that onemust simply use quantum chromodynamics (QCD), the knowntheory of strong interactions to obtain this phase diagram. Dueto a lack of time, we will not dwell on any other model but follow


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5.2. QGP from QCD 221

the latter option in the next lecture.

5.2 QGP from QCD

5.2.1 Path integral formalism

Let us begin by recalling that the partition function

ZQCD = Tr exp

HQCD BNB

T

contains all the information we are looking for. All that one hasto do is to choose the appropriate complete set of states and com-pute the trace. For QCD, we can begin from the gauge invariantLagrangian LQCD , construct HQCD and obtain Z. Unfortunately,this turns out to be too complicated and intractable. First of all,one has to worry about gauge dependence : HQCD has to be de-fined in a gauge specific manner. Then, the trace must be carriedover gauge independent physical states, by appropriate projectors.A neater solution is to reformulate the problem as a functional in-tegral of the gauge invariant euclidian Lagrangian. One can thenexploit its similarity with T = 0 field theory and employ essen-tially the same techniques to obtain thermal expectation value of a given observable .

For simplicity, let us demonstrate the idea of reformulationwith a quantum mechanical example: Set = 0 and consider the

hamiltonian

H =

p2

2m + V(x) in stead of

HQCD . Employing theeigenstates of the position operator, x|x = x|x to evaluate thetrace, the partition function can be written as:

Z = Tr eH =

x| eH |x. (5.15)

Let = n with small and n large : lim 0. The operatorin (5.15) can then be written as a product of n terms involving

eH. Inserting the completeness relation below,dx|xx| = 1, x|x = (x x) (5.16)


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(n 1) times in between the terms of this product for xi, i =1, n 1 and defining x0 = xn = x in (5.15), Z becomes

Z = n

i=1dxi xi1| e

H

|xi

=

ni=1

dxi xi1| eV(x)/2 ep2/2m eV(x)/2|xi.

(A symmetric (Weyl) ordering of operators has been used above.)Let |p denote the momentum eigenket, p|p = p|p. Using thecorresponding completeness and orthonormality relations,

p|p = (p p),

dp |pp| = 1,

2n times, we obtain

Z = n

i=1

dxi dpi dpi xi1| pi e(V(xi1)+V(xi))/2

pi|xi pi| ep2/2m |pi

=

ni=1

dxi dpi exp p2i

2m+

V(xi1) + V(xi)2

ipi xi xi12

. (5.17)

Finally, taking the continuum limit 0, one sees that xi x(t),pi p(t) and 1 (xi xi1) x(t). Thus Z is given by

Z =

[dp]

x(0)=x()[dx] e

0

dt[ip(t)x(t)H[x(t),p(t)] (5.18)

Remarks :

1. The integrals over x and p,

dx and

dp, are to be thoughtof as functional integrals. At each t, all possible values ofthese functions are allowed.

2. The integration over p is unrestricted but that over x hasperiodic boundary condition x(0) = x(). This arises dueto the trace in Eq.(5.15).


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3. There are no operators in Eq.(5.18) anymore. All variablesare classical.

One can choose to do the pi integration before sending

0.For our hamiltonian this amounts to simply completing the squareand then doing the gaussian integrals in pi:

dpi exp

p

2i

2m+ ipi(xi xi1)

=

m

2exp

m

2(xi xi1)2

.

Therefore, in the limit 0, the partition function is

Z =

x(0)=x()

paths eS, (5.19)where, S =

0

d

mx2

2+ V(x)

. (5.20)

Remarks :

1. Z is now like an Euclidean path integral, being the sum overall possible classical paths of the action S with a Boltzmannweight eS.

2. T 0, Z reduces to usual zero temperaturepath integral.

As is usually done in field theory, one can define a partitionfunction with a source j() :

Z(, j) =

DxeS()+0

j()x()d

Differentiate twice with respect to j and set j = 0 to obtain thepropagator:

1

Z()

2Z(, j)

j(1)j(2)

j=0

=1

Z()

Dx x(1)x(2) eS (5.21)


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Recall that A = Tr AeH/Tr eH = 1Z() Tr A exp(H)(since = 0 for us here). We will now show how this is related to

the propagator in (5.21). Since x(t) = eiHt x eiHt , one can write

x(i) = eH x eH.Defining

T(x(i1)x(i2)) =

x(i1) x(i2), for 1 > 2,x(i2) x(i1), for 2 > 1.

one can show that

i) T(x(i1)x(i2)) is the RHS of Eq.(5.21);ii) T(x(i)x(i)) = T|x(0)x(i);

iii) ( ) = (), where () = T(x(i)x(0).Problem 3: Show the above.

Problem 4: Choose V(x) = 12 2x2 corresponding

to a harmonic oscillator and set m = 1 to make aconnection to field theory.

Z(, j) =

pbc

Dx exp

0d

1

2x2 +

1

22x2 j()x()

=

pbc

Dx exp

0d

1

2x(0)

d

2

d2+ 2

x()

+

0

j()x()d= Z() exp

1

2

ddj()K(, )j()

,

where K(, ) is the inverse of d2d2 + 2

, i.e.,

d2

d2+ 2

K(, ) = ( ) (5.22)

Using the definition of , one obtains K(, ) = F( ). Thesubscript F stands for harmonic oscillator (or equivalently Free


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Field Theory). Solving (5.22), subject to the boundary condition( ) = () leads to

F() = 12 [1 + n()]e + 12 n()e

, (5.23)

with 1/n() = exp(||) 1 corresponding to the Bose-Einsteindistribution.

5.2.2 Spectral function

The spectral function is a useful tool for extracting physical in-formation in a variety of situations including those involving de-viations from thermal equilibrium. Let us see how it arises in oursimple example.

Let us define two new two-point functions

D>(t, t) = x(t) x(t) D(t, t)

Here time t is complex in general. Insert the complete set ofeigenvectors of H in between x(t) and x(t)

D>(t, t) =1

Z()

n

n|eHx(t)x(t)|n, (5.24)

to obtain

D>(t, t) =1

Z()

n,m

eEnn|eiHt x(0)eiHt |m

m|eiHt x(0)eiHt |n (5.25)=

1

Z()

n,m

eEneiEn(tt)|n|x(0)|m|2eiEm(tt)

For Im(t t) 0, D>(t, t) is well behaved and definedwhereas for Im(t t) 0, the same is true of D


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Further,

D>(t, t) =1

ZTreHx(t)x(t)

= 1Z

Tr x(t + i)eHx(t)

=1

ZTr eHx(t)x(t + i)

= D(k0) D(k0) = (1 + f(k0))(k0)

and D 0, (k0) is the sign function

iii)

dk0 (k0)(k0) = 1.

When x(t) is the position operator for harmonic oscillator, onecan use

x(t) =12

(aeit + aeit)

to show F = 2(k20 2) which obeys all the three properties

i) , ii), iii) above. Moreover, F is temperature dependent.


