Functional renormalization group approach in gauge theories

Functional renormalization group approach ingauge theories

P. M. Lavrov

Tomsk State Pedagogical University

Dubna, SQS’2013, August 2, 2013

P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 1 / 36

Based on

PML., I.L. Shapiro, JHEP 06 (2013) 086; arXiv:1212.2577 [hep-th]


Content

Introduction

Faddeev-Popov method

Functional renormalization group (FRG) approach

Generating functionals and FRG equation

Vacuum functional

Gauge dependence of effective average action

New approach

Conclusions


Introduction

The recent development of Theoretical Physics is related to thenon-perturbative aspects of quantum properties of dynamical systems(critical phenomena in statistical physics, confinement, IR regime in QCD,etc).

One of the most promising approaches is related to different versions ofWilson renormalization group approach [Wilson (1971); Wilson, Kogut(1974); Polchinski (1984)].

In connection with Quantum Field Theory the qualitative idea of suchapproaches was discussed by Polchinski [Polchinski (1984)] and can beformulated as follows: regardless we do not know how to sum up theperturbative series, in some sense there is a good understanding of thefinal output of such a summation for the propagator of the quantum field.An exact propagator is supposed to have a singe pole and also providesome smooth behavior in both UV and IR regions.


Introduction

Possible form of a cut-off dependent propagator

R(p2/Λ20)

p2 −m2, R(x) = 1 , x < 1;R(x)→ 0, x→∞ (x = p2/Λ2

0)

The cut-off dependence of the vertices can be derived from the generalscale-dependence of the theory which can be established by means of thefunctional methods. To get this kind of propagators one can add to theaction of given theory of the new action which is quadratic in thecorresponding field(s) and contains the regulator function(s).

A compact and elegant formulation of the non-perturbativerenormalization group has been proposed by Wetterich in terms ofeffective action [Wetterich (1991,1993)]. The method was calledfunctional renormalization group (we shall use abbreviation FRG) for theeffective average action.


Introduction

Application of the FRG approach to gauge theories has been proposed byWetterich and Reuter [Wetterich, Reuter (1993, 1994)]

Many aspects of gauge theories in the framework of FRG has beendiscussed with success, but there is still one important question whichremains unsolved. The consistent quantum description of gauge theorieshas to provide the on-shell independence on the choice of the gauge fixingcondition. In a consistent formulation, such an independence should holdfor the S-matrix elements and, equivalently, for the on-shell effectiveaction.

There is a good general understanding that the construction of FRG startsfrom the propagator, which is not a gauge invariant object and, inparticular, always depends on the choice of gauge fixing condition. Byitself, this fact can not yet serve a negative for consistent formulation ofthe theory (see consistent formulation of gauge theories). However, thecomplete analysis of whether this general difficulty leads to problems atthe level of S-matrix, was not done.


Introduction

As far as the source of the problem with gauge non-invariance is theintroduction of the scale-dependent propagator, it is clear that this is theaspect of the theory which should be reconsidered first.

The known theorems about gauge-invariant renormalizability [Voronov,PML, Tyutin (1982); Gomis, Weinberg (1996)] tell us that the exacteffective action should be BRST-invariant. We will show that it is not thecase for the FRG approach.

Therefore, the problem is just to find the way to implement this invariancewhen one takes into account the regulator functions. The proposal whichwe present here is to use an old idea of [Jackiw (1974)] about introductionof composite operators and their extension for gauge theories [PML(1991)]. Following this line, we will introduce the regulator functions ascomposite operators and show that, in this case, the BRST symmetry ismaintained at quantum level and, as a consequence, the S-matrix in thetheory is gauge invariant and well defined.


Introduction

Remarks about presentation:

We restrict here ourself to the case of Yang-Mills theories althoughwe can consider arbitrary gauge theories as well.

For a better understanding of the gauge dependence in the FRGapproach it is useful to remind the main features of theFaddeev-Popov method [Faddeev, Popov (1967)] for Yang-Mills fieldsincluding the demonstration of the gauge invariance of the vacuumfunctional, the Slavnov-Taylor identity and the on-shell invariance ofeffective action.


