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

HJE Four-loop renormalization of QCD with a reducible K.G. Chetyrkin 0 1 2 4 M.F. Zoller 0 1 3 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 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 - elds 2 2 1 Introduction 2 Introduction 2.2.1 Direct four-loop calculation in the Feynman gauge with massive HJEP06(217)4 tadpoles Indirect four-loop calculation using three-loop massless propagators computation of the gauge group factors The behaviour of Green's functions w.r.t. a shift of the renormalization scale is described by the anomalous dimensions of the elds and parameters of the theory, which enter the Renormalization Group Equations (RGE). For QCD the full set of four-loop renormalization constants and anomalous dimensions was presented in [2]. The results for the four-loop QCD -function [3, 4] and the four-loop quark mass and eld anomalous dimensions had already been available [5{7].1 In this paper we consider a model with a non-abelian gauge group, one coupling constant and a reducible fermion representation, i.e. any number of irreducible fermion representations. The -function for the coupling this model was computed in an earlier work [1]. Here we provide the remaining Renormalization Group (RG) functions in full dependence on the gauge parameter . Apart from completing the set of renormalization constants and the RGE of the theory, which is important in itself, the gauge boson and ghost propagator anomalous dimensions serve another purpose. These quantities are essential ingredients in comparing the momentum dependence of the corresponding propagators derived in non-perturbative calculations 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 for the model and the computed RG functions We will also repeat how the special case of QCD plus Majorana gluinos in the adjoint representation of the gauge group can be derived from our more general results. Then we will present analytical results for the four-loop 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]). { 1 { anomalous dimensions of the gauge boson, ghost and fermion eld as well as the ones for the ghost-gluon vertex, the fermion-gluon vertex and the fermion mass in Feynman gauge for compactness. The renormalization constants and anomalous dimensions for a generic gauge parameter can be found in machine readable form as supplementary material to this article. of the gauge group is given by HJEP06(217)4 (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) with the structure constants f abc. We have one quadratic Casimir operator CF;r for each fermion representation, de ned through and CA for the adjoint representation. The dimensions of the fermion representations are given by dF;r and the dimension of the adjoint representation by NA. The traces of the di erent representations are de ned as TF;r ab = Tr T a;rT b;r = Tiaj;rTjbi;r: At four-loop level we also encounter higher order invariants in the gauge group factors which are expressed in terms of symmetric tensors daR1a2:::an = 1 X n! perm Tr T a (1);RT a (2);R : : : T a (n);R ; LQCD = + 4 1 a G G a mq;r q;r q;r + gs q;rA=aT a;r q;r ; with the gluon eld strength tensor G a The index r speci es the fermion representation and the index q the fermion avour, q;r is the corresponding fermion eld and mq;r the corresponding fermion mass. The number of fermion avours in representation r is nf;r for any of the Nrep fermion representations. The generators T a;r of each fermion representation r ful ll the de ning anticomuting 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; { 2 { sentation, R = A, where Tbac;A = i f abc. where R can be any fermion representation r, noted as R = fF; rg, or the adjoint repre An important special case of this 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. Here we have Nrep = 2 and nf;1 = nf ; TF;1 = TF ; CF;1 = CF ; ng~ 2 ; nf;2 = TF;2 = CA; CF;2 = CA; (2.7) HJEP06(217)4 should also be compensated by a factor 12 . the factor 12 in front of the number of gluinos ng~ being a result of the Majorana nature of the gluinos (see e.g. [19]). This can be understood in the following way: it