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5.2.3 Neutral scalar fields

We can use the experience gained in the previous subsection tomake a leap to neutral scalar field theory, defined by the lagrangiandensity:

L = 12

1

2m22 U(), (5.28)

where the potential U() = g3 + 4. Since = = L

, the

hamiltonian density H = L can be written as

H = 2

2+

1

22 + 1

2m22 + U(). (5.29)

Note now the similarity with the harmonic oscillator exampleconsidered above:

p, V(x)

the rest. The partition

function for the neutral scalar field theory is thus

ZN S =

(0,x)=(,x)

D exp

0d

d3x L(, )

, (5.30)

where the euclidean lagrangian density is

L(, ) =

2+ 2 + m22 + U()

. (5.31)

Here we have let t i giving rise to (t)2

()2

.For U() = 0, one has free scalar field theory corresponding tofree bosons. Since direction is compact and since (0, x) =(, x), p0 is discrete, with allowed values given by p0 = 2nT.By integrating , by parts and using periodic b.c. of thefields , one gets

Z =

D exp

0d

d3x

1

2D

,

where D =2

2+ 2 + m2


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Therefore the gaussian functional integration can be done for-mally:

Z = const det1

2

D,or, ln Z = 1

2ln detD + C

= 12

ln TrD + C,

where C is a constant.

Problem 5: This problem is an explicit calculationof ln detD.

Show that

ln Z = V d3p(2)3

12 ln(1 e ) ,where 2 = p2 + m2.

Using Fourier transform of (, x) evaluate E(T = 0)and show it to be infinite. The vacuum energy can,however, be explicitly subtracted using the freedom tochoose the scale for energy.

Perturbative expansion: Unlike above, the potential U() isnot zero in general. For U() = 0, i.e. for a general L one needsa method of computation of Z or some approximation scheme.If the coupling is sufficiently small, one can use a perturbativeexpansion in it. Denoting the free action density above as S0,which is quadratic in field , and the interaction part as SI, thepartition function can be expanded as below.

Z =

D eS0+SI =

D eS0

l=0

1

l!SlI

=

DeS0

1 +1

DeS0

l=1

DeS0SlI

.


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Or, equivalently:ln Z = ln Z0 + ln ZI.

The first term ln Z0 has been evaluated above. The second term

can be expanded as a power series. For simplicity, let us consider,SI = U() = 4. The expansion then is governed by powers of. The lowest order term is

ln Z1 = D eS0

DeS0

0d

d3x 4(x, )

. (5.32)

In order to evaluate it, let us write down the Fourier expansion ofthe field in momentum space.

(x, ) =

1

V T

12

n pei(p.x+n)n(p) , (5.33)

where we have assumed a finite spatial volume V, thus discretiz-ing the three momenta as well in addition to the energy (which,recall, has to be discrete due to T = 0). The action S0 is thentransformed to

S0 = 12T2

n,p

(n2 + p2 + m2)n(p)n(p) , (5.34)

and correspondingly D eS0 is:

n,p

dn(p) exp

1

2T2(n

2 + p2 + m2)n(p)n(p)

. (5.35)

Note that the periodic boundary condition, (, x) = (0, x) wasused above in writing S0. The Fourier transform of the four fieldsin the numerator yields sums over ni, i =1,2,3 and of exp(

inj t)

and similar sums over pi multiplied by4

j=1nj (pj). The integra-tions over x and lead to Kronecker -constraints which eliminatesome sums over energy and momenta. One finally obtains (in thelimit V ),

ln Z1 = 3V

T

n

d3p

(2)3D0(, )

2, (5.36)


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where D0(, ) = (n2 + p2 + m2)1 is the free boson propaga-tor. Diagrammatically, the above equation can be represented asshown in Fig.(5.4).

ln Z = 31

Figure 5.4: Diagrammatic representation of Eq.(5.36).

The closed loop in the figure denotes a factor of energy-summation and momentum integrations, the vertex stands for afactor of

, and the lines denote the free boson propagators.

As can be guessed from Fig.5.4, one can devise Feynman rulessimilar to T = 0 and use them for higher order calculations. Let usdefine general propagator D(x1, 1; x2, 2) = (x1, 1) (x2, 2),Its Fourier transform is D(n, p) = 2n(p)n(p). Since thefull action S = S0 +SI contains

2n(p)n(p) with D0 = (2n +p2 + m2) as coefficient, one sees that

D(n, p) = 2 D10

ln Z = 2D20

D0 ln Z. (5.37)

Define self energy (n, p) by

D(n, p) = 12n +p

2 + m2 + (n, p)

=1

D10 + =

D01 + D0 .

Then (5.37) and (5.38) together imply that

(1 + D0)1 = 2D0 D0 ln Z = 1 + 2D0

D0 ln ZI (5.38)

Using (5.38) the self energy can be computed in powers of : =

l=1 l. The first and second order terms 1 and 2


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satisfy

1

D01 = 1 + 2

D0

D0ln Z1

D02 + D01D01 = 2D0 D0 ln Z2 .

Recall that diagrammatically, ln Z1 is as shown in Fig. 5.4. Takingits derivative with respect to D0 will break one loop, leading tothe diagram in Fig. 5.5, which stands for

= 2ln Z 1 0D

Figure 5.5: Diagrammatic representation of 1.

1 = 12T

n

d3p

(2)31

2n + 2

. (5.39)

Problem 6: Show that the evaluation of Eq.(5.39)leads to

1 = 12 d4p

(2)4

1

p24 +p2 + m2 +12 d3p

(2)3

1

1

e 1 ,where the first term is temperature independent andquadratically divergent.

The above is the canonical divergence we are familiar in scalarfield theories. The usual T = 0 couterterm can be employed heretoo to eliminate it: add 12 m

22. The remaining finite part ren1 T2 on dimensional grounds alone. Its being nonzero for T = 0is an important feature of T = 0 field theory, used in variousapplications to cosmology etc.


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Seeing the connection of 1 with ln Z1, it is no surprise thatit contributes to pressure

P = TV

ln Z = T4290

48

+ .The pressure decreases due to the lowest order correction at O().Let us make some general remarks, based on this example. It istrue in general that (i) no new ultra-violet divergences arise atT = 0 or( = 0); the only divergences to be encountered are theT = 0 ones which can be handled in the usual way and (ii)theinteractions change pressure, energy , etc. in a manner similar toabove and related to 1, 2 etc.

5.2.4 Infra-red problems

As noted above, there are no additional ultra-violate divergencesat finite temperature or density. However, the finite temepraturefield theory has severe infra-red problems in general. In order toillustrate these in our simple example considered in this chapter,let us consider the massless case: m = 0. As we will argue be-low, the resultant infra-red divergences, when treated adequately,change the naive expectation for the next correction to the pres-sure P2. Instead of being of O(

2) it turns out to be of O(3/2),making the 2 term the next one in the perturbative expansion.

Qualitatively, one can understand this by inspecting ln Z2 dia-grammatically. Fig. 5.6 shows the two diagrams which contributeto it.

2ln Z = +

Figure 5.6: Diagrammatic representation of ln Z2.


