Bödeker’s effective theory: From Langevin dynamics to Dyson–Schwinger equations

Annals of Physics 324 (2009) 2108–2145

Contents lists available at ScienceDirect

Annals of Physics journal homepage: www.elsevier.com/locate/aop

Bödeker’s effective theory: From Langevin dynamics to Dyson–Schwinger equations Claus Zahlten, Andres Hernandez *, Michael G. Schmidt * Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany

a r t i c l e

i n f o

Article history: Received 13 March 2009 Accepted 7 April 2009 Available online 3 May 2009 Keywords: Sphaleron Non-equilibrium Schwinger–Dyson equations Bodeker’s effective theory

a b s t r a c t The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order jkj  g 2 T. Such a situation is typical for the processes of electroweak baryon number violation in the early Universe. Bödeker has derived an effective theory that describes the dynamics of the soft field modes by means of a Langevin equation. This effective theory has been used for lattice calculations so far [G.D. Moore, Nucl. Phys. B568 (2000) 367. Available from: ; G.D. Moore, Phys. Rev. D62 (2000) 085011. Available from: ]. In this work we provide a complementary, more analytic approach based on Dyson–Schwinger equations. Using methods known from stochastic quantitation, we recast Bödeker’s Langevin equation in the form of a field theoretic path integral. We introduce gauge ghosts in order to help control possible gauge artefacts that might appear after truncation, and which leads to a BRST symmetric formulation and to corresponding Ward identities. A second set of Ward identities, reflecting the origin of the theory in a stochastic differential equation, is also obtained. Finally, Dyson–Schwinger equations are derived. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Lattice calculations are an ideal tool to extract with a minimum of theoretical prejudice a specific piece of information from a given theory. However, in a sense, they are kind of a ‘black box’ that give the answer but hide the way how that answer comes about.

* Corresponding authors. Fax: +49 6221 54 9333. E-mail addresses: [email protected] (C. Zahlten), [email protected] (A. Hernandez), M.G.Schmidt@ thphys.uni-heidelberg.de (M.G. Schmidt). 0003-4916/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2009.04.006

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2109

The aim of this work is to provide a complementary, more analytic approach to the non-perturbative physics encoded in Bödeker’s effective theory [1]. The emphasis thereby lays not primarily on the accuracy of the results where it is hardly possible to beat the lattice calculations. Our aim is to provide a tool for a deeper understanding of what is really going on in the non-perturbative sector of hot nonabelian gauge theory and during creation of baryon number. In particular, it allows for an analytic study of the sphaleron rate [2,3]

C  lim lim

V!1 t!1

hðN CS ðtÞ  NCS ð0ÞÞ2 i Vt

ð1Þ

Such a deeper understanding is not only important for baryogenesis. Magnetic screening and the corresponding identification of a magnetic mass are of quite general theoretical interest with applications also in the field of the quark–gluon plasma [4–7]. We base our analysis on Bödeker’s effective theory despite the fact that Bödeker has also derived a generalised Boltzmann–Langevin equation which is valid to all orders in ½logð1=gÞ1 [8], of which Bödeker’s effective theory is merely the leading logarithmic approximation and the existence of other more general approaches, e.g. [9]. We choose this approximation because of the tractability of the analytic approach within this framework. The more general Boltzmann–Langevin equation not only is far more complicated, but is also not renormalisable by power counting [10]. On the other hand, the effective theory with which we deal here is ultraviolet finite, and is known to still be valid at next-to-leading logarithmic order provided one uses the next-to-leading logarithmic order colour conductivity r [11]. The key idea of this work is rather simple. Bödeker’s effective theory has the form of a Langevin equation. It is well-known from stochastic quantisation that a Langevin equation can be recast in the form of a path integral [12–14]. This path integral then can be reinterpreted as the functional integral formulation of an Euclidean quantum field theory with some ‘strange’ action. In this way, one gains access to all the powerful methods developed in QFT. Specifically, it is possible to derive the Dyson–Schwinger equations of the theory, offering an approach to the non-perturbative sector that is independent from, and complementary to, the existing lattice studies. On the way to this goal a couple of obstacles have to be overcome. These are mostly related to the peculiar role played by gauge invariance in the context of stochastic quantisation and Bödeker’s effective theory. A thorough understanding of this role proves to be essential in pursuing our aim. The outline of this work is as follows. Section 2 is devoted to the transcription of Bödeker’s theory in path integral form. From this path integral one could proceed to derive the Dyson–Schwinger equations, and in principle, could gain access to the non-perturbative sector of the theory. At the end of the day, however, one will be forced to rely on a certain truncation scheme to extract any concrete results from the equations. This truncation may introduce a possible gauge dependence and thus may render the results worthless. To keep control over the gauge dependence, it is therefore necessary to generalise Bödeker’s equation from A0 ¼ 0 gauge to a more general class of gauges before applying the formalism. Gauge fixing in a stochastic differential equation is quite delicate. One has to make sure not to destroy the Markovian nature of the equation. Applying methods developed in the context of stochastic quantisation [15], we introduce a gauge fixing term into Bödeker’s equation thereby achieving the desired upgrade to a general class of flow gauges. In Section 3, we argue that any physically reasonable truncation of the Dyson–Schwinger equations requires the introduction of gauge ghosts. In the full, untruncated theory gauge ghosts are not necessary, which is generally true in stochastic quantisation [13,15]. As was shown in Ref. [15], gauge ghosts can be introduced in stochastic quantisation in order to establish a gauge BRST symmetric formulation. We carry out this program in the case of Bödeker’s theory and derive the Ward–Takahashi identities corresponding to the gauge BRST symmetry of the action. These should be respected by the truncations to be used. The gauge Ward identities are not the only restrictions to be observed. A second class of Ward identities exist, that are related to the characteristic structure of the theory reflecting its origin in a

2110

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

stochastic differential equation. This characteristic structure can as well be expressed in the form of a BRST symmetry by introducing another kind of ghost fields, referred to as equation of motion (EOM) ghosts in this work. Introducing the gauge ghosts, however, destroys this second stochastic BRST symmetry. Nevertheless, it does not change the physical contents of the theory. The stochastic BRST symmetry is only an elegant way to express this structure. By directly referring to the underlying physics, it is still possible to derive the corresponding stochastic Ward identities. They provide a second set of restrictions to be imposed on the truncations. In Section 4, we derive the Dyson–Schwinger equations of Bödeker’s effective theory. In combination with the gauge and stochastic Ward identities of Section 3, this constitutes an independent approach to the non-perturbative dynamics of the soft, non-abelian gauge fields encoded in Bödeker’s effective theory. In Section 5, we summarise and discuss our results. Appendix A shows the explicit calculation of some of the Jacobians encountered in this work. We have included this Appendix in order to make our presentation more self-contained. The Feynman rules corresponding to our field theoretic transcription of Bödeker’s effective theory are listed in Appendix B. Finally, in Appendix C, we present explicit identities for the lower n-point function following from the general Ward identities. 2. Path integral formulation of Bödeker’s theory 2.1. Transcription to a path integral in A0 ¼ 0 gauge According to Bödeker’s effective theory the dynamics of the soft modes of the gauge field is described to leading logarithmic order by the Langevin equation [1]

Dab  Bb þ rA_ a ¼ fa

ð2Þ

which is written in A0 ¼ 0 gauge and where f is a gaussian white noise stochastic force. The stochastic force field incorporates the influence of higher momentum modes and has the correlator

D

E fai ðt; xÞfbj ðt 0 ; x0 Þ ¼ 2rT dij dab dðt  t 0 Þ dD1 ðx  x0 Þ

ð3Þ

reflecting its gaussian white noise character. Here and in the following, the number of spacial dimensions is D  1 ¼ 3, however, we leave D unspecified to allow for dimensional regularisation later. The only physical parameters entering Eqs. (2) and (3), and therefore the effective theory, are the temperature T, the colour conductivity r, and the self-coupling of the gauge field hidden in the definition of abc the covariant derivative Dab ¼ dab r  gf Ac . The procedure of reformulating a Langevin equation like Eq. (2) in the form of a field theoretic path integral is well-known [12–14]: According to Eq. (2), the gauge field evolves, starting from certain initial conditions, under the influence of the stochastic force. An arbitrary observable of the theory then is defined by some functional of the gauge field F½A and given by the expectation value of that functional with respect to the possible realisations of the stochastic force

hF½Ai ¼

Z

Df F½As ½f.½f ¼

Z

  Z 1 D1 dt d x fa ðt; xÞ  fa ðt; xÞ Df F½As ½f exp  4rT

ð4Þ

Here we have denoted by As ½f the solution of Eq. (2) for a specific choice of the stochastic force and the given initial conditions. To proceed and recast the effective theory of the gauge field into a form resembling the path integral formulation of an ‘ordinary’ quantum field theory we would rather like to have a path integral running over the gauge field than running over the stochastic force. This can be achieved by inserting unity in an appropriate way. In fact, one has

1¼

Z

DE dðE  fÞ ¼

Z

DA Det ði:c:Þ

  dE½A dðE½A  fÞ dA

where we choose the functional E½A as the left-hand side of Eq. (2)

ð5Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

Ea ½A ¼ Dab  Bb þ rA_ a

2111

ð6Þ

The invertibility of E½A is essential to justify the change of variables in Eq. (5). It follows from the parabolic nature of the expression and from the restriction to those gauge field configurations in the second path integral satisfying the initial conditions. Because Eq. (5) holds independently of f, it can be inserted into the path integral (4). The delta function then assures that only those gauge field configurations contribute to the integral that obey E½A ¼ f. Due to our choice of E½A, however, this is identical to the condition A ¼ As ½f. Thus, after inserting the delta function we may replace As ½f in the path integral simply by the integration variable A, and we are left with

hF½Ai ¼

Z

Df .½f

  dE½A DA Det dðE½A  fÞ F½A dA ði:c:Þ

Z

ð7Þ

Moreover, the restriction to field configurations obeying a specific set of initial conditions can be dropped if these initial conditions are specified at t ¼ 1. This is a consequence of their transversal component always being damped and the fact that any longitudinal contribution drops out whenever a gauge invariant observable is calculated. In case of a gauge variant quantity, however, a damping of the longitudinal component can be achieved by introducing an additional gauge fixing term into the Langevin equation [13]. This will be necessary anyway in the following section in order to generalise from A0 ¼ 0 gauge. Henceforth, we will therefore drop the restriction on the path integration in Eq. (7). At this point, one has two choices. One possibility is to proceed by doing the f integral with the help of the delta function. This results in a theory containing only the gauge field (and perhaps some additional ghost fields to be introduced later), however, at the expense of rather complicated interactions: the functional E½A shows up as argument of the gaussian probability distribution, and since E½A contains terms up to A3 , the action would inherit vertices of up to sixth order. To avoid this situation, we instead choose to introduce an additional auxiliary field k to represent the delta function

dðE½A  fÞ ¼

Z

 Z  D1 Dk exp i dt d x ka  ðEa ½A  fa Þ

ð8Þ

In this way, one can still perform the f integral that becomes gaussian, thereby eliminating the stochastic force field from the theory. One obtains

hF½Ai ¼

Z

  dE½A F½A eS½A;k DADk Det dA

ð9Þ

with

S½A; k ¼

Z

dx



rT ka  ka  ika  Ea ½A



ð10Þ

The determinant in Eq. (9) need not be taken into account since it can be shown to be a constant in dimensional regularisation (see Appendix A for an explicit calculation). We could, nevertheless, introduce a ghost representation of the determinant referring to the corresponding ghost fields as equation of motion (EOM) ghosts in the following. As a benefit of doing so the action (10) would be endowed with a BRST symmetry, allowing to easily obtain a kind of Ward identities (so-called stochastic Ward identities) reflecting the origin of the theory in a stochastic differential equation. Since it is desirable to obtain as many non-perturbative identities as possible in order to find a judicious ansatz for the truncation of the Dyson–Schwinger equations, introducing EOM ghosts, at first, seems the natural way to proceed. However, there is another type of Ward identities related to gauge invariance. Unfortunately, the gauge ghosts to be introduced to obtain these gauge Ward identities will break the stochastic BRST symmetries. So, instead of introducing EOM ghosts now, we will later introduce gauge ghosts in order to obtain the gauge Ward identities. The stochastic Ward identities will be derived without the help of a BRST symmetry by directly referring to the fundamental structure of the theory that reflects its origin in a stochastic differential equation.

2112

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

For now, absorbing the constant determinant in the measure, we are left with

hF½Ai ¼

Z

DADk F½A eS½A;k

ð11Þ

where the action S is given by Eq. (10). 2.2. Upgrading to

j gauge

Bödeker’s theory is written in A0 ¼ 0 gauge, and so is our transcription as field theoretic path integral so far. At the end of the day, however, we will be forced to use an approximation to solve the non-perturbative equations obtained, e.g. Dyson–Schwinger equations, and this approximation might introduce gauge artefacts into the calculation. In order to allow some control over the gauge dependence of the results, we need to base our derivations on a reformulation of Bödeker’s equation in a more general gauge. In [15], Zinn-Justin and Zwanziger have shown that adding a term to Eq. (2) that is tangent to the gauge orbit

Dab  Bb þ rðA_ a þ Dab vb ½AÞ ¼ fa

ð12Þ

has no effect on expectation values of gauge invariant objects of the form F½A. This is not the most general modification of Eq. (2) which leaves expectation values of gauge invariant objects unchanged [16], but it suffices for our purposes. As long as va ½A contains no time derivatives, the added term has no effect in calculations of gauge invariant objects. We can reformulate this fact in a different way: Since the non-abelian electric field is given by Ea ¼ A_ a  Dab Ab0 , one may rewrite Bödeker’s equation in the compact form

Dab  Bb  rEa ¼ fa

ð13Þ a0

a

which then may be interpreted in any of the so-called flow gauges A ¼ v ½A with no time derivatives allowed inside the functional va ½A. The restriction that va ½A does not contain time derivatives plays a more substantial role in our context than in the context of stochastic quantisation which was the object of Zinn-Justin and Zwanziger: In stochastic quantisation the time variable describes a fictitious time that is introduced only as a device to reinterpret a given Euclidean quantum field theory as the limit of a stochastic process for large values of the fictitious time [13]. Absence of time derivatives in stochastic quantisation therefore means absence of derivatives with respect to fictitious time and does not pose any restrictions to usual time derivatives. In our context, on the contrary, time is the real, physical time and the restrictions above narrow down the class of possible gauges leading to a well defined Langevin equation. Moreover, because of the different role of the time variable, we also have a component of the gauge field that is associated with the t variable of the Langevin equation. In stochastic quantisation this is not the case because t is fictitious and the time associated with A0 is just the zero component of the Euclidean x vector. To cope with this different structure, to some extent will demand a generalisation of the proof of Zinn-Justin and Zwanziger. In effect, we not only have to prove that gauge invariant objects of the form F½A are left invariant by the introduction of the term va ½A, as was shown in [15]. Instead we have to prove the following: Given Bödeker’s equation in the form (13) and a gauge invariant functional F½A0 ; A, then any choice of a flow gauge leads to the same result. Or put in different words, calculating hF½v½A; Ai by means of the equation (Eq. (12)) gives always the same value, independent of v½A. We now proceed in a similar manner to [15]. Let us consider the left-hand side of Eq. (12) where we add a small variation of the va ½A term. We evaluate this expression for a gauge field that is subject to an arbitrary, infinitesimal gauge transformation A0a ¼ Aa þ Dab xb and find

h i abc D0ab  B0b þ rðA_ 0a þ D0ab vb ½A0  þ D0ab dvb ½A0 Þ ¼ ðdab þ gf xc Þ Dbd  Bd þ rðA_ b þ Dbd vd ½AÞ  b @x þ rDab þ ½H½Axb þ dvb ½A @t

