Functional renormalization group and its
applications to gauge theories
Gergely Fejős

Eötvös University, Budapest
Institute of Physics

Functional Renormalization Group at RIKEN 2023
21 January, 2023

Gergely FejősFunctional renormalization group and its applications to gauge

Outlineaaa
Wilsonian and Functional renormalization

Field theoretical RG vs. the FRG

Gauge symmetry violation in the FRG

Applications

Summary

Gergely FejősFunctional renormalization group and its applications...

Motivation
Quantum field theories are extensively used in particle physics,
condensed matter physics, statistical mechanics...
−→ 3 + 1 dim. continuum theories−→ thermal field theories−→ dimensionally reduced effective models
Central objects that one is looking for:hφ1 φ2 ...φn i correlation functions

Gergely FejősFunctional renormalization group and its applications...

Motivation
Quantum field theories are extensively used in particle physics,
condensed matter physics, statistical mechanics...
−→ 3 + 1 dim. continuum theories−→ thermal field theories−→ dimensionally reduced effective models
Central objects that one is looking for:hφ1 φ2 ...φn i correlation functions
Most successful method: perturbation theory−→ Taylor-expansion based on some small parameters−→ asymptotic series, does not necessarily show any
aaaaconvergence
Non-perturbative methods are necessary!
−→ Functional Renormalization Group (FRG)

Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
At the critical point in 2nd order phase transitions: masslessmodes invalidate PT (IR singularities)
Idea of the Wilsonian Renormalization Group:−→ momentum-shell integration: introduce anaaaaintermediate scale k and define
aaaaφ(~p ) = φ< (~p )Θ(k − |~p |) + φ> (~p )Θ(|~p | − k)
Z
Z
Z
< >
−S[φ]
<
Z =
Dφe
= Dφ
Dφ> e −S< −S> −Smix [φ ,φ ]Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
At the critical point in 2nd order phase transitions: masslessmodes invalidate PT (IR singularities)
Idea of the Wilsonian Renormalization Group:−→ momentum-shell integration: introduce anaaaaintermediate scale k and define
aaaaφ(~p ) = φ< (~p )Θ(k − |~p |) + φ> (~p )Θ(|~p | − k)
Z
Z
Z
< >
−S[φ]
<
Z =
Dφe
= Dφ
Dφ> e −S< −S> −Smix [φ ,φ ]φ> integral is performed using pert. theory
Z<
Z = Dφ< e −S[φ ]<>S → S − log <e −[S> ]−Smix [φ ,φ ]>Result: Λ → k, mΛ2 → mk2 , gΛ → gk + new vertices
Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
Rescaling dimensionful quantities with k−→ flow equations for individual coupling constants:(i)k∂k m̄k2 = βm (m̄k2 , ḡk , ...)

(i)

(i)

k∂k ḡk = βg (i) (m̄k2 , ḡk , ...)

Fixed points: statistically self-similar behavior→ no relevant length scale! (2nd order phase transitions)
Identification of relevant and irrelevant directions→ close to a fixed point only the relevant ones matter→ finite T transition: one relevant direction

Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
Rescaling dimensionful quantities with k−→ flow equations for individual coupling constants:(i)k∂k m̄k2 = βm (m̄k2 , ḡk , ...)

(i)

(i)

k∂k ḡk = βg (i) (m̄k2 , ḡk , ...)

Fixed points: statistically self-similar behavior→ no relevant length scale! (2nd order phase transitions)
Identification of relevant and irrelevant directions→ close to a fixed point only the relevant ones matter→ finite T transition: one relevant direction
The WRG has had great success in−→ describing universality in 2nd order transitions−→ predicting critical exponents−→ understanding the concept of effective theories

Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
FRG generalizes the idea of the WRG: fluctuations are taken
into account at the level of the quantum effective action
Z
ZR
Z [J] = Dφe −(S[φ]+ Jφ) ⇒ Γ[φ̄] = − log Z [J] − J φ̄Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
FRG generalizes the idea of the WRG: fluctuations are taken
into account at the level of the quantum effective action
Z
ZR
Z [J] = Dφe −(S[φ]+ Jφ) ⇒ Γ[φ̄] = − log Z [J] − J φ̄Introduction of a flow parameter k and inclusion offluctuations for which q & k
RRk2
aa Zk [J] = Dφe −(S[φ]+ Jφ)
1

R

φRk φ
Rk(q)

aaaaaaaa×e − 2−→ regulator: mom. dep. mass
aaaaterm suppressing low modes−→ not only sharp cutoff!