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When n = 0 in the middle loop of the first diagram, it be-comes 21 dp p2 (since the middle loop has two propagatorsand a p2dp from the 3-momentum integration) and is thus di-

vergent. This divergent contribution is absent at T = 0 since1 T2. The nonzero 1 acts like an effective thermal mass.Although we chose mbare = 0, the non-vanishing 1 at T = 0yields m1loop

T. Thus, if we could exploit the existence of

the 1-loop thermal mass in the propagators in 2, the infra-reddivergence encountered above can be tackled. At the order oneis computing, the replacement of bare propagators by the 1-loopones is, of course, permitted. This infra-red problem keeps com-ing at higher orders. In fact at O(N) the divergent terms are dp p2(N1), making it more and more severe. It turns out thatthe offending terms can be resummed to all orders which leads to.

P =2T4

90

1 15

8

2

+15

2

2 32

+ . . .

.

In order to understand the origin of the 3

2 term better, let usimagine redrawing the diagram in Fig. 5.6 and draw the loops atthe edges much smaller than that in the middle. The smaller loopscan be then seen as decorations of the central scalar propagatorloop. We know that ln Z = ln Z0 + ln Z1 + ln Z2 . . . is an infiniteseries with increasing order in , or equivalently with diagramsincreasing in number of loops. Now, we saw above that

ln Z2 21 d3

p(p20 + p

2) (1 D0)2d3p.

At the next order, one has to add one more bubble on thebig loop at the center (one more petal on the flower). Mathe-matically, this means one multiplies the above expression by anextra (1 D0). Taking all such diagrams into account, leads to asum: T

n=2

d3p

(2)3

N

[D01]NN

where the combinational factor

N actually arises as a product of (N 1)!/2 (number of ways ofarranging the petals) and 1/N! from the expansion. This series,


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which is diagarammatically expressed as shown in Fig. 5.7, can besummed to

n d3p(2)3 ln[1 + 1D0] 1D0.

21 1

314

+ +

Figure 5.7: Diagrammatic representation of diagrams from ln ZN.

Note that 1 = vac1 +

T=01 , with a quadratically (c

2) di-vergent contribution in the first term which has to be cancelled bya counter term m2. A careful evaluation including the counterterms modifies the summed result for the series as below.

Series

n

d3p

(2)3

ln[1 + T=01 D0] T=01 D0

=

n p2dp

22

ln

1 +

T2

2n +p2

T

2

2n +p2

= 3

2 T3

n

x2dx

ln

1 +1

x2 + y2n

1

x2 + y2n

.

In the above, we have changed variables: x = p/

T and yn =n/

T.

An alternative, perhaps better, way to see how the 3/2 comesabout is from the of Fig.5.5. At the Nth order, N 1 loopswill contribute with the same number of external legs. This canbe achieved by adding the diagram successively on the loop todecorate its loop by petals.


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Since in the massless (m = 0) limit,

1 = 12Tn d3p

(2)31

2n

+p2

12T

n

d3p

(2)3D0(, p),

the graph at the N th order is

12T

n

d3p

(2)3

1(1p)2n +p

2

N 12n +p

2.

Sum over all N can be done by recognising that this is justan expansion like (k2 + k20)

1 = k2k20/k2N. Hence the

expression for the full resummed propagator becomes =

12Tn d3p1

2n +p2 + 1 . Using 1 = T2

and taking n = 0with p 0, one has

T

p2dp

p2 + T2

T

tan1

pT

0

,

giving back the non-analytic behaviour in coupling.

Similar infra-red problems also occur in QED and QCD, within fact a similar solution in form of a resummation of a class ofdiagrams to all orders. Note the alternating sign in the expressionabove. Even this is repeated in gauge theories, leading to poorerconvergence of the series for P, etc. in QCD.

Finally, renormalization group equations lead us in the scalarfield theory to R = 2

2 [9ln(/M)]1 at leading order. Therenormalized coupling R 0 as M/ 0. M can be taken as aconstant c times the temperature T and the corresponding canbe used in P above. As (T /) 0,

P = Pideal

1 15

8

2

9ln(/cT)

+

15

2

2

9ln(/cT)

32

+

,

where the ideal gas pressure Pideal for this case of scalar fieldtheory is 2T4/90.


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5.3 QGP from QCD II

The treatment of the previous lecture can be generalized to QCD

to obtain ZQCD in terms of its basic fields, namely, quark andgluon fields. The generalization, however, has to face many keytechnical problems. These are related to the gauge invariancethe theory must have, the fermionic nature of the quarks and thenon-abelian nature of the gluons. We will begin by consideringthe simpler example of the U(1) gauge theory first to address theproblems related to the first issue and then gradually move on tothe quark and gluon fields.

5.3.1 Quark and gluon fields

The U(1) gauge theory of the photon field A(x) is defined by the

Lagrangian

L = 14

d4x FF

, (5.40)

where the field tensor F(x) = A(x) A(x). Let us notethat an A eA scaling changes L L/e2. We will prefer thelatter form for the non-abelian gauge theories later. As is wellknown, the theory has a gauge invariance:

A(x) = A(x) + (x), (5.41)

where (x) is an arbitrary scalar function. Thus all four A(x) are

not independent of degrees of freedom. In fact, there are only two,as expected of a massless vector field. Choosing the A0(x) = 0gauge, the lagrangian above can be written as

L =1

2

d4x ((0 A)

2 B2)

=1

2

d4x (E2 B2)

L now has a residual gauge invariance,

Ai(x, t) = Ai(x, t) + i(x), (5.42)


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5.3. QGP from QCD II 237

where the permissible gauge functions are time-independent. Us-ing the standard method, one finds the canonical momenta fromthe L above.

i =LAi = Ai = Ei. (5.43)

Using these the hamiltonian density can be obtained as

H = iEi Land H =

d3x H = 1

2

d3x (E2 + B2).

It may be commented that one need not have fixed the gaugeA0(x) = 0. This would have lead to a gauge field dependenthamiltonian density

H =1

2 (E2

+ B2

) + E A0. (5.44)Integrating by parts and using the Maxwells equation, one couldstill obtain the same hamiltonian in the pure photonic theory(without sources).

Having identified the fields Ai, i = 1, 2 and 3, and the cor-responding canonical momenta Ei, one can impose the canonicalcommutation

[ Ej (x, t), Aj (x, t)] = i ij (x y),in order to define the quantum theory and then Z = Tr exp(H)defines its partition function. Of course, the trace must be takenover only physical states (or equivalently gauge invariant states)

Problem 7: Show that [H, E] = 0, i.e. H and E can be simultaneously diagonalized.

