The Effective Average Action Beyond First Order

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by  H. Ballhausen

The Effective Average Action Beyond First Order
∗
H. Ballhausen
Institute for Theoretical Physics, Heidelberg University
Philosophenweg 16, 69120 Heidelberg, Germany
A derivative expansion of the effective average action beyond first order yields renormalization
group functional flow equations which are used for the computation of critical exponents of the
Isinguniversalityclass. Thecriticalexponentν inD=3isconsistentwithhigh-precisionmethods.
The effective average action Derivative expansion beyond first order
TheeffectiveaverageactionΓ [ϕ](see[1]and[2,3,4,5]) In a second order derivative expansion of the effective
k
4 isafunctionalofthefieldsϕ. Itinterpolatesbetweenthe average action there are contributions from up to four
0 classicalactionS atsomemicroscopicultravioletscaleΛ nonvanishing momenta. These are parametrized by the
0 and the effective action Γ: ΓΛ = S, limk→0Γk = Γ. Its effectiveaveragepotentialandonerespectivelythreelin-
2
dependence onthe momentumscalek is describedby an earlyindependentwavefunctionrenormalizationsinfirst

n exact renormalization group functional flow equation: respectively second order:
a
8 J ∂kΓk = 12Tr{(Γ(k2)+Rk)−1∂kRk} Γk =R dD+xZ{Uk((ρρ)) 1∂ ϕ∂µϕ
0v3 1 wwtliumhimtekhr→ecruΛetΓsRop(kff2ke)cftuisntcotthiotehanneswdeficiltoeihlnmddtsqh2→feaun0pndRrcotkRipo(ekqnr2at)dliee>sdneoli0rtmieveskan→tasi0uverRminkoogmf=teΓhn0ke-, +++ZZZCDBA,,,,kkkk(((ρρρ)))····4244111!!!!ϕ∂ϕµµ∂ϕϕµ∂∂∂µµµ∂ϕϕµ∂∂ϕνν∂ϕϕν∂∂∂νννϕϕϕ+O(∂6)}
→ ∞
7 above limit properties of Γk. Universal properties of the This truncation has been the starting point for a recent
0
renormalizationgroup flow, however,are independent of investigation in [6]. There an additional field expansion
3
0 the actual choice of Rk. of the potential and the wave function renormalizations
3 The flow of Γ(2) and higher n-point-functions can be de- in powers of ρ is done. Here we keep the full field de-
0 k pendence butinsteadgoonlyonestepbeyondfirstorder
h/ rived from the flow of Γk by functional derivation, e.g. andtakeonlyUk,ZA,k andZD,k intoaccountdue tothe
p-t ∂kΓ(k2) =1TTr{rΓΓ(k3()42)((ΓΓ(k(22))++RRk))−−32∂∂kRRk} following reasoning:
e − 2 { k k k k k} WhileZ andZ directlycontributetoΓ(2) andthus
h A,k D,k k
(3) (4) (5) enter all flow equations as q2 and q4 terms in the de-
: wherethe flowofΓ wouldinvolveΓ andΓ andso
v k k k nominators of the integrands, Z and Z only give
on. Inordertoendupwithaclosedsystemoffunctional B,k C,k
i
X flow equations one needs to do a finite truncation of Γ . additive contributions to the four-point-vertex and the
k
three-point-vertex respectively. Also, their flow requires
r
a In the following we concentrateona one-componentreal ∂ Γ(4) and∂ Γ(3). ThisiswhyweneglectZ andZ
k k k k B,k C,k
scalar field ϕ : IRD IR. There are the invariants in this paper which corresponds to a truncation beyond
→
ρ= 12ϕϕandρq2 = 12∂µϕ∂µϕfeaturingtheO(1)symme- first and below second order.
try of the Ising universality class. Here the lowest order
truncation of the effective average action reads We hope that this improvesthe criticalexponentν com-
pared to first order calculations. On the other hand, the
Γ = dDx U (ρ)+ 1∂ ϕ∂µϕZ +O(∂2)
k R { k 2 µ k } anomalous dimension η is related to the momentum de-
pendence of the wave function renormalization and will
in terms of the effective average potential U (ρ) and a
k
probably suffer from the missing contributions.
