functional renormalization group approach in gauge theories
TRANSCRIPT
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]
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 2 / 36
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
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 3 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 4 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 5 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 6 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 7 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 8 / 36
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
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 9 / 36
Faddeev-Popov method
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
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 10 / 36
Faddeev-Popov method
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 11 / 36
Faddeev-Popov method
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)].
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 12 / 36
Faddeev-Popov method
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)
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 13 / 36
Faddeev-Popov method
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
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 14 / 36
Faddeev-Popov method
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
)}
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 15 / 36
Faddeev-Popov method
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 16 / 36
Faddeev-Popov method
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
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 17 / 36
Faddeev-Popov method
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 18 / 36
Faddeev-Popov method
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 19 / 36
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 .
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 20 / 36
Generating functionals and FRG equation
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 .
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 21 / 36
Generating functionals and FRG equation
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 .
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 22 / 36
Generating functionals and FRG equation
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 .
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 23 / 36
Generating functionals and FRG equation
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
).
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 24 / 36
Generating functionals and FRG equation
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
}.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 25 / 36
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)}
.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 26 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 27 / 36
Gauge dependence of effective average action
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 .
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 28 / 36
Gauge dependence of effective average action
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 29 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 30 / 36
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 .
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 31 / 36
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 .
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 32 / 36
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).
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 33 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 34 / 36
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.
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 35 / 36
Thank youfor attention!
P.M. Lavrov (Tomsk) Functional renormalization group approach in gauge theories Dubna 2013 36 / 36