Faddeev-Popov method[Faddeev, Popov (1967)]

Yang-Mills action[Yang, Mills (1954)]

SYM (A) = −1

4F aµνF

aµν , F aµν = ∂µAaν − ∂νAaµ + fabcAbµA

cν

Gauge invariance

δSYM = 0, δAaµ = Dabµ ξ

b, Dabµ = δab∂µ + facbAcµ

Faddeev-Popov action

SFP (Φ) = SYM + Sgf + Sgh = SFP (A) + χaBa + CaKabCb

ΦA = (Aaµ, Ba, Ca, Ca)

Kab =δχa

δAcµDcbµ



For more popular gauges in Yang-Mills theories the function χa is chosenas

Landau gaugeχa = ∂µAaµ

One parameter linear gauge

χa = ∂µAaµ +ξ

2Ba

The Faddeev-Popov operator

Kab = ∂µDabµ = δab∂µ∂µ + facb∂µ ·Acµ



BRST symmetry

[Becchi,Rouet,Stora (1975),Tyutin (1975)]

δBSFP (Φ) = 0

δBAaµ(x) = Dab

µ Cb(x)µ

δBCa(x) =

1

2fabcCb(x)Cc(x)µ

δBCa(x) = Ba(x)µ

δBBa(x) = 0

µ is a constant Grassmann parameter, µ2 = 0. Due to the Noethertheorem there exists conserved charge, the BRST charge QB.Corresponding BRST operator, QB, defines the physical space states,QB|phys >= 0.



Let δBΦA = sΦAµ

Nilpotency of the BRST transformations

s2Aaµ = sDabµ C

b = 0

s2Ca = sBa = 0

s2Ba = 0

s2Ca = s1

2fabcCbCc = 0

It leads to very important property of the BRST operator QB to benilpotent, Q2

B = 0. This allows effectively to study the unitarity problem[Kugo, Ojima (1979)].



Generating functional of Green’s functions

Z(j) =

∫DΦ exp

{ i~

(SFP (Φ) + jaµA

aµ)}

jaµ(x) are external sources to fields Aaµ(x)Generating functional of connected Green’s functions

Z(j) = exp{ i~W (j)

}.

Generating functional of vertex functions (effective action)

Γ(A) = W (j)− jA

Aaµ(x) =δW

δjaµ(x),

δΓ

δAaµ(x)= −jaµ(x)



Consider the vacuum functional Z(0) ≡ Zχ constructing for a givenchouse of gauge χa = 0

Zχ =

∫DΦ exp

{ i~SFP (Φ)

}.

and make use an infinitesimal change of gauge fixing functionχa → χa + δχa corresponding to the gauge χa + δχa = 0. Using changeof the variables of integration in the form of BRST transformations we canprove

Gauge independence of vacuum functional

Zχ+δχ = Zχ



The next consequence of gauge invariance of SYM is the Slavnov-Tayloridentity [Taylor (1971), Slavnov (1972)]

jaµ < Dµab >j + < Ba δχa

δAcµDµcb >j +facd < Ca∂µDcb

µ Cd >j= 0

where < G >j means vacuum expectation value in presence of sources

< G >j=

∫DΦ G(Φ) exp

{ i~

(SFP (Φ) + jA

)}



To discuss other features of the Faddeev-Popov quantization it is veryuseful to introduce additional sources to all fields ΦA = (Aaµ, Ba, Ca, Ca)

JA = (jaµ, σa, ηa, ηa)

and extended generating functional

Z(J) =

∫DΦ exp

{ i~

(SFP (Φ) + JΦ)

}.

It is clear that

Z(J)∣∣σ=η=η=0

= Z(j).

Slavnov-Taylor identity in terms of Z = Z(J)

Jaµ∂µ δZ

δηa+ ηa

δZ

δσa− i~facb

(Jaµ

δ2Z

δJcµδηb

+1

2ηa

δ2Z

δηcδηb

)≡ 0.