ð14Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2113

Here we have used

D0ab  B0b ¼ ðdab þ gf

abc

xc Þ Dbd  Bd b ab @ x _b

abc A_ 0a ¼ ðdab þ gf xc Þ A þ D

ð15Þ ð16Þ

@t

i.e. the product Dab  Bb transforms covariantly whereas the transformation of A_ a has a covariant and non-covariant contribution. In the same way we have split the transformation of va ½A into a covariant and non-covariant part: Starting from

Z

D1

dva ½Aðt; xÞ

dAbi ðt; yÞ

ð17Þ

xc Þ vb ½Aðt; xÞ þ ½H½Axa ðt; xÞ

ð18Þ

va ½A0 ðt; xÞ ¼ va ½Aðt; xÞ þ

d

y

dAbi ðt; yÞ

we have indeed

va ½A0 ðt; xÞ ¼ ðdab þ gf bi

where dA ¼

Dbc i

abc

c

x has been used and we have introduced the abbreviation

½H½Axa ðt; xÞ ¼

Z

d

D1

y

dva ½Aðt; xÞ dAbi ðt; yÞ

c ðDbc i x Þðt; yÞ  gf

abc b

v ½Aðt; xÞ xc ðt; xÞ

ð19Þ

Note that the functional derivatives in Eqs. (17) and (19) are only with respect to a spacial variation because va ½A does not contain any time derivatives (otherwise we would also have to integrate over time). Let us give the explicit form of this somewhat frightening expression for H½Ax in the case of the choice va ½A ¼  j1 r  Aa . One simply obtains

½H½Axa ðt; xÞ ¼ 

1

j

ðDab  rxb Þðt; xÞ

ð20Þ

abc

ð21Þ

Finally, Eq. (18) leads to

D0ab vb ½A0  ¼ ðdab þ gf

xc Þ Dbd vd ½A þ Dab ½H½Axb

where it was used that x is infinitesimal and of course

D0ab dvb ½A0  ¼ Dab dvb ½A

ð22Þ

because dv is infinitesimal itself. Let us now come back to Eq. (14) and its meaning. Suppose the gauge field, before the gauge transformation has been performed, was a solution of Bödeker’s equation with the va ½A term present, but without the additional dva ½A term. In other words, the original gauge field was a solution of Eq. (12). We can then replace the first square bracket on the right-hand side of Eq. (14) by the stochastic force and find

 b @x D0ab  B0b þ rðA_ 0a þ D0ab vb ½A0  þ D0ab dvb ½A0 Þ ¼ f0a þ rDab þ ½H½Axb þ dvb ½A @t

ð23Þ

This means, if we subject the original gauge field to an arbitrary, infinitesimal gauge transformation with parameter x, then the gauge transformed field will be a solution of Eq. (23), i.e. of the original equation with v replaced by v þ dv and the stochastic force transformed in the same way as the gauge field . . .but with an ugly additional term on the right-hand side. However, one can play a dirty trick: What was said so far was true for an arbitrary gauge transformation. But if we demand x to be a solution of

@ xb þ ½H½Axb þ dvb ½A ¼ 0 @t

ð24Þ

then the square bracket on the right of Eq. (23) will vanish and we finally arrive at

D0ab  B0b þ rðA_ 0a þ D0ab vb ½A0  þ D0ab dvb ½A0 Þ ¼ f0a

ð25Þ

2114

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

However, there is a certain subtlety that we want to draw attention to. To clarify this point, let us once again repeat the line of reasoning: Starting with a gauge field being solution of

Dab  Bb þ rðA_ a þ Dab vb ½AÞ ¼ fa

ð26Þ

we search for a gauge transformation x that obeys

@ xa þ ½H½Axa þ dva ½A ¼ 0 @t

ð27Þ

(and we can always find such an x because (27) is a linear, inhomogeneous equation with given inhomogeneity dva ½A). Then the gauge field transformed with this x, A0a ¼ Aa þ Dab xb , is a solution of the original equation with v replaced by v þ dv and the stochastic force also transformed by the same x

D0ab  B0b þ rðA_ 0a þ D0ab vb ½A0  þ D0ab dvb ½A0 Þ ¼ f0a

ð28Þ

The subtle point is the following: The original gauge field A is a solution of Eq. (26) and thus depends on the stochastic force f, of course. But A is an input of Eq. (27) that determines x. Therefore, x via A too depends on f. As a consequence of this, f0 inherits a non-trivial dependence on f: The stochastic force f0 not only depends on f because it is the gauge transform of f, but also because the gauge transformation itself depends on f

f0a ¼ ðdab þ gf

abc

xc ½fÞ fb

ð29Þ

s

We denote by A ½f; v; Aini  the solution of Eq. (12) for the specific realisation f of the stochastic force term and initial conditions Aini . Correspondingly, let As ½f; v þ dv; Aini  denote the solution of this equation with v replaced by v þ dv and for the same stochastic force and initial conditions. We can then express the contents of Eq. (28) in this new notation

As ½x f; v þ dv; x Aini  ¼ x As ½f; v; Aini 

ð30Þ

where the superscript x indicates gauge transformation with the special parameter x corresponding to the solution on the right-hand side via Eq. (27). After these preparations we can now show that gauge invariant expectation values hF½A0 ; Ai are independent of the choice of va ½A. To this end, let us write the gauge invariant observable as functional of the non-abelian electric and magnetic field

Ea ¼ A_ a  Dab Ab0 1 abc Ba ¼ r  Aa þ gf Ab  Ac 2

ð31Þ

We then have

hF½E; Bivþdv ¼

Z

Df0 .½f0  F ½Evþdv ½A; Bvþdv ½AA¼As ½f0 ;vþdv;A0

ini



ð32Þ

with

Eavþdv ½A ¼ A_ a  Dab vb ½A  Dab dvb ½A

ð33Þ

and Bvþdv ½A ¼ Bv ½A as in Eq. (31). Changing variables according to Eq. (29), one obtains

hF½E; Bivþdv ¼

Z

 x  d f Df Det .½x f F ½Evþdv ½A; Bvþdv ½AA¼As ½x f;vþdv;A0ini  df

ð34Þ

We now use independence on the initial conditions, the transformation property (30), gauge invariance of .½f and finally the fact that the determinant is unity (shown in Appendix A). This all together leads to

hF½E; Bivþdv ¼

Z

Df .½f F ½Evþdv ½x A; Bvþdv ½x AA¼As ½f;v;Aini 

Taking into account the transformation properties (16), (21) and (22), we find

ð35Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

Eavþdv ½x A ¼ ðx Ev ½AÞa  Dab

 b @x þ ½H½Axb þ dvb ½A ¼ ðx Ev ½AÞa @t

2115

ð36Þ

Bavþdv ½x A ¼ ðx Bv ½AÞa and thus

hF½E; Bivþdv ¼

Z

Df .½f F ½x Ev ½A; x Bv ½AA¼As ½f;v;Aini  ¼ hF½E; Biv

ð37Þ

because F½E; B is a gauge invariant functional. Consequently, we have shown that Bödeker’s equation in A0 ¼ 0 gauge

Dab  Bb þ rA_ a ¼ fa

ð38Þ

can equivalently be formulated in any flow gauge

Dab  Bb þ rðA_ a þ Dab vb ½AÞ ¼ fa

ð39Þ a

without any time derivatives allowed inside the functional v ½A. We will henceforth use the special choice Aa0 ¼ va ½A ¼  j1 r  Aa and refer to it as j gauge. This is a natural choice for va ½A, since it has the lowest order in A, preserves colour invariance, and with j > 0 the term Dab vb ½A provides a globally restoring force along gauge orbits [21], while at the same time having the correct dimensions. 3. BRST symmetric action and Ward–Takahashi identities We have argued that in order to derive any reliable statements from our theory, it is essential to gain some control over the gauge dependence possibly introduced by the truncation of the Dyson– Schwinger equations. This was our main motivation to generalise Bödeker’s equation from A0 ¼ 0 gauge to a more general class of flow gauges. In addition to this, the corresponding introduction of a gauge-fixing force has a welcome side-effect: It solves at the same time the problem of undamped longitudinal components of the initial gauge field configuration. However, the detection of an unphysical gauge dependence is not what we really want; in fact, we would rather like to avoid it. The ultimate goal is to construct a truncation scheme that is physically reasonable and does not (or, realistically speaking, only slightly) violate the gauge symmetry. To this end, we need identities expressing the gauge symmetry on the level of n-point functions, i.e. we need the Ward–Takahashi identities of the theory.1 Any physically reasonable truncation will have to respect these identities. Besides this conceptual importance, we may also hope that some of the Ward identities to be derived in the following will be of some practical use in solving the Dyson–Schwinger equations: In ordinary QCD, for instance, the full gluon propagator in covariant gauge is restricted to being purely transversal as a consequence of the Ward identities. This leads, of course, to a great simplification in the Dyson–Schwinger equations of QCD. In this section, we study three different kinds of non-perturbative identities: gauge Ward identities, i.e. Slavnov–Taylor identities; stochastic Ward identities; and ghost number conservation. 3.1. Constructing a BRST symmetric action In Section 2.2, we saw that Eq. (12) transforms covariantly only under a restricted class of gauge transformations. Obtaining the gauge Ward identities with this restriction turns out to be rather cumbersome. Instead, we will raise to life the gauge parameter x by introducing into the theory an additional (Grassmann valued) field that realises the constraint on the gauge transformations. The resulting action will be endowed with a BRST symmetry, and we will be able to obtain the gauge Ward identities in a straight-forward manner.

1 In the non-abelian context, these identities are often referred to as Slavnov–Taylor identities. However, following the terminology of Ref. [15], we denote these identities as gauge Ward identities in analogy to the stochastic Ward identities also encountered in this work.

2116

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

Setting dva ½A to zero in Eq. (24) we see that Eq. (12) transforms covariantly under gauge transformations which obey

@ xb þ ½H½Axb ¼ 0 @t

ð40Þ

Note that the introduction of va ½A does not restrict the gauge group any further than it already would be. Even without the extra term, the gauge transformations would have to be restricted in order for Eq. (12) to be gauge covariant. The restriction in Eq. (40) can be taken into account in the path integral in the following way. Define a term ca ½x; A from the left-hand side of Eq. (40), which for our choice of the functional va ½A takes the form

ca ½x; A ¼

@ xa 1 ab  D  r xb @t j

ð41Þ

Perform a change of variables from c to x in the following Grassmann integral representation of unity

1¼

Z

Dc dðcÞ ¼

Z

Dx

 a  @x 1 d  Dab  rxb @t j Det dcd½xx;A 1

ð42Þ

Since the determinant is Grassmann even, it no longer depends on x and it can be pulled out of the integral. The determinant is a constant, and can be calculated in a similar manner to the determinant in Eq. (9); see Appendix A for an explicit calculation. Inserting the integral representation of the Grassmann delta function

dðcÞ ¼

Z

 exp Dx

Z

 a ðxÞ ca ðxÞ dx x

 ð43Þ

and absorbing the constant determinant into the measure, we find the identity

1¼

Z

 exp DxDx

Z

   @ 1  a ðxÞ dab  Dab  r xb ðxÞ dx x @t j

ð44Þ

which holds independently of the gauge field A. Therefore, it can be inserted into the analogous of the path integral representation of the generating functional, Eq. (11), based on the generalised version of Bödeker’s Eq. (12). This leads to

Z½J ¼

Z

  Z  exp S½A; k; x; x   þ dx Ja ðxÞAa ðxÞ DADkDxDx

ð45Þ

with the action now given by

  ¼ SðDÞ ½A; k þ SðGGÞ ½A; x; x  S½A; k; x; x where S

ðDÞ

ð46Þ

½A; k is the generalised contribution of the dynamical fields

SðDÞ ½A; k ¼

Z

dx







1 rT ka  ka  ika  Dab  Bb þ r A_ a  Dab r  Ab j

 ð47Þ

and

 ¼ SðGGÞ ½A; x; x

Z

 1 a ab _aþ x  D  rxb  ax dx  x

j

ð48Þ

. is the new contribution containing the gauge ghosts x and x 3.2. Gauge Ward identities The Slavnov–Taylor identities can be derived by noting that the action (46) is invariant under the following BRST transformation

2117

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

de Aa ðxÞ ¼ Dab ðxÞ exb ðxÞ;

abc

exc ðxÞxb ðxÞ  a ðxÞ ¼ gf exc ðxÞx  b ðxÞ þ ierDab ðxÞ  kb ðxÞ de k ðxÞ ¼ gf exc ðxÞk ðxÞ; de x abc

a

de xa ðxÞ ¼ 12 gf

ð49Þ

abc

b

where e is a constant Grassmann parameter. It is convenient to introduce the finite BRST operator s  is defined as (left) derivative such that the result of s acting on a functional of the fields A, k, x and x with respect to the parameter e of the variations in Eq. (49). We thus have

 ¼ sF½A; k; x; x

@  de F½A; k; x; x @e

ð50Þ

or conversely

  ¼ e sF½A; k; x; x  de F½A; k; x; x

ð51Þ

From Eq. (50) one finds the following representation

s¼

 d d d d  aÞ dx ðsAai Þ ai þ ðskai Þ ai þ ðsxa Þ þ ðs x a dxa dx dk dA

Z

ð52Þ

with the finite BRST transforms of the fundamental fields given by Eq. (49)

sAa ðxÞ ¼ Dab ðxÞ xb ðxÞ;

sxa ðxÞ ¼ 12 g f

abc

xc ðxÞxb ðxÞ abc  a ðxÞ ¼ g f abc xc ðxÞx  b ðxÞ þ irDab ðxÞ  kb ðxÞ ska ðxÞ ¼ g f xc ðxÞkb ðxÞ; sx

ð53Þ

The BRST operator s has two essential properties: it annihilates the complete action (46)

 ¼ 0 sS½A; k; x; x

ð54Þ

  under the BRST transformation (49), and it’s nilpotency expressing the invariance of S½A; k; x; x

s2 ¼ 0

ð55Þ

Using the operator s, we now define the generating functional in the following way

 Z   exp S½A; k; x; x   þ dx Aa  JaA þ ka  Jak þ xa J ax þ x  a J ax þ IasA  sAa DADkDxDx   a ð56Þ þIask  ska þ Iasx sxa þ Iasx sx

Z½J; I ¼

Z

 , sA and sk together with their sources J x , J x , IsA , Isk are Grassmann odd, the remaining Note that x, x quantities Grassmann even. We proceed to vary the fields in Eq. (56) according to Eq. (49). The Jacobian of such a transformation is unity due to Eq. (53) as can be seen from the explicit calculation in Appendix A. We also know   ¼ S½A0 ; k0 ; x0 ; x  0 . In addition, the that the action is invariant under this change of variables S½A; k; x; x source terms of the BRST transformed fields are invariant due to the nilpotency of s and the fact that the variations are s-transforms themselves, e.g. de A0 ¼ esA0 . Only the source terms of the fundamental fields are not invariant and transform according to

Aa  JaA ¼ A0a  JaA þ de A0a  JaA ¼ A0a  JaA þ e sA0a  JaA

ð57Þ

and likewise for the other fields. Thus, under the change of variables (49), the integrand in Eq. (56) is simply reproduced with all fields replaced by their primed counterparts and an additional factor

exp

 Z

e

   0a J ax dx sA0a  JaA þ sk0a  Jak þ sx0a J ax þ sx

 ð58Þ

generated by the transformation of the fundamental source terms, Eq. (57). Because odd we have

exp

 Z



 a J ax e dx sAa  JaA þ ska  Jak þ sxa Jax þ sx

 

¼1þe

Z

e is Grassmann

   0a J ax dx sA0a  JaA þ sk0a  Jak þ sx0a J ax þ sx

2118

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

Inserted back into the path integral Eq. (56), the one just gives Z½J; I, which cancels the left-hand side of the equation. Hence, we obtain