has been shown in [20] that one can treat Majorana fermions by rst drawing all possible Feynman diagrams and choosing an arbitrary orientation (fermion ow) for each fermion line. Then Feynman rules are applied in the same way as for Dirac spinors, especially one can use the same propagators p= im for the momentum p along the fermion ow and p=i m for p against the fermion ow. Closed fermion loops receive a factor ( 1). One then applies the same symmetry factors as for scalar or vector particles, e.g. a factor 12 for a loop consisting of two propagators of Majorana particles. For this work we generate our diagrams using one Dirac eld for all fermions, i.e. we produce both possible fermion ows in loops unless they lead to the same diagram. The latter case is exactly the one where the symmetry factor 12 must be applied. The rst case means that the loop was double-counted which By adding counterterms to the Lagrangian (2.1) in order to remove all possible UV divergences we arrive at the bare Lagrangian expressed through renormalized elds, masses and the coupling constant: LQCD;B = a 1 Z(4g)gs2 f abcAb Ac 2 1 2 3 2 1 4 1 1 (2.8) Z(q;r) i 2 ! mq;rZm(q;r)Z(q;r) 2 were we have already used the fact that Z = Z(2g). 3 Due to the Slavnov-Taylor identities all vertex renormalization constants are connected and can be expressed through the renormalization constant of the coupling constant and { 3 { the renormalization constants of the elds appearing in the respective vertex: Zgs = Z(3g) Z(2g) 1 3 Zgs = 1 qZ(4g) Z(2g) 3 Zgs = Z(ccg) 1 Zgs = Z(q;r) 1 3 2 ; 1 3 3 ; 3 2 Z(2c)qZ(2g) (a) = a2 dda za(1)(a) : "a + (a) ; (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) and the fact that = 3 and the -function of the model. from the corresponding anomalous dimension, a nite and usually more compact quantity, (2g). Using (2.17) one can reconstruct renormalization constants { 4 { In the MS-scheme using regularization in D = 4 2" space time dimensions all renormalization constants have the form where a = 16gs2 2 . From the fact that the bare parameter aB = Zaa 2" (with Za = Zg2s ) does not depend on the renormalization scale one nds Given a renormalization constant Z the corresponding anomalous dimension is de ned as (a; ) = 2 d log Z(a; ) d 2 = a X n=1 (n)( ) an : From the de nition of anomalous dimensions (2.16) it follows that (a; ) = ("a (a)) d log Z(a; ) da 3 The 1-particle-irreducible Feynman diagrams needed for this project were generated with QGRAF [21]. We compute Z 2 3 3 (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 from the 1PI corrections to a Green's function with an insertion of one operator and an external fermion line of type (q; r). We used two di erent methods to calculate these objects, rst a direct four-loop calculation in Feynman gauge with massive tadpoles and then an indirect method where four-loop objects are constructed from propagator-like three-loop objects to derive the full dependence on the gauge parameter := 1 . Direct four-loop calculation in the Feynman gauge with massive tadpoles counterterm M2 ZM(22g) Aa Aa 2 = 0 (Feynman gauge) the topologies of the diagrams were identi ed with the C++ programs Q2E and EXP [22, 23]. In this approach all diagrams were expanded in the external momenta in order to factor out the momentum dependence of the tree-level vertex or propagator, e.g. q q q2g were projected onto scalar integrals, using e.g. q q for the gluon self-energy. Then the tensor integrals q4 as well as gq2 as projectors for the gluon self-energy. After this we set all external momenta to zero since the UV divergent part of the integral does not depend on nite external momenta. We then use the method of introducing the same auxiliary mass parameter M 2 in every propagator denominator [24, 25]. Subdivergencies / M 2 are cancelled by an unphysical gluon mass restoring the correct UV divergent part of the diagrams. This method was e.g. used in [3, 4, 26{30] and is explained in detail in [31]. For the expansions, application of projectors, evaluation of fermion traces and counterterm insertions in lower loop diagrams we used FORM [ 32, 33 ]. The scalar tadpole integrals were computed with the FORM-based package MATAD [34] up to three-loop order. At four loops we use the C++ version of FIRE 5 [35, 36] in order to reduce the scalar integrals to Master Integrals which can be found in [4]. Technical details of the reduction are described in the previous paper [ 29 ]. 