Jhep09(2014)078

JHEP09(2014)078 - β Springer July 9, 2014 3, the result : August 10, 2014 ≥ /N Expansion, : L September 12, 2014 , : , we find a finite ra- Received 10.1007/JHEP09(2014)078 Accepted 3) Published y doi: − agrams. For , we calculate these scaling di- receive finite-size wrapping and L egral. It matches with an earlier g, on which we have reported ear- oken — even in the ’t Hooft limit. Z ndent due to the non-vanishing (2 rections, TBA and Y-system equa- = 4 SYM theory, the scaling dimen- ζ -nvri¨tzut-Universität Berlin, N Published for SISSA by ∼ 2 scalar fields [email protected] L , -deformed i 1 3 γ SYM theory at leading wrapping L ] composed of = 4 L 1 − Z L . 3 N 1405.6712 The Authors. Gauge-gravity correspondence, AdS-CFT Correspondence, 1 c

In the non-supersymmetric = 2, where the integrability-based equations yield infinity , [email protected] L -deformed i [email protected] Institut Mathematik für und Institut Physik, für IRIS Humbold Zum Gebäude, Grossen Windkanal 6, 12489E-mail: Berlin, German order Open Access Article funded by SCOAP Keywords: Abstract: Jan Fokken, Christoph Sieg and Matthias Wilhelm A piece of cake:γ the ground-state energies in Renormalization Group ArXiv ePrint: sions of the operators tr[ prewrapping corrections in the ’tmensions to Hooft leading limit. wrapping order In directly this from paper Feynman di tions. At is proportional to the maximallyresult transcendental obtained ‘cake’ from int the integrability-based cor Lüscher tional result. This result isfunction renormalization-scheme of an depe induced quartic scalarlier. double-trace This couplin explicitly shows that conformal invariance is br JHEP09(2014)078 8 ]. ]. 1 5 8 10 5 4 14 16 16 18 ject = 4 – = 4 form 2 N N part of a , a simple ]. Like the 0 such that 6 πβ n the ’t Hooft → − = YM -deformed i g 3 γ 3 — are applied to γ 1 , 2 = , ], which is obtained by unin and Maldacena [ 2 1 γ essed. = 1 = and the three Cartan charges i original correspondence [ i 1 , lity in the γ i γ ft, T-duality (TsT) transforma- r-scalar) couplings of the γ r when connected to external states, and CFT correspondence [ – 1 – / -deformation, which is a special case of the exactly β = 4 SYM theory ). They depend on the ]. They give rise to finite-size effects, which are the main sub 8 N N and the Yang-Mills coupling constant is kept fixed. In this limit, the string theory becomes free background. This breaks the SO(6) isometry group to its 5 N S = 4 SYM theory classified by Leigh and Strassler [ → ∞ 2 YM 3 × g N Cartan subgroup. On the gauge-theory side, phase factors de 5 N ≥ = = 2 3 q L -deformed λ L i γ U(1) -deformation). This theory was proposed as the field-theory ], where i × = 4 SYM theory, also these deformations are most accessible i 7 γ 2 q N ). In the limit of equal deformation parameters U(1) 3 factor of the AdS × , q 1 5 2 q = 1) supersymmetry is restored, and one obtains the setup of L Non-planar non-vacuum diagrams may, however, become plana 3.1 Generic case 3.2 Special case , q 1 1 N q ( and in the gauge theory non-planar vacuum diagrams are suppr the Yukawa-type (fermion-fermion-scalar) and F-term (fou ( U(1) SYM theory with gauge group SU( the S the ’t Hooft coupling SYM theory ( non-supersymmetric example of the AdS The gauge theory becomes themarginal deformations (real) of undeformed (planar) limit [ tions — each depending on one of the real parameters applying a three-parameter deformation toOn both the sides string of theory the side, three consecutive T-duality, shi thus may contribute in the ’t Hooft limit [ In this paper, we provide a field-theoretic test of integrabi 1 Introduction and summary B Tensor identities C Renormalization of composite operators D Renormalization-scheme dependence 3 Finite-size corrections to the ground state energies A The action of Contents 1 Introduction and summary 2 Deformation-dependence of diagrams of this work. JHEP09(2014)078 ] ] 19 16 = 4 (1.1) CFT / N ] and [ ubsectors sions and 15 ) separately. ] and then in the r parent 1.1 8 lculation of [ ] therein. In the ons, respectively. 