Exploiting the property above, the trace over physical statescan be defined as follows. Gauss law states that the divergence ofthe electric field is given by the charge density. Thus any eigen-state | of the hamiltonian satisfies

E| = (x)|, (5.45)


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which can be used to select the gauge invariant states. So a pro-jection operator P into gauge invariant states is given by a -functional for the Gauss law :

P =D(x) exp id3x(.E (x))(x) . (5.46)

Since P| = |phys, the constraint of summing over only phys-ical states in Z can now be easily handled: Z = Tr P exp(H),with trace now on all possible states. Note we have used the com-mutation relation above as well as the relation P2 = P to writeZ. As in the harmonic oscillator example above, the partitionfunction can now be seen to be

Z =

D A(x) A|eHP| A. (5.47)

Proceeding as in the simple quantum mechanical example, can be subdivided to obtain a product of factors exp(H). Sincethe projection operator can be raised to Nth power trivially, itcan be associated with each such factor. The |EE| and |AA|completeness relations can be inserted each time step and thecanonical momenta E can be eliminated in favour of A. The finalresult is

Z =

D DAi exp

0

d3x d{12

( B2+(A )2)+i(x, t)(x)}

.

(5.48)We have used the relation

exp

i

Ed3x

= exp

i

E d3x

,

provided (x) = 0 as r . Renaming (x, t) as A0(x, t), thepartition function can be written in an explicitly gauge invariantfashion:

Z =

pbc

DA(x) exp

0

d

d3x [L + i(x)A0(x)]

. (5.49)


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Again it is the trace in the definition of Z which results in theperiodic boundary conditions (pbc) on the gauge fields:

A

(x, 0) = A

(x, )

= 1, 2, 3, 4. (5.50)

For = 0, i.e., the sourceless theory, ZU(1) is, as may have been

anticipated, the functional integral of exp(0

d

d3xL) over thegauge fields A(x). Note that all four components of the vectorpotential appear above and Z is explicitly gauge invariant. Thefull set of allowed gauge trasformation has to maintain the pbc,A(x, 0) = A(x, ).

Problem 8: Find out all independent gauge transfor-mation subject to the boundary condition pbcabove.

It is straightforward to see that in the limit T 0, the partition function reduces to the usual generating functionalof the U(1) gauge theory. We will employ this simple example toillustrate one important concept below.

Let us now consider sources in the U(1) theory and choose = (x x0), which corresponds to a point source. Substitutingin (5.48), one has

Z() =

DA eSE L(x0), (5.51)

where we have defined

L(x0) = exp

i 0

d A0(x0, )

. (5.52)Using (5.51) and (5.48) and the definition of thermal expectationvalue, one obtains,

L(x0) = Z()Z( = 0)

. (5.53)

Problem 9: Show that L above is gauge invariant.


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One can further show that T lnL(x) = (Fq(x0)F0), whereFq(x0) is the free energy of a point charge (or an abelian quark)and F0 is the free energy of vacuum. Taking now a pair of opposite

charges, i.e.,

(x) = (x x0) (x y0) ,

one can similarly define L(x0)L(y0) which is related to the freeenergy of the pair of charges, Fqq(x0 y0), in the same manner.

We will now generalize the discussion above to a non-abeliangauge theory as a first step of going towards QCD. Recall that anSU(N) gauge theory describes a theory of N2 1 gluons. An im-portant distinction the SU(N) theory has over the U(1) exampleis that it is a fully interacting theory since the gluons have self-coupling. The case ofN = 3 colours corresponds to QCD (without

quarks). In general, the same considerations as U(1) go throughexcept that the electric field E Ea with colour index a whichruns from 1 to (N2 1) for an SU(N) gauge theory. Similarly,for the magnetic field B Ba and the charge density a etc.The non-abelian version of the Gauss law is D E = , where wehave used the generators of the SU(N) group in the fundamentalrepresentation, Ta, to define matrices E = EaTa, = aTa etc.and Dij = ij + (Tb)ij Ab with i, j running from 1 to N. Itis easy to check that the Gausss law does not commute with Hin this case. So trace in Z has to average over all a for a. Infact, it would otherwise be itself gauge dependent. Except for this

complication, all the above considerations go through to yield thepartition function of the Yang-Mills theory,

ZY M =

A(x,0)=A(x,)

DA(x, )exp

0

d

d3x LY M

. (5.54)In view of the deceptively simple looking equation above, let

us remind ourselves that the fields A are actually NN matricesand many complicated interaction terms are hidden in (5.54). Thefree energy of a static quark is given again by T ln Z = (Fq (x0)


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F0) but with L given by

L(x0) = 1NTr Texpi

0

d Aa0 (x0, ) Ta . (5.55)L is called the thermal Wilson loop or the Polyakov loop. If allquarks are confined, one expects the free energy of a single quarkto be infinite, Fq(x0) = , leading to L(x) = 0, whereas theexistence of a free quark with finite free energy (amounting to itsdeconfinement) implies L = 0. The free energy of a pair of quarkantiquark pair, located respectively at x0 and y0, is given by thecorrelation function L(x0)L(y0), as before.

In order to reach the final goal of writing down the partition

function of full QCD, one needs to introduce quarks in (5.54).Bringing in the quarks necessitates handling of another technicalproblem. Quarks obey Fermi-Dirac statistics, and the correspond-ing operators obey anticommutation relations. Introduction of thesum over all possible paths for them has to take these propertiesinto account. I will not deal with the details of this here but sim-ply state that one uses anticommuting Grassmann variables todefine the fermionic path integrals. Details about the Grassmannvariables, and their calculus can be found in standard references.

The Lagrangian density for the free quarks (or fermions) isgiven by

L = (i0 t

+ i. m). (5.56)

One can easily work out the canonical momentum to be =L/ = i, thus suggesting to use , = 0 as independentvariables in the path integrals. The hamiltonian density is givenby

H = L = ( i + m). (5.57)

Using it, and the expression for the number density, the partition


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function Z = Tr exp ((H N)/T) can be written as

Z = apbcD D exp

0

dd3x (0 + i m + 0) ,(5.58)

where apbcdenotes antiperiodic boundary conditions on the fields:

(x, 0) = (x, ) and (x, 0) = (x, ).

These arise due to the trace again but are antiperiodic for thefermionic fields. This leads to the quantized energy sums in thiscase to run over

n = (2n + 1)T , < n < .

Note that n = 0 for any n and the lowest value is 0 = T. Thisshould be contrasted with the bosonic case discussed earlier, where0 = 0. Defining i

for = 1, 2, 3, one has {, } = 2as may be expected in a Euclidean theory. The resultant quarkpartition function is

Z =

D D exp

0

d

d3x( + m + 0)

.(5.59)

The integration over Grassmann variables can be done to obtainZ = det D or alternatively, ln Z = Tr ln D, where D stands forthe Dirac matrix sandwitched in and in (5.59). For the freefermion case, the trace can be evaluated by going to the Fourierspace and one obtains

ln Z = 2V

d3p

(2)3[ + ln(1 + e()) + ln(1 + e(+))] ,

(5.60)where 2 = p2 + m2.

Problem 10: Obtain Eq.(5.60) from Eq.(5.59).


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Remarks:

The energy density obtained from (5.60) diverges quarti-cally. As for bosons, this zero point energy needs to be

subtracted. This can be easily done by using the freedomto shift the origin of the energy density scale and thus bydefining it to be identically zero at T = 0.

