huber gaugefixingaspects

8/12/2019 Huber GaugeFixingAspects

1/157

SU(2)

SU(N)


2/157


3/157


4/157


5/157


6/157

MAG


7/157

d


8/157


9/157


10/157

d
http://cpc.cs.qub.ac.uk/summaries/AECT_v1_0.html


11/157


12/157

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,


13/157

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)


14/157

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


15/157

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,


16/157

= 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


17/157

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


18/157

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]).


19/157

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) =


20/157

=i br (Ar) cr (Drscs) =i br(Ar) cr Mrs cs.

s c


21/157

DrsA,(p2) =rs

gpp

p2

p2

p4 + 2 N g24,

N

1/p2

1/p4

Vphys


22/157

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,


23/157

|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).


24/157

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


25/157

cA(p2)

cc(p2)

DA(p2) =dA (p2)A, Dc(p2) =dc (p2)c.

dA dc


26/157

:=c=A/2.

0.5953 . . .

1/2


27/157

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


28/157

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)


29/157


30/157


31/157

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.


32/157

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


33/157

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,


34/157

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).


35/157

=

i-1/2 -1/2 -1/3! -1/2 -1/3!

i=

i

i=

i

i=

i


36/157

-1

=

-1

-1/2 -1/2 -1/2 -1/3! -1/2 - -1/2


37/157

[]k

[]k

k

k

k0

k

[]

S

[]k

S

k

k

Wk[J] =ln Zk[J] =ln

D eS[]+Jii

12iR

kijj .


38/157

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 =


39/157


40/157

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


41/157


42/157

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


43/157

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


44/157

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


45/157

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


46/157

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.


47/157

k k = + 12

1

kRk

k

k


48/157

c 0

Dc(p

2)p21 p200

Dc(p

2)


49/157


50/157


51/157

= +...

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


52/157

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


53/157

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


54/157

v,max =12

i

mii.


55/157

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


56/157

(a) (b) (c) (d)

1

2nbi1...ir

j

jki1...ir

j = 0.

i1...ir = 0.


57/157


58/157

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


59/157

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

.


60/157

A

DabA,(p2) =ab

gpp

p2

cA(p

2)

p2 +ab

ppp4

.

(A)(A)


61/157

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.


62/157

= 0, 2

= 1

c0.

A0, c0, A+ 2c0.

A= 0

c = 0

c = 0

A+ 2c = 0 A = c = 0 A c


63/157

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


64/157

1

2

(2AV

A

V)

nAV + dressed

vertices

ndi1...ir ki1...irAV

0.

ki1...irAV

i1 ir

AV


65/157


66/157


67/157


68/157


69/157


70/157

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,


71/157

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.


72/157

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


73/157

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

,


74/157


75/157


76/157


77/157

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.


78/157

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


79/157


80/157

1

2

ab

a b

1

2

ab

a b

AABB

AAcc

AABB

AAcc

nA


81/157

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.


82/157

(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 .


83/157

02LG:= A=2c

c 1 A 1 1/p4

0LG1.

1/p6

2

LG

A >1 LG >1/2 MAG

1

MAG > 1


84/157

AABB

AAcc

BBcc

AABB

AAcc

BBcc

MAG


85/157


86/157

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


87/157

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


88/157


89/157

:=

{A

|A= 0, M >0

}

M=D

A = 0

Dabcc (k)

Dabcc (k) =ab

k2ccc(k) =(M1)ab.


90/157

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


91/157

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.


92/157

= 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


93/157

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


94/157

= 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


95/157

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


96/157

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


97/157

AAV

e,

e.

V

V


98/157


99/157


100/157

i j1

i j1

A A

ij

V A

ij

ab

ab

A

V

dabc

fabc

V V

abcd, acbd, adbc, fabefcde, facefbde, dabcdA .


101/157

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.


102/157

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).


103/157

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


104/157


105/157

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


106/157

A+V 0.

AV

AAV

V

V

AV

V

V

U

V

AV

V


107/157

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


108/157

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


109/157

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


110/157

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,


111/157

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


112/157

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.


113/157

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


114/157

A

V

A =2c

U

V

AV > /2

AV AV =/2


115/157


116/157

SU(3)

SU(2)

SU(N > 2)


117/157


118/157


119/157


120/157

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.


121/157

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


122/157

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 .


123/157

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


124/157


125/157

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 .


126/157

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.


127/157


128/157

:=

,

:=

.

Z[,] =

D[]eS[,]++


129/157

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}


130/157

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.


131/157


132/157

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 .


133/157

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 .


134/157

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


135/157
http://cpc.cs.qub.ac.uk/summaries/AECT_v1_0.html


136/157


137/157

a

b

i

S

12

S

12

S

1

6

S

+S

12

S


138/157


139/157

ab


140/157


141/157

AABB


142/157


143/157


144/157


145/157


146/157


147/157
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)90625-4


148/157
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/0202202


149/157
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.78.074501


150/157
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/0812.0654


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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.77.094510


152/157


153/157
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/0907.2193


154/157


155/157
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.2007.10.073


156/157


157/157
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

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