#### Four-loop renormalization of QCD with a reducible fermion representation of the gauge group: anomalous dimensions and renormalization constants

Received: April
Four-loop renormalization of QCD with a reducible
K.G. Chetyrkin 0 1 2 4 5
M.F. Zoller 0 1 3 5
Open Access 0 1 5
c The Authors. 0 1 5
0 Luruper Chaussee 149 , 22761 Hamburg , Germany
1 Wolfgang-Gaede-Stra e 1 , 76131 Karlsruhe , Germany
2 II Institut fur Theoretische Physik, Universitat Hamburg
3 Institut fur Physik, Universitat Zurich (UZH)
4 Institut fur Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT)
5 [36] A.V. Smirnov , FIRE5: a C
We present analytical results at four-loop level for the renormalization constants and anomalous dimensions of an extended QCD model with one coupling constant and an arbitrary number of fermion representations. One example of such a model is the QCD plus gluinos sector of a supersymmetric theory where the gluinos are Majorana fermions in the adjoint representation of the gauge group.
Perturbative QCD; Renormalization Group
Contents
1 Introduction 2 Notation and de nitions 2.1
QCD with several fermion representations
Technicalities
Conclusions
Introduction
Direct four-loop calculation in the Feynman gauge with massive
computation of the gauge group factors
already been available [5{7].1
on the gauge parameter .
on the lattice, with perturbative results (see e.g. [11{18]).
This paper is structured as follows:
rst, we will give the notation and de nitions
1Recently, the ve-loop QCD
-function has been obtained for QCD colour factors [8] as well as for a
generic gauge group [9] (see, also, [10]).
gauge parameter
this article.
Notation and de nitions
QCD with several fermion representations
of the gauge group is given by
fermion representation, de ned through
di erent representations are de ned as
ab = Tr T a;rT b;r
= Tiaj;rTjbi;r:
which are expressed in terms of symmetric tensors
daR1a2:::an =
Tr T a (1);RT a (2);R : : : T a (n);R ;
LQCD =
Nrep nf;r
r=1 q=1
a )2 + @ ca@ ca + gsf abc @ caAb cc
with the gluon eld strength tensor
a = @ A
@ Aa + gsf abcAb Ac :
is the corresponding fermion
eld and mq;r the corresponding fermion mass. The number
relation of the Lie Algebra corresponding to the gauge group:
hT a;r; T b;ri = if abcT c;r
Tiak;rTkaj;r = ij CF;r;
sentation, R = A, where Tbac;A =
nf;1 = nf ;
TF;1 = TF ;
CF;1 = CF ;
nf;2 =
TF;2 = CA;
CF;2 = CA;
should also be compensated by a factor 12 .
p=i m for p against
Dirac eld
for all fermions, i.e. we produce both possible fermion
ows in loops unless
and the coupling constant:
LQCD;B =
1 Z(2g) @ A
1 Z(3g)gsf abc @ A
1 Z(4g)gs2 f abcAb Ac 2
Nrep nf;r
r=1 q=1
q;rA=aT a;r q;r ;
+ Z(2c)@ ca@ ca + Z(ccg)gsf abc @ caAb cc
Zgs = Z(3g) Z(2g)
1 3
Zgs =
Z(a; ) = 1 + X1 z(n)(a; )
(D)(a) =
(a) = a2 dda za(1)(a) :
"a + (a) ;
and the fact that
and the -function of the model.
(2g). Using (2.17) one can reconstruct renormalization constants
2" space time dimensions all
renormalization constants have the form
not depend on the renormalization scale
one nds
(a; ) =
2 d log Z(a; )
= a
@z(1)(a)
From the de nition of anomalous dimensions (2.16) it follows that
(a; ) = ("a
d log Z(a; )
d log Z(a; )
| is described
by its anomalous dimension, i.e.
B = Z
QGRAF [21]. We compute Z
(2c), Z(2g) and Z(q;r) from the 1PI self-energies of the elds
Aa , c and q;r as well as Z(ccg) and Z(q;r) from the respective vertex corrections and Zm(q;r)
1 1
counterterm M2 ZM(22g) Aa Aa
2
tex or propagator, e.g. q q
were projected onto scalar integrals, using e.g. q q
for the gluon self-energy. Then the tensor integrals
as well as gq2 as projectors for the
gent part of the integral does not depend on
nite external momenta. We then use the
restoring the correct UV divergent part of the diagrams.
are described in the previous paper [29].
