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8/12/2019 Huber GaugeFixingAspects
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SU(2)
SU(N)
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MAG
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d
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d
http://cpc.cs.qub.ac.uk/summaries/AECT_v1_0.html8/12/2019 Huber GaugeFixingAspects
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LQCD = q( /D+m)q+ LY M, LY M=1
2tr{FF}.
LQCD
D
D=+i g A.
LY M
LY M
F
F=A A+i g [A, A].
A T
r
[Tr, Ts] =ifrstTt,
tr{TrTs}= Tfrs,
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Tf=
12
SU(N)
A= ArTr
F=FrT
r,
Fr=Ar Ar g frstAsAt.
LY M=14
FrFr
.
AU (x) =U(x) A(x) U(x)1 +
i
g(U(x)) U(x)
1,
U(x)
U(x) =ei g (x)
(x)
(x) =r(x)Tr.
A
r
Ar
+A
r
= A
r
r
g frst
s
A
t
= A
r
Drs
s
,
Drs
Drs =rs+g f
rstAt.
SU(3)
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SU(N)
SU(3)
SU(N)
SU(2)
SU(2)
SU(N)
G2
N
A
(x) = L
(0A(x))=F0,
0
[A(x), (y)] =i g(x y).
0
O[A] :={A|A=AU }.
AU
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1 = [A] DU (f[AU]),
f[A] = 0 DU
[A]
A
U
U
[A]
rs
[A] =detfr[A(x)]
s(y) =:det Mrs(x, y),
M(x, y)
c
c
detMrs(x, y) =
D[cc]e
dxdy cr(x)Mrs(x,y) cs(y).
A = 0 0
Lgf= 12
(Ar(x))
2
dycr(x) Mrs(x, y) cs(y).
LY M LY M+ Lgf,
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= 0
exp(
12
(Ar)
2)
0
Mrs(x, y) =Mrs(x)(x y) =Drs(x y).
A= 0
A= 0A D != 0 M != 0.
M(A)
:={A|A= 0, M(A)> 0}.
A = 0
A = 0
M(0) =
A= 0
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A1 A2
A3
A3 = A1+ (1 )A2, 01.
R[A] :=1
2
dxA(x)A(x)
R[A] = dxAr(x)Ar(x) = dxAr(x)(Drs(x)
s(x)) =
=
dxAr(x)(r(x) g frstAt(x)s(x)) =
=
dx(r(x)A
r(x)).
A
r
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2R[A] = dx(r(x)(Drs(x)s(x)) = dx r(x)Mrs(x)s(x).
Mrs
R[A]
R[A]
R[A]
c
b
Dbr e
dx(ibrfr[A]) =N (fr[A]).
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f[A] = 0
f[A] =i b/2
Dbr e dx(ibrfr[A]+ 2 brbr) =N e 12 (f[A])2
N
= 0
f[A] = 0
s Ar=Drscs, s cr =1
2g frstcsct,
s cr =i br,
s br = 0.
br
i (A
r)/
s2 = 0
c
s2A = 0 s c c
b
s c= i b
s b= 0
f[A]
c
s(cf[A])
f[A] =A
Lgf=s(crf[A]r) =s(crAr) =
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=i br (Ar) cr (Drscs) =i br(Ar) cr Mrs cs.
s c
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DrsA,(p2) =rs
gpp
p2
p2
p4 + 2 N g24,
N
1/p2
1/p4
Vphys
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QB|= 0,
QB
V0
|> 0, |= 0.
|
|= QB|, | / Vphys.
V0 Vphys Vphys
Vphys
Hphys =V
phys/V0.
|
QB| = 0
| |
|
QB|= 0
|> 0
|
|
|k, N k
N
|1=|k, N, |2= QB|1=|k, N 1,
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|3=|k,N, |4= QB|3=|k,(N+ 1).
|1 |2 |3 |4 |1 |3 |2 |4
QB|= 0
Ja :=Fa+ {QB, Dab cb}, Ja = 0.
Qa :=Ga +Na :=
d3x
iF
a0i+ {QB, Dab0 cb}
,
Ga
Na
iF
a0i
Ga
Ga
Na
Qa
Na
a
gfabcAcc
b
uabb
uab
dx ei p(xy)Dae ce (x)gfbcdAd(y)cc(y)=
gppp2
uab(p2).
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Na
(Dab cb)(x)
(Dab cb)(x)
x0= (ab +uab)b(x),
uab uab(0) =ab.
iFa0i
uab
ab
Dabc (p2) =ab cc(p
2)
p2
u
cc(0) = 1
1 +u.
u=1
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cA(p2)
cc(p2)
DA(p2) =dA (p2)A, Dc(p2) =dc (p2)c.
dA dc
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:=c=A/2.
0.5953 . . .
1/2
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10-4
10-3
10-2
10-1
100
101
102
p2[GeV
2]
10-4
10-3
10-2
10-1
100
Z(p
2)
scaling
decoupling
10-4
10-3
10-2
10-1
100
101
102
p2[GeV
2]
100
101
102
G(p
2)
scaling
decoupling
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u=
1
(t)
(t) :=
d3x
d4p
(2)4ei x p(p2)0,
(p2
)
D(p2 = 0) =
d4xD(x y) =
dt(t).
D(p2 = 0) = 0
(t)
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i i
S[] =1
2!Srsrs 1
3!Srstrst 1
4!Srsturstu.
Srs Srst Srstu
Z[J] =
D[]eS+iJi =eW[J],
Ji i Z[J]
W[J]
W[J]
[] =W[J] + iJi.
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J
i:=iJ= W[J]Ji =Z[J]1 D[]ieS+jJj .
Ji =[]
i.
Ji1in
Ji1in :=
[]
i1 in .
Ji1in
J
Dij i1in
Dij :=DJ=0ij ,
i1in :=
J=0
i1in.
DJij
DJij :=2W[J]
JiJj=
2[]
ij
1.