Actually there also exists a quadratic -dependent diver-gence in the energy density as well as the number densityin the free fermion theory. This can be attributed to thezero point number subtraction. However, unlike the caseabove, the prescription to eliminate it cannot be related toany physical principle but appears to be arbitrary.

Putting all the elements above together, and recognising that

gauge interaction can be introduced in the fermionic action in(5.59) by substituting the covariant derivative Da defined earlierin place of the ordinary derivative, the partition function for QCDis

ZQCD =

bc

DAD D expS(A, , ; g, mf, f, T) ,

(5.61)where bcdenotes collectively the boundary conditions on the quark(antiperiodic) and the gluon fields (periodic) :

A(x, 0) = A(x, 1/T) (pbc),

(x, 0) =

(x, 1/T) (apbc),

(x, 0) = (x, 1/T) (apbc).The QCD action S is given by

S =

1/T0

dt

d3x

14

Fa Fa

+

f

[(D + mf) f f 0 f] ,

(5.62)and thermal expectation value of a physical observable is givenby

= 1Z

DA D D eS. (5.63)


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The index f runs over flavours: up, down, strange, charm etc.and the corresponding mass and chemical potentail are denotedby mf and f respectively.

Comments :1. The (baryonic) chemical potential appears to couple like a

constant electromagnetic potential, A = (1, 0, 0, 0). This isa helpful mnemonic in diagramatics but one should be wearyof attaching a physical significance to it since the action doesnot have any corresponding gauge invariance.

2. Integrating the fermions, the flavour contribution is seen to

be ln Z 6

f=1det Df. Ignoring heavy quarks in Z, since the

masses of charm, bottom and top quarks are very large com-pared to the temprature scales we will need (a posteriori), Zis determined by u, d and s quarks. Ifm = mu = md = ms,then the QCD partition function has SU(3) flavour symme-try.

5.3.2 Symmetries and order parameters

It is useful to study the symmetries of the QCD partition function(5.61) in various limits and define corresponding order parametersto check whether i) the vaccum respects a given symmetry orbreaks it spontaneously and ii) increasing the temperature/densityhas any effects on the breaking.

In the limit of mq for all the flavours, the quarks areeffectively frozen. In this quenched approximation to QCD, thepartition function reduces to that of an SU(3) gauge theory. Thefermionic boundary conditions then become irrelevant, resultingin an extra global Z(3) summetry in that case. For a generalSU(N) gauge theory, the origin of the extra Z(N) symmetry canbe understood as below. Under V SU(N), the gauge fieldstransform as

A = V AV1 + iV V1,

along with A(x, 0) = A(x, ),


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where A = Aa t

a are N N matrices. The periodic boundarycondition restricts the set of allowed gauge transformations to

V(x, 0) = zV(x, ), z

Z(N). (5.64)

Problem 11: Derive the above condition.

One would have, of course, expected the set of allowed gaugetransformations to be periodic at T = 0 but it turns out to havea further global Z(N) symmetry. It is straightforward to see thatunder the Z(N), L(x0) L(x0), where

L(x0) =1

NTr V(x0, 0) Texp

0

Aa0 tad

V(x0, )= zL(x0).

If this Z(N) global symmetry is exact, then the above transfor-mation property implies the thermal average of the Polyakov loopor thermal Wilson loop, L(x0) = 0. On the other hand, if itis broken spontaneously by the vaccum, L(x0) = 0, which is asignal of deconfinement as we saw earlier, since

L = exp [ (Fq(x)/T)] ,Fq(x) = L = 0 confinement,Fq(x) < L = 0 deconfinement.

Thus a deconfinement phase transition with L as its or-der parameter is synonymous to the spontaneous breaking of theglobal Z(N) symmetry the theory has in the quenched approx-imation. Since L(x)L(y) is the correlation function of staticquarks, its behaviour can tell us about the nature of the phases,it being a measure of the free energy of the quark and antiquarkpair. In a confining phase, we expect VQQ = x, and hence,

LL exp(|r|/T), where r is the distance between the QQpair. With increasing temperature T, the tension (T) of thestring connecting them decreases, leading to a a decrease in thecorrelation function as well.


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The deconfinement phase transition between the confiningphase, where we exist and no free quarks are observed and thephase at high temperatures, where free quarks can exist, appears

to be similar to that in spin models with Z(N) symmetry. Theorder parameter for the former, L, plays the role of magneti-zation M which is the order parameter for the latter. If for agiven number of colours, N, the deconfinement phase transition issecond order one can appeal to universality and expect to obtainpredictions for various critical exponents:

L |T Tc| as T Tc ,

LLc |T Tc| as T Tc ,

Cv |T Tc| as T Tc,

T/

|T

Tc

| as T

Tc .

In addition to the order parameter which vanishes in the mannerprescribed above, its susceptibilty , the specific heat Cv and thecorrelation length diverge at Tc as per the indices of the corre-sponding observables of the Z(N) spin/Potts models. In case of afirst order phase transition, one observes a discontinuous turn-onfor the order parameter, and the dimensionality of space (three inour case) governs the divergences above.

If the fermion masses are finite, then the Z(N) global symme-try no longer exists: with (x, 0) = (x, ), one obviously hasV(x, 0) = V(x, ) only and L is no longer required to be zeroat any T. It is thus strictly not an order parameter for confine-ment/deconfinement in full QCD. One may, however, continue touse the limiting case as a guide inside the m T phase diagram.For large enough quark masses, one can imagine them as havingbeen tuned down from infinity. If the deconfinement phase tran-sition is first order at m = , such a tuning down will result ina line of first order phase transition in the m T phase diagram.The strngth of the transition, the latent heat, will decrease as thequark mass decreases. For the second order case, however, thetransition ceases to exist inside the m T diagram as a result ofsuch a tuning.


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In the opposite limit of vanishing quark masses, the QCD la-grangian has another extra symmetry. Its expected SU(3) flavoursymmetry of for equal u, d and s quark masses is enhanced to

SU(3)L SU(3)R U(1)A U(1)B symmetry when all thesequarks are massless. Here the subscripts L and R denote leftand right moving quarks obtained by the projection operatorsP = (1 5)/2. Physically, this is the statement that for freemassless spin half particles, spin projections are intrinsic prop-erties of the particles which cannot be changed by any Lorentztransformations.

Problem 12: Show this.

QCD thus has for equal quark masses: mf = m,

1. a U(1)B symmetry corresponding to the conservation ofbaryon number,

2. an SU(Nf) flavour symmetry leading to the conserved quan-tum numbers (by strong interaction) such as isopsin andstrangeness.