2.2.2 Indirect four-loop calculation using three-loop massless propagators The case of a generic gauge parameter is certainly possible to treat in the same massive way but calculations then require signi cantly more time and computer resources.2 As a result we have chosen an alternative massless approach which reduces the evaluation of any L-loop Z-factor to the calculation of some properly constructed set of (L 1)-loop massless propagators [38{41]. As is well-known (starting already from L = 2 [42]) calculation of L-loop massive vacuum diagrams is signi cantly more complicated and time-consuming than the one of corresponding (L 1)-loop massless propagators. The approach is easily applicable for any Z-factor except for Z3 [2]. The latter problem is certainly doable within the massless approach but requires signi cantly more human e orts in resolving rather sophisticated combinatorics.3 On the other hand, one could 2Nevertheless, it has been done recently along theses lines in [37] for the case of one irreducible fermion representation. 3Very recently the problem has been successfully solved in two radically di erent ways [8] and [ 9 ]. { 5 { ψ ψ (a) (d) R2 R3 R3 R1 R2 R1 Aa μ R4 R4 ¯ ψ ¯ ψ (b) ψ (e) R2 R1 Aa μ R1 R2 R3 R3 ¯ ψ Ab ν (c) ψ (f) c a R1 R2 R1 ¯ ψ c¯ b restore the full -dependence of Z3 from all other renormalization constants and from the fact that the charge renormalization constant Zg is gauge invariant [2, 37]. As Zg in QCD with fermions transforming under arbitrary reducible representation of the gauge group has been recently found in [1] we have proceeded in this way. For calculation of 3-loop massless propagator we have used the FORM version of MINCER [43]. 2.2.3 computation of the gauge group factors The calculation of the gauge group factors was done with an extended version of the FORM package COLOR [44] already used and presented in [1]. We take the following steps: 1. For the generation of the diagrams in QGRAF [21] we use one eld A for the adjoint representation (gauge boson) and one eld for all the fermion representations. This has the advantage that we do not produce more Feynman diagrams than in QCD. Each fermion line in a diagram gets a line number and is treated as a di erent representation from the other fermion lines. Since we compute diagrams up to fourloop order we need up to four di erent line representations R1; : : : ; R4 (see gure 1) and Tiaj;R4 = T4(i,j,a). Each fermion loop gets assigned a factor nf . 2. The modi ed version of COLOR [1, 44] then writes the generators into traces Tr T a1;R : : : T an;R = TRfRg(a1,...,an); (R = R1; : : : ; R4) (2.19) which are then reduced as outlined in [44] yielding colour factors expressed through traces TFfRg, the Casimir operators cFfRg and cA, the dimensions of the representations dFfRg and NA. { 6 { 3. Now we change from fermion line numbers R1; : : : ; R4 to four explicit physical fermion representations r by substituting each of the line numbers R1; : : : ; R4 by the sum over all representations r = 1; : : : ; 4. An example of the substitution of fR1; : : : ; R4gcolour factors with those of the physical representaions in a one-loop diagram is Nf*TF1 ! nf;1TF;1 + nf;2TF;2 + nf;3TF;3 + nf;4TF;4: (2.20) At higher orders this subtitution becomes much more involved.4 Diagram (a) from gure 1 now corresponds to a sum of 44 = 256 diagrams with explicit fermion representations. This lengthy representation of our results is needed for the renormalization procedure, since e.g. a one loop counterterm to the gluon self-energy, computed from a diagram with only R1, must be applied to all the fermion loops in gure 1 (a,b,d,e). This is most conveniently achieved if each fermion-loop is considered a sum over all physical fermion representations just as it is considered a some over all (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 diagrams with