fieldindependentwavefunctionrenormalizationZ . The
k
effective averagepropagator
Substituting U := U , Z := Z and Z := 2!ϕϕZ
k 0 A,k 1 4! D,k
Γ(2)(q2)+R (q2)=U′(ρ)+2ρU′′(ρ)+q2Z +R (q2) the truncation thus reads:
k k k k k k
gives the flow of the effective average potential Γ = dDx U(ρ)
k
R {
+Z (ρ) 1∂ ϕ∂µϕ
∂kUk(ρ)= 12R (2dπD)qDUk′(ρ)+2ρUk′′∂(ρk)R+kq2Zk+Rk(q2) +Z01(ρ)·· 221!!∂µµ∂µϕ∂ν∂νϕ+O(∂4)}
and ∂kZk can be determined from ∂q2∂kΓ(k2)(q2)|q2=0. and the next step is to derive ∂kU, ∂kZ0 and ∂kZ1:
2
Renormalization group flow equations Scale invariant flow equations
Expanding the fields ϕ(x) = ϕ¯+χ(x) in small fluctua- As we are interested in critical phenomena of second or-
tions χ(x) = eiqxχ(q) in momentum space around der phase transitions corresponding to fix points in the
Pq
a constant background ϕ¯ yields the n-point-functions renormalization group flow, we write the flow equations
Γ(n)(q ,...,q )= δ δ Γ= δ δ Γ : in explicitely scale invariant form.
1 n δϕ(q1)···δϕ(qn) δχ(q1)···δχ(qn) |χ=0
Γ(2)(q, q)=U′+2ρU′′+q2 Z +q4 Z Tothisendonesubstitutesz (σ)=Z (ρ)/Z (ρ )(where
0 1 0 0 0 k
Γ(2)(p,−−p)=U′+2ρU′′+p2··Z0+p4··Z1 ρuk=iskt−hdeUr,utnn=ingLmnikn,imηu=m o∂fULn(ρZ)()ρ, σ)=etcZ.0(sρukch)kt2h−aDtρ,
Γ(3)(p,q, p q)= − − t 0 k
Γ(3)(p+q−, −q, p)=√2ρ 3U′′+2ρU′′′
′ ′ ′′
++((pp24++−(2ppq−2)(p+q)q+2)3·pZ20′q{2+2(pq)q2+q4)·Z1′ } ∂−tucc0=·II(201−2+η)u +(D−2+η)σu
Γ(4)(p,q, q, p)=3U′′+12ρU′′′+4ρ2U′′′′ − 1· 2
+(p2−+q−2) (Z′ +2ρZ′′) ′
+(p4+4p2q·2+0q4)·(Z01′ +2ρZ1′′) ∂+tz40σ=c0ηc0z·+((D−2+−η)0σ+DzD0 J40+ D2K41)
+8σc c (0+1DI0 2+ DJ1+ 2K2)
These are inserted into 0 1· D 3 − D 4 D 4
+4σc c (1+2DI1 4+ DJ2+ 2K3)
∂kΓ= 12R (2dπD)pDΓ(2)(p∂,k−Rp()p+2)R(p2) +8σc10c12··(0+DD3DI331−− 4+DDDJ442+ DD2K443)
∂kΓ(2)(q, q) +4σc1c2·(4+D8DI32− 12+D2DJ43+ D4K44)
=R (2d−πD)pD(ΓΓ(2(3)()p(p2,)q+,−Rp(−p2q)))Γ2((3Γ)((2p)+((qp,+−qq),2−)p+)R∂(k(pR+(pq2)2))) +c4σc2Ic02·(4+D6DI33− 8+DDJ44+ D2K45)
1 dDp Γ(4)(p,q,−q,−p) ∂kR(p2) −c4·42I1
− 2R (2π)D (Γ(2)(p2)+R(p2))2 − 5· 2
and yield the flow equations ∂ z =(2+η)z +(D 2+η)σz′
t 1 1 − 1
+2σc c ( K0)
∂kU =∂kΓ +4σc0c0·( 0+ 2DJ0+ 8+D K41)
∂kZ0 = ∂∂q2 ∂kΓ(2) |q2=0 +2σc01c11··( 0+D2DI30−− 10+DD4DJ441+ 20DD+D K442)
∂ Z = 1 ∂ ∂ ∂ Γ(2) +4σc c ( 0+ 2DI0 8+ 6DJ1+ 36+18D+D2K2)
k 1 2∂q2∂q2 k |q2=0 0 2· D 3 − D 4 D(D+2) 4
+4σc c ( 4+ 8DI1 32+ 8DJ2+ 64+30D+D2K3)