14 and in [ -deformation. If, in i 2 1 γ planar limit. In the i , between planar single- ations that capture the = 3 6= 2 med theories in terms of ] corresponding to operational ]. symptotic regime, can be 3, these are the wrapping β that are protected in the 15 factor which is determined - and 10 ns that are not rooted in the ≥ [ ]. β in picture. In the asymptotic ation hain picture) are determined ple but still yield highly non- L s were obtained up to eleven 17 lobal charges, operator mixing eformed theory such a diagram ] for ]. These findings can directly rporated by introducing twists , j gram approach in [ , the dilatation operator can be 23 10 3 pacetime non-commutative field theories , rmations via noncommutative Moyal-like er matrix [ 2 , ) with 1.1 = 1 = 2 case of the operators ( called critical wrapping order. By a direct L L ] and in particular chapter [ – 2 – = 13 2, they are given by , i, j K ] ≥ j ]. The obtained result, as well as the deformed gravity φ L ] with the integrability found in the original AdS 1 11 − ]. 12 L ]. Beyond the asymptotic regime, finite-size corrections , ) 21 i 20 φ 10 )-charge of the external fields alone. This relation was used 3 ] or, alternatively, a twisted S-matrix [ , q 2 16 = tr[( , q 1 -deformation, the single-impurity operators of the SU(2) s 1 β L, q O -deformation share certain important properties with thei i γ ] can easily be incorporated into the explicit three-loop ca = 4 SYM theory; in particular, they affect the anomalous dimen ], is compatible [ CFT-integrability in [ 18 which start at loop order 1 / N , based on the corrections, Lüscher Y-system and TBA equati 4 ]. β - and 5 β ]. These results were successfully reproduced in [ = 4 SYM theory expression [ 22 ] to determine the one-loop dilatation operator of the defor A simple test of the claimed integrability, also beyond the a The There are, however, important properties of the deformatio ] is adapted to the deformed theories, implementing the defo N The twisted Bethe ansatz can be derived from a twisted transf This relation follows when a theorem formulated by Filk for s Their general properties were first analyzed in the Feynman dia 10 = 4 SYM theory but acquire anomalous dimensions in the 9 3 2 4 -products [ and they correspond toregime, single-magnon states their in anomalous theby dimensions spin-cha the (energies dispersion in relation the of spin-c the twisted Bethe ansatz [ cannot occur. Thus,trivial the results. calculations In become the relativelyare sim of this type. For generic lengths addition, such operators are determined uniquely by their g ∗ twisted boundary conditions [ context of AdS for generic and the all-loop argumenthave to of be [ taken into account. For the operators ( be verified in thedeformation Feynman [ diagram approach, where the modific into the boundary conditions of the asymptotic Bethe ansatz performed by analyzingN the spectrum of composite operators from the order and ( the background [ integrability-based approach, the deformation can be inco trace Feynman diagrams of elementary interactions:is in given the d by its undeformed counterpart multiplied by a phase in [ correspondence, see the review collection [ undeformed in [ hence integrability, forcing us to discuss the corrections, Feynman diagram calculation atloops [ this order, explicit result SYM theory. Oneasymptotic regime, of i.e. these inobtained the is directly absence claimed from of to its finite-size undeformed effects be counterpart integrability via in a rel the JHEP09(2014)078 . . - ]. of 1 ii , β 29 2 ii F o be (1.3) (1.4) (1.2) O δQ ) theory N is vanishing 1 loops. For ) is replaced 1 , r double-trace − N 2 expansion: the L O N -deformation with - -deformation with = ) is logarithmically β β has no fix-points in pping in addition to ), a further type of K , 1.1 ii he N ii F ) oop dilatation operator 4 Q , g . ut the contributions start β ] ( i ) theory, this coupling is at φ O ping, which is caused by the ar fields, it generates the i ]. At one loop, the dilatation N cit summation never applies. φ ) or U( + 24 ounter-term contribution N hes in the undeformed theory when a 2 r corrections, TBA and Y-system ] tr[ ) i , ii ¯ φ ) i for the ground-state energy of the TBA [ ii F ] reproduce the latter result for ¯ φ Q +2 i 10 γ ) tr[ ] for the operators ( + ( . The function operators already at ]. In the SU( i ii ± − 16 i ii F ]. It reads L 24 γ ∼ +1 while it is non-vanishing already at one loop in i 2 26 δQ γ – 3 – 5 ( ], sin + + 3 1 2 = 2. The anomalous dimension of i + i 27 ii ) theory. Although this double-trace coupling has a ∓ ], such that this theory is conformal. As explained in L γ ii F directions. This regularization extends to the ground state N 2 25 = Q 6 5 . Hence, this type of double-trace coupling is running, ( ± g i 3 γ =1 4 sin ], the role of this double-trace coupling in the SU( i ]. X = 2. 