J
Dij ij
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D[]
S
i+Ji
eS+jJj =
S
i
i=/Ji
+Ji
Z[J] = 0.
Z[J]
eW[J]
eW[J]
Ji
eW[J] =
W[J]
Ji+
Ji
eW[J]
S
i i= W[J]Ji + Ji+Ji= 0.
W[J]
W[J]/Ji i /Ji
Ji=
jJi
j=
Ji
W[J]
Jj
j=
2W[J]
JiJj
j=DJij
j.
Si
i=i+DJij/j
+[]
i= 0,
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S
i=Siss 1
2!Sistst 1
3!Sirststu.
i i + D
Jij
j
j
ij =ij,
iDJjk =
i
2[]
j k
1=
=
2[]
j m
1 3[]
min
2[]
nk
1=DJjm
JminD
Jnk,
iJj1jn =
[]
ij1 jn= Jij1jn.
(M M1) = 0
Mij = D
Jij
M1ij = Jij
[]
i=Siss 1
2Sist(st+D
Jst)+
13!
Sistu(stu+ 3sDJtu+D
JsvD
JtwD
Jux
Jvwx).
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=
i-1/2 -1/2 -1/3! -1/2 -1/3!
i=
i
i=
i
i=
i
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-1
=
-1
-1/2 -1/2 -1/2 -1/3! -1/2 - -1/2
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[]k
[]k
k
k
k0
k
[]
S
[]k
S
k
k
Wk[J] =ln Zk[J] =ln
D eS[]+Jii
12iR
kijj .
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12
iRkijj =12 ddq
(2)dRkab(q)a(q)b(q),
Rkab
k0
k
q2 < k2 k2
q2 > k2
k[] =Wk[J] +Jii 12
iRkijj
i=Wk[J]
Ji=i.
k
S
[]
k
S
[]
Rk
[]k
k
kk[] = kWk[J] W
k[J]
JikJi+
Jik
i 12
ikRkijj =
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k k = + 12
1
kR
k
= 2
ij
1
2 k,Jmn+R
kmn
1kR
kmn=
1
2
iGJmr
k,Jrjs G
JsnkR
kmn=
=1
2GJmt
k,Jtiu G
Jur
k,Jrjs G
JsnkR
kmn+
1
2GJmr
k,Jrjs G
Jst
k,Jtiu G
JunkR
kmn+
1
2GJmr
k,JirjsG
JsnkR
kmn.
GJijl
=
k,Jij +Rkij
1l
=GJimk,JmlnG
Jnj ,
i
k,J
j1jn = k[]
ij1 jn = k,J
ij1jn.
Rk
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D(i)ab (p) =P
(i)ab
c(i)(p2)p2
,
P
(i)ab c
(i)
(i)
c(i)
c(i),IR (p2) =d(i)
(p2)
(i)
.
d(i)
(i)
(i)
(i) < 0
(i) > 0
(i) > 1/2
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1/(p
k)2
ddk
(2)d
k21 (k p)22 = (4)d2 d2 +1 d2 +2 d2 1 2
(1) (2) (d+1+2)p2 d
2+1+2 .
1
2
d/2 +1+ 20 d/2 +10 d/2 + 20
F4
n3
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A c AAA Acc AAAA
p
d1A (p2)1A =p2 (p2)1+AAA+2ALgloop + (p2)1+Acc+2cLghloop
(p2)1+AAAA+3ALsunset (p2)1+2AAA+4ALsquint.
dA
L
L
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1
=
1
1
2
1
2
+ 1
6
1
2
AAAA+ 2A
AAcc+ 2c
AAAAA+ 3A A2AAA+ 4A.
AAA 0
AAA
0AAA+ 3A3A A0.
1/p4
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Rk
Rk
Rkij(p2) = 0ij(p
2)(p2 k2),
k
2
p2 k2 k 0
Rkij (p2) = 0ij(p
2)r(p2/k2).
G1ij (p2) = 0ij(p
2)(1 +r(p2/k2))
p
kij(p2) =Gmt
ktiuGur
krjs GsnkR
kmn+
1
2Gmr
kirjsGsnkR
kmn.
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k k = + 12
1
kRk
k
k
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c 0
Dc(p
2)p21 p200
Dc(p
2)
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= +...
A A A A
1
B
k k
= +...
A B
BA
A B
BA
C Dk k
A BAABB, AABB2ABCD+C+ D.
ABCD+1
2(A+B+C+ D)0,
m= 2n 4
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A
B C
k
k + . . .
A
B C
E
=
D D
=
AB
C
A
B
C
CA
B
D Ek
k + . . .
A
B
C
prim. div.ABC 0,
1
2 i ki1...iri i0 .
A
B
l
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v
ni i mi i
nbi...k i
k n
di...k i
k
v =ld
2+
i
ni(i 1) +
vertices,r3
ndi1...ir(i1...ir+ci1...ir)+
+
vertices,r3
nbi1...irci1...ir cv.
ci...j/i...j i j
d
d = 4
d nd
nb
r
r= 3
ndAAA
AAA+
1
2
+ndAcc
Acc+
1
2
+
1
2nbAAA+
1
2nbAcc.
v = 12
i
mii+
vertices,r3
ndi1...ir
i1...ir+
1
2
i
ki1...iri i
+
+
vertices,r3
nbi1...ir
1
2
i
ki1...iri i
.
nbi1...ir
v
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v,max =12
i
mii.
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i=
1
2 i mii+ vertices ni1...ir 1
2 i ki1...iri i+ dressed
vertices
ndi1...iri1...ir ,
i
ni1...ir n
di1...ir
nbi1...ir
1
2 iki1...iri i
0.
nbi1...ir
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(a) (b) (c) (d)
1
2nbi1...ir
j
jki1...ir
j = 0.
i1...ir = 0.
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1
2A+c0, 3
2A0, 2A0,
A c
1
2A+c0, A0.
