Gluon Polarization Tensor in external field in SU(3) theory New Physics and Quantim Chromodynamics at External Conditions 2011 May 5 Dnipropetrovsk M. Khandoga, V. Skalozub
Dec 31, 2015
Gluon Polarization Tensor in external field in SU(3) theory
New Physics and Quantim Chromodynamics at External Conditions 2011
May 5
Dnipropetrovsk
M. Khandoga, V. Skalozub
Introduction
• Magnetic field of order is spontaneously generated in QCD vacuum at high temperatureSuperdaisy resummations: A. O. Starinets, A. V. Vshivtsev, V. Ch. Zukovskii. Phys. Lett. B 322, 403 (1994)Lattice simulations: N.O. Agasian (2003), V. Demchik (2008).
• Cosmological proof: Fermi-LAT Collaboration found out, that relict intergalactic magnetic fields of order ~ 10-15 G are observed (Science, Vol. 328. no. 5979, pp. 725 – 729, April 2010).
• Peripheral collisions of heavy ions: magnetic field is generated by parts of nuclei, travelling by sides
4 2gB g T
SU(2)-gluodynamics in external field
2 21 1( )
4 2L F A D
ab acb cD A [ ] ( ) ( ) ( ) ( )a a a abc a aF A A x A x A x A x
[ [ ], [ ]]ab abc cD A D A F
- QCD Lagrangian
, - Ghost fields
Background gauge
31 2( ) .a aB x x B
( ) ( ) ( ).a a aA x B x Q x
Field potential А(х) is divided into external field B(x) and quantum fluctuations Q(x):
External field is chosen in the following form:
2 33 3 3
3 33
2
1 1 1( )
4 4 21 1
( )2 21 1
( )4 2
ab b ab b
bc c b bc ab b ab b c b
abc abe b c d e ab bgh
L F F D Q D Q F Q
gf F Q Q gf D Q D Q Q Q
g f f Q Q Q Q D Q L
Lagrangian in background gauge:
Ghost Lagrangian3 3 .ac ac a cD gf B
Charged basisSince external field is directed along 3rd axis in the color space, it is
convenient to introduce the following basis, which is called charged:
3Q Q )(2
1 21 iQQW
3
* 3
.
.
D igB
D igB
3 3 * 3 3
3 33
2 3 3 33
1 1 1 1( )( ) ( )( )
4 2
( ) ( )
1 1( 2 )
2 2 gh
L Q Q W W D W D W Q Q
ig F Q W W igQ W W W W
g à W W W W Q Q W W F Q L
3 3 3
* *
( )
( )
Q Q Q
W D W D W
W D W D W
Ã
SU(3)-gluodynamics in external field
• Spatial structure remains unchanged• Now we have 8 degrees of freedom instead of 3
which results in 8 gauge particles. • One more external field is added, it has same
spatial orientation and directed along 8th axis in color space.
3 3 8 81 2( ) ( )a a aB x x B B
SU(3)-gluodynamics Lagrangian in background gauge
à 1 21
1( ),2
W Q Q
4 52
1( ),2
W Q Q
,Q Q Q
Let’s switch to charged basis:
1 21
1( ),2
W Q Q 6 7
3
1( ),2
W Q Q
1 3 ,D igB 32,3 8
3.
2 3
BD i g B
,Ã abcf - SU(3) group structure constants.
Neutral gluons sectorNeutral gluons do not interact with each other. We can write interaction
Lagrangians of both neutral gluons as a combination of SU(2)-like Lagrangians:
3 3 int(3 ) int(3 ) int(3 )free I II IIIL L L L L
8 8 int(8 ) int(8 )free II IIIL L L L Every interaction Lagrangian has a structure, identical to SU(2) case.
Thus the polarization operator of neutral gluons in SU(3) theory can be brought to SU(2) case, already researched by M.Bordag, V. Skalozub, Phys. Rev. D 75, 125003 (2007)
3 3 8 (2) 3 (2) 3 8
(2) 8 3
1 1 3( , , ) ( , ) ( , )
4 2 2
1 3 1( , ),
4 2 2
Q Neut SU Neut SU
Neut SU
k B B k B B k B B B
k B B B
8 3 8 (2) 3 8 (2) 8 33 1 3 3 3 1( , , ) ( , ) ( , ).
2 2 2 2 2 2Q Neut SU Neut SUk B B k B B B k B B B
In the recent paper (V.Skalozub, A. Strelchenko (2004)) it was found out, that two fields are generated. After reaching the deconfinement phase two fields are generated:
3 2 3 2
8 2
30,9989 .
2
g TB
Spontaneous generation of magnetic fields at high temperature
3 2
3 20,2976 ,
g TB
0 3 8( ) , ,dB T T T B B
But after reaching some temperature only one field remains: 0 dT T
3 22
0 3 82
1 1( ) (1 ) ; 0.
4 2
g TB T T B B
Hence the behavior of field-dependant quantities differs significantly at high temperature. Let’s illustrate it on Debye mass.