As m 0, these are enhanced to SU(Nf) SU(Nf) chiralsymmetry and U(1) U(1) symmetry. If the interactions respectthe symmetry, i.e., the vaccum is chirally symmetric, then theorder parameter for this symmetry, the chiral condensate, ,must vanish. In addition, one expects the hadrons to be par-

ity degenerate since the L- and R-quarks do not transform intoeach other. Since this is not seen experimentally, one is lead topostulate that the chiral symmetry is broken dynamically by theQCD vacuum and = 0. A consequence of such a breakingof a continuous symmetry is the prediction of massless Goldstonebosons. Since m2, m

2K m2N, these can be identifed as the

Goldstone bosons. These do have masses though : m2 = 0 andm2K > m

2. Even this can be understood as due to the small ex-

plicit breaking of the chiral symmetry by nonzero up, down andstrange quark masses. It is natural to expect that the magnitudeof the quark masses will govern the Goldstone masses. One can


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show that m2 mu + md and m2K ms, assuming mu, md < ms.The quark masses are, in fact, obtained using these relations.

At finite temperature, one naturally expects to be a func-tion of T. If at T TCH, = 0, one may have a chiral sym-metry restoring phase transition (PT). As for the deconfinementphase transition above, the chiral condensate is a strict order pa-rameter for the PT only in the massless limit for the quarks.Depending the number of flavours, i.e., the chiral symmetry, thistransition may be a first order one. It is expected to be so forNf 4. Again as one tunes up the masses from zero, one oughtto expect a line of first order chiral phase transitions in that caseand no chiral transition for nonzero quark masses in the secondorder chiral transition for massless quarks. The m-T diagram thusgenerically has lines of first order deconfinement and chiral tran-

sitions coming into from m = and zero respectively. The realworld has small unequal up and down quark masses in the rangeof a few MeV and a moderately heavy strange quark of abouthundred MeV. For both analytical as well as numerical methods,it turns out easier to obtain information for the real world fromeither limits discussed above. The questions one therefore has toclarify are the nature of the transitions in the limiting cases, theflow of the lines inside the diagram, whether they merge or cross,and finally which line is close to the real world, and thus whetherone has chiral, deconfining or both phase transitions at high tem-peratures/densities. In addition to the order parameters above,one can also investigate various other physical quantites, such as

the energy density , the pressure P, the number density n, thequark number susceptibilities , etc. These can be obtained fromthe partition function (5.61) by employing the canonical formul,

e.g., = T2

Vln Z

T .

5.3.3 Free energy in pertubation theory

As must be clear from the discussion of the previous section, oneneeds to evaluate ln Z and/or the thermal expectation values ofthe physical observables and the order parameters to extract the


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information on QCD phase diagram. One very straightforwardand familiar method is to use the small coupling, g, expansion.This perturbative approach works along the lines we discussed

earlier, provided one takes adequate care of the additional tech-nical problems of gauge invariance and fermions. The pressure inthis approximation can be written down order by order as

P/T4 = P0(1 + P1 + P2 + P3 + P4 + . . . )

with P0 = ideal gas terms = pq+g,

P1 = g2 5Nf + 1272

,

P2 = +g3

12c2,

P3 = g4 ln g2 c3,

P4 = g4

c4P5 = +g

5 c5.

Here the cis are numerical constants which have been evaluatedand their exact values can be found in the literature cited at theend. The leading term is that of an ideal gas of appropriate num-ber of quarks and gluons, as expected. The series is not a simpleg2 expansion like at zero temperature but has all powers of g andeven log g. The g3 and g5 terms arise due to the nontrivial massthe electric gluons have : mel gT, with a proportionality con-stant decided by the number of colours and flavours. These termsalso come with positive sign, making the series alternating in signbetween successive terms. A natural consequence would be its ex-pected convergence for very small values of the coupling g. SinceQCD is an asymptotically free theory, its coupling can be expectedto be small for asymptotically large temperatures/densities:

g2(T) =1

2b0 ln(T /T)

2b12b30

lnln(T /T)

ln (T /T)2 ,

where b0 and b1 are the first two coefficients of the QCD -function. For T >> T, g

2(T) small and use of perturbationtheory is justified. This yields Pideal as the pressure for T ,


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which was the ansatz we used in the Bag model case. The Pisprovide corrections to it, and the region of validity of the weakcoupling expansion is also where the ansatz can be used safely.

Various criteria of stability of a series can be used to figure itout. Naively, one expects the difference in the absolute values ofp(gn+1) and p(gn) should be small. When they become compara-ble, the alternating sign of these terms would induce instabilites.Putting in the known values of the constants cis, it turns out thattypically (P Pideal)/Pideal 12% for a stable series, leading toan estimate of T 108 MeV or so up to which the series is valid.This is, of course, way above any temperature of interest for anyapplications in the heavy ion physics or even the early Universe(due to the dominance of other physics at such high temperatures,such as electroweak symmetry breaking etc.).

5.3.4 Hierarchy of scales in QCD at finite tempera-

ture

In principle, the pressure, P, and any other thermodynamic quan-tity could be computed to any order in g using the diagrammaticapproach discussed above. One simply has to draw all the alloweddiagrams, obtain the expressions for them using the Feynman rulesand evaluate them. However, as was first shown by Linde, an in-surmountable barrier occurs at O(g6) due to the severe infra-redproblems of QCD, which are in fact general for any non-Abelian

gauge theory at nonzero temperature. Consider the (l + 1) loopdiagram shown in Fig. 5.8. Note this is possible only due to theself-couplings of gluons, and thus occurs exclusively in non-abeliangauge theories. Since it has 2l three-gluon vertices, one gets a fac-tor g2lp2l. Its 3l propagators yield (p2 + m2)3l, where we haveintroduced an infra-red cut off m, anticipating problems for smallmomenta p 0. The presence of (l +1) loops implies (l +1) inte-grals, and a factor,

T

d3pl+1

. Putting all the factors together,the diagram is proportional to g2l Tl+1

d3l+3p p2l/(p2 + m2)3l.

Dimensionally, it can be worked out to be T4. Since no extra ultra-violet divergences arise due to nonzero temperature, one can take


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T as the upper limit for the integrals. It can be shown that forl 2, the contribution of the diagram is g2lT4. For l = 3, it is of

dp/p type and is g6 T4 ln

Tm

. For l > 3, it is g6 T4(g2T/m)l3,

as can be seen already on dimensional grounds. We know thatthe electric gluons acquire a Debye mass gT but at that orderthe magnetic ones do not. If they do so at the next order, i.e.,if m O(g2T), then for l 3 all diagrams contribute at thesame order, O(g6). This is, of course, a breakdown of perturba-tive method where one usually expects the higher order diagramsto contribute only at higher order of g. Of course, this is not en-tirely a new phenomenon, since the perturbative expansion of thelast section already had terms coming from an infinite set of dia-grams which give rise to the terms with odd powers in g. The newfeature, however, is that the all the diagrams in the infinite set

contribute at the same order, making a perturbative computationunfeasible.

l+121 l

Figure 5.8: A 2l-th order diagram.