A = 0
c0 A = 0
A+ 2c = 0 c= 0
c= 0
A +2c= 0 c= 0
A= 0
A + 2c= 0
:=c
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1
=
1
d
d/2 2
d
A+ 2c+
d
2 2 = 0.
m
2n
m,2n=m2
A n c = (n m),
d
d
m,2n=m2
A n c+
d
2 2
1 m
2 n
= (n m)+ (1 n)
d
2 2
.
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A
DabA,(p2) =ab
gpp
p2
cA(p
2)
p2 +ab
ppp4
.
(A)(A)
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e 12 (A)
2
A,long
A = A,long =2c
A,long = 0 A =2c = 0
Lga = 14
F2+ 1
2(A)
2 +
2 1
2
2fabcfadecbcccdce+
2
(Dc)c 1 2 (c)Dc.
c =c, c =c.
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= 0, 2
= 1
c0.
A0, c0, A+ 2c0.
A= 0
c = 0
c = 0
A+ 2c = 0 A = c = 0 A c
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i=i
A
V
v = 12
i=A,V
mii+1
2nAV(2AV A V)+
+ vertices,r3
ndi1...ir i1...ir+12
i=A,Vk
i1...iri i+
vertices,r3n
bi1...ir 12
i=A,Vk
i1...iri i .
nAV
AV := 2AV A V
AV
i+1
2
j
j mj nbi1...ir
1
2
j
ki1...irj i
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1
2
(2AV
A
V)
nAV + dressed
vertices
ndi1...ir ki1...irAV
0.
ki1...irAV
i1 ir
AV
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fabcBcBb=
fabcBbB
c= 0
Dab :=ab+g f
abcBc+g fabiAi.
s Ar=Drscs.
r
i
a
s Ba=Dabcb Daici =(Dabcb g fabcBbcc g fabiBbci), s Ai=Dij cj Diaca =(ci g fiabBacb).
s ca =12
g fabccbcc g fabicbci, s ci =12
g fiabcacb,
s ca =i ba, s ci =i bi,
s ba = 0, s bi = 0.
frstfruv +frsufrvt +frsvfrtu = 0,
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fabifacj +fabjfaic = 0,
fabcfadi +fabdfaic +fabifacd = 0,
frabfrcd +fracfrdb +fradfrbc = 0.
RMAG=1
2 dx Ba(x)B
a(x).
DabBb= 0.
SMAG = s
dx ca(DabB
b i
2ba) =
=
dx (
2baba +ibaDabB
b+ c
aDab Dbcc
c g fabica(DbcBc)ci
g fbcdcaDabB ccd g2 fabifcdiBbBccacd (1 )g fabicaBbci),
Dab
Dab :=ab+g f
abiAi.
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SR=s
dx(1
2 g fabicacbci 1
4 g fabccacbcc) =
=
dx(i g f abibacbci i1
2g fabcbacbcc +
1
4g2fabifcdicacbcccd+
+14
g2fabcfadicbcccdci +18
g2fabcfadecbcccdce)
SU(2)
SMAG,R =
dx
caDabD
bcc
c g fabica(DbcBc)ci g fbcdcaDabBccd
g2 fabifcdiBbB
cc
acd
(1
)g fabicaBbc
i + 1
2
(DabBb)
2+
+2
8g2fabcfadecbcccdce
g fabi(Dac B
c)c
bci
2g fabc(Dad B
d)c
bcc +
2g2fabifacdcbcicccd+
+1
4g2fabifcdicacbcccd +
1
4g2fabcfadicbcccdci +
1
8g2fabcfadecbcccdce
.
, , 0 = 1
= =
SMAG,R
AAcc
BBcc
ABcc
cccc
Acc
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Bcc
U(1)N1
A
i= 0
b
Sdiag =s
dx ci(A
i i
2bi) =
= dx ibi(Ai i 2bi) + ci(ci g fabiBacb) .
Sdiag =
dx
1
2(A
i)
2 + ci(ci g fabiBacb)
.
ci ci +1g fabiBacb.
S=SY M+SMAG,R+S
diag,
SMAG,R =SMAG,R
==
=
dx
caDabD
bcc
c g fbcdcaDabBccd
g2 fabifcdiBbBccacd + 1
2(Dab B
b)
2 +
8g2fabcfadecbcccdce
12
g fabc(Dad Bd)c
bcc +1
4g2fabifcdicacbcccd +
1
8g2fabcfadecbcccdce
,
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DijA (p2) =ij
cA(p2)
p2
gpp
p2
+ ij
ppp2
,
DabB(p2) =ab
cB(p2)
p2
g (1 )pp
p2
,
Dabc (p2) =ab cc(p
2)
p2 .
cA(p2)
p20= dA (p2)A,
cB(p2)
p20= dB (p2)B ,
cc(p2)
p20= dc (p2)c.
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SU(2)
A+B0, A+c0, B+c0, 2B0,2c0, A+1
2B0, A+1
2c0.
SU(N >2)
1
2A+
1
2B+c
0,
1
2A+
3
2B
0,
3
2B
0,
1
2B+c
0.
SU(N)
A+B0, A+c0, B0, c0.
SU(N >2)
SU(2)
SU(N)
A =B =c = 0
B = 0
B = 0 B
AABB
AAcc
A + c = 0 A
c B = 0 A = c = 0
A = 0 AABB c
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1
2
ab
a b
1
2
ab
a b
AABB
AAcc
AABB
AAcc
nA
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A2A1
A3
A4
AnA1
AnA
B1
B2
BnB1
BnB c1
c2
cnc1
cnc
...
......
A
B
c
AABB
AAcc
nB nc
(nA, nB, nc) =
1
2(nA nB nc)MAG
nA+nB+ nc
.
12
MAGAB20, 12
MAGAc20,
3
2 MAGBBB0, 3
2MAGBcc0.
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(nA, nB, nc) =12(nA nB nc+)MAG (nA+nB+ nc .