2 ) theory [ 31 27 L g , 2 YM N N g 26 denotes the (undetermined) tree-level coupling which has t = 4 − ii ii ii F ii F Q Q β ), this occurs only at ), quantum corrections induce the running of a quartic scala ] for explicit one- and two-loop calculations, respectively N 28 is the effective planar coupling constant and 1.1 , it can contribute at the leading (planar) order in the large -deformation with either gauge group SU( = 4 SYM theory is essentially the same if the gauge group SU( i λ 1 π N 3 and cyclic identification γ 4 , √ N 2 ] and [ , ) theory without tree-level double-trace coupling [ = ), this is no longer the case after the theory is deformed. In t 24 g N ) gauge group. It is an open problem how to incorporate prewra ], we have incorporated the prewrapping effect into the one-l ], which then captures the complete one-loop spectrum of the N = 1 ], it was found that the divergent ground-state energy vanis In the N i 27 See [ Such a divergence was encountered earlier in the expressions 10 30 5 6 where function the operators ( In combination with the self-energy counter term of the scal In [ regulating twist is introduced in the AdS SU( divergent when evaluated at the U( operator and twisted asymptotic Bethe ansatz of [ of [ detail in our recent works [ for to all loop orders in the SU( underlying mechanism is theone same loop as order in earlier, the i.e. case it of can wrapping, affect b length- wrapping into the integrability-based approachequations. of Lüsche In fact, the present TBA result of [ the supersymmetric deformations [ the perturbative regime of small While the double-trace coupling occurs in the action [ included in the action since one-loop corrections induce a c its non-vanishing IR fix point value [ In [ In this expression, where throughout this paper Einstein’s convention of impli by U( gauge group U( coupling, which breaks conformal invariance [ can be understood inabsence terms of of the the U(1) finite-size mode effect in of the prewrap SU( prefactor of JHEP09(2014)078 , ) ’. 1 L, ] in 1.5 (1.5) (1.6) (1.7) O 33 t with 3), such ] diverge − and enters 33 L ) depends on 1 N (2 ] for a detailed ζ 1.7 26 3, the calculation ], but solely from = 0, the remaining , 33 ≥ ii . of the operators ( 3) ii F L ii tional to the maximally Q ii F − -function Q or ζ d a counter-term contribu- level [ L oop order earlier than the e-impurity operators in order to match the definitions analogy to the double-trace (2 ̺ β re. They have the properties ± i ortional to the tree-level cou- 2 ζ γ g ) was confirmed. Note, however, that 2 3 1 and from the TBA and Y-system -deformation, they do not receive = 2, the equations of [ into = 4 SYM theory and are uniquely − i 1.2 − ) has a prefactor of − an orbifolded theory, which were obtained γ L L − i L N L matches the leading-order expression γ , 1.2 2 2 ] At L L O ) − 8 i sin i γ 2, these operators are given by 2 φ + i ) allow for a test of the claimed integrability Lγ γ ≥ -deformation; see our paper [ 2 2 and a factor i 1.1 L – 4 – γ = 2, we obtain the following result for the planar g sin -deformation is ‘conformally invariant in the planar limit is running, the two-loop term of ( sin = tr[( ], even in the planar limit conformal invariance is broken by i L 4 γ + i L ii 26 g ). In this paper, we demonstrate this at an explicit example. 2 ii F O -deformation mentioned above. Lγ ]. We thank Stijn van Tongeren for this comment. 32 1.2 For Q ] and hence to the Riemann β 2 34 9 − 35 -deformation. In the ii sin β ii F L 3, their anomalous dimensions were determined in [ ). Our result for 2 Q g 2 ). Since ≥ g 1.4 64 By a slight generalization of the notion, we also associate i 1.2 L − 7 = 4 = ), i.e. they are protected in the 2 of ( O L ii γ 1.1 O -deformation, also the coupling ( ], the running of the double-trace coupling ( ) and entirely originates from prewrapping. For ii F γ β 32 ] from integrability. 