0¼

Z

e DADkDxDx

Z

   a J ax exp fð. . .Þg dx sAa  JaA þ ska  Jak þ sxa J ax þ sx

ð59Þ

where the dots represent the exponential in Eq. (56). This has to be true for any e and thus the expression without e has to vanish itself. Changing the BRST transformed fields for functional derivatives with respect to their sources, we find the following identity

"

Z

dx

J ai A ðxÞ

d dIai sA ðxÞ

þ

J ai k ðxÞ

# d d a þ Jx ðxÞ a Z½J; I ¼ 0 þ J x ðxÞ a dIsx ðxÞ dIsx ðxÞ dIai sk ðxÞ d

a

ð60Þ

Finally, let us transcribe this relation in an identity for the generating functional of one-particle irreducible (1PI) correlation functions. To this end, we first express it by the generating functional of connected correlation functions W½J; I ¼ ln Z½J; I. In terms of W½J; I the relation (60) reads

"

Z

dx J ai A ðxÞ

dW½J; I dIai sA ðxÞ

þ J ai k ðxÞ

dW½J; I dIai sk ðxÞ

þ Jax ðxÞ

# dW½J; I dW½J; I a þ J ¼0 ðxÞ  x dIasx ðxÞ dIasx ðxÞ

ð61Þ

To define the generating functional of one-particle irreducible correlation functions, we introduce the usual expectation values for the fields in the presence of the external sources

Aai ðxÞ ¼ dW½J;I ; xa ðxÞ ¼  dW½J;I dJa ðxÞ dJ ai ðxÞ x

A

ai

k ðxÞ ¼

dW½J;I ; dJ ai k ðxÞ

 a ðxÞ ¼  dW½J;I x dJa ðxÞ

ð62Þ

 x

The minus signs in the case of the ghost fields are a consequence of our definition of the generating functional, Eq. (56), where we ordered the sources to the right of the fundamental fields. Assuming that the relations (62) can be solved for the sources J, we can define the 1PI generating functional C as the Legendre transform of W½J; I with respect to the sources J. The sources of the BRST transformed fields are not Legendre transformed and play the role of spectators only. With the definition

 ; I ¼ C½A; k; x; x

Z

   a J ax  W½J; I dx Aa  JaA þ ka  Jak þ xa J ax þ x

ð63Þ

one finds dC dAai ðxÞ

¼ J ai A ðxÞ;

dC dxa ðxÞ

¼ J ax ðxÞ

dC dkai ðxÞ

¼ J ai k ðxÞ;

dC  a ðxÞ dx

¼ J ax ðxÞ

ð64Þ

and also dC dIai ðxÞ sA

¼  dIdW ; ai ðxÞ

dC dIasx ðxÞ

¼  dIdW a ðxÞ

dC dIai sk ðxÞ

¼  dIdW ; ai ðxÞ

dIasx  ðxÞ

dC

¼  dIdW a ðxÞ

sA

sk

sx

ð65Þ

 sx

which may be used to reexpress the gauge Ward identity (61) in terms of C

Z

"

# dC dC dC dC þ ¼0 þ ai þ dx  a ðxÞ dIasx ðxÞ dxa ðxÞ dIasx ðxÞ dx dk ðxÞ dIai dAai ðxÞ dIai sA ðxÞ sk ðxÞ dC

dC

dC

dC

ð66Þ

3.3. Stochastic Ward identities  , all of which were We have included in Eq. (56) the auxiliary field k and the ghost fields x and x not strictly necessary, but rather were included so as to facilitate our work. They could, in principle, be integrated out and we would be left with Eq. (11), except that we have now also introduced sources

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2119

for the extra fields, as well as for the BRST transformed ones. This would suggest that there could be some sort of relations for the generating functional in Eq. (56) resulting from our choice to include the extra fields and sources. To derive these relations for Bödeker’s effective theory, one starts from the generating functional (56), including sources of the fundamental as well as the (gauge) BRST transformed fields. Inserting the action and the BRST transforms according to Eqs. (46)–(48) and (53) with the definitions (6) and (41) in use, the generating functional Z½J; I may be written

Z½J; I ¼

Z

Z

h

 abc dx rTka  ka þ ika  Ea ½A  iJak þ igf xb Icsk  rDab Ibsx 

 1  a ca ½x; A þ J ax  gf abc xb Icsx þ Aa  JaA þ xa J ax þIasA  Dab xb þ Iasx gf abc xc xb þx 2  exp DADkDxDx

ð67Þ  have been collected. Because the exponent is quadratic in the former where terms multiplying k and x and linear in the latter, both of these fields can be integrated. One obtains Z   Z 1 0a 0a 1 abc ð68Þ Z½J;I ¼ DADx dðc0 Þ exp dx  E  E þ Aa  JaA þ xa J ax þIasA  Dab xb þ Iasx gf xc xb 4rT 2 with the new functionals E0 and c0 defined as

E0a ½x; A; Jk ; Isk ; Isx  ¼ Ea ½A  iJak þ igf

abc

xb Icsk  rDab Ibsx c0a ½x; A; Jx ; Isx  ¼ ca ½x; A þ Jax  gf abc xb Icsx

ð69Þ ð70Þ

Hence, when restricting to vanishing sources JA ¼ IsA ¼ 0 and J x ¼ Isx ¼ 0 the exponent becomes purely quadratic in E0 . Defining for brevity

Z 1 ½Jk ; J x ; Isk ; Isx  ¼ Z½JA ¼ 0; Jk ; J x ¼ 0; J x ; IsA ¼ 0; Isk ; Isx ¼ 0; Isx 

ð71Þ

we have

Z 1 ½Jk ; J x ; Isk ; Isx  ¼

Z

  Z 1 dx E0a  E0a DADx dðc0 Þ exp  4rT

ð72Þ

where E0 and c0 both depend on A and x as indicated in Eqs. (69) and (70). Thus, it is quite natural to attempt a change of variables from A and x to E0 and c0 . The Jacobian can be calculated in a similar manner as the Jacobian of Eq. (42), and again can be shown to be a constant. The resulting integral is gaussian and evaluates to a constant functional Z 1 leading to

Z 1 ½Jk ; J x ; Isk ; Isx  ¼ const:

ð73Þ

or likewise for W 1 ¼ ln Z 1

W 1 ½Jk ; J x ; Isk ; Isx  ¼ const:

ð74Þ

As a consequence, any combination of functional derivatives with respect to sources chosen from the class fJk ; J x ; Isk ; Isx g yields zero when acting on the full generating functionals and evaluated for vanishing sources:

d d d W½J; IjJ¼I¼0 ¼ 0  d|fflfflfflfflfflfflfflfflfflfflfflfflffl . . . d .ffl.{zfflfflfflfflfflfflfflfflfflfflfflfflffl . d . .ffl}. any combination

ð75Þ

of Jk ; J x ; Isk ; Isx with the same relation holding for derivatives of Z½J; I. To obtain a corresponding identity for the 1PI generating functional C, note that due to Eq. (74) one has on the submanifold defined by the vanishing of the four sources JA ; IsA ; J x and Isx

2120

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

dW 1

k ðxÞ J ¼ I ¼ 0 ¼ ai ¼0 sA A dJ k ðxÞ J x ¼ Isx ¼ 0 ai

and

dW x ðxÞ

J ¼ I ¼ 0 ¼  a 1 ¼ 0

A sA dJ x ðxÞ J x ¼ Isx ¼ 0 a

ð76Þ

So C could at most depend on A, x, Isk and Isx . However, from Eq. (64) we have

dC dAai ðxÞ

dC ¼ J ax ðxÞ dxa ðxÞ

¼ J ai A ðxÞ;

ð77Þ

and therefore C may not depend on A or x anymore. The same conclusion can be reached for the Isk and Isx by looking at Eqs. (65) and (75). Therefore, C must be a constant, this leads to

d d d  ; IjJ¼I¼0 ¼ 0 C½A; k; x; x  d|fflfflfflfflfflfflfflfflfflfflfflfflffl . . . d .ffl.{zfflfflfflfflfflfflfflfflfflfflfflfflffl . d . .ffl}. any combination

ð78Þ

of A; x; Isk ; Isx which is the equivalent of the stochastic Ward identity (75) in terms of the 1PI generating functional C. 3.4. Ghost number conservation We will discuss one last symmetry of the action (46). The action is invariant under the global transformation

xa ðxÞ ¼ eia x0a ðxÞ  0a ðxÞ  a ðxÞ ¼ eia x x

ð79Þ

 to the transforof the ghost and anti-ghost fields. In addition to this, subjecting the measure DxDx mation (79), i.e. to

 0a ðx1 Þ; eia x0a ðx2 Þ; eia x  0a ðx2 Þ; . . .Þ  a ðx1 Þ; xa ðx2 Þ; x  a ðx2 Þ; . . .Þ ¼ ðeia x0a ðx1 Þ; eia x ðxa ðx1 Þ; x

ð80Þ

one finds

 ¼ DxDx

Y

 a ðxn Þ ¼ ½dxa ðxn Þ dx

a;n

Y  0a ðxn Þ Jðx0 ; x  0Þ ½dx0a ðxn Þ dx

ð81Þ

a;n

with the Jacobian

0

eþia

B B 0 B B 1 0 0 J ðx ; x Þ ¼ det B 0 B B 0 @ .. .

0

0

0

eia

0

0

0

eþia

0

0 .. .

0 .. .

eia .. .



1

C C C C C¼1 C C A .. .

ð82Þ

Hence, the measure is also invariant under the transformation (79)

 ¼ Dx0 Dx 0 DxDx

ð83Þ

Together with the invariance of the action, this symmetry leads to ghost number conservation, which poses another restriction on the form of the generating functionals and their derivatives. Indeed, taking the parameter a in Eq. (79) to be infinitesimal and performing the corresponding change of variables

xa ðxÞ ¼ x0a ðxÞ þ iax0a ðxÞ  0a ðxÞ  0a ðxÞ  ia x  a ðxÞ ¼ x x in the defining path integral (56) of the generating functional Z½J; I yields

ð84Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2121

 Z  Z    exp ia dx xa J ax  x  a J ax þ IasA  sAa þ Iask  ska þ 2Iasx sxa exp fð. . .Þg Z½J; I ¼ DADkDxDx Here we have already renamed the primed symbols again to unprimed ones after the change of variables has been completed. As before, the dots represent the original exponent as it occurs in Eq. (56). Using the fact that a is assumed to be infinitesimal, we can expand the first exponential and replace any fields that appear by functional derivatives acting on the exponential (after interchanging the order of the ghost and anti-ghost field and their corresponding sources leading to a minus sign in either case). The derivatives can finally be pulled out of the functional integral and we obtain

Z

"

# d d d d d a ai ai a dx J x ðxÞ a  J x ðxÞ a þ IsA ðxÞ ai Z½J; I ¼ 0 þ Isk ðxÞ ai þ 2Isx ðxÞ a dJx ðxÞ dJx ðxÞ dIsx ðxÞ dIsA ðxÞ dIsk ðxÞ a

ð85Þ

Again, the definition W½J; I ¼ ln Z½J; I implies that the same identity holds for the generating functional W½J; I of connected correlation functions

Z

" dx J ax

# dW a dW ai dW ai dW a dW ¼0  J þ I þ I þ 2I  x sA sk sx dJax dJ ax dIasx dIai dIai sA sk

ð86Þ

where we have suppressed the space-time argument x and the dependence of W on the sources J and I. This identity in turn can easily be translated to the corresponding restriction on the 1PI generating  ; I. By means of Eqs. (62), (64) and (65) one finds functional C½A; k; x; x

Z

" dx

# dC a dC a dC ai dC a dC  þ Iai ¼0 x  x þ I þ 2I sA sk sx a dxa dx dIasx dIai dIai sA sk

ð87Þ

This concludes our derivation of non-perturbative identities for the generating functional (56). Explicit forms of these identities for lower n-point functions are shown in Appendix C. 4. Dyson–Schwinger equations To derive the Dyson–Schwinger equations, we observe that the path integral of a functional derivative vanishes, i.e.

Z

D/

d F½/ ¼ 0 d/ðxÞ

ð88Þ

for any functional F½/. Hence, in the case of Bödeker’s theory, we obtain four different equations by  into the generating inserting a functional derivative with respect to each of the fields A, k, x or x functional

 Z   exp S½A; k; x; x   þ dx Aa  JaA þ ka  Jak þ xa J ax þ x  a J ax DADkDxDx   a þ IasA  sAa þ Iask  ska þ Iasx sxa þ Iasx sx

Z½J; I ¼

Z

ð89Þ

4.1. General Dyson–Schwinger equations  ) equations 4.1.1. Ghost (x) and anti-ghost (x Starting from the identity

0¼

Z

 DADkDxDx

d exp fð. . .Þg dxa ðxÞ

where the dots represent the exponent of Eq. (89), gives

ð90Þ

2122

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

0¼



Z

1 g  b ðxÞ  f abc x  b ðxÞ r  Ac ðxÞ þ J ax ðxÞ þ r  IasA ðxÞ _ a ðxÞ þ Dab ðxÞ  rx  x DADkDxDx j j

i abc  c ðxÞ exp fð. . .Þg þ gf IbsA ðxÞ  Ac ðxÞ  Ibsk ðxÞ  kc ðxÞ þ Ibsx ðxÞ xc ðxÞ þ Ibsx ðxÞ x ð91Þ

Expressing the fields by derivatives acting on the exponential, one obtains

  1 dZ g d2 Z  f abc @ j b  Z r  IasA ðxÞ ZJax ðxÞ ¼ @ t þ D a j dJx ðxÞ j dJ x ðxÞ dJ cjA ðxÞ " # dZ dZ dZ dZ abc bj bj b b þ gf IsA ðxÞ cj þ Isx ðxÞ c þ Isk ðxÞ cj þ Isx ðxÞ c dJ x ðxÞ dJ x ðxÞ dJ A ðxÞ dJ k ðxÞ

ð92Þ

or in terms of W

" #   1 dW g abc d2 W dW dW J ax ðxÞ ¼ @ t þ D f @  þ  r  IasA ðxÞ j j dJax ðxÞ j dJ bx ðxÞ dJcjA ðxÞ dJbx ðxÞ dJ cjA ðxÞ " # dW dW dW dW abc bj b b IsA ðxÞ cj ðxÞ ðxÞ ðxÞ þ I þ gf þ Ibj þ I  sk sx sx dJcx ðxÞ dJcx ðxÞ dJ A ðxÞ dJ cjk ðxÞ

ð93Þ

Transcription to the 1PI generating functional C yields

" #   dC 1 g d2 W  a ðxÞ  f abc @ j  b ðxÞ Acj ðxÞ  r  IasA ðxÞ  þ D x x ¼  @ t dxa ðxÞ j j dJ bx ðxÞ dJcjA ðxÞ h i abc bj b b cj c  c ðxÞ þ gf IsA ðxÞ Acj ðxÞ þ Ibj  ðxÞ x sk ðxÞ k ðxÞ  I sx ðxÞ x ðxÞ  Isx

ð94Þ

The anti-ghost equation is obtained in a similar manner and reads

" #   dC 1 g abc d2 W cj 0 a b þ D x ðxÞ þ f @ x ðxÞA ðx Þ  ¼ @ t j  a ðxÞ dx j j dJ b ðxÞdJ cj ðx0 Þ x

A

 gf

abc b Isx ðxÞ

xc ðxÞ

ð95Þ

x0 ¼x

where x0 is set to x after the space-time derivative is carried out, i.e. the derivative acts on the argument of J bx ðxÞ only. 4.1.2. Auxiliary field (k) equation To deduce the auxiliary field equation from

0¼

Z

 DADkDxDx

d dkai ðxÞ

exp fð. . .Þg

ð96Þ

 ¼ we need, among other things, the functional derivative of the action S½A; k; x; x  . However, in the present case the corresponding expression becomes rather SðDÞ ½A; k þ SðGGÞ ½A; x; x cumbersome. As for the Feynman rules in Appendix B, we want to use a symmetrised kA2 and kA3 vertex. The k dependence of the action spreads out over the three contributions to the dynamical action ðDÞ ðDÞ ðDÞ Sð DÞ ½A; k ¼ S0 ½A; k þ Sint;3 ½A; k þ Sint;4 ½A; k. The corresponding derivatives can be written in the form ðDÞ

dS0 ½A; k ai

dk ðxÞ

h

r i ¼ 2rTkai ðxÞ  i dij ðr@ t  DÞ þ 1  @ i @ j Aaj ðxÞ

j

ðDÞ

dSint;3 ½A; k ai

dk ðxÞ

¼

h

i h i 1 rh ij 0 ðigÞf abc 1  d @ k  dik @ j þ 2 dij @ k  dik @ 0j 2! j

h ii

jk 0 kj þ d @ i  d @ i Abj ðxÞAck ðx0 Þ 0 x ¼x

ðDÞ dSint;4 ½A; k ai

dk ðxÞ

¼

1 2 bj ck dl ðig Þ V abcd ijkl A ðxÞA ðxÞA ðxÞ 3!