Due to presence of the second-order derivatives of Z the Slavnov-Tayloridentity is presented in non-local form. Fortunately there exists apossibility to present the Slavnov-Taylor identity in local form using a trick[Zinn-Justin (1976)] connected with introduction of the set of externalsources KA = (Ka

µ, La, La, Na) (ε(KA) = εA + 1) to the BRST

transformation, sΦA, and the extended generating functional of Green’sfunctions

Z(J,K) =

∫DΦ exp

{ i~

(S(Φ,K) + JΦ

)}, S(Φ,K) = SFP (Φ) +KsΦ.

Zinn-Justin equation

δBSFP (Φ) = 0 ⇒ δS

δΦA

δS

δKA= 0.

Slavnov-Taylor identity in the local form

JAδZ(J,K)

δKA≡ 0



The generating functional of vertex Green’s functions (effective action), Γ,is introduced through the Legendre transformations of W = −i~lnZaccording to

Γ(Φ,K) = W (J,K)− JAΦA, ΦA =δW

δJA.

Then

δΓ

δΦA= −JA,

δΓ

δKA=

δW

δKA.

Slavnov-Taylor identity in terms of Γ

δΓ

δΦA

δΓ

δKA≡ 0.



The effective action Γ is the main object and depends on gauges. Theequation describing the gauge dependence of effective action Γ = Γ(Φ,K)under variation of gauge has the form

δΓ = − δΓ

δΦA

δ

δKAδψ(Φ),

where the notations

δψ(Φ) = Caδχa(Φ), ΦA = ΦA + i~(Γ−1)ABδlδΦB

were used. The matrix Γ−1 is inverse to the matrix Γ with elements

ΓAB =δlδΦA

( δΓ

δΦB

), (Γ−1)ACΓCB = δAB.

Gauge independence

δΓ

δΦA= 0 → δΓ = 0.


FRG approach[ Reuter, Wetterich (1993, 1994)]

The main idea of the FRG is to use instead of Γ an effective averageaction, Γk, with a momentum-shell parameter k, such that

limk→0

Γk = Γ .

For the Yang-Mills theories it was suggested to modify the Faddeev-Popovaction with the help of the specially designed regulator action Sk

Sk(A,C, C) =1

2Aaµ(Rk,A)abµνA

bν + Ca(Rk,gh)abCb .

Regulator functions Rk,A and Rk,gh obey the properties

limk→0

(Rk,A)abµν = 0 , limk→0

(Rk,gh)ab = 0 .

BRST non-invariance

δB Sk(A,C, C) 6= 0 .



The generating functional of Green’s functions, Zk, is constructed in theform of the functional integral

Zk(J,K) =

∫DΦ exp

{ i~[SFP (Φ) + Sk(Φ) + JΦ +KsΦ

]},

where, for the sake of uniformity, we used notation Sk(Φ) insteadSk(A,C, C), despite Sk does not depend on fields Ba

Slavnov-Taylor identity

JAδZkδKA

− i~{

(Rk,A)abµνδ2ZkδjbνδK

aµ

+ (Rk,gh)abδ2ZkδηaδLb

−

−(Rk,gh)abδ2ZkδηbδLa

}≡ 0 .



The generating functional of vertex functions in the presence of regulators(the effective average action), Γk = Γk(Φ,K), satisfies the functionalintegro-differential equation

exp{ i~

Γk(Φ,K)}

=

∫Dϕ exp

{ i~

[SFP (Φ + ϕ) + Sk(Φ + ϕ) +

+Ks(Φ + ϕ) − δΓk(Φ,K)

δΦϕ]}

.

The tree-level (zero-loop) approximation corresponds to

Γ(0)k (Φ,K) = SFP (Φ) + Sk(Φ) + KsΦ .



Slavnov-Taylor identity in terms of Γk = Γk(Φ,K)

δΓkδΦA

δΓkδKA

−{

(Rk,A)abµν Abν δΓkδKa

µ

+ (Rk,gh)ab CaδΓkδLb

− (Rk,gh)abCbδΓkδLa

}−i~

{(Rk,A)abµν

(Γ

′′−1)bν A δ2

l ΓkδΦA δKa

µ

+ (Rk,gh)ab(Γ

′′−1k

)aA δ2l Γk

δΦA δLb

−(Rk,gh)ab(Γ

′′−1k

)bA δ2l Γk

δΦA δLa

}≡ 0 .