Nevertheless, for sufficiently small coupling, g 1, one has ahierarchy of scales at finite temperature: T > gT (electric Debyemass) > g2T (magnetic Debye mass). Note that the temperauresat which this is really valid are very large, T 108 MeV, sincethe running coupling s(T) 1/4. One can imagine integratingout the degrees of freedom at the successive scales and obtain asequence of effective theories at high temperatures. Thus inte-grating out fields up to the scale T, the usual LQCD (T = 0) goesover to a three (spatial) dimensional theory, L3dgauge+higgs, whileintegrating up to the scale gT, yields a three dimensional gaugetheory, L3dgauge. This is the essence of the (perturbative) dimen-


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sional reduction program, which has also been extended to obtainthe pressure for temperatures where naive perturbation theoryfails.

5.4 QGP and lattice QCD

In our goal to obtain predictions for quark-gluon plasma from thebasic underlying theory, QCD, using its partition function,

ZQCD =

bcD D DA exp

0

d

d3x LQCD

,the next step is to introduce a space-time lattice as a tool forcomputations. In the equation above, bc denote collectively the

boundary conditions on the fields due to the trace in Zas before:antiperiodic for the quark fields , and periodic for the gaugefields A.

Just as ordinary integration can be done by taking a limit ofsums on discrete points in the range of integration, lattice QCDcould be thought of as an attempt to do the functional integrals inZ by discretizing the space-time over which the fields are defined.Indeed, the formally but imprecisely defined functional integralsin Z can actually be given a precise meaning using the lattice.The lattice need not be regular but is usually chosen to be so.Let N and N denote the number of points in the spatial and

temporal (equivalently inverse temperature) direction and let a,a denote the corresponding lattice spacings. Then the volume Vand temperature T are given by V = N3 a

3 and T

1 = = Na.One, of course, needs to take a continuum limit at the end ofthe calculation, somewhat like the ordinary integration where theinterval between the points has to shrink to zero. The number oflattice points N, N as a, a 0 in this limit, keeping thevolume and the temperature fixed. In addition, we need to takeV for getting QCD thermodynamics.

Recall that we needed a regularization to compute momen-tum integrals in perturbation theory. A regulator is essential


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5.4. QGP and lattice QCD 253

in quantum field theory. Space-time lattice provides one suchregularization. It restricts the range of allowed momenta to/a pi /a. So lattice can be used for doing calculations inperturbation theory as well. But its great strength lies in b eingable to handle large couplings. In fact, large g2 is in a sense trivialon lattice, with easy demonstration of, e.g., confinement of quarksin this limit. The problem here is the a 0 limit which has tobe taken since there is no experimental evidence for a discretespace-time.

A space-time lattice breaks Lorentz symmetry. Nevertheless,its great virtue is the ability to define a gauge invariant theoryand perform all computations in a gauge invariant way, not alwaysfeasible with many other regularizations. One should try to definethe quark and gluon fields and their action so as to preserve asmany remaining continuum symmetries as p ossible. As we shallsee shortly, one crucial symmetry is the chiral symmetry whichturns out to be very nontrivial on the lattice.

5.4.1 Lattice fermions

Let us first consider free fremions, defined by the usual La-grangian:

LEuclidian = (+ m)., (5.65)Here we have already chosen the four-dimensional euclidean spaceand correspondingly gamma matrices satisfy {, } = 2along with the relation = . The factors of i which usu-ally appear have therefore got absorbed in the spatial matricesand in the definition of imaginary time. The 5 matrix also ishermitean:

5 = (1234)

= 5.Introduce now an L4 lattice and place spinors (n) and (n)

on each of its site n = (n1, n2, n3, n4). A derivative can then nat-urally be approximated by = [(n + a) (n a)]/2a.This is not the only way to discretize it. Indeed, one could takeasymmetric forward or backward derivatives, which are, however,not (anti)hermitean. One can add further terms to it whichhave higher powers of a. These are usually used to improve


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the behaviour of lattice fermions to make them mimic continuumfermions. As a consequence of this discretization the Dirac actionbecomes:

SF = n,l

(n)Mn,l(l), (5.66)

where

Mn,l =1

2a3

(l,n+a l,na) + a4m n,l. (5.67)

The Grassmann variables can be integrated out to yield the freefermion partition function,

Z= det M. (5.68)

Absorbing the a

3

factor in the equation above by rescaling thefermion fields to have no dimensions, and defining a Fourier trans-form of one can solve the free Dirac equation on the lat-tice: (k) = n (n)exp(2ik n/L), with its inverse definedby (n) =

k(k)exp(2ik.n/L)/L4. Then the lattice Dirac

equation,

(n + a) (n a)+ 2am (n) = 0, (5.69)simplifies to

i

sin ka + am = 0. (5.70)

. This clearly has the correct continuum limit. As a 0,sin ka ka, and one obtains i k + m = 0, as expected.

Note, however, that there are other allowed solutions as well.To see this, one rewrites the momentum integral (shown below fora specific dimension ) as follows. Let r = k /a for a given, giving dr = dk. Then

/a/a

dk =

/2a/a

dk +

/2a/2a

dk +

/a/2a

dk =

/2a/2a

(dk + dr), (5.71)


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where we have substituted r in the first and third integrals andrearranged terms. Thus as a 0, k and r both will contribute.This is the famous fermion doubling problem on the lattice. In

fact, due to the four dimensions, we have 24

= 16 fermions. Un-der the k r transformation, the sin term also changes sign,leading effectively to a transformation. Each of thedoubled fermion has thus a different set of -matrices. (with dif-ferent signs). This leads to alternating sign for their 5, and thuschiralities in pairs. The lattice keeps chirality exact at all stagesof cut-off when the bare mass m = 0.

Many solutions have been proposed for the fermion doublingproblem of lattice QCD. They all share the common feature ofhaving some undesirable property. Indeed, thanks to a no-go the-orem by Nielsen and Ninomiya, we know that the solution willinvolve either non-locality (discretization of the derivative withmany, possibly infinite, terms) or loss of chirality or loss of reality(non-hermitean Hamiltonian). In spite of the strong enthusiasmfor a solution, called the overlap fermions, we will discuss here thetwo most widely used options.

1. Wilson Fermions: Wilson in his seminal paper on latticegauge theories proposed to modify the Dirac matrix byadding a second derivative term which acts like an extramass for the undesired (15) doublers. His modified Diracmatrix (for the dimensionless fermion fields) is:

Mpn =1

2 [(r + )p+a,n + (r )pa,n]+ (ma + 4r)p,n, (5.72)

where r is a parameter usually taken to be unity. In thatcase, 12 (1 ) is a projection operator. Thus only part ofthe spinor propagates. Note also that even for m = 0 theaction has no chiral symmetry for any nonzero r.

The Wilson fermions break all chiral symmetries completely.They are recovered only in the continuum limit. This isspecially bad for any finite temperature studies where chiral


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symmetry restoring transition may be crucial. There havebeen such investigations but the results may still not be reli-able due to the smallness of temporal lattices or equivalently

coarse lattice spacings.