1)-loop massless
than the one of corresponding (L
1)-loop massless propagators.
e orts in resolving rather sophisticated combinatorics.3
On the other hand, one could
treated as a di erent representation R1; : : : ; R4.
computation of the gauge group factors
representation (gauge boson) and one eld
for all the fermion representations.
Tr T a1;R : : : T an;R
tions dFfRg and NA.
Nf*TF1 ! nf;1TF;1 + nf;2TF;2 + nf;3TF;3 + nf;4TF;4:
gure 1
(massless) fermion
avours.5 The factors involving daF1;ra2a3a4 , daF1;ra2a3 , da1a2a3a4 and
A
da1a2a3 appear only at four-loop level and do hence not interfere with lower order
A
sentations r, e.g.6
nf;2TF;2
nf;3TF;3
nf;4TF;4:
tions Nrep.
Results
combinations of nf;1, TF;1, cF;1, etc.
treated as massless for their computation.
x3 nfx;44TFy;11TFy;22TFy;33TFy;44CFz1;1CFz2;2CFz3;3CFz4;4 in a function
tors over all representations r.
d(A4A) =
A ; d(F4A);i =
FA;r = daFb;rcddabcd
daFb;icddabcd
NA
we give the results for
= (1
) can be found
as supplementary material to this article.
anomalous dimension according to (2.16)
From the gauge boson
eld strength renormalization constant Z3(2g) we compute the
3 CA + X 4
243 CA2 + X nf;iTF;i (4CF;i + 5CA) ;
46CF3;i + CACF2;i
2 CA;
CACF;j
+d(F4F);ij
+ X nf;inf;j TF;iTF;j CF2;j
X nf;inf;jnf;kTF;iTF;jTF;k
4 + 110 5
+ d(F4A);i
3 + 126 4
CF;i + CA
CF;iCF;j
3 + 120 5
X nf;inf;jTF;iTF;j;
5 + X nf;i TF;iCA
86 3 + 69 4
+ d(F4A);i (48 3
From the fermion eld strength renormalization constant Z2(q;r) we nd
= CF;r;
CACF;r
2 CF;r
9CF;r +
X nf;inf;jTF;iTF;j;
CA2CF2;r
CACF;i
+160 5)
+ 400 3
+ 848 3
1440 5
+ X nf;i TF;iCF;r 3CF2;i + CF;rCF;i (62
+ 214 3 + 66 4
190 3 + 170 5)
12 4 + CACF;r
112 3 + 24 4
3 + 25 4 + 80 5
+ 128 d~(4)
X nf;inf;jTF;iTF;jCF;r
X nf;inf;jnf;kTF;iTF;jTF;k
for the anomalous dimension of a representation r fermion eld.
= CF;r + CA;
CACF;r +
X nf;iTF;i 2CF;r + 6 CA ;
6CF;rCF;i + 9CF2;r
CACF;i
+ X nf;inf;jTF;iTF;j
CACF;r
+CACF2;i
+CACF2;r
+ 400 3
+ 848 3
1440 5
CA2CF2;r
+ 214 3 + 66 4
190 3 + 170 5)
+ X nf;i TF;i 3CF;rCF2;i + CF2;rCF;i (62
CACF;rCF;i
112 3 + 24 4 + 160 5
102 3 + 63 4
1365691 119
3 +25 4 +80 5
+ 128 d~(4)
d(F4A);i (48 3
+ X nf;inf;jTF;iTF;j CF;rCF;j (44
+24 4 + CACF;r
+ X nf;inf;jnf;kTF;iTF;jTF;k
{ 10 {
2 CA;
3 C2;
= 3 CF;r;
CACF;r
X nf;inf;jTF;iTF;jCA2 250423
X nf;iTF;i CF;r + CF;i (45
48 3) + CA
X nf;inf;jTF;iTF;j;
+ 336 3 + CACF3;r
+ 316 3
CA2CF2;r
+152 3
+ X nf;i TF;iCF;r CF2;i
+CACF;r
+d~(F4F);ri (64
CF;r + CA
+ 160 3
X nf;inf;jnf;kTF;iTF;jTF;kCF;r
240 3)
+ 296 3
CF;rCF;i (38
CACF;i
592 3 + 264 4
224 3 160 5
480 3)
X nf;inf;jTF;iTF;j CF;j
160 3 + 96 4
{ 11 {
We checked that the well known relations
= 2 1(ccg)(a; )
= 2 1(q;r)(a; )
on the gauge parameter
= 1
in the anomalous dimensions. This dependence cancels
Conclusions
pendence on the gauge parameter .
Acknowledgments
Open Access.
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