SU(2)
0
1
SU(N >2)
2
3
MAG:=A = B =c0
(nA, nB, nc) =1
2(nA nB nc)MAG nA+nB+ nc
(nA, nB, nc) =1
2(nA nB nc+)MAG (nA+nB+ nc .
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02LG:= A=2c
c 1 A 1 1/p4
0LG1.
1/p6
2
LG
A >1 LG >1/2 MAG
1
MAG > 1
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AABB
AAcc
BBcc
AABB
AAcc
BBcc
MAG
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AAcc
AABB
SU(N)
tabij1 =abij, tabij2 =f
aicfbjc, tabij3 =dabijA ,
dabijA
dabijA =16
TaT(bTiTj)
.
fajcfbic
tabij2
MAG
ISS(a,b,c,e,f;p2) :=
ddq
(2)d
ddr
(2)d
(q2)a[(r)2]b[(p
q
r)2]c(2p q)e(2p r)f
{e, f} N e= 0
X(p2, MAG)
X(p2, MAG)
MAG
= 0
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0.2 0.4 0.6 0.8 1.0
MAG
1
1
2
3
1
0.75
0.25
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.2
0.4
0.6
0.8
1.0
MAG
MAG
MAG
MAG = 0.739908
= 100
MAG 0.74
0.5
0.9
2.2 MAG0.74
MAG= 1 = 3.359
MAG
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:=
{A
|A= 0, M >0
}
M=D
A = 0
Dabcc (k)
Dabcc (k) =ab
k2ccc(k) =(M1)ab.
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ccc(k)
Dabcc (k) =ab
k21
1 (k, A) .
(k, A)
k
(0, A)< 1
(0, A)
(0, A)
A
(1 (0, A)) =
d2 i e
i (1(0,A))
i .
4
n
tr(n) > 0
h(x) = lim(x)
dy
Dac(x)2(x)
M1
ab(x, y)
Dbc(y)
2(y)
.
(x)
(0, A)
dx h(x)< d 4(N2 1)V,
d
N
V
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dx h(x) =d 4(N2 1)V.
Snonlocal =SF P+Sh,
SF P =SY M+Sgf,
Sh =
dx (h(x) 4d(N2 1)),
SY M=1
4 dxFrF
r,
Sgf=
dx
i br(A
r) + c
r Mrs cs
.
4d(N2 1)
Sh
ab
ab ab
ab
s ab =ab , s
ab = 0,
sab = ab, s
ab = 0.
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= 1, . . . , d
a, b= 1, . . . , N 2
1
Sh
D[]e
dx (ac Mabbc+2 g fabcAa(bc bc)) =(det M)d(N
21)e(4 g2 faceAa(M
1)cdfbdeAb),
D[]e
dxac M
bc =(det M)d(N
21)
eSh =
D[]e
dx (ac Mabbc+2 g fabcAa(bc bc)ac Mbc d 4 (N21)).
Slocal =SF P+SGZ,
SGZ=
dx
acM
abbc +2 g fabcAa(
bc bc) ac M bc d 4 (N2 1)
.
SBRST =
d x g f abeac
(Ded c
d)bc
.
ab(x)ab(x) g
dy (M1)ac(x, y)fcde
(Ddfc
f(y))eb(y)
,
s(ac Mabbc) =
acM
abbc ac Mabbc +g fabeac
(Ded cd)bc
.
SBRST c
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c
Slocal +SBRST
2
s
2 g fabcAa(bc bc)
=2 g fabc
(Dad cd)(bc bc) +Aabc
.
F2
g fabcAa(bc bc)= 2 d 2 (N2 1)
2
[0]
2 = 0,
[0]
e[0] =
D[Acc]eSlocal.
c
c
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= 0
c
c
1/k4
U
V
Slocal =SF P+SGZ,
SGZ=SU+SV +SU V M, SU=
1
2
dx Uac M
ab Ubc ,
SV =
1
2 dx Vac Mab Vbc +i g 22fabc dx AaVbc , SU V =
1
2i gfabc
dx Uad V
bd A
c,
U
V
= 1
2(U+i V) , =
12
(U i V) .
A = 0 LU V
A
V
U
U
V
c
c
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U
D[cc]ec M c =det M,
D[]e M = (detM)d(N21),
D[U]e 12U M U = (detM)d2 (N21).
= 0
V
V
V
S=
dx
1
4FaF
a+
1
2(A)
2 acMab bc+
+1
2Vac M
ab Vbc +i g 2
2fabcAaVbc
+1
2i gfabcUad V
bd A
c
,
1
d2
(N21)+1
1
N2
1
N
A
V
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c c 1
d(N2 1)
d(N2 1) U d/2(N2 1)
V d/2(N2 1)
d/2(N2 1) + 1
c
c
U
V
N2 1
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AAV
e,
e.
V
V
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i j1
i j1
A A
ij
V A
ij
ab
ab
A
V
dabc
fabc
V V
abcd, acbd, adbc, fabefcde, facefbde, dabcdA .
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dabcdA =16
TaT(bTcTd) ,
Ta
SU(N)
fadefbce
fabefcde
facefbde
dabc
A
V
1
d2
(N2 1 ) + 1
ca
ca
ab ab U
ab
//U
//U
//U
1
c
c
V
= AA AV
V A
V V ,
AA,ac =acp2cA(p
2)P+ ac 1
cA(p
2)pp,
V V,abcd =acbdp2cV(p
2)g,
AV,cab =fcabi p2cAV(p
2)g.
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cij(p
2)
P P=
g
pp/p
2
cA(p2) = 1
cV(p2) = 1
cAV(p2) =g 2/p2
()1 =D =
DAA DAV
DV A DV V
=
1
AAV V (AV)2
V V AVV A AA
,
DAA,ab =ab 1
p2P
cV(p2)
cA(p2)cV(p2) + 2N c2AV(p
2),
DV V,abcd = 1
p21
cV(p2)acbdg fabefcde 1
p2P
2c2AV(p2)
cA(p2)c2V(p
2) + 2N c2AV(p2)cV(p2)
,
DAV,abc =i fabc1p2
P
2cAV(p
2)
cA(p2)cV(p2) + 2N c2AV(p
2).