1.2 δQ ] nevertheless claims that the 33 32 in ( are defined in ( ii -deformation. For generic lengths ± i i ii F γ γ Q ]. Operators even simpler than those in ( In this paper, we determine the planar anomalous dimensions Hence, as already explained in our paper [ Note that one has to absorb a factor of 2 into These results are formally the same as their counterparts in 33 7 9 8 and they correspond tomentioned ground above states ( in the spin-chain pictu corrections from the twisted Bethe ansatz at the asymptotic two-loop term can be tracedtion back involving to wrapping diagrams only an where Already at one loop,pling it receives a contribution which is prop transcendental ‘cake’ integral of [ can be reduced to only four Feynman diagrams. They are propor the planar spectrum ofcritical the wrapping order. theory via a finite-size effect one l the running of theIn double-trace the coupling later ( workthe [ author of [ equations up to next-to-leadingin wrapping a order. similar fashion as those of the at leading wrapping order directly from Feynman diagrams. F finite-size effects.the For integrability-based approach as corrections Lüscher in the discussion in the context ofcoupling the in AdS/CFT correspondence. the In prewrapping. obtained in [ they are even protected in the anomalous dimension: determined by their global charges. In contrast to the singl and conformal invariance is broken in the of [ earlier up to leading wrapping order in [ that we find JHEP09(2014)078 - 3 β for = 2 . In , we ̺ C D ) for a L 1.7 arbitrary. ii -deformation is ii F ) in the con- ) and formu- i γ Q 1.1 1.5 based approach, = 0, while in the 11 In the dimensional . ̺ inimal subtraction (MS) π 10 ve 3 case and the ]. ). = 0 and erators ( uxiliary identities for the ≥ 32 ̺ + ln 4 the renormalization of the 1.3 = 2. invariance. If the resulting cted by prewrapping in the erators. In appendix L E ], respectively. is indicated by the parameter L γ he resulting modified descrip- , we present the action of the 37 mension, we can associate the , or with the expression ( can be reconstructed from the t calculational effort. Section − A ensitive to the non-conformality malous) scaling dimensions of gauge- − i ormation dependence. Since the so the non-conformal theory in a γ = -deformation. 2 -deformation. If the corresponding ̺ i i γ given in ( γ , we analyze the diagrams which de- 12 sin e renormalization-scheme independent. The 2 o the claim of [ ii + i ] from the integrability-based equations the . We refer the reader to appendix ii F γ and the scheme, i.e. the parameter B . We identify a subclass of them which )). Such a description must exist if this 33 2 Q ii K N MS) scheme of [ ii F sin ) therefore explicitly shows that the ) in the ) has the same functional dependence on the = 2 states with the running of a contributing Q – 5 – ∝ 1.7 = 2 single-impurity operator ( 1.5 L 1.7 ii L ). The first step is to find a finite and correct ii F Q 1.7 -function for DR) scheme we have β , respectively. In appendix = 0, or 3.2 as the one found in [ ii i ii F ) at any loop order γ are derived in appendix Q and = 2 result ( 1.5 3 L 3.1 DR schemes are the supersymmetric analoga of the widely used m = 2 ground-state operator ( L in the planar anomalous dimension ( ] and the modified minimal subtraction ( ], we have proposed the following test of the integrability- ̺ 36 -deformation (with gauge group SU( ], we have formulated necessary criteria for states to be affe 26 β 27 This paper is organized as follows. In section In [ The DR and In [ It is a well-known fact that in a conformal field theory the (ano -deformation as well as our notation and conventions. Some a 11 12 10 i , and it is proportional to the theory is integrable as claimed.tion The to second the step is to apply t The integrability-based description might thenfixed capture scheme. al Further tests of prewrapping-affectedwould states be s required to check if this is indeed the case. calculation in section case in subsections scheme of [ γ a short review ofdiscuss the the renormalization renormalization-scheme theory dependence emerging of a composite op termine the planarlate anomalous restrictive dimensions criteria of forcontributions the them from to composite the have op deformation-independent adeformation-dependent diagrams non-trivial ones, def this drastically reduces the in the following cases: particular choice of theparticular, tree-level the coupling two-loop contributiondeformation in parameters ( integrability-based description for the formal reduction (DR) scheme used in the main part of this paperwhich we involves ha the equations still yield anpreviously infinite encountered result divergence for in the the anomalous di invariant composite operatorspresence are of observables and are henc modified dimensional reduction ( contains the main part of the calculation, which treats the ̺ the chosen renormalization scheme. This scheme dependence double-trace coupling