00Gergely FejőskqFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
Scale-dependent effective action:ZΓk [φ̄] = − log Zk [J] −−→ k ≈ Λ: no fluctuations−→ k = 0: all fluctuations

1
J φ̄ −
2Z
φ̄Rk φ̄⇒ Γk=Λ [φ̄] = S[φ̄]⇒ Γk=0 [φ̄] = Γ[φ̄]The scale-dependent effective
action interpolates between
classical- and quantum
effective actions
The trajectory depends on Rk
but the endpoint does not

Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
Flow of the effective action is described by the Wetterich
equation:Z
1
1(2)∂k Γk =Tr [(Γk + Rk )−1 (p, q)∂k Rk (−q, −p)] =2 qp
2

Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
Flow of the effective action is described by the Wetterich
equation:Z
1
1(2)∂k Γk =Tr [(Γk + Rk )−1 (p, q)∂k Rk (−q, −p)] =2 qp
2
Slightly different form: [∂˜k acts only on Rk ]
Z
1
1 X
(2)
∂k Γk =
∂˜k Tr log[Γk + Rk ] = ∂˜k2
2Gergely FejősFunctional renormalization group and its applications...

Wilsonian and Functional Renormalization
Flow of the effective action is described by the Wetterich
equation:Z
1
1(2)∂k Γk =Tr [(Γk + Rk )−1 (p, q)∂k Rk (−q, −p)] =2 qp
2
Slightly different form: [∂˜k acts only on Rk ]
Z
1
1 X
(2)
∂k Γk =
∂˜k Tr log[Γk + Rk ] = ∂˜k2
2
Scale dependence of the proper vertices are described by
one-loop diagrams [propagators are dressed!]−→ exact equation!
Advantage: flows are directly accessible in any dimension
aaaaaaaaaaabut approximation is needed
Gergely FejősFunctional renormalization group and its applications...



Field theoretical RG vs. the FRG

How is the Wilsonian (and Functional) RG related to the field
theoretical RG?
Considering the Euclidean φ4 theory in 4d:
(2)

Γ

(4)

Γ

2

λ
+ ... = m +222(0) = m + δm −

Z

1

q

~ 2 + m2q

3λ2
+ ... = λ −2(0) = λ + δλ −

+ ...

Z

1

q

(~
q 2 + m2 )2+ ...

Minimal substracion:λδm (µ) = −22δλ(µ) =

Gergely Fejős

Z 

q
Z
2
3λ

2

q

1
µ2 − m 2+~ 2 + µ2 (~q
q 2 + µ2 )2



1

(~
q 2 + µ2 )2Functional renormalization group and its applications...

Field theoretical RG vs. the FRG
Finite 2 and 4-point functions:µ2 − mµ2
1
1−−~ 2 + mµ2
~ 2 + µ2 (~q
q 2 + µ2 )2
q q

Z 
3λ2
1
1
Γ(4) (0) = λµ −+ ...
−2 q (~
q 2 + mµ2 )2 (~q 2 + µ2 )2

Γ(2) (0) = mµ2 +

λ
2

Z 


+ ...

Integrand of the fluctuation contributions cuts off at q ∼ µ−→ mµ2 and λµ plays the role of mk2 and λk !
−→ Why? ⇒ mk2 and λk refer to an effective action whereaaaaaaaaaaaa fluctuations are included where q & k
Renormalization scale µ in the field theoretial RG
aaaaaaaaaaaaaaaaa≡
Separation scale k in the Wilsonian (Functional) RG

Gergely FejősFunctional renormalization group and its applications...

Field theoretical RG vs. the FRG
Two approaches to obtain the β functions:−→ βm2 : k∂k mk2−→ βλ : k∂k λk

or µ∂µ mµ2
or µ∂µ λµ

Field theoretical RG:µ∂µ mµ2 = −λµ 2(µ − mµ2 ),2µ∂µ λµ =3λ2µ2

Wilsonian (Functional) RG:k∂k mk2 = −λkk4,
2 k 2 + mk2

k∂k λk =3λ2k
k42 (k 2 + mk2 )2

Two results agree but only close to the Gaussian fixed point!
−→ Wilsonian (Functional) RG is more general!

Gergely FejősFunctional renormalization group and its applications...

Field theoretical RG vs. the FRG
Why do we use the field theoretical RG after all?
−→ it does not necessarily introduce a momentum cutoffaaaa(dim. reg., Pauli-Villars, etc.)
aaaa=⇒ gauge symmetry survives−→ QCD in the UV and QED in the IR works fineaaaa(small couplings)

Gergely FejősFunctional renormalization group and its applications...