Debye mass
Sometimes it is convenient to use an inverted quantity:
If electrical potential is surrounded by plasma, it has a limited reach:
.)( Dr
r
er
QrV
DD
mr
1
In QFT Debye screening is caused by vacuum polarization. Debye mass can be obtained from polarization operator:
).0,0,,( 4442 kkBTmD
In finite-temperature QCD there is a well-known result:
.)63
( 222 TgNN
m fcD O. Kalashnikov (1984)
Debye mass of neutral gluons
)))(2
4775.0
2
8859.0(
2
3(
3
2 32222
)( 3 BT
iT
gTgmQD
)))(2
4775.0
2
8859.0(
2
3(
3
2 2222
)( 8 T
iT
gTgmQD
38
3
1
2
3BB 38
3
1
2
3BB
Debye mass slightly grows at high temperature:
8
2 42 2 2
0 2( )( ) (1 (0,8222 0,4431) 0,1289 )
3 (2 )dD Q
g gm T T T g T i
8
2 42 2 2
0 2( )( ) (1 (0,5799 0,3866) 0,3982 )
3 (2 )D Q
g gm T T g T i
3
2 42 2 2
0 2( )( ) (1 (0,7574 0,4082) 0,0853 )
3 (2 )dD Q
g gm T T T g T i
3
2 42 2 2
0 2( )( ) (1 (0,3486 0,1638) 0,4289 )
3 (2 )D Q
g gm T T g T i
Vertex term of the Lagrangian3 3 3
1 1 1 1 1 1
8 3 8 32 2 2 2 2 2
8 3 8 33 3 3 3 3 3
( )
3 1 1( ) ( )( )
2 3 3
3 1 1( ) ( )( )
2 3 3
ig Q W W Q W W Q W W
i g Q Q W W Q Q W W W W
i g Q Q W W Q Q W W W W
1
2abc a b cgf Q Q Q
1 2 3 1 2 3
2 3 1 2 3 1
3 2 1 3 2 1
2(
2
)
ig W W W W W W
W W W W W W
W W W W W W
aW aW
bW
cWa b c
Charged gluons sectorIn SU(3) theory charged gluons do interact with each other:
int(3 ) int(8 ) int( ) ,WII WII free II II I II IIIL L L L L
int(3 ) int(8 ) int( ).WIII WIII free III III I II IIIL L L L L
SU(2) case was researched in paper by M. Bordag and V. Skalozub Phys. Rev. D 77, 105013 (2008)
For polarization operators of charged gluons we get
int(3 ) int( ) ,WI WI free I I II IIIL L L L
3 8 (2) 3 8 (2) 3 8
int 3 8
3 1 3 1 1 3( , , ) ( , ( )) ( , ( ))
4 2 2 4 2 2
( , , ),
IIw Char SU Char SU
Char
p B B p B B B p B B B
p B B
3 8 (2) 3 int 3 8( , , ) ( , ) ( , , ),Iw Char SU Charp B B p B B p B B
3 8 (2) 3 8 (2) 3 8
int 3 8
1 1 3 3 1 3( , , ) ( , ( )) ( , ( ))
4 2 2 4 2 2
( , , ).
IIIw Char SU Char SU
Char
p B B p B B B p B B B
p B B
Charged gluons Debye mass
Expressions for Debye mass:
38
3
1
2
3BB
38
3
1
2
3BB
Dependence on temperature:
2 42 2 2
0 2( )( ) (1 (0,8419 0,4567) 0,0915 )
3 (2 )I dD W
g gm T T T g T i
2 2 2 2 3
( )
2 3 0.8859 0.4775( ( )( ))
3 2 2 2ID W
m g T g i BT T
2 2 2 2 3
( )
2 3 0.8859 0.4775( ( )( ))
3 2 2 2IID W
m g T g i BT T
2 2 2 2 3
( )
2 3 0.8859 0.4775( ( ))
3 2 2 2IIID W
m g T g i BT T
2 42 2 2
0 2( )( ) (1 (0,7128 0,3794) 0,0179 )
3 (2 )II dD W
g gm T T T g T i
2 42 2 2
0 2( )( ) (1 (0,8562 0,1794) 0,1128 )
3 (2 )III dD W
g gm T T T g T i
2 42 2 2
0 2( )( ) (1 (0,2965 0,1459) 0,0671 )
3 (2 )ID W
g gm T T g T i
2 42 2 2
0 2( )( ) (1 (0,4821 0,1786) 0,2465 )
3 (2 )IID W
g gm T T g T i
2 42 2 2
0 2( )( ) (1 (0,3579 0,1421) 0,5872 )
3 (2 )IID W
g gm T T g T i
Conclusions
• Gluon polarization operator in external field is obtained in SU(3) case. Significant differences with SU(2) gluodynamics are observed.
• The spontaneously generated external field appears to reduce Debye mass.
• Obtained result may be used for further research, finding gluon spectra and magnetic masses.