ð97Þ

ð98Þ ð99Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2123

where V abcd ijkl is defined in Eq. (B.24). One then obtains in terms of the 1PI generating functional

h

h

i r i ig rh ij 0 d @ k  dik @ j ¼ 2rTkai ðxÞ  i dij ðr@ t  DÞ þ 1  @ i @ j Aaj ðxÞ  f abc 1  2! j j dk ðxÞ " # 2 h i h ii d W bj ck 0 þ2 dij @ k  dik @ 0j þ djk @ 0i  dkj @ i þ A ðxÞA ðx Þ ck 0 dJ bj A ðxÞdJ A ðx Þ x0 ¼x " # 2 3 2 ig abcd d W d W dl bj ck dl þ 3 A  ðxÞ þ A ðxÞ A ðxÞ A ðxÞ V ijkl ck dl 3! ðxÞ dJ ck dJbj dJ bj A ðxÞ A ðxÞ dJ A ðxÞ dJ A ðxÞ h iA abc ci bi b c  gf Isk ðxÞ x ðxÞ þ irIsx ðxÞ A ðxÞ þ ir @ i Iasx ðxÞ dC ai

ð100Þ

4.1.3. Gauge field (A) equation Finally, coming to the gauge field equation

0¼

Z

d

 DADkDxDx

dAai ðxÞ

exp fð. . .Þg

ð101Þ

and using the derivatives ðDÞ

dS0 ½A; k

h

r i ¼ i dij ðr@ t  DÞ þ 1  @ i @ j kaj ðxÞ

j

ai

dA ðxÞ ðDÞ

dSint;3 ½A; k ai

dA ðxÞ

¼ igf

abc

h

i h i rh ij 0 d @ k þ djk ð@ i þ @ 0i Þ þ 2 djk @ 0i þ dij ð@ k þ @ 0k Þ  1

j

ii

 dik @ 0j þ dik ð@ j þ @ 0j Þ kbj ðxÞAck ðx0 Þ h

ðDÞ dSint;4 ½A; k ai

dA ðxÞ

¼

ð102Þ

x0 ¼x

1 2 bj ck dl ðig Þ V dabc lijk k ðxÞA ðxÞA ðxÞ 2!

ð103Þ ð104Þ

where the symmetry of V abcd ijkl has been exploited, together with

 dSðGGÞ ½A; x; x dAai ðxÞ

¼

g

j

 b ðxÞ @ i xc ðxÞ f abc x

ð105Þ

one arrives at

dC ai

dA ðxÞ

h

r i ¼  i dij ðr@ t  DÞ þ 1  @ i @ j kaj ðxÞ 2

"

j

3

ig dabc d W d2 W d2 W þ 2 dl Ack ðxÞ þ kdl ðxÞ bj V bj bj ck 2! lijk dJdl dJ k ðxÞ dJ A ðxÞ dJ A ðxÞ dJck k ðxÞ dJ A ðxÞ dJ A ðxÞ A ðxÞ  i h

h i h i r ij 0 abc bj ck dl jk 0 jk 0 d @ k þ d ð@ i þ @ i Þ þ 2 d @ i þ dij ð@ k þ @ 0k Þ  1 þk ðxÞ A ðxÞ A ðxÞ  igf j " # h ii d2 W ck 0 ik 0 ik 0 bj þ k ðxÞ A ðx Þ  d @ j þ d ð@ j þ @ j Þ ck 0 dJ bj k ðxÞ dJ A ðx Þ x0 ¼x " # 2 h i g d W abc bi  b ðx0 Þxc ðxÞ þx  f abc @ i b þ gf IsA ðxÞxc ðxÞ þ irIbsx ðxÞkci ðxÞ ð106Þ c 0 j dJ x ðx Þ dJ x ðxÞ x0 ¼x 

4.2. Explicit equations for lower n-point functions 4.2.1. Definitions and general relations Concerning the propagators, mixing will occur between the gauge field A and the auxiliary field k, resulting in four possible propagators from the gauge/auxiliary field sector that can be combined into

2124

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

one matrix propagator. These are completed by the propagator of the gauge ghosts. Altogether, we define the full (connected) propagators as

d2 W½J; I

¼ A ðxÞ A ðyÞ ¼ ai

c dJ A ðxÞ dJ bj A ðyÞ J¼I¼0

D E d2 W½J; I

ðkAÞab ðx; yÞ ¼ kai ðxÞ Abj ðyÞ ¼ ai Gij

c dJk ðxÞ dJ bj A ðyÞ J¼I¼0

D E d2 W½J; I

ðkkÞab Gij ðx; yÞ ¼ kai ðxÞ kbj ðyÞ ¼ ai

c dJk ðxÞ dJ bj k ðyÞ J¼I¼0

 a  d2 W½J; I

ðxÞab b  ðyÞ c ¼ G ðx; yÞ ¼ x ðxÞ x

dJ ax ðxÞ dJbx ðyÞ J¼I¼0

ðAAÞab Gij ðx; yÞ

ðAkÞab

D

ai

E

bj

ð107Þ

ð108Þ

ð109Þ

ð110Þ

ðkAÞba

and Gij ðx; yÞ ¼ Gji ðy; xÞ of course. In graphical representations we denote the gauge field by curly lines, the auxiliary field by double curly lines and the gauge ghosts by dotted lines. Thus, the full propagators are represented by

and finally

Besides the propagators, we have to set out our definition for the self-energies. To this end, let us summarise the two left-hand equations of (64) in the form

J ai F ðxÞ ¼

 ; I dC½A; k; x; x

ð111Þ

dF ai ðxÞ

where the index F stands for any of the fields A or k. Taking the functional derivative of this equation  with respect to J bj G ðyÞ, where again G 2 fA; kg, then yields (observing that A, k, x and x are functionals of the sources J and I) ab

ij

d d dFG dðx  yÞ ¼

Z

dz

" dAck ðzÞ

d2 C

þ

dkck ðzÞ

d2 C

ai ck dJ bj G ðyÞ dk ðzÞ dF ðxÞ #  c ðzÞ dxc ðzÞ d2 C dx d2 C þ bj þ ai c dJ G ðyÞ dxc ðzÞ dF ai ðxÞ dJ bj G ðyÞ dx ðzÞ dF ðxÞ ck ai dJ bj G ðyÞ dA ðzÞ dF ðxÞ

ð112Þ

Thus, using Eq. (62) to express the first factor in each term as a second derivative of W and finally setting the sources to zero leads to ab ij

d d dFG dðx  yÞ ¼

Z

" dz

ðGAÞbc Gjk ðy; zÞ



ck ai dA ðzÞdF ðxÞ d2 C

J¼I¼0

ðGkÞbc þ Gjk ðy; zÞ



ai ck dk ðzÞdF ðxÞ d2 C

3 5

ð113Þ

J¼I¼0

Here, the definitions (107)–(109) have been used and the terms involving ghost and anti-ghost fields have vanished due to ghost number conservation.

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2125

In the following we will often encounter multiple derivatives of the generating functionals W and C evaluated for vanishing sources. Let us therefore introduce a shorthand notation where we indicate the fields with respect to which the derivatives are taken as superscripts. Possible Lorentz or colour indices as well as space-time arguments appear in the order of the fields they belong to. For instance, we abbreviate

C

 Þabc ðkAx ðx; y; zÞ ij



¼ ai

 c ðzÞ J¼I¼0 dk ðxÞ dAbj ðyÞ dx d3 C

ð114Þ

In the case of W, we also use the fields as superscripts though the derivatives are taken with respect to the corresponding sources, of course. In this new notation, Eq. (113) reads

dab dij dFG dðx  yÞ ¼

Z

ðGHÞbc

dz Gjk

ðHFÞca

ðy; zÞ Cki

ðz; xÞ

ð115Þ

where H is a summation index running over the fields A and k. This equation expresses the fact that the matrix propagator of the gauge/auxiliary field sector

0 ^ ab ðx; yÞ ¼ @ G ij

ðkkÞab

Gij

ðx; yÞ

ðAkÞab

Gij

ðkAÞab

Gij

ðx; yÞ

ðAAÞab

ðx; yÞ Gij

ðx; yÞ

1 A

ð116Þ

is inverse to the matrix

0 ^ ab ðx; yÞ ¼ @ C ij

ðkAÞab CðkkÞab ðx; yÞ Cij ðx; yÞ ij ðAAÞab CðAkÞab ðx; yÞ Cij ðx; yÞ ij

1 A

ð117Þ

^ ab ðx; yÞ is determined via constructed of the second derivatives of C. Consequently, the self-energy P ij the relation ðFGÞab ðFGÞab CðFGÞab ðx; yÞ ¼ ðD1 Þij ðx; yÞ þ Pij ðx; yÞ ij

ð118Þ

ðFGÞab

where ðD1 Þij ðx; yÞ are the components of the inverse free propagator of perturbation theory (see Appendix B, Eqs. (B.6)–(B.9)), and where F; G 2 fk; Ag as before. Analogously, taking the derivative with respect to J bx ðyÞ of

J ax ðxÞ ¼

 ; I dC½A; k; x; x dxa ðxÞ

ð119Þ

and performing the same manipulations as described above leads to

dab dðx  yÞ ¼ 

Z

dz GðxÞ bc ðy; zÞ

d2 C

a c  ðzÞ dx ðxÞ dx

ð120Þ

J¼I¼0

Hence, we define the self-energy of the gauge ghosts via

h

CðxxÞ ab ðx; yÞ ¼  ðD1 ÞðxÞ ab ðx; yÞ þ PðxÞ ab ðx; yÞ 

i

ð121Þ

with the free inverse propagator ðD1 ÞðxÞ ab ðx; yÞ given in Eq. (B.18). In our graphical representations we denote self-energies and other one-particle irreducible quantities by open circles. Though generally we are using three-vectors, in the Fourier transformation we use four-vector notation

f ðxÞ ¼

Z

D

d k ð2pÞD

eikx f ðkÞ

ð122Þ

with ikx ¼ ik0 t þ i k  x. The proper vertex functions in momentum space are basically given by the Fourier transforms of the various functional derivatives of the 1PI generating functional C. However, due to translational invariance of the theory, all these Fourier transforms contain a delta function

2126

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

expressing momentum conservation at the vertex. It is therefore convenient to pull these delta functions out of the definitions of the vertex functions. In this way, the latter become functions of one momentum variable less than indicated by the number of external legs. For instance, we define  GÞabc ðxx

ð2pÞD dD ðk1 þ k2 þ k3 Þ Cj

ðk1 ; k2 Þ ¼

Z

 GÞabc ðxx

dx dy dz eik1 xik2 yik3 z Cj

ðx; y; zÞ

ð123Þ

or equivalently

Cðj xxGÞabc ðx; y; zÞ ¼ 

Z

D

D

d k1 d k2 D

ð2pÞ ð2pÞD

 GÞabc ðxx

eik1 ðzxÞik2 ðzyÞ Cj

ðk1 ; k2 Þ

ð124Þ

 GÞabc ðxx

Here, the two arguments of the proper vertex function Cj ðk1 ; k2 Þ refer to the (incoming) momenta along the ghost lines leaving and entering the vertex in this order. The choice of the N  1 momenta that are used as arguments of a vertex with N external legs is, of course, arbitrary and thereby a source of possible confusion. We therefore explicitly list the definitions of the other relevant vertex functions used in this work

CðFGHÞabc ðx; y; zÞ ¼  ijk

Z

D

D

d k2 d k3 D

ð2pÞ ð2pÞD

ðFGHÞabc

eik2 ðxyÞik3 ðxzÞ Cijk

ðk2 ; k3 Þ

ð125Þ

with k2 and k3 denoting the incoming momenta along the G and H line, respectively, and ðFGHKÞabcd Cijkl ðx; y; z; wÞ ¼ 

Z

D

D

D

d k2 d k3 d k4 ð2pÞD ð2pÞD ð2pÞD ðFGHKÞabcd

 eik4 ðxwÞ Cijkl

eik2 ðxyÞik3 ðxzÞ ðk2 ; k3 ; k4 Þ

ð126Þ

with incoming momenta k2 , k3 , k4 along the G, H and K line. Note the minus signs in the last two equations. The definitions above are chosen in such a way that they reduce at leading order to the corresponding vertices of the Feynman rules, i.e.

Cðj xxAÞabc ðk1 ; k2 Þ ¼ 

C C

ig

j

j

f abc k2 þ   

ð127Þ

ðkAAÞabc ðk2 ; k3 Þ ¼ g V abc ijk ðk2 ; k3 Þ þ    ijk 2 abcd ðkAAAÞabcd ðk ; k ; k Þ ¼ ig V ijkl þ    2 3 4 ijkl

ð128Þ ð129Þ

4.2.2. DSE for PðxÞ ðkÞ Let us again start with the ghost equations, being much simpler than the equations for the gauge/ auxiliary field sector. By taking the derivative of Eq. (95) with respect to xb ðyÞ, one finds evaluated for vanishing sources

d2 C

 a ðxÞ dxb ðyÞ dx

  1 g ade d d2 W½J; I

¼ d @ t þ D dðx  yÞ þ f @ j

j j dxb ðyÞ dJdx ðxÞ dJ ejA ðx0 Þ x0 ¼x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} J¼I¼0 ðD1 ÞðxÞ ab ðx; yÞ ab

J¼I¼0

ð130Þ

Comparing to the definition of the self-energy of the gauge ghosts in Eq. (121) then leads to the relation

PðxÞ ab ðx; yÞ ¼

g

j

f ade @ j

d d2 W½J; I

dxb ðyÞ dJdx ðxÞ dJ ejA ðx0 Þ x0 ¼x

ð131Þ

J¼I¼0

for the gauge ghost self-energy. If we carry out the functional derivative with respect to xb ðyÞ, four terms arise because any of the sources JA , Jk , J x and J x depends on x. However, due to ghost number conservation three of these terms vanish when the sources are set to zero and one is left with2 2 It should be clear that x0 is set to x only after the space-time derivative is carried out. In order to avoid an extensive use of brackets we decided to assume in this and similar cases some thoughtfulness on the part of the reader.