In orderto be able to perform the q2-derivatives one has 1 2· D 3 − D 4 D(D+2) 4
toexpandthedenominatorcontaining(p+q)2. LetN = +2σc2c2·(16+D22DI32− 72+D12DJ43+ 96D+4(D2D++2)D2K44)
Γ(2)(p2)+R(p2) and M =Γ(2)((p+q)2)+R((p+q)2): c I0
− 5· 2
1 = 1 (2pq+q2)N˙ + (2pq+q2)2N˙2 1(2pq+q2)2N¨ +... where for abbreviation we have used the constants
M N − N2 N3 − 2 N2
Then the mixed scalar products (pq) can also be elimi- w =u′+2σu′′
nated: c =w′
0
′
21R (2dπD)pDf(p2,q2)·(pq)2 = 21R (2dπD)pDf(p2,q2)p2Dq2 cc12 ==zz01′
12R (2dπD)pDf(p2,q2)·(pq)4 = 21R (2dπD)pDf(p2,q2)D(pD4q+42) cc3 ==zw′′++22σσzw′′′′
12R (2dπD)pDf(p2,q2)·(pq)k =0 for all odd k c45 =z01′ +2σz01′′
This finally gives the explicit flow equations:
and the momentum integrals
∂kU =1·IhN1i Inm(w,z0,z1,η)[r]:= 21R (2dπD)pD (w+p(2pz20)+mp4∂˜zt1r+r)n
∂kZ0 =4ρC02·Ih− NN˙4 + D4 pN2N5˙2 − D2 pN2N4¨i+... Jnm(w,z0,z1,η)[r]:= 21R (2dπD)pD (p2()wm+(pz20z+0+2pp24zz11++rr˙))n∂˜tr
∂kZ1 =2ρC02·Ih2NN˙52 − NN¨4i+... Knm(w,z0,z1,η)[r]:= 12R (2dπD)pD(p2()wm+p22(zz00++2pp42zz11++rr)˙n)2+1∂˜tr
Ihf(p2)i= 21R (2dπD)pD f(p2) ∂kR(p2) − 21R (2dπD)pD ((wp+2)pm2z(02+z1p+4zr¨1)+∂˜rt)rn
′′ ′′′
C0 =3U +2ρU which will be discussed next:
3
Momentum integrals Critical exponents
The momentum integrals I, J and K can be computed Starting with a quartic potential u = 1(ρ κ )2 the
2 − Λ
efficiently in case of a linear cutoff. For the choice [7, 8, flow equations are discretized on a grid and numerically
9, 10, 11] R =Z (k2 q2)Θ(k2 q2) one has solved for different values of κ . During the evolution
k k Λ
− −
towards k 0, the running minimum of the potential
r =(1 p2) Θ(1 p2) ( u′(κ) = 0→) may either end up in the massless sponta-
− · −
r˙ = Θ(1 p2) (1 p2) δ(1 p2) neouslybrokenphase(m=0,κ )orinthemassive
− − − − · − symmetric phase ( m2 k2u′(0),→κ∞=0 ).
r¨=2δ(1 p2)+(1 p2) δ′(1 p2) ∼
− − · −
∂˜tr = kZ∂kkkR2k =(2−η+p2η)·Θ(1−p2) cThhaeraccrtietirciazeldκbcylefiaxdspotointassoefcoanlldcoourpdleirngpshaansedtvraannissihtiionng
and I, J and K are reduced to a single integral H: masses. Finetuningκ aroundκ thenyieldsthecritical
Λ c
exponent ν through the scaling law m T T ν and
c
Inm =(2−η)Hnm the establishedproportionality|κΛ−κc|∼∼||T−−Tc||. The
+ηHm+1 critical exponent η may be identified with the fix point
n
∗
η of the anomalous dimension.