Field theoretical RG vs. the FRG
Why do we use the field theoretical RG after all?
−→ it does not necessarily introduce a momentum cutoffaaaa(dim. reg., Pauli-Villars, etc.)
aaaa=⇒ gauge symmetry survives−→ QCD in the UV and QED in the IR works fineaaaa(small couplings)
Wilsonian (Functional) RG by definition contains amomentum cutoff−→ several generalizations in the market−→ construction of a gauge invariant RG flow equation−→ typically very complicated, hard to use them for practicalcomputations that go beyond usual perturbative treatments
Goal here: practical computations

Gergely FejősFunctional renormalization group and its applications...

Gauge symmetry violation in the FRG
Abelian gauge (Ai ) theory, N scalar fields (φa )(Di = ∂i − ieAi , Fij = ∂i Aj − ∂j Ai )
Z h
i
λ
1
Fij Fij + (Di φa )† Di φa + m2 φ†a φa + (φ†a φa )2 + Sgf
S=
4
6
Gauge symmetry:δφa (x) = ieΘ(x)φa (x),δAi (x) = −∂i Θ(x)Quantum theory: gauge symmetry is encoded in the
Ward-Takahashi identities [ΦT = (Ai , φ†a , φa )]ZδΓ[Φ̄]
δZ [J] = 0 ⇒hδΦα (x)i α− hδSi = 0δΦ̄ (x)xR
Covariant gauge fixing: δS = ξi p Θ(−p)p 2 pi Ai (p)→ project the master eq. onto different operators: WTIs
Gergely FejősFunctional renormalization group and its applications...

Gauge symmetry violation in the FRG
Projecting onto ∼ Āi : WTI of the Ai Aj vertex
11pi Γij (p) = − p 2 pj ⇒ Γij (p) = (−δij p 2 + pi pj )(1 − Π(p)) − pi pjξξ−→ only the transverse part receives quantum corrections

Gergely FejősFunctional renormalization group and its applications...

Gauge symmetry violation in the FRG
Projecting onto ∼ Āi : WTI of the Ai Aj vertex
11pi Γij (p) = − p 2 pj ⇒ Γij (p) = (−δij p 2 + pi pj )(1 − Π(p)) − pi pjξξ−→ only the transverse part receives quantum corrections
Pr. onto ∼ φ†a φb : WTI between φ†a φb and Ai φ†a φb verticese Γab (q + p) − Γab (q) = pi Γiab (p, q)−→ scalar and charge rescaling factors agree: Zφ = Ze

Gergely FejősFunctional renormalization group and its applications...

Gauge symmetry violation in the FRG
Projecting onto ∼ Āi : WTI of the Ai Aj vertex
11pi Γij (p) = − p 2 pj ⇒ Γij (p) = (−δij p 2 + pi pj )(1 − Π(p)) − pi pjξξ−→ only the transverse part receives quantum corrections
Pr. onto ∼ φ†a φb : WTI between φ†a φb and Ai φ†a φb verticese Γab (q + p) − Γab (q) = pi Γiab (p, q)−→ scalar and charge rescaling factors agree: Zφ = Ze
Projecting onto ∼ Āi1 Āi2 ...Āin−1 : WTI of the n-point A vertex
pi Γi,i1 ,i2 ,...,in−1 (p, q1 , q2 , ...) = 0

Gergely FejősFunctional renormalization group and its applications...

Gauge symmetry violation in the FRG
In the FRG formalism the WTIs change:
ZZ
ZZ
1
1
T
S −→ S +Φ Rk Φ, Γ −→ Γk +
Φ̄T Rk Φ̄
22
The modified Ward-Takahashi identities (mWTI) are
generated via
ZZZδΓ[Φ̄]1 TαhδΦ (x)i α− hδSi = hδ
Φ Rk Φi2δ Φ̄ (x)
x
Z
ZZ
1 T
δ
− hδΦα i αΦ̄ Rk Φ̄2δ Φ̄The gauge field does not contribute to the rhs:ZZZ
δΓk [Φ̄]hδΦα (x)i α− hδSi = ie Θ(−p) hφ†a (q)φa (q + p)ic
δ Φ̄ (x)pqx× [Rk (p + q) − Rk (q)]Gergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
LPA ansatz for the effective action with optimal regulator:[Rk (q) = (k 2 − q 2 )Θ(k 2 − q 2 )]