and hencevalue the is, breakdown however, of finite, chances conformal are high that it coincides 2 Deformation-dependence of diagrams In this section, wecomposite analyze operators the ( diagrams which contribute to not conformally invariant in the planar limit — in contrast t deformation. They can be straightforwardly generalized to JHEP09(2014)078 - L = ∗ O and n s, i.e (2.1) arent i . For , ). Re- φ n L . The A } A.2 = )-charge. c . )-charge of 3 ξ 2 loops, the 3 ) , q R R loops, respec- − , q 2 external fields 2 ξ 2 th double-trace L ] it reads L ξ A , q ,...,R , q P 1 h color structure R 1 1 38 ≤ q ≥ ce of the connected q R s the phase factor of ) to multi-trace dia- K K ∗ · · · ∗ ∈ { 2.1 of the composite opera- 2 = 4 SYM theory. L R ion of Filk’s theorem for A gram of elementary inter- r and ( correspondingly each piece O onsists of the operator ains all information on the ∗ ion ( N 1 and Z eformation-independent ones 1 dence. Only these diagrams -product defined in ( n the external fields − A ndependent. In the asymptotic ∗ ). Moreover, their deformation- L ) are protected in the undeformed Φ( , where i . At least for ¯ ≥ . In each piece, 2.1 φ L +2 +1 1.5 R become identical scalar fields obey the condition R 2 R O K A A A ξ Z R R , . . . , c = 1 ,...,A = 4 , the resulting subdiagram is a direct product ], and in the formulation of [ 1 connected single-trace pieces of the subdiagram ξ – 6 – L 9 c A (albeit occurring in different linear combinations). planar N O c i 2, these deformation-independent diagrams are the ) L − 1 1 x R ( − A L A i R ¯ φ ]. A ) such that the respective subdiagram has the double-trace , the renormalization constant ≤ 1, all fields of the operator can also interact in a single non- L L ) C interact, where the ... = i − K ) ¯ i φ = 1 ¯ L φ ] can contribute. They are associated with finite-size effect +2 +1 1 R -deformation. x 2 R R L ) are thus protected as in the parent ( β ], since their individual trace factors carry net ( R ) ≥ i -deformation is given by its counterpart in the undeformed p A A A i ] tr[( ¯ φ β ¯ 1.5 φ L 27 ) K ) x i ( φ L ] tr[( - and L 1 loops, also diagrams containing connected subdiagrams wi i hO become the respective anti-scalar fields = 1 and ) ]. For -def. i γ − i ξ c φ R γ planar L R 2 ) external fields i 2 loops, each such piece is a planar single-trace diagram wit ) directly applies to each of the ≥ are depicted as gray-shaded regions. The operator Φ extract ¯ φ ξ ( is determined by the same diagrams that yield the UV divergen − ξ R R K 2.1 1 1 R L R 2 − L A A ,...,A ) R connected pieces, which we label by i O As mentioned in the introduction, a planar single-trace dia As reviewed in appendix At A φ ≤ +1 and c R i ,..., only contributions to the renormalization constant K theory times a phase factor which is determined from the orde actions in the tively. These diagrams are not captured by relation ( composite operators ( products then reduce to ordinaryindividually products and the yielding subdiagram Φ as =regime, a 1, whole and i.e. is for deformation-i loop orders φ planar piece ( color structure tr[( the external fieldsspacetime-noncommutative alone. field theories This [ relation is based on the adapt Each diagram contributing to the connected Green function c with the prewrapping and wrapping corrections at tr[( structure tr[( of Green function A dependence cannot be determinedgrams from formulated the in extension [ of relat lation ( theory as well as in the tors of elementary interactions. In this case, and a subdiagram ofdeformation the dependence. elementary If interactions, we which remove cont have to be evaluated explicitly.can be The reconstructed using contribution the from fact that the the d operators ( contains all diagrams with a non-trivial deformation depen where the arbitrary planar elementary interactions betwee 1 its argument, which is determined by the non-commutative JHEP09(2014)078 (2.2) (2.3) g condi- . ]. Hence, rs carries diagram is 27 i i ¯ ¯ φ φ . According to the L formed. Given such O Z e does no longer have a ntain couplings or contri- the vertices at their ends in loops that wrap around . prewrapping effect already ps at least one gauge-field und the wrapping loop can he rhs. do not contain such pe of vertices. Such a path must hence be contained in wrapping loop that is purely as the same dependence on e, from the definition of the diagrams can be decomposed ), as discussed in [ ependent diagrams. This