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

Journal of High Energy Physics, Jun 2017

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. The renormalization constants of the gauge boson (gluon), ghost and fermion fields are analytically computed as well as those for the ghost-gluon vertex, the fermion-gluon vertex and the fermion mass. All other renormalization constants can be derived from these. Some of these results were already produced in Feynman gauge for the computation of the β-function of this model, which was recently published [1]. Here we present results for an arbitrary ξ-parameter.

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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. References [INSPIRE]. [hep-ph/9703278] [INSPIRE]. [1] M.F. Zoller, Four-loop QCD -function with di erent fermion representations of the gauge group, JHEP 10 (2016) 118 [arXiv:1608.08982] [INSPIRE]. [4] M. Czakon, The Four-loop QCD -function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE]. 583 (2000) 3 [hep-ph/9910332] [INSPIRE]. theories, Phys. Lett. B 373 (1996) 314 [hep-lat/9512003] [INSPIRE]. 61 (2000) 114508 [hep-ph/9910204] [INSPIRE]. Rev. D 60 (1999) 094509 [hep-ph/9903364] [INSPIRE]. Nucl. Phys. Proc. Suppl. 83 (2000) 159 [hep-lat/9908056] [INSPIRE]. [INSPIRE]. [arXiv:1012.3135] [INSPIRE]. [arXiv:1111.3023] [INSPIRE]. Rev. D 92 (2015) 074505 [arXiv:1302.5943] [INSPIRE]. [hep-ph/9611355] [INSPIRE]. interactions, Nucl. Phys. B 387 (1992) 467 [INSPIRE]. [INSPIRE]. [INSPIRE]. diagrams, hep-ph/9905298 [INSPIRE]. s) to the Lett. B 344 (1995) 308 [hep-ph/9409454] [INSPIRE]. 402 [hep-ph/0211288] [INSPIRE]. loops, JHEP 05 (2004) 006 [hep-lat/0404003] [INSPIRE]. loops, JHEP 03 (2017) 020 [arXiv:1701.07068] [INSPIRE]. [INSPIRE]. momentum expansion, Nucl. Phys. B 397 (1993) 123 [INSPIRE]. NIKHEF-H-91-18 [INSPIRE]. [6] K.G. Chetyrkin , Quark mass anomalous dimension to O( S4), Phys . Lett . B 404 ( 1997 ) 161 [7] K.G. Chetyrkin and A. Retey , Renormalization and running of quark mass and eld in the [8] P.A. Baikov , K.G. Chetyrkin and J.H. Kuhn, Five-Loop Running of the QCD coupling constant , Phys. Rev. Lett . 118 ( 2017 ) 082002 [arXiv:1606.08659] [INSPIRE]. [9] F. Herzog , B. Ruijl , T. Ueda , J.A.M. Vermaseren and A. Vogt , The ve-loop -function of Yang-Mills theory with fermions , JHEP 02 ( 2017 ) 090 [arXiv:1701.01404] [INSPIRE]. [10] T. Luthe , A. Maier , P. Marquard and Y. Schroder, Towards the ve-loop -function for a general gauge group , JHEP 07 ( 2016 ) 127 [arXiv:1606.08662] [INSPIRE]. [11] H. Suman and K. Schilling , First lattice study of ghost propagators in SU(2) and SU(3) gauge [12] D. Becirevic et al., Asymptotic scaling of the gluon propagator on the lattice , Phys. Rev . D [18] V.G. Bornyakov , E.M. Ilgenfritz , C. Litwinski , V.K. Mitrjushkin and M. Muller-Preussker , [21] P. Nogueira , Automatic Feynman graph generation , J. Comput. Phys . 105 ( 1993 ) 279 [22] T. Seidensticker , Automatic application of successive asymptotic expansions of Feynman [23] R. Harlander , T. Seidensticker and M. Steinhauser , Complete corrections of O( [25] K.G. Chetyrkin , M. Misiak and M. Mu nz, -functions and anomalous dimensions up to three loops, Nucl . Phys . B 518 ( 1998 ) 473 [hep-ph/9711266] [INSPIRE]. [26] Y. Schr oder, Automatic reduction of four loop bubbles , Nucl. Phys. Proc. Suppl . 116 ( 2003 ) [27] F. Di Renzo , A. Mantovi , V. Miccio and Y. Schr oder, 3 - D lattice QCD free energy to four [28] K.G. Chetyrkin and M.F. Zoller , Three-loop -functions for top-Yukawa and the Higgs self-interaction in the Standard Model , JHEP 06 ( 2012 ) 033 [arXiv:1205.2892] [INSPIRE]. [29] M.F. Zoller , Top-Yukawa e ects on the -function of the strong coupling in the SM at four-loop level , JHEP 02 ( 2016 ) 095 [arXiv:1508.03624] [INSPIRE]. [30] K.G. Chetyrkin and M.F. Zoller , Leading QCD-induced four-loop contributions to the -function of the Higgs self-coupling in the SM and vacuum stability , JHEP 06 ( 2016 ) 175 [43] S.A. Larin , F.V. Tkachov and J.A.M. Vermaseren , The FORM version of MINCER, [44] T. van Ritbergen , A.N. Schellekens and J.A.M. Vermaseren , Group theory factors for Feynman diagrams , Int. J. Mod. Phys. A 14 ( 1999 ) 41 [hep-ph/9802376] [INSPIRE].


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K. G. Chetyrkin, M. F. Zoller. Four-loop renormalization of QCD with a reducible fermion representation of the gauge group: anomalous dimensions and renormalization constants, Journal of High Energy Physics, 2017, 1-15, DOI: 10.1007/JHEP06(2017)074