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AA
AV
V V
D,abcd = (,abcd )
1 =abcd c(p2)
p2 .
cij(p2)
IR= dij(p
2)ij .
cA(p
2
)cV(p2
) + 2N c2AV(p
2
)
c2AV > cAcVA+V >2AV
cAcV > c
2AV2AV > A+V
c2AVcAcVA+V = 2AV
c2AVcAcVA+V = 2AV
V
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1
2
V V
V V
A A
ij
A A
V V
V V
ij
AA
V V
V V
AV V
V +1
22V 1
2(A+ 2V) 1
2
d
2 2
0 2V A d
2+ 20.
A
2V V + 2AV d2
+ 20 2AV V + d2 2.
AA
V V
A2V + d2 2 +AV V A+V V +d
2 2
AV V0 AV V
V +d2 2A+V >2AV V + d
2 2,
0> 0.
AV
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A+V 0.
AV
AAV
V
V
AV
V
V
U
V
AV
V
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V
:= c = = U = V
A+ 2=d
2 2
2n
m
2n,m= (n m)+ (1 n)
d
2 2
.
0.595 353
(1 +)(2 +)12(3 + 4(2 +)) = 1,
= 1
98 93
1201 = 0.595 353
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A A
A V
A V
ij
A A
V V
A V
ij
V A
V V
A V
ij
V A
AV V AAA AAV
AAA = 3V,
AAV =AV + 2V.
AV
AV
dAV
dV
dA
V
AV
dAV
dV dA g
2/dAd2V
0.0267784
AV
AV
V V
AV V
AAV
dAd
2V (dAd
2V)
2
AV AV V V
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fabc
(d 1)(p2)AV+1dAV =g2 N ddq(2)d
dAV((p+q)2
)1+V
(q2
)1+AV
(p2
q2
(p q)2
)2p2dAd2V
.
dAV
1 =0.0513093(1 AV)(0.595353 +AV)
(1.40465 AV)(1 +AV) 0.0370825(AV)(0.595353 +AV)
(1.40465 AV)(2 +AV)
0.0513093(
AV)(1.59535 +AV)
(0.404647 AV)(2 +AV) 0.0210772(
1
AV)(0.595353 +AV)
(1.40465 AV)(3 +AV) 0.0247217(1 AV)(1.59535 +AV)
(0.404647 AV)(3 +AV) 0.0513093(1 AV)(2.59535 +AV)
(0.595353 AV)(3 +AV)
0.0237307(2 AV)(0.595353 +AV)(1.40465 AV)(4 +AV)
0.0210772(2 AV)(1.59535 +AV)(0.404647 AV)(4 +AV)
0.0370825(2 AV)(2.59535 +AV)(0.595353 AV)(4 +AV) +
0.0513093(2 AV)(3.59535 +AV)(1.59535 AV)(4 +AV) .
AV
2AV > V +A =V +d/2 2 0.0668776
0.981386
AV
V V
V +AV +AV V =V +AV A V =AV.
AV
AV
1 AV 2AV >V +d/2 2 V >
d2
V
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1.0 0.5 0.5 1.0 1.5 2.0
AV
0.5
0.5
1.0
1.5
AV y
U
V
A+V = 2AV
detC =
cAcV + 2N c2AV = (p
2)2AVdetD= (p2)A+VdetD
DAA,ab =ab 1
p2P
dV(p2)A
1
detD,
DV V,abcd = 1
p21
cV(p2)
acbdg fabefcdeP 2d
2AV
detD
,
DAV,abc
=i fabc1
p2PdAV
(p2)AV
2
detD .
detD
A=A,
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V =V, AV =
AV.
AV = 2AV A V =2AV +A+V = 0.
V V
A
AV V
:= V =, A+ 2=d
2 2, AV V =A = 0.
AV =
2+
d
4 1.
A
AV V
2n,m= (n m)+ (1 n)
d
2 2
,
m
n
AV > /2 AV =/2
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dij
AV
AV V V
2
= 2
A+V = 2AV
dAdV =2 N d2AV.
D
detC=cAcV + 2N c2
AV
=
=(p2)A+V(dAdV + 2Nd2AV +dD(p
2)D +. . .) =
=(p2)A+V(dD(p2)D +. . .).
D D >0
V V
DAA,ab =ab 1
p2
PdV
(p2
)A
1
detD
,
DV V,abcd = 1
p21
cV(p2)
acbdg fabefcdeP 2d
2AV
detD
1
p21
cV(p2)
fabefcdeP 2d
2AV
detD
,
DAV,abc =i fabc 1p2
PdAV
(p2)AV
2
detD.
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A=A D, V =V D,
AV =AV D.
AV = 2AV A V = 0
D
V V AV V
V +V 12
(A+ 2V)0,A+ 2V +D0.
V V
AA
V V
VV +A+AV V ,A+ 2V + 2D0,
AV V0 AV V
D
D
DA+ 2V 2D,D
2D.
D D = 0
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A
V
A =2c
U
V
AV > /2
AV AV =/2
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SU(3)
SU(2)
SU(N > 2)
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d
d
d
ABC
3ABC+A+B+ C+
d
22
ABC+
1
2
(A+B+ C) +d
41
0.
A B
d
2 2
AABB,
AABB2ABCD+C+D+ d2 2.
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d
= +...
A
B
C
A
B
C
A
BCk k
ABCD+1
2(A+B+ C+ D) + 2
d
4 1
0.
d
4 1
(r 2) +i1...ir+
1
2
j
ki1...irj i0,
r = 3, 4
ABCDABCDEE+E+ d2 2,
AABBCC2ABCDE+D+E+ d2 2.