de- e cuts each closed path along (or a product thereof). Thus, i i 1.5 φ φ + −→ . This restricts prewrapping contributions i i ¯ ¯ φ φ L O – 7 – ) stand for matter-type fields. Since the gauge- 2.3 , ) is deformation-dependent, so are these prewrapping 1.2 i i φ φ = ], this coupling can only contribute if one of its trace facto thereby generating the double-trace structure. By imposin i 27 ¯ φ i i ¯ ¯ φ φ )-charge as the operator 3 ) is the only source of prewrapping contributions to , q -deformation cannot affect the operators ( 2 β 1.2 , q ) can be applied to it (or each of its factors) showing that the 1 q ), the central solid lines in ( 2.1 2.2 external fields = 2. Since the coupling ( i i We can now prove that the diagrams of the second class are unde Subdiagrams associated with wrapping contributions conta Subdiagrams associated with prewrapping contributions co L φ φ L composition reads tions on the wrapping loops,into the sum two of classes all one wrapping-type sub of which contains only deformation-ind a diagram, we removeaccording all to gauge-field propagators and replace the The diagrams in the firstmade class on out the of rhs. matter-type containbe at fields, least built i.e. one only a from closedis matter-type path depicted propagators as running joining a aro a solid in cycle. closed any ty loop, Diagrams inpropagator i.e. the occurs. in second This all class is on closed represented t by paths the along wiggly the lines. wrapping loo criteria developed in [ the wrapping loopswrapping at loop; least instead, once. itrelation is The ( a resulting planar single-traceundeformed. diagram diagram henc All deformation-dependentthe wrapping first diagrams class. As in ( boson interactions arethe undeformed, deformation the parameters resulting assecond diagram the class h original it one. follows immediately Furthermor that the above procedur contributions. butions to the propagatorpresent that in are the of double-tracethe coupling type. ( The the same ( to JHEP09(2014)078 , ) e B 1.5 (3.1) , L L , P L 1 P L P 1 − P − L L L 1 1) 2 L 1) 1 − 2 grams are of wrapping − ) given in appendix + ) are protected in the 2( + + i ]2( grams are non-vanishing. L 1.5 + B.3 i , , L ) ) Lγ T d dashed lines, respectively. L L he composite operators ( T Lγ ) nd all prewrapping diagrams, ), ( L L ) i P P i P P ρ cos + − i i ρ L L cos )( B.1 − i ) ) )(˜ formation-independent) diagrams i i Lγ Lγ − i † ij ji ji ij † iLγ ρ ˆ ˆ ρ Q Q ) occurs. These diagrams have to be iLγ − ( ( (( cos cos L L 3 3 2 e 2 e =1 =1 1.2 j j tr tr[((˜ X X N N L L L L L L L L 2 2 YM YM N N N N N N g g L L L L L L 4 4 2 2 2 2 2 2 YM YM YM YM YM YM – 8 – − − g g g g g g ======and the identities ( 2 2 2 2 from Feynman diagrams. Specializing the previous A L 3 3 1 3 1 1 3 1 = ) is drawn as the central dot. All these diagrams depend on K 3 1.5 ≥ L L L L , the only diagrams which can be affected by the deformation ar L L 1 1 1 1 = − − − − L L L L K ) = ) = ) = ) = L L L L ( ( ( ( 3, prewrapping is absent and all deformation-dependent dia ¯ ˜ S S F F ≥ L type with a matter-typeUsing wrapping the loop. conventions in Only appendix four of these dia At where scalar and fermionicThe fields composite are operator represented ( by solid an they evaluate to evaluated explicitly. The contributions fromcan all other be (de reconstructedundeformed from theory. the condition that the3.1 operators ( Generic case to leading wrapping order In the following, we determine the anomalous dimensions of t 3 Finite-size corrections to the ground state energies discussion to wrapping diagrams with a single matter-typei.e. wrapping diagrams loop in a which the double-trace coupling ( JHEP09(2014)078 - ) L ), is β P L 3.4 3.2 (3.3) (3.7) (3.6) (3.4) (3.5) (3.2) P -pole. ε 1 ) in the . , . L given in ( 1.5 L 3) P P L . − P − i L L 2 +1 , . The integral P 1 Lγ L (2 ε , such that it cancels 2 ζ 2 3) β , which corresponds to +1 − − 3 sin 1 , . 1 L πβ − i L 2 + − i with a fermionic wrapping − − 2 he operators ( (2 L − Lγ es on the fermionic wrapping = 0 = 0 sh, as can be easily seen from es are both either external or L Lγ ζ = is given by a simple 2 al while the other is contracted 1 s is indeed the case. Then, ( 2 -independent diagrams as 3 ensors . It is given by 3 cos γ BC i 1 2 L L L i BC ˜ ρ sin + i − ρ O ε 1 − = N L Z L L 2 Lγ L L CA i i CA N 2 2 2 γ YM ˜ ) ρ ρ g L 1 )] π 2 YM 4 cos = − i g =1 =1 4 4 L (4 − ( 1 2 X X − C C γ ˜ 16 Lγ F = − i ∝ ∝ ) yields the contribution of all deformation- 2 − ) = ) contain two particular configurations of the – 9 – L =0 Lγ ) + B B 3.1 sin ± i P L 3.1 -loop order then reads γ = 1 ( ) and using the explicit result for +