The matrix (Γ′′−1k ) is inverse to the matrix Γ

′′k with elements

(Γ′′k)AB =

δlδΦA

( δΓkδΦB

),

(Γ

′′−1k

)AC(Γ

′′k

)CB

= δAB .



FRG flow equation for Γk

∂tΓk = ∂tSk + i~{1

2∂t(Rk,A)abµν

(Γ

′′−1k

)(aµ)(bν)+ ∂t(Rk,gh)ab

(Γ

′′−1k

)ab},

∂t = kd

dk.

Usually the functional RG approach is formulated in terms of the functionalwhich does not depend on sources K. Since the equation does not containderivatives with respect to K, we can just put KA = 0 and arrive at

∂tΓk = ∂tSk + i~{1

2∂t(Rk,A)abµν

(Γ

′′−1k

)(aµ)(bν)+ ∂t(Rk,gh)ab

(Γ

′′−1k

)ab},

Γk = Γk(Φ) = Γk(Φ,K = 0

).



In the condensed notations

∂tΓk = ∂tSk + i~{1

2Tr[∂t(Rk,A)

(Γ

′′−1k

)]A− Tr

[∂t(Rk,gh)

(Γ

′′−1k

)]C

},

where we took into account the anticommuting nature of the ghost fieldsCA and defined

Tr[∂t(Rk,gh)

(Γ

′′−1k

)]C

= − ∂t(Rk,gh)ab(Γ

′′−1k

)ab,

Tr[∂t(Rk,A)

(Γ

′′−1k

)]A

= ∂t(Rk,A)abµν(Γ

′′−1k

)(aµ)(bν).

The conventional presentation of the FRG flow equation in terms of thefunctional Γk = Γk − Sk reads

∂tΓk = i~{1

2Tr[∂tRk,A

(Γ

′′k +Rk,A

)−1]A−

−Tr[∂t(Rk,gh)

(Γ

′′k +Rk,gh

)−1]C

}.


Vacuum functional

The vacuum functional, corresponding to the given gauge function χa inthe Faddeev-Popov action

Zk,χ = Zk(0, 0) =

∫DΦ exp

{ i~(SFP + Sk

)}.

By construction, the regulator functions in the FRG approach do notdepend on gauge χa and therefore the action Sk is gauge independent.Let us consider an infinitesimal variation of gauge χ→ χ+ δχ andconstruct the vacuum functional corresponding to this gauge

Zk,χ+δχ =

∫DΦ exp

{ i~

(SFP + Sk + Ca

δ δχa

δAcµDcbµ C

b + δχaBa)}

.


Vacuum functional

In the last functional integral we make a change of variables in the form ofthe BRST transformations but trading the constant Grassmann-oddparameter θ to a functional Λ = Λ(Φ). Sk is not invariant, with thevariation given by

δSk = Aaµ(Rk,A)abµν DνbcCcΛ +

1

2Ca(Rk,gh)ab f bcdCcCdΛ−

−Ba(Rk,gh)abCbΛ.

Choosing Λ in a natural way, Λ = i~−1 Ca δχa , then

Zk,χ+δχ =

∫DΦ exp

{ i~(SFP + Sk + δSk

)},

then, for any value k 6= 0, one has

gauge dependence

Zk,χ+δχ 6= Zk,χ.



The investigation of gauge dependence is based on a variation of thegauge-fixing function, χa → χa + δχa, which leads to the variation of theFaddeev-Popov action SFP and consequently of the generating functionalZk = Zk(J,K). Introducing the functional δψ = Caδχa, the gaugedependence can be presented in the form

δZk =i

~

∫DΦ

δ δψ

δΦA

δ(KsΦ)

δKAexp

{ i~[SFP (Φ) + Sk + JΦ +KsΦ

]}.