2127

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

PðxÞ ab ðx; yÞ ¼

g

j

Z

f ade @ j

" dv

d2 C

dx

#

d3 W

x

b ðyÞ d  c ðvÞ

dJ cx ðvÞ dJ dx ðxÞ dJejA ðx0 Þ

ð132Þ x0 ¼x J¼I¼0

Finally, we express the connected three-point function by its 1PI counterpart  FÞabc ðxx

Wj

ðx; y; zÞ ¼

Z

0

0

ðFGÞcc0

du du du GðxÞ a a ðu; xÞ GðxÞ bb ðy; u0 Þ Gjj0 0

00

 GÞa0 b0 c0 ðx x

ðz; u00 Þ Cj0

ðu; u0 ; u00 Þ

ð133Þ

where F represents one of the fields k or A and G is a summation index taking these two values. The shorthand notation used here was introduced in Eq. (114). Note that the order of the ghost and anti  ghost fields in Eq. (133) is changed from W ðxxFÞ to CðxxGÞ and that the (full) gauge ghost propagator is  GðxÞ ab ðx; yÞ ¼ W ðxxÞ ab ðx; yÞ, as defined in Eq. (110). Now, inserting relation (133) into Eq. (132), using the property (120) of the two-point functions and 0 0 0

0 0 0

Cðj0xxGÞa b c ðu; u0 ; u00 Þ ¼ Cðj0xxGÞb a c ðu0 ; u; u00 Þ 



ð134Þ

yields the Dyson–Schwinger equation

PðxÞ ab ðx; yÞ ¼ 

Z

0

00

ðAGÞee0

du du Gjj0

ðx; u00 Þ

g

j

 GÞd0 be0 ðxx

0

f ade @ j GðxÞ dd ðx; u0 Þ Cj0

ðu0 ; y; u00 Þ

ð135Þ

Using the definition for the momentum space proper vertex Eq. (123), we transform to momentum space

PðxÞ ab ðkÞ ¼ 

Z

D 0

d k ig D

ð2pÞ

j

0j

ðAGÞee0

f ade k Gjj0

0

 GÞd0 be0 ðxx

ðk  k Þ GðxÞ dd ðk Þ Cj0 0

0

0

ðk ; kÞ

ð136Þ

The structure of the Dyson–Schwinger equation (136) is illustrated in Fig. 1. In Eq. (136) the field index G has a summation index taking the values G ¼ k and G ¼ A. In the graphical representation of Eq. (136) such a summation is symbolised by a solid line. This short-hand notation will become even more important in the other Dyson–Schwinger equations to follow. Thus, the right-hand side of Fig. 1 stands for two individual diagrams. Above we have deduced the Dyson–Schwinger equation of the gauge ghost self-energy from the general anti-ghost equation (95). A complementary relation can be obtained from the ghost equation  b ðyÞ of Eq. (94), one obtains (94). By taking the derivative with respect to x

PðxÞ ab ðkÞ ¼ 

Z

D 0

d k ig ð2pÞ j D

j

ðGAÞe0 e

f dbe k Gj0 j

0

 GÞad0 e0 ðxx

ðk  k Þ GðxÞ d d ðk Þ Cj0 0

0

0

ðk; k Þ

ð137Þ

4.2.3. DSE for PðkkÞ ðkÞ We come now to the Dyson–Schwinger equations of the gauge/auxiliary field sector. Taking the derivative with respect to kbj ðyÞ of the auxiliary field equation (100) yields after setting the sources to zero

Fig. 1. DSE of the gauge ghost self-energy, Eq. (136). Filled circles denote full propagators. Empty circles are used for oneparticle irreducible quantities, i.e. self-energies and proper vertices. The solid line represents a summation of one graph with the line replaced by a gauge field and a second diagram with an auxiliary field instead.

2128

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145 ðkkÞab

d C

ai bj dk ðxÞ dk ðyÞ 2

J¼I¼0

ðx; yÞ ðD1 Þij

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 2

ig acde d d3 W

ab ij ¼ 2rT d d dðx  yÞ  V iklm bj

ck dl em 3! dk ðyÞ dJ A ðxÞ dJA ðxÞ dJ A ðxÞ J¼I¼0 h

i ig rh ik 0 il i h ik  f acd 1  d @ l  d @ k þ 2 d @ l  dil @ 0k 2! j

h ii d d2 W

kl 0 lk þ d @i  d @i

bj ck dl 0 x0 ¼x dk ðyÞ dJ ðxÞ dJ ðx Þ A

A

ð138Þ

J¼I¼0

Thus, comparing to Eq. (118) one reads off the self-energy component

i ig acd h

rh ik 0 il i h ik 1 f d @ l  d @ k þ 2 d @ l  dil @ 0k 2! j

h ii d d2 W

kl 0 lk þ d @i  d @i

bj ck dl 0 x0 ¼x dk ðyÞ dJ A ðxÞ dJ A ðx Þ

PðkkÞab ðx; yÞ ¼  ij

J¼I¼0

2

ig acde d d3 W

V iklm bj 

ck dl em 3! dk ðyÞ dJ ðxÞ dJ ðxÞ dJ ðxÞ A

A

A

ð139Þ J¼I¼0

To evaluate Eq. (139), we have to calculate the remaining functional derivatives and finally transform into momentum space. Let us start with the k derivative of the connected two-point function. Because we will encounter similar expressions also in the Dyson–Schwinger equations of the other self-energy components, it is useful to generalise a bit and do the work once and for all. Thus, with F, G and H chosen from the set fk; Ag, we find by means of the chain rule and using ghost number conservation, together with the identities (64)



bj ck dl 0 dF ðyÞ dJ ðxÞ dJ ðx Þ d2 W

d

G

H

¼

Z

" dv

d2 C

#

d3 W

ck dl 0 dF bj ðyÞ dK em ðvÞ dJem K ðvÞ dJ G ðxÞ dJ H ðx Þ

J¼I¼0

J¼I¼0

The field index K in this equation is summed over the two values k and A. Expressing the connected three-point function by its one-particle irreducible counterpart ðFGHÞabc

W ijk

ðx; y; zÞ ¼ 

Z

0

ðGG0 Þbb0

ðFF 0 Þaa0

00

du du du Gii0 ðHH0 Þcc0

 Gkk0

ðx; uÞ Gjj0

ðF 0 G0 H0 Þa0 b0 c0

ðz; u00 Þ Ci0 j0 k0

ðy; u0 Þ

ðu; u0 ; u00 Þ

ð140Þ

and exploiting the relation (115) then leads to the identity



bj ck dl 0 dF ðyÞ dJ ðxÞ dJ ðx Þ d2 W

d

G

H

¼ 

Z

0

00

ðGG0 Þcc0

du du Gkk0

ðHH0 Þdd0

ðx; u0 Þ Gll0

ðFG0 H0 Þbc0 d0

ðx0 ; u00 Þ Cjk0 l0

ðy; u0 ; u00 Þ

ð141Þ

J¼I¼0

Again, doubled field indices are summed over k and A (which we will assume from now on in all relevant cases). Finally, transforming into momentum space and inserting the definition of the threepoint vertex function (125) yields



dF bj ðyÞ dJ ck ðxÞ dJ dl ðx0 Þ d2 W

d

G

H

J¼I¼0

¼

Z

D

d k

D 0

d k

D

D

ð2pÞ ð2pÞ

ðHH0 Þdd0

 Gll0

0

0

ðGG0 Þcc0

eikðxyÞ eik ðxx Þ Gkk0 0

ðFG0 H0 Þbc0 d0

ðk Þ Cjk0 l0

0

0

ðk  k Þ 0

ðk  k; k Þ

ð142Þ

Analogously, one can derive a general expression for the fourth functional derivative in Eq. (139). Using the chain rule as above, exploiting ghost number conservation and the identity (115), translat-

2129

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

ing connected into one-particle irreducible quantities as in Eq. (140) and finally introducing the momentum space vertex functions

(125) and (126) leads to "



bj ck dl em dE ðyÞ dJ ðxÞ dJ ðxÞ dJ ðxÞ d3 W

d

F

G

H

¼

Z

Z

D

d k

ð2pÞ

J¼I¼0

eikðxyÞ

D

0

00

00

0

 ðk  k

D 0

0 0

ðHG0 Þed0

 ðk  k Þ Gml0

0

ðFG0 Þcd0

0

00

0

0

ðGG0 Þdd0

 ðk  k  k Þ Gll0 0

ðGH0 Þde0

00

0

0

0

0

ðHF 0 Þec0

0

ðFF 0 Þcc0

ðk  k; k Þ þ Gmk0 ðL0 G0 H0 Þh0 d0 e0

00

ðk Þ Cjk0 r0

ðHH0 Þee0

ðk ÞGmm0 #

ðGF 0 Þdc0

ðk Þ Cs0 l0 m0

ðEF 0 K 0 Þbc0 g 0

0

0

ðL0 G0 H0 Þh0 d0 e0

00

ðEF 0 K 0 Þbc0 g 0

0

0

ðk  k; k Þ þ Glk0

ðk Þ Cs0 l0 m0

ðk Þ Cjk0 r0 00

ðL0 K 0 Þh0 g 0

00

00

ðFH0 Þce0

ðk  k Þ Glm0

 ðk  k ; k ÞGs0 r0 0

ðEF 0 K 0 Þbc0 g 0

0

ðk Þ Cjk0 r0 00

 ðk  k ; k ÞGs0 r0  ðk  k Þ Gkl0

0 0 0

0

ðk  k Þ Gkm0 ðL0 K 0 Þh0 g 0

00

ðGG0 Þdd0

0

ðk  k Þ Gll0

G H Þh d e CðL s0 l0 m 0

ðL0 K 0 Þh0 g 0

00

0

ðFF 0 Þcc0

þ Gkk0

D

ð2pÞ ð2pÞ

 ðk  k ; k ÞGs0 r0 00

d k

D

00 ðHH0 Þee0 Þ Gmm0 ðk Þ

0

D 00

d k

00

0

ðk  k; k Þ þ Gkk0 ðEF 0 G0 H0 Þbc0 d0 e0

ðk Þ Cjk0 l0 m0

00

 ðk þ k  k; k ; k Þ;

ð143Þ

Exploiting the identities (142) and (143) one can now readily obtain the Dyson–Schwinger equation of the PðkkÞ self-energy component from Eq. (139). One finds

1 2

PðkkÞab ðkÞ ¼  ij

Z

D 0

d k

0

D

ðAG0 Þcc0

0

ðgÞV acd ikl ðk  k ; k Þ Gkk0

ð2pÞ Z D 0 D 00 1 d k d k 2 ig V acde  iklm D 2 ð2pÞ ð2pÞD ðAF 0 Þcc0

 Gkk0

ðAG0 Þdd0

0

ðk  k Þ Gll0

ðL0 G0 H0 Þh0 d0 e0

00

0

00

ðAH0 Þee0

ðL0 K 0 Þh0 g 0

0

0

ðk  k Þ Gmm0 00

0

ðAH0 Þdd0

ðk  k Þ Gll0

ðkG0 H0 Þbc0 d0

0

ðk Þ  Cjk0 l0

0

0

ðk  k; k Þ

00

ðk Þ ðkF 0 K 0 Þbc0 g 0

0

0

ðk  k ; k ÞGs0 r0 ðk Þ Cjk0 r0 ðk  k; k Þ  Cs0 l0 m0 Z D 0 D 00 1 d k d k 0 00 0 00 2 ðAF 0 Þcc0 ðAG0 Þdd0 ðAH0 Þee0 ig V acde ðk  k  k Þ Gll0 ðk Þ Gmm0 ðk Þ  iklm Gkk0 6 ð2pÞD ð2pÞD ðkF 0 G0 H0 Þbc0 d0 e0

 Cjk0 l0 m0

0

00

0

00

ðk þ k  k; k ; k Þ

ð144Þ

V acde iklm

where we have used the symmetry of the vertex in the last three pairs of indices to combine the first three terms arising from Eq. (143) into one. We have illustrated Eq. (144) in Fig. 2. 4.2.4. DSE for PðkAÞ ðkÞ Taking the derivative of Eq. (100) with respect to Abj ðyÞ instead of kbj ðyÞ and afterwards setting the sources to zero leads to the Dyson–Schwinger equation for the PðkAÞ self-energy component, namely ðkAÞab

d2 C

bj ai dk ðxÞ dA ðyÞ

J¼I¼0

ðD1 Þij

ðx;yÞ

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ h i r ¼ i dab ðþr@ t  DÞ dij þ ð1  Þ @ i @ j dðx  yÞ

j

i ig acd h

rh ik 0 il i h ik d @ l  d @ k þ 2 d @ l  dil @ 0k  1 f 2! j

h ii d d2 W

kl 0 lk þ d @i  d @i

bj ck dl 0 x0 ¼x ðxÞ dJ ðx Þ dJ dA ðyÞ A A J¼I¼0

2

ig acde d d3 W

 V

dl em

3! iklm dAbj ðyÞ dJ ck A ðxÞ dJ A ðxÞ dJ A ðxÞ J¼I¼0

2

2 ig acdb d W

 dðx  yÞ ck V

2! iklj dJ A ðxÞ dJ dl A ðxÞ J¼I¼0

ð145Þ

Reading off the self-energy component by comparing with Eq. (118), and using Eqs. (142) and (143) one arrives at

2130

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

Fig. 2. Dyson–Schwinger equation of the PðkkÞ self-energy component, Eq. (144).

PðkAÞab ðkÞ ¼  ij 

1 2

Z

1 2

D 0

d k

ð2pÞ Z

0

D

0

ðAG0 Þcc0

ðgÞV acd ikl ðk  k ; k Þ Gkk0

D 0

d k

D 00

d k

D

ð2pÞ ð2pÞ ðL0 G0 H0 Þh0 d0 e0

00

ðAF 0 Þcc0

2

ig V acde iklm Gkk0

D 0

00

0

0

ðAG0 Þdd0

ðk  k Þ Gll0

ðL0 K 0 Þh0 g 0

0

ðAH0 Þdd0

ðk  k Þ Gll0

ðAF 0 K 0 Þbc0 g 0

0

ðAG0 H0 Þbc0 d0

0

ðk Þ  Cjk0 l0 ðAH0 Þee0

00

ðk  k Þ Gmm0 0

0

0

ðk  k; k Þ

00

ðk Þ

0

 Cs0 l 0 m 0 ðk  k ; k Þ Gs0 r0 ðk Þ Cjk0 r0 ðk  k; k Þ Z D 0 D 00 1 d k d k 0 00 0 00 2 ðAF 0 Þcc0 ðAG0 Þdd0 ðAH0 Þee0 ig V acde ðk  k  k Þ Gll0 ðk Þ Gmm0 ðk Þ  iklm Gkk0 6 ð2pÞD ð2pÞD ðAF 0 G0 H0 Þbc0 d0 e0

0

00

0

00

 Cjk0 l0 m0 ðk þ k  k; k ; k Þ Z D 0 1 d k 2 ðAAÞcd 0 ig V abcd ðk Þ  ijkl Gkl 2 ð2pÞD which is depicted in Fig. 3.

Fig. 3. Dyson–Schwinger equation of the PðkAÞ self-energy component, Eq. (146).