i L 2 F i i + cos def Lγ , = K( to C.12 2 L ) in ( + i ) + L L non-def O L , L O P ( sin L ( Z Lγ i i ˜ δ Z ¯ F O L S 2 − Z . Its diagrammatic representation and its UV divergence cos g δ L , ) + = 64 P A A + L L 2 − ( ) and N S 3 1 def L , = dimensions the operator K extracts all poles in ( L L 2 YM K[ non-def L ε F , O g 2 O L − Z γ O δ L − = 0, the above expression has to be independent of = 4 = Z 1 δ − i − def γ = 4 L , L , = 1 + = D O ] L πβ L Z O δ P − 35 Z The negative sum of the poles of ( Inserting this expression into ( The diagrams = + i with the remaining deformation-independent diagrams. Thi γ The renormalization constant Already at this point, the vanishing of the divergences for t we obtain the anomalous dimension deformation provides a non-trivial check: for directly determines the contribution from the deformation dependent diagrams to the renormalization constant loop, the scalar fieldscontracted of with at the least composite two operator.the adjacent Yukawa following Such vertic contractions diagrams of vani the corresponding coupling t Yukawa vertices. The scalar fieldsloop of always play any a two different role adjacentwith in vertic the the diagram: composite one is operator. extern In all other possible diagrams free of subdivergences, and hence its overall UV divergence the scalar ‘cake’ integral where in read [ JHEP09(2014)078 , . ′ 1 λ π I 4 √ 13 = 2 (3.8) (3.9) ]. (3.10) (3.11) L = depen- 33 and g 1 I ) at 1.5 in order to match ± i γ , ) , ′ 1 into 1 I L 1) . Their net contribution apping diagram involving apping effect contributes ), the remaining deforma- n the integrals 2 om prewrapping diagrams xplicitly keeping also finite − NαI self-energy correction of the deformed theory and in the , 3.9 α on f the operator ( 2 YM ) ity-based equations in [ . , g 6 g . 1 ( = + 2( I ] 1 ii 1 O and a factor I I I) diagrams and their counter terms. ii F ) g + K[ α 2 ii NQ (2) O ii F Z 2 YM (1 + δ is the external momentum. Since the com- g − 2 ν NQ + ( p ) − 2 ε 2 YM – 10 – (1) O − g = , Z has to drop out of the final result, and this serves 2(1 into the effective planar coupling constant δ 1 = 2 α F π Q 2 NI Np (1) O 2 2 YM YM Z = 1 + g g δ 2 ) as subdiagram. It reads = = O = 0. The one-loop contribution to the renormalization con- Z 1.2 F ii Q ii F = 2 Q is specified below. All other planar one-loop diagrams are in L 1 I ]. 33 ) is therefore given by is the gauge-fixing parameter and 3.8 α = 2, there are also deformation-dependent contributions fr The only deformation-dependent one-loop diagram is a prewr The two-loop calculation requires the one-loop diagram ( Note that one has to absorb a factor of a factor of 2 into L 13 -deformation where the double-trace coupling ( where the superscript in parenthesis denotes the loop order stant ( already at one loop, we split the renormalization constant o in addition to those from wrapping diagrams. Since the prewr as It exactly matches the expression found from the integrabil tion-independent one-loop one-particle-irreducible (1P They occur as subdiagrams and henceterms. we have The to evaluate 1PI them diagrams e scalar of operator fields renormalization respectively and read the vanishes since the compositeβ operator is protected in the un posite operators are gaugeas invariant, a check of our calculation. The above expressions depend o 3.2 Special case where we have absorbed powers of 4 where where the integral At the definitions of [ dent of the deformation according to the discussion in secti JHEP09(2014)078 ) or ii = 2 3.11 ii F ression (3.13) (3.14) (3.12) L Q , , ε 1 ) and ( ) , originating α µ ) 3.9 ε ( of the respective ibution from the s ( O 2 -poles coming from (1 + , 2 ε 1 P + ) g )) t mass 2 β ontribute. First, there = 2. Second, there are − , 2 , β − p ave to inject an external L ccur in a scalar product πµ = ε 1 1 -dependence starts in the ) 4 2 ) D µ β the composite operator into − α ution from all diagrams that − α r to render it dimensionless in )Γ( in at least one coupling ( 14 + ln the om the one of the deformation- α . ntially given by setting non-def α h alters the , G E (1 + 1 2 − 2 γ I − (1) O ] , g ator renormalization and the one of the self 2 D − D ) δJ − 40 1) + ε , ( = = )Γ( − )Γ( + 2 O 39 β 2 D -dependence is postponed