The final form

δZk =i

~JA

δ

δKAδψ(~i

δ

δJ

)Zk +

[(Rk,A)abµν

δ2

δjaµδjbν

+

+(Rk,gh)abδ2

δηaδLb− (Rk,gh)ba

δ2

δηaδLb

]δψ(~i

δ

δJ

)Zk .



In terms of the effective average action

δΓk = − δΓkδΦA

δ

δKAδψ(Φ)− i~Sk;A

( i~

(Φ− Φ)) δ

δKAδψ(Φ),

where ΦA is defined as

ΦA = ΦA + i~(Γ

′′−1k

)AB δlδΦB

.

We see that if on-shell is defined in the usual way, we obtain

gauge dependence even on-shell

δΓkδΦA

= 0 → δΓk 6= 0

This result shows that the gauge dependence represents a serious problemfor the FRG approach in the standard conventional formulation.


New approach

Our approach is based on an idea of the composite fields [PML (1991)] toformulate the FRG framework for gauge theories. The idea is to use such afields to implement regulator functions. Consider the regulator functions

L1k(x) =

1

2Aaµ(x)(Rk,A)abµν(x)Abν(x) ,

L2k(x) = Ca(x)(Rk,gh)ab(x)Cb(x) .

Now we introduce external scalar sources Σ1(x) and Σ2(x) and constructthe generating functional of Green’s functions for Yang-Mills theories withcomposite fields

Zk(J,K; Σ) =

∫DΦ exp

{ i~[SFP (Φ) + JΦ +KsΦ + ΣLk(Φ)

]}.

where ΣLk = Σ1L1k + Σ2L

2k.


New approach

We arrive at the new version of the FRG flow equation for the generatingfunctional Zk = Zk(J,K; Σ),

∂tZk =~i

{1

2Σ1 ∂t(Rk,A)abµν

δ2Zkδjaµ δj

bν

+ Σ2 ∂t(Rk,gh)abδ2Zkδηa δηb

}.

[Jackiw (1974)]

Γk(Φ,K;F ) = Wk(J,K; Σ) − JAΦA − Σi

[Lik(Φ) + ~F i

],

where

ΦA =δWk

δJA, ~F i =

δWk

δΣi− Lik

(δWk

δJ

), i = 1, 2.

δΓkδΦA

= −JA − ΣiδLik(Φ)

δΦA,

δΓkδF i

= −~Σi .


New approach

Let us introduce the full sets of fields FA and sources JA according to

FA = (ΦA, ~F i) , JA = (JA, ~Σi).

From the condition of solvability of equations

δFC(J )

δJBδlJA(F)

δFC= δBA .

One can express JA as a function of the fields in the form

JA =(− δΓkδΦA

− δΓkδF i

δLik(Φ)

δΦA, − δΓk

δF i

)and, therefore,

δlJB(F)

δFA= −(G

′′k)AB ,

δFB(J )

δJA= −(G

′′−1k )AB .


New approach

FRG flow equation

∂tΓk = − i

2Tr{ δΓkδF 1

∂t(Rk,A)(G′′−1k )

}A

+ iTr{ δΓkδF 2

∂t(Rk,gh) (G′′−1k )

}C,

when we used usual traces in the sectors of vector Aaµ and ghost Ca

fields.

The gauge dependence

δΓk = − δΓkδΦA

δ

δKAδψ(Φ)− 1

~δΓkδF i

Lik,A (Φ− Φ)δ

δKAδψ(Φ).


New approach

Let us define the mass-shell of the quantum theory by the equations

δΓkδΦA

= 0 ,δΓkδF i

= 0 .

Then the gauge independence of the effective action followsΓk = Γk(Φ,K;F ) on-shell.

Gauge and k independence on-shell

δΓk = 0, ∂tΓk = 0.


Conclusions

Investigation of gauge dependence of Green’s functions within thefunctional renormalization group approach was given.

It was shown the gauge dependence of effective action even on shell.It means that S-matrix depends on gauge. In particular, vacuumexpectation values of gauge invariant operators such as F aµνF

aµν dodepend on gauge.

It was proven that a consistent formulation of gauge theories withinthe FRG approach does exist.


Thank youfor attention!


Documents

Functional renormalization group approach in gauge theories