ð146Þ

2131

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

4.2.5. DSE for PðAkÞ ðkÞ From the gauge field equation (106) one obtains by taking the derivative with respect to kbj ðyÞ for vanishing sources ðAkÞab

ðD1 Þij

d C

dAai ðxÞ dkbj ðyÞ

ðx;yÞ

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ h

r i @ i @ j dðx  yÞ ¼ i dab ðr@ t  DÞ dij þ 1 

2

J¼I¼0

j

h

i h i rh ik 0 kl acd d @ l þ d ð@ i þ @ 0i Þ þ 2 dkl @ 0i þ dik ð@ l þ @ 0l Þ  igf  1 j

h ii d d2 W

il 0 il 0  d @ k þ d ð@ k þ @ k Þ

bj ck dl 0 x0 ¼x dk ðyÞ dJ ðxÞ dJ ðx Þ A

k

J¼I¼0

2

ig eacd d d3 W

 V mikl bj

em ck dl 2! dk ðyÞ dJ k ðxÞ dJ A ðxÞ dJ A ðxÞ J¼I¼0

2

ig bacd d2 W

 V jikl dðx  yÞ ck

2! dJ A ðxÞ dJ dl A ðxÞ J¼I¼0

g d d2 W

 f acd @ i bj

0 c d 0 j dk ðyÞ dJ x ðx Þ dJ x ðxÞ x ¼x

ð147Þ

J¼I¼0

As in Eq. (146), we have again a self-energy, a tadpole, and terms of the type in Eqs. (142) and (143), but we also have a new term involving gauge ghosts. It can be calculated in a similar way to the previous cases, and comes out to



bj c d 0 dE ðyÞ dJ  ðx Þ dJ ðxÞ d2 W

d

x

x

¼

Z

D

d k

D 0

d k

D

ð2pÞ ð2pÞD

J¼I¼0

0

0

eikðxyÞ eik ðxx Þ GðxÞ c c ðk Þ

0

0

 EÞd0 c0 b ðxx

 GðxÞ dd ðk  k Þ Cj 0

0

0

0

ðk  k; k Þ

ð148Þ

With this, and the previous identities, Eq. (147) can be written as

PðAkÞab ðkÞ ¼  ij

Z

D 0

d k

ð2pÞD

ðkAÞcc0

0

ðgÞV cda kli ðk ; kÞ Gkk0

ðkAH0 Þbc0 d0

0

0

ðAH0 Þdd0

ðk  k Þ Gll0

0

ðk Þ

0

ðk  k; k Þ  Cjk0 l0 Z D 0 D 00 d k d k 0 0 00 00 2 ðAF 0 Þcc0 ðAG0 Þdd0 ðkAÞee0 ig V eacd ðk  k Þ Gll0 ðk  k Þ Gmm0 ðk Þ  mikl Gkk0 D D ð2pÞ ð2pÞ ðL0 G0 AÞh0 d0 e0

00

0

00

ðL0 K 0 Þh0 g 0

0

ðkF 0 K 0 Þbc0 g 0

0

0

ðkAK 0 Þbc0 g 0

0

0

ðk  k ; k Þ Gs0 r0 ðk Þ Cjk0 r0 ðk  k; k Þ  Cs0 l0 m0 Z D 0 D 00 1 d k d k 0 0 00 00 2 ðkAÞec0 ðAG0 Þcd0 ðAH0 Þde0  ig V eacd ðk  k Þ Gkl0 ðk  k Þ Glm0 ðk Þ mikl Gmk0 2 ð2pÞD ð2pÞD ðL0 G0 H0 Þh0 d0 e0

00

0

00

ðL0 K 0 Þh0 g 0

0

 Cs0 l0 m0 ðk  k ; k Þ Gs0 r0 ðk Þ Cjk0 r0 ðk  k; k Þ Z D 0 D 00 0 0 1 d k d k 0 00 0 00 2 ðAF Þcc ðAG0 Þdd0 ðkAÞee0  ig V eacd ðk  k  k Þ Gll0 ðk Þ Gmm0 ðk Þ mikl Gkk0 D D 2 ð2pÞ ð2pÞ ðkF 0 G0 AÞbc0 d0 e0

0

00

0

00

ðk þ k  k; k ; k Þ  Cjk0 l0 m0 Z D 0 0 0 d k ig cda  kÞd0 c0 b 0 0 0 0 0 ðxx f ðk  k Þi GðxÞ c c ðk Þ GðxÞ dd ðk  k Þ Cj ðk  k; k Þ þ ð2pÞD j Z D 0 1 d k 2 ðAAÞcd 0 ig V bacd ðk Þ  jikl Gkl 2 ð2pÞD A graphical representation of this identity can be found in Fig. 4.

ð149Þ

2132

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

4.2.6. DSE for PðAAÞ ðkÞ ðAAÞab ¼ 0, one has in this case Finally, we come to the pure gauge field component PðAAÞ . Because ðD1 Þij ðAAÞab ðx; yÞ ij

P



¼ ai

dA ðxÞ dAbj ðyÞ d2 C

ð150Þ J¼I¼0

and thus one obtains from Eq. (106) the final identity

PðAAÞab ðkÞ ¼  ij 

Z

D 0

d k

Z

ðkAÞcc0

0

D

ð2pÞ

ðgÞV cda kli ðk ; kÞ Gkk0

D 0

d k

D 00

d k

ðAF 0 Þcc0

2

D

ð2pÞ ð2pÞD

ig V eacd mikl Gkk0

ðAkÞdd0

0

ðk  k Þ Gll0 0

ðAG0 Þdd0

ðk  k Þ Gll0

ðL0 G0 AÞh0 d0 e0

00

0

00

ðL0 K 0 Þh0 g 0

ðAG0 H0 Þh0 d0 e0

00

0

00

ðAkÞh0 g 0

0

ðAAkÞbc0 d0

0

ðk Þ  Cjk0 l0 0

ðkAÞee0

00

0

0

ðk  k; k Þ 00

ðk  k Þ Gmm0 ðk Þ

ðAF 0 K 0 Þbc0 g 0

0

0

 Cs0 l 0 m 0 ðk  k ; k Þ Gs0 r0 ðk Þ Cjk0 r0 ðk  k; k Þ Z D 0 D 00 1 d k d k 0 0 00 00 2 ðkAÞec0 ðAG0 Þcd0 ðAH0 Þde0 ig V eacd ðk  k Þ Gkl0 ðk  k Þ Glm0 ðk Þ  mikl Gmk0 2 ð2pÞD ð2pÞD 0

ðAAkÞbc0 g 0

0

0

 Cs0 l 0 m 0 ðk  k ; k Þ Gs0 r0 ðk Þ Cjk0 r0 ðk  k; k Þ Z D 0 D 00 1 d k d k 0 00 0 00 2 ðAF 0 Þcc0 ðAG0 Þdd0 ðkAÞee0 ig V eacd ðk  k  k Þ Gll0 ðk Þ Gmm0 ðk Þ  mikl Gkk0 2 ð2pÞD ð2pÞD ðAF 0 G0 AÞbc0 d0 e0

0

00

0

00

 Cjk0 l0 m0 ðk þ k  k; k ; k Þ Z D 0 0 0 d k ig cda  AÞd0 c0 b 0 0 0 0 0 ðxx f ðk  k Þi GðxÞ c c ðk Þ GðxÞ dd ðk  k Þ Cj ðk  k; k Þ þ ð2pÞD j Z D 0 d k 2 ðkAÞec 0 ig V eacb ðk Þ  mikj Gmk ð2pÞD

Fig. 4. Dyson–Schwinger equation of the PðAkÞ self-energy component, Eq. (149).

ð151Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2133

which completes our derivation of the Dyson–Schwinger equations in Bödeker’s effective theory. 5. Discussion and outlook In this work we have constructed an analytic approach to the non-perturbative physics encoded in Bödeker’s effective theory [1]. Our approach is based on Dyson–Schwinger equations and allows for an investigation of the non-perturbative dynamics of soft, non-abelian hot gauge fields that is independent of the existing lattice studies of Bödeker’s theory [17,18] (see Fig. 5). The basic starting point is to transform Bödeker’s Langevin equation into a path integral. From this path integral, in principle, one could deduce the Dyson–Schwinger equations. However, it would hardly be avoidable to introduce an uncontrolled gauge dependence when finally truncating these equations. To control this gauge dependence, we therefore enlarged the system by the introduction of gauge ghosts (which is optional in stochastic quantisation). This enlarged system is endowed with a BRST symmetry reflecting the gauge invariance; and we have derived the corresponding Ward– Takahashi identities. A consistent truncation of the Dyson–Schwinger equations is achieved if the gauge and ghost sectors are truncated in accordance with these identities. We also derived a second class of restrictions, so-called stochastic Ward identities known from stochastic quantisation [15]. These reflect the characteristic structure of the path integral action induced by its origin in a stochastic differential equation. Finally, we have deduced the Dyson–Schwinger equations of the theory. They contain, in principle, the possibility of (finite!) vertices coupling auxiliary fields to gauge ghosts or gauge field/auxiliary field vertices with more than one auxiliary field, both of which are not present at tree level. Whether these vertices are really non-zero, will be an interesting question to be decided by an implementation of our formalism. In combination with the gauge and stochastic Ward identities given in Eqs. (C.25)–(C.27), the Dyson–Schwinger Eqs. (136), (137), (144), (146), (149) and (151) provide all the necessary tools for an analytic study of the non-perturbative physics encoded in Bödeker’s effective theory. In particular, it can be used to study the sphaleron rate Eq. (1), where N CS in terms of the gauge field takes the form

Fig. 5. Dyson–Schwinger equation of the PðAAÞ self-energy component, Eq. (151).

2134

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

NCS ðt 2 Þ  N CS ðt1 Þ ¼

Z

t2

t1

dt

Z

3

d x

g2 a a E B ðxÞ 8p 2 i i

ð152Þ

Restricting to the lowest correlators, we are then interested in the unequal time correlators hEai ðx1 ÞEbj ðx2 Þi, hEai ðx1 ÞBbj ðx2 Þi and hBai ðx1 ÞBbj ðx2 Þi. The first one should approach a delta function, while the second one should be subleading [19]. It would be a good test of our ansatz if we could (roughly) reproduce the factor in front of the sphaleron rate [17,18]. We close this discussion with a few comments on what is to come. Since for the hot sphaleron rate we are interested primarily in the infrared behaviour of the theory, the first thing to be done moving forward is determining the appropriate relation between the anomalous dimensions for k0 and j kj2 . This can be done by investigating the limit when k0 ! 0, and comparing with the anomalous dimension in Yang–Mills theory in three dimensions [20]. One can also analyse the importance of the ansatz for the vertex functions by comparing with time-independent stochastic quantisation [21]. Acknowledgments A.H. is supported by CONACYT/DAAD, Contract No. A/05/12566. M.G.S. gratefully acknowledges very useful discussions with Tomislav Prokopec during the first part of this project. Appendix A. Calculation of Jacobians Throughout this work, there appear several times Jacobians as products of change of variables. As is well known from the literature [12], we have claimed that they are constants and have generally absorbed them in the measure. To make this work more self-contained, we provide here a derivation of this claim. In order to simplify the expressions, we will suppress the colour and space indices until it becomes necessary. The first Jacobian that we encountered was in Eq. (5), where the following expression appears

Det



dE dA

 ðA:1Þ

with

!  1 dE½A @ 1 dK½A @ 1 @ dK½A ¼ þ ¼ 1þ dA @t 2 dA @t 2 @t dA

ðA:2Þ

where K contains all the terms in the left-hand side of Eq. (12) without time derivatives. The kernel of the operator ð@=@tÞ1 is constrained by causality to be Hðt2  t1 Þ. We then have

Det

!   1 dE 1 @ dK½A ¼ const:  Det 1 þ dA 2 @t dA



ðA:3Þ

with

"   #ab Z 1 1 @ dK½A 1 dK a ½Aðt00 ; xÞ 00 ðt; x; t 0 ; x0 Þ ¼ dt Hðt  t00 Þ i b 0 2 @t dA 2 dAj ðt ; x0 Þ

ðA:4Þ

ij

Since K contains no time derivatives, the functional derivative produces a delta function in the time variable, i.e.

dK ai ½Aðt 00 ; xÞ dAbj ðt 0 ; x0 Þ

¼ dðt 00  t0 Þ

dx K ai ½Aðt 0 ; xÞ dx Abj ðt0 ; x0 Þ

ðA:5Þ

where we have introduced the symbol dx to denote a variation with respect to the x dependence only. Hence, we find

2135

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

"   # 1 1 @ dK½/ 1 dx K ai ½/ðt 0 ; xÞ ðt; x; t0 ; x0 Þ ¼ Hðt  t 0 Þ 2 @t d/ 2 dx Abj ðt0 ; x0 Þ

ðA:6Þ

ab

Coming back to Eq. (A.3) and using Tr lnð. . .Þ ¼ ln Detð. . .Þ in addition to the series expansion of the logarithm, the determinant takes the form

"  #n ) (   1 1 X dE½A ð1Þnþ1 1 @ dK½A Det Tr ¼ const:  exp dA @t dA n 2n n¼1

ðA:7Þ

The trace in this expression can be evaluated with the help of Eq. (A.6). One obtains

"  #n Z 1 dx K ai11 ½Aðt 2 ; x1 Þ @ dK½A D1 D1 ¼ dt1    dt n d x1    d xn Hðt 1  t 2 Þ @t dA dx Aai 2 ðt 2 ; x2 Þ

Tr

2

 Hðt 2  t3 Þ

dx K ai22 ½Aðt3 ; x2 Þ dx Aai33 ðt 3 ; x3 Þ

   Hðtn  t1 Þ

dx K ainn ½Aðt1 ; xn Þ dx Aai11 ðt 1 ; x1 Þ

ðA:8Þ

and thus

"  #n Z 1 @ dK½/ ¼ dt 1    dtn Hðt 1  t 2 Þ Hðt2  t 3 Þ    Hðtn  t1 Þ f n ðt 1 ; t2 ; . . . ; t n Þ @t d/

Tr

ðA:9Þ

if we set

fn ðt 1 ; t2 ; . . . ; tn Þ ¼

Z

d

D1

D1

x1    d

xn

dx K ai11 ½Aðt 2 ; x1 Þ dx K ai22 ½Aðt 3 ; x2 Þ dx Aai22 ðt 2 ; x2 Þ

dx Aai33 ðt3 ; x3 Þ



dx K ainn ½Aðt1 ; xn Þ dx Aai11 ðt1 ; x1 Þ

for abbreviation. Unless n ¼ 1, however, the expression (A.9) vanishes for any function fn . Therefore, only the first term of the sum in Eq. (A.7) survives and we finally arrive at

(

)   Z dE½A 1 d K a ½Aðt; xÞ

Det Hð0Þ dt dD1 x x i a ¼ const:  exp dA 2 dx Ab ðt; x0 Þ x0 ¼x

ðA:10Þ

Our next task is to calculate the functional derivative of K ai ½A. To this end, it is easiest to write it down in components which clarifies the structure

h

i h

i 1 a r r abc 1  Abi @ j Acj þ 2Acj @ j Abi þ Abj @ i Acj þ 1  @ i @ j  dij D Aaj K i ½A ¼ gf 2 j j þ g 2 f abc f bde Acj Adj Aei

ðA:11Þ

Obviously, the first term, i.e. the term quadratic in the gauge field, does not contribute to the functional derivative with respect to Aai because it always produces a dab or dac that is contracted with the structure constants f abc in front of the square bracket. The linear term, on the other hand, only contributes a constant that can be absorbed into the constant in Eq. (A.10). Thus, we only have to take care of the third order term which leads to

    Z dE½A g2 D1 Det ¼ const:0  exp C A ðD  2Þ Hð0Þ dD1 ð0Þ dt d x Aa ðt; xÞ  Aa ðt; xÞ dA r

ðA:12Þ

where f acd f bcd ¼ C A dab as usual. However, in dimensional regularisation dD1 ð0Þ gives zero as a consequence of the general rules of D-dimensional integration, and the determinant is simply a constant. We came across another determinant in Eq. (34)

 x  d f Det df One finds

ðA:13Þ

2136

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

dx fai ðt; xÞ bj

df ðt

0

; x0 Þ

h

d

acd

¼ dab dij dðt  t 0 Þ dD1 ðx  x0 Þ þ gf

bj

df ðt

0

; x0 Þ

i

xd ½fðt; xÞ fci ðt; xÞ

ðA:14Þ

and thus because x is infinitesimal

" #  x  Z h i d f d D1 acd ci d Det x ½fðt; xÞ f ðt; xÞ ¼ 1 þ dt d x gf t 0 ¼t df dfai ðt0 ; x0 Þ