2. Staggered fermions: Also referred to as Kogut-Susskindfermions, these are most often used in finite temperaturestudies as they do have some chiral symmetry for m = 0.However, for nonzero lattice spacing, they mix the flavourcomponents with the spin and thus do not have a clear def-inition for quark flavour. One uses single component inplace of the four component to define their Dirac matrix,

Mpn =1

2

(1)x1+...+x1 (p+a,n pa,n)

+ map,n. (5.73)

Taking the continuum limit for these fermions needs a lit-tle care in interpretation. One can show that 16 on a 24

hypercube give rise to quarks of four flavours, each havingthe usual four component spinors. Thus in the continuum,these fermions describe Nf = 4 flavour always. How to getthe 2 + 1 flavours, observed in nature? One takes appropri-ate root of the determinant (e.g., 1/4th root for one flavour),which is justified from a perturbation theory viewpoint.

Problem 13: Show that in momentum space,

Mk = ma + i

sin ka + r

(1 cos ka),

and the r-term kills 15 zeroes ofMk but gives the usualrelation in the continuum limit.

5.4.2 Gauge fields

Once quarks on the space-time lattice have been introduced, thegluon (gauge) fields can be introduced by simply demanding in-variance under rotation by SU(N) to be a local symmetry. Let


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(x) (x) = Vx(x) be such a local gauge rotation withVx SU(N) as usual. Then the quark action (m)M(n) be-comes VmMVn under this rotation. Introduce Ux SU(N)as gauge field associated with a directed link from x to x + , asshown in Fig. 5.9. Let U x denotes the link in opposite direction,as shown in the same figure. Defining further gauge transforma-tion of the link fields by U

x = VxUx V

x+, one can ensure that the

fermion action is invariant under it. One needs a gauge invariantgluon action for the fields to be dynamical. The above transfor-mation of the link (gauge) variables implies that a gauge invariantgluon action will result only from closed loops of U, as shown inFig. 5.9. Defining the smallest such square loop on a lattice tobe a plaquette, one sees that the product along the directed pathgives rise to a matrix

UP = Ux U

x+ U

x+U

x+.

It is easy to check that Tr UP is gauge invariant, leading to anaction for the gluons,

SG =

all plaquettes P

1 1

NRe Tr Up

. (5.74)

xU

Ux

Plaquette

Figure 5.9: Gluon fields and action on a space-time lattice.


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Problem 14: Does Eq. (5.74) reduce to the usualFF action in the continuum limit?

(Hint: define a vector potential by Ux = expigaAa(x)t

a

such that x = (i +j)a/2 and show that it does so, pro-vided = 2N/g2.)

Combining all the ingradients so far from Eqs. (5.66) and(5.74) along with either Eq. (5.72) or (5.73), one obtains the fullexpression for the partition function of QCD at finite temperatureon a space-time lattice :

Z=

bc

dUx

d(x) d(x) eSFSG , (5.75)

where bc denote the boundary conditions on gluon and quarkfields,

Ux,0 = Ux,1/T,

(x, 0) =

x,1

T

, (x, 0) =

x,

1

T

.

Since the functional integral over fermions involve non-commutingGrassmann variables, one usually integrates the fermions out, us-ing

d d exp

mMmnn

= det M. (5.76)

Assuming the fermionic determinant to be positive, which it is forstaggered fermions, and using the identity det M = exp(Tr ln M),the QCD partion function (5.75) for staggered fermions can bere-written as

Z=

bc

dUx exp

SG + 1

8NfTr ln M

M

. (5.77)

The factor of Nf/8 allows one to use the staggered fermions foran arbitrary number of flavours. As mentioned above, Nf = 4 inthe continuum limit for these fermions for which the factor is halfto account for the presence of both M and M in Eq. (5.77). Thefermion determinant can be shown to be real but not positive forthe Wilson fermions. Thus a factor ofNf/2 and M from Eq. (5.72)in Eq. (5.77) gives the QCD partion function for Wilson fermions.


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5.4.3 Calculational techniques

From the partion function defined in the previous subsection, allphysical quantities of interest can be obtained as thermal expecta-tion values of appropriate operators. One needs to choose a suit-able calculational technique to proceed further. There are variousoptions available for the lattice QCD partion function (5.77) inview of its similarity with the conventional statistical mechanicssystems such as the Ising model. We discuss some of these below,pointing out their domain of applications.

1. The strong coupling expansion is valid for g2 orequaivalently 0. The gluonic Boltzmann factorexp

(1 1n

Re Tr Up)

can be expanded in this limit.

At = 0, all the gauge variables Ux

are totally uncon-strained and random and the partition function is dominatedby quark contributions. I will refer you to the standard textbooks, cited at the end, for details. In this limit, one can

(a) show that VQQ = r for mq such that a2 = ln(/2N), signalling confinement of quarks,

(b) show for staggered fermions that the chiral condensate, = 0 and the pion mass, m2 mq, signalling dy-namical breaking of the chiral symmetry by the QCDvacuum,

(c) calculate the hadron spectrum analytically to find it inqualitative agreement with the experiments,

(d) show that SU(3) gauge theory has a first order decon-finement phase transition and SU(2) a second orderone.

One cannot show confinement in this limit in a theory withdynamical quarks. Since mesons can pop out of the vacuumfor sufficiently large energies, the expected potential is any-way VQQ = r for very large r. One thus has no suitableorder parameter to probe confinement.


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It may be instructive to elaborate a little on (d). In thelimit of large quark masses, the so-called quenched limit,QCD becomes a pure gauge theory (or Yang-Mills theory).

Thus strong coupling limit of the quenched QCD providesa physical picture of the deconfinement transition which isuseful to understand. One proceeds by constructing an effec-tive action for the order parameter by integrating out otherfields (which can be done in this limit). The definition ofthe order parameter in subsection 5.3.1 can be transcribed

to the lattice to introduce (x) =

Nt=1

U4 (x, t). Using this

the effective action is defined by introducing a functional-function as below.

eS

eff

(L)

= DUeSGx (L(x) (x)). (5.78)It is easy to see that in general the effective action will be,

Seff(L) =

d3xV(L(x))+

d3x d3y L(x) S2(xy)L(y)+ .

(5.79)One uses strong coupling expansion and mean field theoryto show that

V(L) = aLL + bL3 + c(LL)2, forSU(3)V(L) = aLL + b(LL)2 + c(LL)3, forSU(2)

(5.80)Further it is possible to show that in the strong couplinga(T) > 0, b(T) > 0 for high temperatures and a(T) increasesas T decreases yielding a first order phase transition forSU(3) while for SU(2), b(T) > 0, c(T) > 0 but a(T) < 0(respectively > 0) for low (high) temperatures, giving a sec-ond order phase transition when it passes through zero.

All this has been obtained with the lattice cut-off in place.One has to remove it. This can be done by successivelyincreasing the order of the expansion but the radius of itsconvergence is finite, making us look for other methods for


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continuum limit. Nevertheless, many qualitative featurescan already be learnt from the strong coupling expansion.

2. The weak coupling is valid for small g and can be identifiedwith the usual perturbation theory. Indeed, lattice is just yetanother regulator for doing this expansion and all the

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