Z 
Γk =

ZA,k
1 2
Ai [−∂ 2 δij + ∂i ∂j (1 − ξk−1 )]Aj + mA,k
Ai Ai
2
2
+Zφ,k ∂i φ†a ∂i φa + iZe,k eAi (φ†a ∂i φa − ∂i φ†a φa )
+

2
Ze,k

Zφ,k


2
Zφ,k mk2 †a a Zφ,k λk †a a 2†
e Ai Ai φa φa +φ φ +(φ φ )
2

2

6

Proposal: try to maintain all WTIs related to the ansatz(1) Ze,k = Zφ,k (2) transverse corrections to the gauge prop.
The ∼ φ†a φb projection leads toZZe,kRk (q)
2= 1 + 2ek ξk2
2
Zφ,kq (q + Rk (q))−→ choose the Landau gauge (ξ = 0)!
Gergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
The ∼ Ai projection leads to two conditions [O(p 0 ), O(p 2 )]
2
mA,k
=−

−4Ne 2Ωd k d−2 ,d(d + 2)ZA,k1
4k d−4−= Ωd e 2ξkξΛd(d + 2)2 → 0, if k → 0)−→ photon mass is irrelevant (mA,k−→ plugging in the flow of ZA,k one gets ξk ≡ 2/(d − 4) 6= 0!
−→ Landau gauge cannot be chosen!

Gergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
The ∼ Ai projection leads to two conditions [O(p 0 ), O(p 2 )]
2
mA,k
=−

ZA,k
1
4k d−4−= Ωd e 2ξkξΛd(d + 2)

−4Ne 2Ωd k d−2 ,d(d + 2)2 → 0, if k → 0)−→ photon mass is irrelevant (mA,k−→ plugging in the flow of ZA,k one gets ξk ≡ 2/(d − 4) 6= 0!
−→ Landau gauge cannot be chosen!

Exactly the same can be obtained from the flow equation:Zhik∂k Γij (p) = k ∂˜k
q

p+q

i

i

p

p

j

p

p

j

q

−→ photon mass flows but dies out in the IR−→ flow of the longitudinal propagator is nonzero
aaaa⇒ it can be compensated with the above choice of ξkGergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
What to do with the discrepancy of ξ?
−→ in d = 4 there is no problem (pert. results are recovered)k∂k λk =54ek4 − 18ek2 λk + (N + 4)λ2k,
24π 2k∂k ek2 =

Nek4
24π 2

In d 6= 4 one way is to enforce Zφ,k = Ze,k by handand choose ξ = 2/(4 − d) at the same time−→ ξ enters the β functions!

1
GF & T. Hatsuda, Phys. Rev. D93, 121701 (2016)
aaaaGF & T. Hatsuda, Phys. Rev. D96, 056018 (2017)
aaaaGF & T. Hatsuda, Phys. Rev. D100, 036007 (2019)
Gergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
What to do with the discrepancy of ξ?
−→ in d = 4 there is no problem (pert. results are recovered)k∂k λk =54ek4 − 18ek2 λk + (N + 4)λ2k,
24π 2k∂k ek2 =

Nek4
24π 2

In d 6= 4 one way is to enforce Zφ,k = Ze,k by handand choose ξ = 2/(4 − d) at the same time−→ ξ enters the β functions!
Example of superconductivity: Abelian Higgs model with
N = 1 complex scalar−→ perturbative results do not allow for an IR fixed point,
aaaabut it does exist (predictions of Monte-Carlo simulations)−→ the above method does produce an IR fixed point1
1
GF & T. Hatsuda, Phys. Rev. D93, 121701 (2016)
aaaaGF & T. Hatsuda, Phys. Rev. D96, 056018 (2017)
aaaaGF & T. Hatsuda, Phys. Rev. D100, 036007 (2019)
Gergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
60
50
40

III

−2
e
k 30

IV

2

20
10
I II

0
0

50

100

150

−
λk

200

250

300

350

2
GF & T. Hatsuda, Phys. Rev. D93, 121701 (2016)
aaaaGF & T. Hatsuda, Phys. Rev. D96, 056018 (2017)
aaaaGF & T. Hatsuda, Phys. Rev. D100, 036007 (2019)
Gergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
Can we save the transversality of the gauge propagator?
hik∂k Γij (p) = k ∂˜kO(p 2 )
Z
(p + 2q)j(p + 2q)i= −e 2 k∂k 2
q + Rk (q) (q + p)2 + Rk (q + p) O(p2 )q
4−d∼ p 2 δij − 1 −
pi pj
2
q

p+q

i

i

Gergely Fejős

p

p

j

p

p

j

q

Functional renormalization group and its applications...