= +...
A A A A
1
B
k k
= +...
A B
BA
A B
BA
C Dk k
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d
A
B C
k
k + . . .
A
B C
E
=
D D
=
AB
C
A
B
C
CA
B
D Ek
k + . . .
A
B
C
AABBCC
AABB C
d
2 2
AABBCC2ABCDE+D+E+ d
2 2,
1
2(AABB C D E)
d
2 2
ABCDE .
12
(A+B+C+ D+E+ . . . )
d
4 1
(r 2)ABCDE ,
r
d/4 1
2(d/41)
d/41
2n62x = 2 x
d/4 1 2 +x = n 2
d
4 1
(r 2) +1
2
j
ki1...irj i0 .
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v v
d
v =ld
2+
i
ni(i 1) +
vertices,r3
ndi1...ir(i1...ir+ci1...ir)+
+
vertices,r3
nbi1...irci1...ir cv.
ni
i
i1 ir ni1...ir d b
vertices,r3
r
mi i l
i1 ir ci1...ir
A
V
nAV
i=i
l=
i=A,Vni+nAV + 1 vertices,r3
ni1...ir .
A
V
ni=1
2
vertices,r3
ki1...iri ni1...ir mi nAV
, i= A, V,
ki1...iri i i1 ir
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nAV = 0
v =
d
2 2
1 1
2
i
mi
1
2
i
mii+
+
vertices,r3
ndi1...ir
d
4 1
(r 2) +i1...ir+
1
2
i
ki1...iri i
+
+ vertices,r3 nbi1...ir d
4 1 (r 2) +12 i ki1...iri i
v,max =
d
2 2
1 1
2
i
mi
1
2
i
mii
i =
d
2 2
1 1
2
j
mj
1
2
j
mj j+1
2nAV(2AV A V)+
+ dressedvertices
ndi1...iri1...ir+
allvertices
ni1...ird
4 1 (r 2) +1
2
j
ki1...irj i .
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i j
jmj = 2 i+
1
2
j
mji 12
nAV(2AV A V)
allvertices
ni1...ir
d
4 1
(r 2) +1
2
j
ki1...irj i
=
dressedvertices
ndi1...iri1...ir .
i+1
2
j
jmj12
nAV(2AV A V)
allvertices
ni1...ir
d
4 1
(r 2) +1
2
j
ki1...irj i
dressedvertices
ndi1...ir
d
2 2
1 1
2
j
ki1...irj
12 j
jki1...irj +12 ki1...irAV (2AV A V).
ki1...irAV
i1 ir ki1...irAV
AV
AAV
d
n
i+1
2
j
j mj nbi1...ir
d
4 1
(r 2) +1
2
j
ki1...irj i
12
(2AV A V)
nAV +
dressedvertices
ndi1...ir ki1...irAV
0.
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:=
,
:=
.
Z[,] =
D[]eS[,]++
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ijkl= ijkl
==0
.
ijkl
j
i k l
r r+DJ,rt
t,
rr+DJ,rt
t
iDJjk =D
Jjm
J,imn D
Jnk,
i DJjk =DJjmJ,mni DJnk,
iJj1jn =
Jij1jn
,
iJj1jn =
Jj1jni
,
={q, c}
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Ai
r
r +
DJ,rt t
SAirs rs SAirs
rs+DJ,rs
.
Aj
AAij =SAirs D
rr Dss
Ajrs
J=0
+
,
AAij =SAirs D
rr D
ss
Ajsr + .
iS
rstu
rstu =Sistu stu S
ritu rtu=
=S
istu stu+Sirtu rtu = 2S
istu stu.
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g(p2 p1)+g(p1 p3)+ g(p3 p2)+ 1
(gp1 gp2 )
,
abc,(0)BBB,(p1, p2, p3) =g i f
abc(2)4(p1+p2+p3) (g(p2 p1)+g(p1 p3)+ g(p3 p2)) ,
iab,(0)Acc,(p3, p1, p2) =g i f
abi(2)4(p1+p2+p3)(p2+p1),
cab,(0)Bcc,(p3, p1, p2) =g i fabc(2)4(p1+p2+p3)
1
2p3+p2
.
ijab,(0)AABB,(p3, p4, p1, p2) =g2(2)4(p1+p2+p3+p4)f
aiefbje
2gg
1 1
gg
1 1
gg
,
iabc,(0)ABBB, (p4, p1, p2, p3) =g
2(2)4(p1+p2+p3+p4) fabefcie (gg gg) +fbcefaie (gg gg ) +fcaefbie (gg gg) ,
abcd,(0)BBBB,(p1, p2, p3, p4) =g2(2)4(p1+p2+p3+p4)
fabrfcdr (gg gg) +fbcrfadr (gg gg ) +fcarfbdr (gg gg) ,
icab,(0)ABcc,(p4, p3, p1, p2) =g2(2)4(p1+p2+p3+p4)g1
2
2fcbefaie fabefcie ,
ijab,(0)AAcc,(p3, p4, p1, p2) =g
2(2)4(p1+p2+p3+p4)2gfaiefbje,
cdab,(0)BBcc,(p3, p4, p1, p2) =g2(2)4(p1+p2+p3+p4)g
fadkfcbk +fackfdbk
,
cdab,(0)cccc,(p1, p2, p3, p4) =g2(2)4(p1 p2+p3+p4)
4 4fabkfcdk + 2fabefcde +fadefbce facefbde .
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ISS(a,b,c;p2) :=
ddq
(2)dddr
(2)d(q2)a[(r)2]b[(p q r)2]c =
=(4)d(p2)d+a+b+c(a +d/2)(b+d/2)(c+d/2)(a b c d)
(a)(b)(c)(a +b+c+ 3d/2) .
p q
p r
q r
(2p q)
ISS(a,b,c,e,f;p2) :=
ddq
(2)dddr
(2)d(q2)a(r2)b[(p q r)2]c(2p q)e(2p r)f.
q r=1
2
(p q r)2 p2 q2 r2 + 2p q+ 2p r .