ðA:15Þ

x0 ¼x

The functional derivative acting on fci produces a dac and therefore does not contribute because the Kronecker delta is contracted with the structure constants. To determine the remaining functional derivative of xd ½f, let us formally integrate Eq. (27)

xa ðt; xÞ ¼ xa ð1; xÞ 

Z

t

00

dt ½H½Axa ðt 00 ; xÞ 

1

Z

t

00

dt dva ½Aðt 00 ; xÞ

ðA:16Þ

1

Since H½A and dva ½A are local functionals in time, this equation for x has a causal character, i.e. xðt; xÞ does only depend on the values of the gauge field Aðt 00 ; xÞ at times t00 < t. On the other hand, Eq. (26) leads to

rAa ðt; xÞ ¼ rAa ð1; xÞ 

Z

t

dt

00

h

Z i Dab  Bb þ rDab vb ½A ðt 00 ; xÞ þ

1

t

00

dt fa ðt 00 ; xÞ

ðA:17Þ

1

and Aðt; xÞ itself only depends on the stochastic force fðt00 ; xÞ for t00 < t. Hence, neither Aðt; xÞ nor xðt; xÞ have a dependence on fðt00 ; xÞ unless t00 < t and in taking the functional derivative of Eq. (A.16), we can restrict the integration range accordingly

dxa ½fðt; xÞ bi

df ðt

0

; x0 Þ

¼

Z

t

t0

dt

00

d½H½Axa ðt00 ; xÞ bi

df ðt

0

; x0 Þ



Z t0

t

dt

00

d dva ½Aðt00 ; xÞ

ðA:18Þ

dfbi ðt0 ; x0 Þ

Evaluating this relation for t ¼ t 0 as in Eq. (A.15) leads to

dxa ½fðt; xÞ

dfbi ðt0 ; x0 Þ

¼0

ðA:19Þ

t¼t 0

The only way to escape this conclusion would be an integrand that is singular in time. However, if dx=df appearing under the integral in Eq. (A.18) was singular, the integrated expression would be finite which again is dx=df. Therefore, dx=df cannot be singular. dA=df on the other hand cannot be singular neither because of the same argument applied to the functional derivative of Eq. (A.17) with respect to f. Thus, we conclude

Det

 x  d f ¼1 df

ðA:20Þ

which completes the proof. During the introduction of gauge ghosts to the path integral, Eq. (42), there appears in our work another Jacobian. We can see that it has the same form as the one we have already calculated, but with

1 a 1 K ½x; Aðt; xÞ ¼  ðDab  rxb Þðt; xÞ 2 j

ðA:21Þ

Hence, we can rely on our general result for the determinant, Eq. (A.10),

(   Z ab

b dc½x; A 1 D1 dx ðD  rx Þðt; xÞ ¼ const:  exp  Hð0Þ dt d x Det

a 0

dx dx x ðt; x Þ j

) ðA:22Þ x0 ¼x

The functional derivative with respect to spacial variations is given by

dx ðDab  rxb Þðt; xÞ abc ¼ ðdab r  gf Ac Þ  rdD ðx  x0 Þ dbd dx xd ðt; x0 Þ

ðA:23Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2137

and thus, evaluated for d ¼ a, gives a constant because the A dependent contribution is set to zero due to the antisymmetry of the structure constants. Note that, this time, we did not have to rely on dimensional regularisation to proof the constancy of the determinant as we had to in the case of DetðdE½A=dAÞ. When we performed the BRST transformation in our derivation of the Ward identities, Eq. (49), one more type of determinant appeared. In general, if xa are Grassmann even and #i Grassmann odd quantities, a mixed change of variables of the form

xa ¼ x0a þ e f a ðx0 ; #0 Þ

ðA:24Þ

#i ¼ #0i þ e /i ðx0 ; #0 Þ with

e being a Grassmann odd parameter leads to a Jacobian J ¼ 1 þ e strðMÞ

ðA:25Þ

In this expression, the matrix M under the super trace is given by

 M¼

A

B

C

D



0 @fa @x0

¼ @ @/

b i

@x0a

@fa  @# 0

1 A

ðA:26Þ

@fa @/i þ @x0a @#0i

ðA:27Þ

i

i  @/ @#0 j

and hence

strðMÞ ¼ trðAÞ  trðDÞ ¼

(See, e.g. [12], Section 1.8.2. Note, however, that in our case e is Grassmann odd which leads to the additional minus signs in the matrix M when e is commuted with the derivative @=@#). In our case, we have two sets of commuting variables, Aai ðxÞ and kai ðxÞ, and two sets of anti-com a ðxÞ. Therefore, the Jacobian is given by muting ones, xa ðxÞ and x

J ¼1þe

Z

" #  0a ðxÞ dsA0ai ðxÞ dsk0ai ðxÞ dsx0a ðxÞ dsx dx þ þ 0ai þ  0a ðxÞ dx0a ðxÞ dx dk ðxÞ dA0ai ðxÞ

ðA:28Þ

However, any of these functional derivatives vanishes as a short glance at the BRST transformed fields in Eq. (53) makes obvious: The derivative always produces a Kronecker delta that is to be contracted with the structure constants. Consequently, the Jacobian of the change of variables (49) is unity. Appendix B. Feynman rules The action, as given by Eq. (46), is

  ¼ SðDÞ ½A; k þ SðGGÞ ½A; x; x  ; S½A; k; x; x

ðB:1Þ

with

    1 dx rT ka  ka  ika  Dab  Bb þ r A_ a  Dab r  Ab j  Z 1 a ab ðGGÞ a a b _ þ x  D  rx   ¼ dx  x  x S ½A; x; x SðDÞ ½A; k ¼

Z

j

ðB:2Þ ðB:3Þ

B.1. The propagators The free, quadratic part of the dynamical action SðDÞ ½A; k can be cast into the following symmetric form reflecting the mixing that occur between the gauge field A and the auxiliary field k ðDÞ S0 ½A; k

¼

Z

1 ^ 1 Þab ðx; yÞ dxdy ðkai ðxÞ; Aai ðxÞÞ ðD ij 2

kbj ðyÞ Abj ðyÞ

! ðB:4Þ

2138

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

with the matrix

0

^ 1 Þab ðx; yÞ ¼ @ ðD ij

ðkkÞab

ðx; yÞ

ðAkÞab

ðx; yÞ ðD1 Þij

ðD1 Þij ðD1 Þij

ðkAÞab

ðx; yÞ

ðAAÞab

ðx; yÞ

ðD1 Þij

1 A

ðB:5Þ

and ðkkÞab

ðx; yÞ ¼ 2rT dab dij dðx  yÞ h

r i ðkAÞab @ @ dðx  yÞ ðD1 Þij ðx; yÞ ¼ i dab ðþr@ t  DÞ dij þ 1  j i j i h

r ðAkÞab ðD1 Þij @ i @ j dðx  yÞ ðx; yÞ ¼ i dab ðr@ t  DÞ dij þ 1  ðD1 Þij

ðAAÞab ðD1 Þij ðx; yÞ

ðB:6Þ ðB:7Þ ðB:8Þ

j

¼0

ðB:9Þ

We denote by non-bold symbols combinations of time and space ^ 1 is symmetric in the following sense dðx  yÞ ¼ dðt x  t y Þ dD1 ðx  yÞ. The matrix D ðFGÞab

ðD1 Þij

ðGFÞba

ðx; yÞ ¼ ðD1 Þji

ðy; xÞ

^ ab ðx; yÞ ¼ @ D ij

DðkkÞab ðx; yÞ ij

e.g.

ðB:10Þ

Hence, the matrix propagator

0

variables,

DðkAÞab ðx; yÞ ij

ðAAÞab DðAkÞab ðx; yÞ Dij ðx; yÞ ij

0D

D

A¼B @D

D

1

E kai ðxÞ kbj ðyÞ E0 ai bj A ðxÞ k ðyÞ

0

kai ðxÞ Abj ðyÞ

E 1

C E0 A A ðxÞ A ðyÞ ai

ðB:11Þ

bj

0

is given by its inverse

Z

D

ðFGÞab

d y Dij

ðGHÞbc

ðx; yÞ ðD1 Þjk

ðy; zÞ ¼ dac dik dFH dD ðx  zÞ

ðB:12Þ

or equivalently ðGHÞbc DðFGÞab ðkÞ ðD1 Þjk ðkÞ ¼ dac dik dFH ij

ðB:13Þ

for the momentum space functions

DðFGÞab ðx; yÞ ¼ ij

Z

D

d k

ðFGÞab

eikðxyÞ Dij ðkÞ ð2pÞD Z D d k ikðxyÞ 1 ðFGÞab ðFGÞab ðx; yÞ ¼ e ðD Þij ðkÞ ðD1 Þij ð2pÞD

ðB:14Þ ðB:15Þ

Note again that though we are most of the time dealing with three-vectors, in the Fourier transform we use four-vector notation, i.e. eikðxyÞ ¼ eik0 ðx0 y0 ÞþikðxyÞ leading to

0

h i1 2 i dab ðirk0 þ k Þ dij  ð1  rjÞ ki kj C ^ 1 Þab ðkÞ ¼ B h i ðD @ A: ij 2 0 i dab ðþirk0 þ k Þ dij  ð1  rjÞ ki kj 2rT dab dij

ðB:16Þ In momentum space, the gauge/auxiliary field propagators are given by:

DðkkÞab ðkÞ ¼ 0 ij DðkAÞab ðkÞ ¼ ij DðAkÞab ðkÞ ¼ ij

idab þirk0 þ jkj2 idab

" "

r dij þ 1 

ki kj

r dij þ 1 

ki kj

j þirk0 þ rj jkj2

# #

j irk0 þ rj jkj2 irk0 þ jkj2 " #   2rTdab r2 ki kj jkj2 DðAAÞab ðkÞ ¼ d þ 1  ij ij j2 r2 k20 þ rj22 jkj4 r2 k20 þ jkj4

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2139

For the gauge ghosts, we have the corresponding contribution to the action, Eq. (B.3), comprises the free part ðGGÞ

S0

 ¼ ½ x; x

  1  a @ t þ D xa dx x

Z

ðB:17Þ

j

and therefore

  1 ðD1 ÞðxÞ ab ðx; yÞ ¼ dab @ t þ D dðx  yÞ

ðB:18Þ

j

or in momentum space

  1 ðD1 ÞðxÞ ab ðkÞ ¼ dab ik0  jkj2

ðB:19Þ

j

Hence the gauge ghost propagator is given by

DðxÞ ab ðkÞ ¼

j dab ijk0  jkj2

B.2. The vertices For the interacting part of the dynamical action (B.2) we have ðDÞ

Sint ½A; k ¼

Z

n h

i r abc dx igf kai 1  Abi @ j Acj þ 2Acj @ j Abi þ Abj @ i Acj j o 2 abc bde ai cj dj ei  ig f f k A A A

ðB:20Þ

Thus, the theory provides a three-point vertex containing one auxiliary and two gauge fields and a four-point vertex of three gauge fields and one auxiliary field. To simplify explicit calculations, it is useful to symmetrise the vertices with respect to the two and three gauge fields in either case. SplitðDÞ ting Sint ½A; k into the contributions corresponding to the 3- and four-point vertex ðDÞ

ð DÞ

Sint ½A; k ¼ Sint;3 ½A; k þ S

ðDÞ int;4 ½A; k

ðB:21Þ

one obtains ðDÞ

Sint;3 ½A; k ¼

ðDÞ

Sint;4 ½A; k ¼

Z

Z

n

1 rh ij bj ck ik ck bj i ðigÞf abc kai 1  d A @kA  d A @jA 2! j i h þ 2 dij Ack @ k Abj  dik Abj @ j Ack h io þ djk Abj @ i Ack  dkj Ack @ i Abj dx

dx

1 2 ai bj ck dl ðig Þ V abcd ijkl k A A A 3!

ðB:22Þ ðB:23Þ

where ace bde ij kl V abcd f ðd d  dil dkj Þ þ f abe f cde ðdik djl  dil djk Þ þ f ade f bce ðdij dkl  dik djl Þ ijkl ¼ f ðDÞ

ðB:24Þ

Observing that there is an additional minus sign because we have S ½A; k in the exponent of the generating functional and noting our conventions of the Fourier transform (B.14) of the propagators, we find for the three-point vertex in momentum space that is symmetrised with respect to the two A fields

2140

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

kA2 vertex

abc  g V abc ijk ðk2 ; k3 Þ ¼ gf

( 

1  rj

  ij k j d k3  dik k2



 k j i i þ2 dij k2  dik k3 þ djk k3  k2

)

Momentum conservation is thereby to be understood. By construction, the object V abc ijk ðk2 ; k3 Þ is symmetric in the last two pairs of indices (and corresponding momenta), i.e. acb V abc ijk ðk2 ; k3 Þ ¼ V ikj ðk3 ; k2 Þ

ðB:25Þ

Analogously, the symmetrised four-point vertex is found to be

kA3 vertex

( ig

2

V abcd ijkl

¼ ig

2

f ace f bde ðdij dkl  dil dkj Þ þ f abe f cde ðdik djl  dil djk Þ )

þf

ade bce

f

ij kl

ik jl

ðd d  d d Þ

where V abcd ijkl was already introduced in Eq. (B.24) and is symmetric in the last three pairs of indices abdc acbd acdb adbc adcb V abcd ijkl ¼ V ijlk ¼ V ikjl ¼ V iklj ¼ V iljk ¼ V ilkj

ðB:26Þ

For the ghost sector, the corresponding interaction term extracted from Eq. (B.3) is given by ðGGÞ

 ¼ Sint ½A; x; x

Z

dx

ðgÞ

j

   a Ac  r xb ¼ f abc x

and leads to the momentum space vertex

 xxAvertex

ig

j

k

f abc k2

Z

dx

ðgÞ

j

 a Ack @ k xb f abc x

ðB:27Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2141

Appendix C. Explicit consequences of identities to lower n-point functions We will now find explicit identities for the lower n-point function from the identities obtained in Section 3. C.1. One-point functions Let us start by explicitly writing down the consequences of Ghost number conservation, Eqs. (85)– (87), to the one-point functions of the theory. Taking the functional derivative of Eq. (86) with respect to one of the sources J x , J x , IsA , Isk or Isx and evaluating for J ¼ I ¼ 0 yields

dW½J;I

dJax ðxÞ J¼I¼0

dW½J;I

ai dI ðxÞ sk

J¼I¼0

¼ 0;

dW½J;I

dJ ax  ðxÞ J¼I¼0

¼ 0; dIdW½J;I

a ðxÞ sx

J¼I¼0

¼ 0;



dW½J;I ðxÞ dIai sA

¼0 J¼I¼0

ðC:1Þ

¼0

The same relations follow for the derivatives of Z½J; I from Eq. (85). On the other hand, Eq. (64) implies

dC

dAai ðxÞ J¼I¼0

¼ 0;

dC

dkai ðxÞ J¼I¼0

¼ 0;

dC dxa ðxÞ

J¼I¼0

¼ 0;

dC  a ðxÞ dx J¼I¼0

¼0

ðC:2Þ

and the combination of Eq. (65) and (C.1) gives dC dIai ðxÞ sA



¼ 0;

J¼I¼0

dC

dIai sk ðxÞ J¼I¼0

¼ 0;

dC

dIasx ðxÞ J¼I¼0

¼0

ðC:3Þ

The last first derivative of C can be computed from the stochastic Ward identities, Eq. (78)

dC

¼0 dIasx ðxÞ J¼I¼0

ðC:4Þ

Thus, all first derivatives of C have to vanish. C.2. Two-point functions The consequences of ghost number conservation to the second derivatives of Z½J; I and W½J; I are summarised in the following table, indicating for any pair of sources whether the corresponding second derivative (evaluated for J ¼ I ¼ 0) is restricted to vanish or not by ghost number conservation