Applications: Abelian gauge theory
Can we save the transversality of the gauge propagator?
hik∂k Γij (p) = k ∂˜kO(p 2 )
Z
(p + 2q)j(p + 2q)i= −e 2 k∂k 2
q + Rk (q) (q + p)2 + Rk (q + p) O(p2 )q
4−d∼ p 2 δij − 1 −
pi pj
2
q

p+q

i

i

p

p

j

p

p

j

q

We may try to regulate the vertex momenta!
~ + α(k ~q̂ − q~ )Θ(k 2 − q 2 )~R α = q→qkk∂k Γij (p) = −e 2

Z
k∂k
q

(p + q)i,Rkα + qi,Rkα (p + q)j,Rkα + qj,Rkα
q 2 + Rk (q)(q + p)2 + Rk (q + p) O(p2 )2∼ p δij − pi pj
if α2 = 2(2 − d)!
Gergely FejősFunctional renormalization group and its applications...

Applications: Abelian gauge theory
Is it justified to regulate the vertex momenta? → yes!
Momentum dependence of the vertex comes from the√derivative coupling: [φa = (s a + iπa )/ 2]Z
e Ai (s a ∂i π a − π a ∂i s a )xZ→ ie Ai (q1 )s a (q2 )π a (q3 )(q3 − q2 )i δ(q1 + q2 + q3 )
q

p+q
i

p

p

j

The
diagrams are generated when
Ai is set to a background field:Z→ ie s a (q2 )π a (q3 )(q3i − q2i )Āi (−q2 − q3 )
i

p

p

j

q

A non-diagonal, background dependent regulator is needed!
Z→ ie s a (q2 )π a (q3 )(q3i,Rkα − q2i,Rkα )Āi (−q2 − q3 )Gergely FejősFunctional renormalization group and its applications...

Applications: non-Abelian gauge theory
Gauge symmetry: [Non-Abelian Higgs model, φ has flavor and color]δφγ (x) = ig Θa T̂ a φγ (x),δAai (x) = −∂i Θa (x) + gf abc Abi Θc

LPA ansatz for the effective action:
Z 
ZA,k a ab 
Γk =
Ai δ
− ∂ 2 δij + (1 − ξk−1 )∂i ∂j Abj2
x
1/2+Zc,k c̄ a (−∂ 2 δ ac − Zg ,k ZA,k gf abc ∂i Abi )c c
2 n−Zφ,k φ†nγ ∂ φγ +1/2

2
Zφ,k mk2 †n n Zφ,k λk †n n 2φγ φγ +
(φγ φγ )62+iZg ,k ZA,k Zφ,k gAai ∂i φ†γ (T̂ a φγ ) − (T̂ a φγ )† ∂i φγ+Zg2,k ZA,k Zφ,k g 2 f abe f cde Aai Abi (T̂ c φγ )† (T̂ d φγ )3/2

+Zg ,k ZA,k gf abc ∂i Aaj Abi Acj
2 g+Zg2,k ZA,k2

4

Gergely Fejősf

abe cdef

Aai Abj Aci AdjFunctional renormalization group and its applications...

Applications: non-Abelian gauge theory
Generalized WTIs: (Slavnov-Taylor identities)−→ formally same as the Abelian but much more complicatedZδΓ[Φ̄]hδΦα (x)i α− hδSi = 0δ Φ̄ (x)x
Examples:−→ Zφ2 A − Zφ and Zφ2 A2 − Zφ are matter independent
aaaa(but not zero!)−→ identical definitions of β function of the gauge coupling− g µ∂µ (δZφ2 A − δZφ − δZA /2)− g µ∂µ (δZφ2 A2 − δZφ − δZA )/2− g µ∂µ (δZ3g − 3δZA /2)− g µ∂µ (δZ4g /2 − δZA )− g µ∂µ (δAc̄c − δZc − δZA /2)
These are all broken due to the regulator!
Gergely FejősFunctional renormalization group and its applications...

Applications: non-Abelian gauge theory
Suggestion: to keep Γk gauge invariant, deal with all the
Slavnov-Taylor identities relevant in the ansatz of Γk !

Gergely FejősFunctional renormalization group and its applications...