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ISS(a,b,c,e,f;p2) =
=d(p2)a+b+c+d+e+f2e+f(a, 2a +e+d/2)(b, 2b+f+d/2)(c, 2c+d/2)
a b c d, 2a + 2b+ 2c+e+f+ 5/2)
max[e,f]
n3,n4,n6,n7,n8=0
(1)n4n82n6(c+d/2, n3+n7)n3!n4!n6!n7!n8!(1 a d/2 e, n3+n4 n8)
P(e, n3+n4, n6)P(f, n7+n8, n6)
(1 b d/2 f, n7+n8 n4)(1 +a +b+c+d, n3+n4+n6+n7+n8).
P
P(a,b,c) := (a/2, b +c/2)(1/2 a/2, b +c/2)
(a, b) :=(a +b)
(a)
P
P
P(e, n3+n4, n6) = 0 e
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a
b
i
S
12
S
12
S
1
6
S
+S
12
S
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ab
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AABB
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http://arxiv.org/abs/hep-ph/0007355http://arxiv.org/abs/hep-ph/9707327http://dx.doi.org/10.1006/aphy.1998.5806http://arxiv.org/abs/hep-ph/9705242http://dx.doi.org/10.1103/PhysRevLett.79.3591http://dx.doi.org/10.1016/0370-2693(93)90726-Xhttp://arxiv.org/abs/hep-lat/0301023http://dx.doi.org/10.1016/S0146-6410(03)90012-3http://arxiv.org/abs/hep-ph/0610365http://dx.doi.org/10.1088/0954-3899/34/7/S02http://dx.doi.org/10.1103/PhysRev.96.191http://dx.doi.org/10.1016/0370-2693(73)90625-48/12/2019 Huber GaugeFixingAspects
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http://arxiv.org/abs/0711.1451http://dx.doi.org/10.1088/1126-6708/2008/02/070http://dx.doi.org/10.1007/BF01614158http://dx.doi.org/10.1016/0003-4916(76)90156-1http://arxiv.org/abs/0812.0580http://dx.doi.org/10.1016/0550-3213(94)90396-4http://arxiv.org/abs/hep-th/9511033http://arxiv.org/abs/hep-th/0109224http://arxiv.org/abs/hep-th/0312324http://dx.doi.org/10.1016/0550-3213(89)90122-3http://dx.doi.org/10.1016/0370-1573(76)90043-0http://dx.doi.org/10.1016/0550-3213(81)90442-9http://arxiv.org/abs/0804.3042http://dx.doi.org/10.1016/j.aop.2008.07.001http://dx.doi.org/10.1016/j.aop.2008.07.001http://arxiv.org/abs/hep-ph/0607293http://dx.doi.org/10.1142/S021773230802700Xhttp://dx.doi.org/10.1142/S021773230802700Xhttp://arxiv.org/abs/0907.5185http://arxiv.org/abs/0810.1987http://dx.doi.org/10.1016/j.aop.2009.07.009http://arxiv.org/abs/hep-ph/02022028/12/2019 Huber GaugeFixingAspects
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http://arxiv.org/abs/0710.2410http://arxiv.org/abs/hep-ph/0510151http://dx.doi.org/10.1016/j.physletb.2005.10.046http://arxiv.org/abs/0904.2380http://arxiv.org/abs/hep-ph/0504110http://dx.doi.org/10.1016/j.cpc.2006.02.005http://dx.doi.org/10.1016/0550-3213(82)90449-7http://arxiv.org/abs/0812.2761http://dx.doi.org/10.1103/PhysRevD.79.074504http://dx.doi.org/10.1103/PhysRevD.79.074504http://arxiv.org/abs/0808.3047http://dx.doi.org/10.1103/PhysRevD.79.014505http://arxiv.org/abs/hep-th/9711070http://dx.doi.org/10.1007/BF02099494http://arxiv.org/abs/1002.1659http://arxiv.org/abs/0811.2760http://dx.doi.org/10.1103/PhysRevD.79.025019http://arxiv.org/abs/hep-ph/0303028http://dx.doi.org/10.1103/PhysRevD.69.016002http://dx.doi.org/10.1007/BF01609471http://dx.doi.org/10.1016/0550-3213(74)90154-0http://arxiv.org/abs/0807.1390http://arxiv.org/abs/0807.1390http://dx.doi.org/10.1103/PhysRevD.78.0745018/12/2019 Huber GaugeFixingAspects
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http://arxiv.org/abs/0704.0722http://dx.doi.org/10.1103/PhysRevD.75.116004http://arxiv.org/abs/hep-lat/0609043http://arxiv.org/abs/hep-lat/0605011http://dx.doi.org/10.1103/PhysRevD.74.014503http://arxiv.org/abs/0801.2762http://dx.doi.org/http://link.aps.org/doi/10.1103/PhysRevD.81.105010http://arxiv.org/abs/hep-th/0412330http://arxiv.org/abs/hep-ph/9604430http://arxiv.org/abs/0904.1873http://dx.doi.org/10.1140/epjc/s10052-010-1371-xhttp://arxiv.org/abs/0903.2193http://dx.doi.org/10.1103/PhysRevD.80.025023http://arxiv.org/abs/hep-ph/0202194http://arxiv.org/abs/0812.2992http://arxiv.org/abs/0812.06548/12/2019 Huber GaugeFixingAspects