Jm =(z 1)Im
n 0− n
+2z Im+1
1 n The critical exponents for the universality class of the
Km =2(z 1)2Im three dimensional Ising model obtained in this way are
n 0− n+1 compared in Table 1 to values obtained in leading order
+8(z 1)z Im+1
0− 1 n+1 ( Z1 =0, Z0 =1, also known as ’local potential approx-
+8z12Inm++12 imation’ ), lowest order beyond leading order ( Z1 = 0,
2z Im Z = const, here called ’lowest order’ ), and full first
− 1 n 0
vD(w+z0+z1)−n order beyond leading order ( Z1 = 0, here called ’first
− order’ ).
where H is a special case of a hypergeometric function
Theexponentsareinagreementwithformercalculations
Hm(w,z ,z ) using the effective average action. For brevity we quote
n 0 1
= 1 dDp (p2)m Θ(1−p2) only results which are basedon simulations as similar as
2R (2π)D(w+p2z0+p4z1+r)n possible. Still they differ in that they either use a field
1 expansion(anexpansionofuandz inpowersofρ)ora
=v dx xa
DR ((w+1)+(z0−1)x+z1x2)n different infrared regulator or both. This might explain
0
the numerical differences.
1
= vD dx xa (1 bx cx2)−n
(w+1)n R · · − −
0 For comparison we also give results from different ap-
1 ∞ k/2 proaches. Anextensivereviewofdifferenttechniquesand
= vD dx xa xk (n+k−l−1)!k!cl bk−2l
(w+1)n R · · P k! P (n−1)! l! (k−2l)! recent results can be found in [13]. We adopt their over-
0 k=0 l=0
all estimates of exponents and errors. In addition we
∞ k/2 1
= vD (n+k−l−1)!cl bk−2l dxxa+k quote some of the newestandprecisestresults fromhigh
(w+1)n kP=0lP=0 (n−1)! l! (k−2l)!R0 temperature expansions, perturbation series at fixed di-
∞ k/2 mension,ǫ-expansionsandMonte Carlosimulationsplus
= vD (n+k−l−1)!cl bk−2l 1
(w+1)n P P (n−1)! l! (k−2l)!a+k+1 some additional references.
k=0l=0
with
Conclusion and outlook
v−1 =2D+1πD/2Γ(D/2)
D
a=m+ D 1 The critical exponent ν has significantly improved com-
2 − pared to first order. This is the first calculation using a
b= 1−z0 linear cutoff that is consistent with high precision meth-
1+w
c= z1 ods. On the other hand, the anomalous dimension is
−1+w sensitiveto the neglectedcontributionsas canbe seenin
The series expansion and reordering is valid and conver- comparison to [6]. Its error has even increased in com-
gent as long as 1 bx cx2 = 0 for x [0;1]. Which is parison to first order.
the case, as the e−ffectiv−e aver6age propa∈gator stays finite Hencetheinclusionoftheneglectedwavefunctionrenor-
at all times. malizations should be the next step.
4
ν η
effective average action to leading order ( this paper, Z =0, Z =1 ) 0.6491 0
1 0
effective average action to lowest order ( this paper, Z =0, Z =const ) 0.6271 0.1120
1 0
effective average action to first order ( this paper, Z =0 ) 0.6255 0.0503
1
effective average action beyond first order ( this paper ) 0.6303 0.0793
effective average action to leading order ( field expansion [11] ) 0.6496 0
effective average action to lowest order ( field expansion [12] ) 0.6262 0.1147
effective average action to first order ( exponential cutoff [14] ) 0.6307 0.0470
effective average action to second order ( exp. cutoff, field exp. [6] ) 0.632 0.033
literature values from the review [13] 0.6301(4) 0.0364(5)
25th-order high-temperature expansion [15], see also [16] 0.63012(16) 0.03639(15)
seven-loop perturbation series at fixed D=3 [17], see also [18] 0.6303(8) 0.0335(6)
five-loop order ǫ expansion with boundary conditions [18] 0.6305(25) 0.0365(50)
Monte Carlo simulation [19], see also [20, 21] 0.6297(5) 0.0362(8)
TABLE 1: Critical exponents of the three dimensional Ising universality class
∗
Electronic address: [email protected]
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