Applications: non-Abelian gauge theory
Suggestion: to keep Γk gauge invariant, deal with all the
Slavnov-Taylor identities relevant in the ansatz of Γk !
Flow of gauge coupling: e.g. use the definition via thegauge-ghost-ghost vertex (does not include matter!)Formally:β(g ) → gk k∂k log Zg ,k
=
=

gk
k∂k
Zg ,k

1/2

Zg ,k ZA,k Zc,k
1/2

ZA,k Zc,k

gk

1/2

1/2
Zg ,k ZA,k Zc,k−!

k∂k (Zg ,k ZA,k Zc,k )gk
gkk∂k Zc,k −k∂k ZA,k
Zc,k
2ZA,k

Use diagrammatics to calculate the red terms!
Gergely FejősFunctional renormalization group and its applications...








Applications: non-Abelian gauge theory

The following diagrams need to be calculated:3

!

pi gf

abc

1/2
k∂k (Zg ,k ZA,k Zc,k )= k ∂˜k

+

!

2 ab

= k ∂˜kp δ k∂k Zc,khik∂k ZA,k p 2 δij − pi pj (1 − ξk−1 ) δ ab = k ∂˜k
+

+

!

+

3

+

GF & N. Yamamoto, JHEP 12 (2019) 069
Gergely FejősFunctional renormalization group and its applications...

Applications: non-Abelian gauge theory
Once again gauge choice is in general not arbitrary!
hi∂k ZA,k p 2 δij − pi pj (1 − ξk−1 ) = (...) p 2 δij − pi pj f (d)For d = 4 the flow is transverse and we get
βg |d=4 = −Gergely Fejős

gk3  11Nf N−c(4π)2 36

Functional renormalization group and its applications...

Applications: non-Abelian gauge theory
Once again gauge choice is in general not arbitrary!
hi∂k ZA,k p 2 δij − pi pj (1 − ξk−1 ) = (...) p 2 δij − pi pj f (d)For d = 4 the flow is transverse and we get
βg |d=4 = −gk3  11Nf N−c(4π)2 3
6

For d = 3 the flow is not transverse → two roots for ξ!
qNf1
ξ± = 1 +±Nf2 − 8Nf + 456
44
Requirement: since at large Nf gauge fields are suppressed,
there should be an IR fixed point → rules out ξ+Gergely FejősFunctional renormalization group and its applications...

Applications: non-Abelian gauge theory
The β-function in d = 3:β(g )|d=3

g3= −gk − k22π

ξ− = 1 +

"

Nf
1
−
4
4

!
#
2Nf
19 16
+ ξ − Nc −
9
45
15

q
Nf2 − 8Nf + 456

Application: color superconductivity (Nc = 3, Nf = 3)−→ β(g ) < 0 for all g > 0−→ no IR fixed point, regardless of the scalar potential!
−→ color superconducting phase transition is of 1st order
Can the flow of the gluon propagator be transverse?
Appropriate vertex regularization?
=⇒ future work!
Gergely FejősFunctional renormalization group and its applications...

Summary
Functional Renormalization Group (FRG) method:−→ generalization of the Wilsonian RG to the effective action−→ regulator: explicit momentum cutoff (separation scale)−→ gauge symmetry gets explicitly violatedGergely FejősFunctional renormalization group and its applications...

Summary
Functional Renormalization Group (FRG) method:−→ generalization of the Wilsonian RG to the effective action−→ regulator: explicit momentum cutoff (separation scale)−→ gauge symmetry gets explicitly violated
In a quantum gauge theory symmetry is encoded in the
Ward-Takahashi identities−→ all WTIs are violated via the regulator
Proposal:
1.) build up an ansatz for the effective action
2.) identify all relevant WTIs
3.) find appropriate gauge choice(s) and/or regulator
aaa functions that recover themGergely FejősFunctional renormalization group and its applications...

Summary
Functional Renormalization Group (FRG) method:−→ generalization of the Wilsonian RG to the effective action−→ regulator: explicit momentum cutoff (separation scale)−→ gauge symmetry gets explicitly violated
In a quantum gauge theory symmetry is encoded in the
Ward-Takahashi identities−→ all WTIs are violated via the regulator
Proposal:
1.) build up an ansatz for the effective action
2.) identify all relevant WTIs
3.) find appropriate gauge choice(s) and/or regulator
aaa functions that recover them
Local Potential Approx. + Abelian gauge theories → OK
Local Potential Approx. + non-Abelian gauge theories−→ transversality of the gluon propagator?
−→ equivalence of choices for the flowing gauge coupling?
Gergely FejősFunctional renormalization group and its applications...