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http://arxiv.org/abs/0809.0258http://arxiv.org/abs/0809.0258http://dx.doi.org/10.1103/PhysRevD.79.031501http://arxiv.org/abs/0804.2371http://arxiv.org/abs/0804.2371http://dx.doi.org/10.1103/PhysRevD.78.094503http://arxiv.org/abs/0710.1968http://arxiv.org/abs/0705.0964http://arxiv.org/abs/0705.0964http://dx.doi.org/10.1140/epjc/s10052-009-1064-5http://arxiv.org/abs/0710.0412http://arxiv.org/abs/0710.0412http://arxiv.org/abs/0712.3517http://dx.doi.org/10.1103/PhysRevLett.100.241601http://arxiv.org/abs/0707.3611http://arxiv.org/abs/0707.3611http://dx.doi.org/10.1103/PhysRevD.77.014504http://arxiv.org/abs/hep-lat/0610053http://arxiv.org/abs/hep-ph/0605224http://arxiv.org/abs/hep-lat/0506007http://arxiv.org/abs/hep-lat/0511056http://dx.doi.org/10.1103/PhysRevD.74.034503http://dx.doi.org/10.1103/PhysRevD.74.034503http://arxiv.org/abs/hep-lat/0312036http://dx.doi.org/10.1016/j.nuclphysb.2004.03.021http://dx.doi.org/10.1016/j.nuclphysb.2004.03.021http://arxiv.org/abs/hep-th/0609009http://dx.doi.org/10.1103/PhysRevD.75.025012http://arxiv.org/abs/0812.4045http://dx.doi.org/10.1140/epjc/s10052-009-1066-3http://arxiv.org/abs/hep-ph/0411052http://arxiv.org/abs/0803.1798http://dx.doi.org/10.1103/PhysRevD.77.0945108/12/2019 Huber GaugeFixingAspects
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http://arxiv.org/abs/hep-ph/0611146http://arxiv.org/abs/hep-th/0512261http://dx.doi.org/10.1016/j.aop.2007.01.007http://arxiv.org/abs/hep-ph/0005122http://arxiv.org/abs/0908.0008http://arxiv.org/abs/0908.0008http://arxiv.org/abs/0808.1491http://arxiv.org/abs/0808.1491http://dx.doi.org/10.1103/PhysRevD.79.014018http://arxiv.org/abs/hep-ph/0603256http://dx.doi.org/10.1103/PhysRevD.75.085015http://arxiv.org/abs/hep-ph/0611191http://dx.doi.org/10.1134/S1063779608070083http://arxiv.org/abs/0907.1828http://dx.doi.org/10.1103/PhysRevD.80.104003http://arxiv.org/abs/1001.5033http://arxiv.org/abs/hep-th/0110054http://dx.doi.org/10.1103/PhysRevD.65.065016http://arxiv.org/abs/hep-th/9605030http://dx.doi.org/10.1103/PhysRevD.57.971http://arxiv.org/abs/0905.4220http://arxiv.org/abs/0907.4229http://dx.doi.org/10.1103/PhysRevD.80.085007http://arxiv.org/abs/1001.2399http://arxiv.org/abs/0906.5492http://dx.doi.org/10.1103/PhysRevD.80.101701http://arxiv.org/abs/cond-mat/0701198http://arxiv.org/abs/0907.21938/12/2019 Huber GaugeFixingAspects
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http://arxiv.org/abs/hep-lat/0410001http://dx.doi.org/10.1103/PhysRevLett.94.132001http://dx.doi.org/10.1016/0550-3213(87)90080-0http://dx.doi.org/10.1016/0370-2693(87)90910-5http://dx.doi.org/10.1016/0370-2693(87)90910-5http://dx.doi.org/10.1103/PhysRevD.32.440http://dx.doi.org/10.1103/PhysRevD.25.2681http://arxiv.org/abs/0710.1164http://arxiv.org/abs/0710.1164http://dx.doi.org/10.1063/1.2823875http://arxiv.org/abs/hep-lat/0205025http://dx.doi.org/10.1103/PhysRevD.66.011503http://arxiv.org/abs/hep-lat/9906024http://dx.doi.org/10.1103/PhysRevD.61.034503http://dx.doi.org/10.1103/PhysRevD.61.034503http://arxiv.org/abs/hep-lat/9906025http://dx.doi.org/10.1103/PhysRevD.61.034504http://dx.doi.org/10.1103/PhysRevD.61.034504http://dx.doi.org/10.1016/0550-3213(85)90562-0http://dx.doi.org/10.1016/0550-3213(82)90454-0http://arxiv.org/abs/0806.3124http://dx.doi.org/10.1016/j.cpc.2008.09.011http://dx.doi.org/10.1016/j.cpc.2008.09.011http://arxiv.org/abs/hep-th/0304134http://dx.doi.org/10.1103/PhysRevD.68.045003http://dx.doi.org/10.1103/PhysRevD.68.045003http://arxiv.org/abs/0710.1054http://arxiv.org/abs/1002.1649http://arxiv.org/abs/1002.1649http://arxiv.org/abs/1003.1668http://arxiv.org/abs/0811.3608http://arxiv.org/abs/0705.3809http://dx.doi.org/10.1016/j.physletb.2007.10.073http://dx.doi.org/10.1016/j.physletb.2007.10.0738/12/2019 Huber GaugeFixingAspects
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http://arxiv.org/abs/hep-lat/9511020http://dx.doi.org/10.1016/0550-3213(96)00177-0http://arxiv.org/abs/0911.4921http://dx.doi.org/10.1016/j.nuclphysb.2009.12.036http://dx.doi.org/10.1016/j.nuclphysb.2009.12.036http://arxiv.org/abs/1003.0086http://arxiv.org/abs/0909.3411http://dx.doi.org/10.1007/JHEP02(2010)009http://arxiv.org/abs/1003.1080http://arxiv.org/abs/0912.5153http://arxiv.org/abs/0906.4257http://dx.doi.org/10.1088/1126-6708/2009/08/110http://arxiv.org/abs/hep-th/0502183http://dx.doi.org/10.1103/PhysRevD.72.014016http://dx.doi.org/10.1103/PhysRevD.72.014016http://arxiv.org/abs/hep-th/9310111http://dx.doi.org/10.1103/PhysRevD.50.6616