ðC:5Þ

 ; I (as well The analogous result for the second derivatives of the 1PI generating functional C½A; k; x; x evaluated for vanishing sources J ¼ I ¼ 0) is

2142

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

ðC:6Þ

In the following, we will often rely on the information summarised in these tables dropping certain terms that are bound to zero by ghost number conservation from our calculations without further notice. To start with, let us recall the gauge Ward identity in terms of the generating functional of connected correlation functions W½J; I. It was found in Eq. (61) to read

Z

" dx

J ai A ðxÞ

dW½J; I dIai sA ðxÞ

þ

J ai k ðxÞ

# dW½J; I dW½J; I a þ J x ðxÞ a ¼0 þ Jx ðxÞ a dIsx ðxÞ dIsx ðxÞ dIai sk ðxÞ

dW½J; I

a

ðC:7Þ

bj Taking second derivatives, a variety of possibilities arise. For instance, choosing d=dJ ai A ðxÞ and d=dJ A ðyÞ yields after setting sources to zero

d2 W½J; I

dJbj ðyÞ dIai ðxÞ A

sA

d2 W½J; I

þ ai

dJ A ðxÞ dIbj sA ðyÞ J¼I¼0

¼0

ðC:8Þ

J¼I¼0

However, due to ghost number conservation both of these terms are zero by themselves. Likewise, the bj b combination of d=dJ ai A ðxÞ with d=dJ k ðyÞ or d=dJ x ðyÞ does not lead to any new relation when ghost number conservation is taken into account. The fourth possibility, however, combining d=dJ ai A ðxÞ and a derivative with respect to J bx ðyÞ, results in the identity



þ ai

b ðxÞ dI ðyÞ dJ

 A sx J¼I¼0



b ai dJ  ðyÞ dI ðxÞ d2 W½J; I

x

sA

d2 W½J; I

¼0

ðC:9Þ

J¼I¼0

that will be further exploited in a moment. Considering the combinations of d=dJ ai k ðxÞ with one of the b ðyÞ or d=dJ ðyÞ again only leads to trivial relations in view of ghost number conservaderivatives d=dJ bj x k b tion. The pairing of d=dJ ai  ðyÞ yields k ðxÞ with d=dJ x



þ ai

b ðxÞ dI ðyÞ dJ

 x k s J¼I¼0

d2 W½J; I

dJb ðyÞ dIai ðxÞ x

sk

d2 W½J; I

¼0

ðC:10Þ

J¼I¼0

However, this relation is a consequence of the two simpler identities

d2 W½J; I

dJb ðyÞ dIai ðxÞ x

sk

J¼I¼0



¼ 0 ai

dJk ðxÞ dIbsx ðyÞ d2 W½J; I

¼0

ðC:11Þ

J¼I¼0

induced by the stochastic Ward identity (75). The remaining possibilities finally, choosing two deriv , or one with respect to x, one to x  again atives with respect to x, two derivatives with respect to x express ghost number conservation only. Hence, up to the level of second derivatives Eq. (C.9) is the only restriction imposed by the gauge BRST symmetry beyond relations that already follow from the stochastic Ward identity or simply are a consequence of ghost number conservation. Implications of

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

2143

the stochastic Ward identities (75) and (78) are most importantly the vanishing of the auxiliary field propagator to all orders ðkkÞab Gij ðx; yÞ

d2 W½J; I

¼ ai

dJ ðxÞ dJbj ðyÞ k

k

¼0

ðC:12Þ

J¼I¼0

or, equivalently, of the ðAAÞ self-energy component ðAAÞab ðx; yÞ ij

P



¼ ai

dA ðxÞ dAbj ðyÞ d2 C

ðAAÞab

 ðD1 Þij

ðx; yÞ ¼ 0

ðC:13Þ

J¼I¼0

where in addition to Eq. (78) it was used that the ðAAÞ component of the inverse free propagator is zero too (cf. Eq. (B.9)). Note that Eq. (C.13) is a special case of the general statement that there are no pure gauge field vertices in the theory: All proper vertex functions of the form ðAA...AÞab...c Cij...k ðx; y; . . . ; zÞ ¼



ai bj ck dA ðxÞ dA ðyÞ    dA ðzÞ dn C

ðC:14Þ

J¼I¼0

vanish as an immediate consequence of the stochastic Ward identity (78). Further implications up to second derivatives (neglecting those only expressing ghost number conservation) are



d2 C

¼ 0

ai ai b b

dx ðyÞ dIsk ðxÞ J¼I¼0 dA ðxÞ dIsx ðyÞ d2 C

¼0

ðC:15Þ

J¼I¼0

together with the equivalent identities (C.11),

dC

dW

¼  ¼0 dIasx ðxÞ J¼I¼0 dIasx ðxÞ J¼I¼0

ðC:16Þ

and for completeness finally



a b dI  ðxÞ dI  ðyÞ d2 C

sx

sx



¼ a

dIsx ðxÞ dIbsx ðyÞ d2 W

J¼I¼0

¼0

ðC:17Þ

J¼I¼0

This last identity, however, does not lead to a simple relation among the lower n-point functions because both of the derivatives act on sources of the BRST transformed fields. In general, to make sense of the above identities we will have to translate the derivatives of the I-type to such with respect to sources of the fundamental fields. For instance, one has

dZ dIai sA ðxÞ

¼ ¼

Z

 b  Dab DADkDxDx i ðxÞ x ðxÞ exp fð. . .Þg

Z

  @i  DADkDxDx

¼ @ i

!!  d d d abc  gf exp fð. . .Þg  dJax ðxÞ dJ ciA ðxÞ dJbx ðxÞ

ðC:18Þ

dZ d2 Z abc þ gf a ci dJ x ðxÞ dJ A ðxÞ dJ bx ðxÞ

where the dots abbreviate the usual exponent of the generating functional as given in Eq. (56). Expressing this identity in terms of W ¼ ln Z yields

dW d2 W dW dW abc þ gf þ ¼ @ i a ai ci dJ x ðxÞ dIsA ðxÞ dJ A ðxÞ dJ bx ðxÞ dJ ciA ðxÞ dJbx ðxÞ dW

Analogously, one obtains after some algebra

! ðC:19Þ

2144

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145

dW dIai sk ðxÞ

¼ gf

d2 W

abc

dJ cik ðxÞ dJ bx ðxÞ

þ

dW

!

dW

ðC:20Þ

dJ cik ðxÞ dJbx ðxÞ

dW 1 abc d2 W dW dW ¼ gf þ c a c dIsx ðxÞ 2 dJ x ðxÞ dJbx ðxÞ dJ x ðxÞ dJbx ðxÞ

! ðC:21Þ

dW dW d2 W dW dW abc ¼ ir@ i ai þ ci  irgf a ci bi dIsx ðxÞ dJ k ðxÞ dJ ðxÞ dJ A ðxÞ dJbi A ðxÞ dJ k ðxÞ k ! d2 W dW dW abc þ gf þ c c dJ x ðxÞ dJbx ðxÞ dJx ðxÞ dJbx ðxÞ

!

ðC:22Þ

With these substitutions Eq. (C.9) translates to

@ i GðxÞ ab ðx; yÞ  ir@ j Gij

ðAkÞab

ðx; yÞ ¼  gf

 AÞbcd ðxx

acd

Wi bcd

þ gf

ðy; x; xÞ  irgf

bcd

ðAAkÞacd

W ijj

ðx; y; yÞ

 AÞcda ðxx Wi ðy; y; xÞ

ðC:23Þ

To further proceed, we express the connected three-point functions by their 1PI counterparts and transform into momentum space. Especially note that we pull out the momentum conserving delta function from the definition of our proper vertices. Hence, only N  1 momentum variables appear  GÞ abc ðxx ðk1 ; k2 Þ where the superscript G is in the argument of a n-point vertex. For instance, we use Cj either the gauge field A or the auxiliary field k and k1 and k2 refer to the (incoming) momenta along ðFGHÞ abc ðk2 ; k3 Þ with the ghost lines leaving and entering the vertex in this order. Accordingly, in Cijk F; G; H 2 fA; kg the two arguments k2 and k3 refer to the incoming momenta along the G and H line, respectively. With these definitions, Eq. (C.23) takes the form

ik GðxÞab ðkÞ þ rk Gij i

j

ðAkÞab

0

ðkÞ ¼ þ GðxÞ b b ðkÞ ðAFÞaa0

 Gii0

ðkÞ

ðAkÞc0 c

ir Gj0 j

Z

D 0

d k

Z

ð2pÞD

gf

acd

gf

bcd

D 0

d k

ð2pÞ

D

ðAGÞdd0

0

ðk ÞGjk0

0

ðAFÞdd0

 FÞc0 b0 d0 ðxx

GðxÞcc ðk ÞGii0

ðk  k Þ Ci0

h

0

0

0

0

 FÞd0 c0 a0 ðxx

GðxÞ c c ðk ÞGðxÞdd ðk  kÞ Ci0 0

ðFGAÞa0 d0 c0

0

ðk  kÞ Ci0 k0 j0

0

i 0 0 ðk  k ;k Þ

0

ðk ;kÞ 0

0

ðk  k ;k Þ ðC:24Þ

The indices F and G in this equation are summation indices taking the two values A and k. However, as we will show now, the stochastic Ward identity leads to a cancellation among some of the terms involved. To this end, let us express also the identities derived from the stochastic Ward identity in the language of full propagators and proper vertex functions. As mentioned above, identity (C.16) relates the normalisations of the gauge ghost and mixed auxiliary/gauge field propagator

gf

abc

Z

D d k h D

ð2pÞ

GðxÞ cb ðkÞ  irGii

ðAkÞcb

i ðkÞ ¼ 0

ðC:25Þ

From the first of Eq. (C.15) one obtains after some relabelling

Z

D 0

d k

ð2pÞD

gf

bcd

ðAkÞc0 c

Gi0 i

0

0

0

0

0

ðk Þ GðxÞ dd ðk  kÞ CðxxAÞd ac i0 ðk  k ; kÞ ¼ 0 0



0

0

ðC:26Þ

from the second equation

Z

D 0

d k

D

ð2pÞ

 ir

Z

gf

bcd

0

 AÞd0 c0 a ðxx

GðxÞ c c ðk Þ GðxÞ dd ðk  kÞ Ci

D 0

d k

ð2pÞD

gf

bcd

0

ðAkÞc0 c

G j0 j

0

ðAkÞdd0

ðk Þ Gjk0

0

0

0

ðk  k ; k Þ ðAkAÞad0 c0

ðk  kÞ Cik0 j0

0

0

ðk  k ; k Þ ¼ 0

ðC:27Þ

C. Zahlten et al. / Annals of Physics 324 (2009) 2108–2145 ðFGHÞabc

2145

ðGHFÞbca

Here we have used Cijk ðk2 ; k3 Þ ¼ Cjki ðk3 ; k2  k3 Þ in accordance with our definition of the vertex functions. Let us now come back to Eq. (C.24), that was found to be the expression of the gauge Ward identity on the level of second derivatives. With the summation index F taking the value A, the second integral in Eq. (C.24) consists of three terms: the one with the two ghost propagators and two copies of the second term corresponding to the two possible values G ¼ k and G ¼ A. The last of these terms is zero because it contains CðAAAÞ . Moreover, the remaining two terms cancel each other due to Eq. (C.27) as a consequence of the stochastic Ward identity. Hence, there is only a contribution of the second integral in Eq. (C.24) for F ¼ k. The first integral, however, contributes for both choices F ¼ k and F ¼ A (and likewise if F is set to k in the second integral, G can still take both values G ¼ k; A). The gauge BRST symmetry therefore leads to the following identity to be obeyed by the full propagators and proper vertex functions of the theory

ik GðxÞ ab ðkÞ þ rk Gij ðkÞ Z D 0 0 0 d k  AÞc0 b0 d0 acd 0 0 0 ðAAÞdd0 ðxx ¼ þGðxÞ b b ðkÞ gf GðxÞ cc ðk Þ Gii0 ðk  k Þ Ci0 ðk ; kÞ D ð2pÞ Z D 0 0 0 d k  kÞc0 b0 d0 acd 0 0 0 ðAkÞdd0 ðxx gf GðxÞ cc ðk Þ Gii0 ðk  k Þ Ci0 ðk ; kÞ þ GðxÞ b b ðkÞ D ð2pÞ Z D 0 h 0 0 d k  kÞd0 c0 a0 bcd 0 0 0 0 ðAkÞaa0 ðxx ðkÞ gf GðxÞ c c ðk Þ GðxÞ dd ðk  kÞ Ci0 ðk  k ; k Þ  Gii0 D ð2pÞ i

j

ðAkÞc0 c

 ir Gj0 j

ðAkÞab

0

ðAAÞdd0

ðk Þ Gjk0

0

ðkAAÞa0 d0 c0

ðk  kÞ Ci0 k0 j0

0

0

ðk  k ; k Þ i 0 0 0 ðAkÞc0 c 0 ðAkÞdd0 ðkkAÞa0 d0 c0 ðk  kÞ Ci0 k0 j0 ðk  k k Þ  ir Gj0 j ðk Þ Gjk0

ðC:28Þ

References [1] D. Bödeker, Phys. Lett. B426 (1998) 351. Available from: ; D. Bödeker, Nucl. Phys. B559 (1999) 502. Available from: . [2] S. Khlebnikov, M. Shaposhnikov, Nucl. Phys. B308 (1988) 885. [3] E. Mottola, S. Raby, Phys. Rev. D42 (1990) 4202. [4] G. Alexanian, V.P. Nair, Phys. Lett. B352 (1995) 435. Available from: . [5] R. Jackiw, S.-Y. Pi, Phys. Lett. B368 (1996) 131. Available from: . [6] J.M. Cornwall, Phys. Rev. D57 (1998) 3694. Available from: . [7] V.P. Nair, Talk given at Fifth International Workshop on Thermal Field Theories and Their Applications, Regensburg, Germany, 10–14 August 1998. Available from: . [8] D. Bödeker, Phys. Lett. B516 (2001) 175. Available from: . [9] D.F. Litim, C. Manuel, Phys. Rev. Lett. 82 (1999) 4981. Available from: ; D.F. Litim, C. Manuel, Nucl. Phys. B562 (1999) 237. Available from: . [10] D. Bödeker, Nucl. Phys. B647 (2002) 512. Available from: . [11] P. Arnold, L.G. Yaffe, Phys. Rev. D62 (2000) 125013. Available from: . [12] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, second ed., Oxford University Press, New York, 1993. [13] P.H. Damgaard, H. Hüffel, Phys. Rep. 152 (1987) 227. [14] P. Arnold, L.G. Yaffe, Phys. Rev. D62 (2000) 125013. Available from: . [15] J. Zinn-Justin, D. Zwanziger, Nucl. Phys. B295 (1988) 297. [16] H. Hüffel, G. Kelnhofer, Ann. Phys. 270 (1998) 231. Available from: . [17] G.D. Moore, Nucl. Phys. B568 (2000) 367. Available from: . [18] G.D. Moore, Phys. Rev. D62 (2000) 085011. Available from: . [19] G.D. Moore, in: C.P. Korthals Altes (Ed.), Proceedings of the SEWM2000 Meeting (Marseille, France, 13–17 June 2000), World Scientific, 2001. Available from: . [20] C. Lerche, L. von Smekal, Phys. Rev. D65 (2002) 125006. Available from: . [21] D. Zwanziger, Time-independent stochastic quantization, DS equations, and infrared critical exponents in QCD. Available from: .