counterterm insertions. They can be treated directly in the next step. 4. After all subdivergencies are cancelled by adding the lower-loop diagrams with counterterm insertions we simplify and generalize the notation. The explicit colour factors are collected in sums of terms built from nf;r, CF;r and TF;r over all physical representations r, e.g.6 nf;1TF;1 ! X nf;iTF;i nf;2TF;2 nf;3TF;3 nf;4TF;4: (2.21) Since we used the maximum number of di erent fermion representations which can appear in any diagram the result is valid for any number of fermion representations Nrep. 3 Results In this section we give the results for the anomalous dimensions of the QCD-like model with an arbitrary number of fermion representations as described above to four-loop level. The number of active fermion avours of representation i is denoted by nf;i. Apart from the Casimir operators CA and CF;i and the trace TF;i the following invariants appear in our 4For this reason it is convenient to collect all combinations Nfx1*TF1x2*CF1x3*TF2x4*CF2x5*TF3x6*CF3x7 *TF4x8*CF4x9 in a function C(x1,...,x9). The factors C(x1,...,x7) are then substituted by the proper combinations of nf;1, TF;1, cF;1, etc. 5Since renormalization constants in the MS-scheme do not depend on masses all fermion avours can be treated as massless for their computation. 6For convenience we collect nfx;11nf;2 x2 nf;3 x3 nfx;44TFy;11TFy;22TFy;33TFy;44CFz1;1CFz2;2CFz3;3CFz4;4 in a function CR(x1,...,x4,y1,...,y4,z1...,44) which are then substituted by the proper sums of colour factors over all representations r. { 7 { dabcddabcd A NA dF;r d(A4A) = A ; d(F4A);i = d~(4) FA;r = daFb;rcddabcd A ; d~(F4F);ri = daFb;rcddaFb;icd ; NA dF;r daFb;icddabcd A ; d(F4F);ij = daFb;icddaFb;jcd ; NA where r is xed and i; j will be summed over all fermion representations. In this section we give the results for = 1 (Feynman gauge), the general case = (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 243 CA2 + X nf;iTF;i (4CF;i + 5CA) ; From the ghost eld strength renormalization constant Z3(2c) we compute 3 3 (2c) (1) (2c) (2) = = 1 2 CA; 2449 CA2 + 5 +CA X nf;inf;jTF;iTF;j CF;j 8315 972 + 86 4 3 + CA X nf;iTF;i CF;i 4 + From the fermion eld strength renormalization constant Z2(q;r) we nd 2 2 2 { 9 { (3.8) (3.9) (3.10) (3.11) (3.12) CF;r for the anomalous dimension of a representation r fermion eld. The fermion eld-gauge boson-vertex renormalization constant Z1(q;r) yields (3.13) (3.14) (3.15) (3.16) 3 + C2 6307 A + for each representation r and the ghost-gauge boson-vertex renormalization constant Z1(ccg) yields (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) Finally, the mass anomalous dimension computed from Zm(q;r) is found to be m m m 3 : (3.25) We checked that the well known relations (a) (a) a a = 2 1(ccg)(a; ) = 2 1(q;r)(a; ) 2 3(2c)(a; ) 2 2(q;r)(a; ) 3 (2g)(a; ); 3 are ful lled with the -function from [1]. This is also true if we include the full dependence on the gauge parameter in the anomalous dimensions. This dependence cancels in the -function. We provide renormalization constants and anomalous dimensions with the full gauge dependence as supplementary material to this article. We compared these fully -dependent results with [37] for one fermion representation and nd full agreement. 4 Conclusions We have presented analytical results for the eld anomalous dimensions 3(2g), 3(2c), 2(q;r), the vertex anomalous dimensions 1(ccg) and 1(q;r) and the mass anomalous dimension (q;r) m in a QCD-like model with arbitrarily many fermion representations and with the full dependence on the gauge parameter . Acknowledgments The work by K.G. Chetykin was supported by the Deutsche Forschungsgemeinschaft through CH1479/1-1 and in part by the German Federal Ministry for Education and Research BMBF through Grant No. 05H2015. The work by M.F. Zoller was supported by the Swiss National Science Foundation (SNF) under contract BSCGI0 157722. Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [INSPIRE]. [hep-ph/9703278] [INSPIRE]. [1] M.F. 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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, 74, DOI: 10.1007/JHEP06(2017)074