to the terms of ε 1 (1) F φ + µ Q )Γ( α, β − 2 ( ) 2 α is the Euler-Mascheroni constant. The arrows β ) 1 π G the 1 π E Γ( – 11 – + (4 ′ 1 γ − (4 I α ) ) is expanded to one loop. . These prewrapping-generated contributions vanish -functions [ Γ( , C.6 G def 1) = , α, β 2 D ε 1 1) = , 2 ( ) , ii O 1 (2 G π ii F ( (1 1 ), which have to be evaluated at δJ 1 2 (4 Q , and hence their contributions cancel as expected when the exp G G , 2 ) ) 3.1 ii (1) g φ 2 2 ]. In the divergent integral D D δ D D ii F ) = ) = − − − − 36 = 4 4 = 2 , δ δQ µ µ 2(2 2(2 α, β α, β p p ( ( def (1) φ , 1 δ 2 G = = non-def 2 , (1) G O 2 p (1) O δJ − δJ = = dimensions [ F ε ). Using these expansions, the counter terms for the diagram 2 is the external momentum and Q ε -deformation and in the undeformed theory. Hence, the contr ( ν = = − β p ). We only have to be careful when extracting the divergence ′ O 1 1 I I At two loops, two types of deformation-dependent diagrams c 3.6 Note that the deformation-independent counter term of oper = 4 14 deformation-independent diagrams can be reconstructeddependent wrapping fr diagrams alone. Theirinvolve sum elementary is single-trace the contrib couplingsin only. ( It is esse in the one of the counter terms for the operator renormalization constant ( where diagrams which are deformation-dependent since they conta read finite terms, while in the finite integral energy are equal, in the secondin diagram indicate the that numerator. the respective The momenta integrals o contain a power of the ’t Hoof the UV divergences. In order to avoid this IR divergence, we h where we have split thedeformation-dependent contributions and to deformation-independent the ones counter term for which are evaluated in terms of the D from a rescaling of the Yang-Mills coupling constant in orde and are explicitly given by cake integral. This integral contains an IR divergence whic order are the wrapping diagrams ( JHEP09(2014)078 2 I artic from ], and (3.19) (3.18) (3.15) (3.17) (3.16) 2 , O 26 16 Z 2 1 ntegral I 2 ) ii ii F . Q and its pole-part ( ] , 2 2 1 ted by contributions I , I N ε 1 , 2 4 YM 1 ) K[ . g , ii ] -dependence is associated + 2 2 ii F D . 2 I 2 = 4 Q ε 1 ins the overall UV-divergence ) 2 − 2 )) K[ p 2 α πµ 1 he planar limit. Concretely, the 3 F F − iven by I 4 i ed from a diagram with a one-loop − This shows that truncating the Q Q γ − , 2 G (1 + 4 2 15 ) ) 1) I + ln 2 1 sin , 1 − π , which cannot be absorbed into a local ) as a subdiagram, one must keep the finite I E 2 ) + to drop out of the final result. i 2 (1 γ (4 p ) γ α 1)(2 α G ii 3.11 2 2 − ) ) ii F − 2 ] = 2 5 − D D , sin Q 1 α − − I 2 2 1 ] (3 ε 1 I 1 – 12 – 2(4 + ( N 4(2 + ( I ii − µ p 2 + − i ii F 2 4 YM I K[ γ I 2 ) g = Q ) ε 2 1 2 α − . The respective 1PI two-loop diagrams read α 2 16 ). Accordingly, the two-loop contribution to 1 and vanishes in Fermi-Feynman gauge. It contributes to the 2 ii N sin I − − , − ii F 4 3.6 + i 4 YM ) (1 + 2 1 α Q = γ g 1 π − 2 2 st ( αI (2(3 in ( , (4 − 2 ] = K[ -pole depending on ln ii ii ii 2 2 ε 1 (2) O ii F ii F ii F ). The counter term of this coupling was determined in [ I = = P Z Q Q Q ] = (4 sin 2 δ 2 2 2 2 I 1.2 F I ) can be traced back to the one-loop renormalization of the qu N N N N Q . It originates from a non-subtracted subdivergence of the i K[ 2 YM 2 = K R[ 4 4 4 YM YM YM g 3.16 g g g 2 O 4 2 2 I into the composite operator. The resulting integral = 2 − − − ν ). Consistency requires that this subdivergence is subtrac ii = = = = p ii F F F F F 3.15 δQ -pole persists, indicating that the contribution originat Q Q Q Q 2 1 ε ] are given by In the diagram involving the scalar self-energy in ( A 2 I 16 15 -pole of the two-loop diagram and is hence required for ε 1 K[ It occurs in onewith the of double-trace the coupling remaining diagrams whose deformation where the operation R subtracts the subdivergence. where the latter replaces contribution that is proportional to action to only single-trace termssubdivergence is inconsistent in — ( even indouble-trace t coupling ( it reads from other Feynman diagrams, such that the result only conta subdivergence. This expression contains a momentum counter term for all diagrams which only involve single-trace couplings is g given in ( JHEP09(2014)078 ] 2 I ] = 2 1 , I (3.22) (3.24) (3.20) (3.23) (3.21) ). This ] 2 1 . . , I 3 case, we = K R[ 1.3 ) , 1 1 6 2 I 1 ≥ g I αI I ( ii L ii F O ) K R[ . def def , , . ii pendent non-1PI 2 2 ) is given by + ii F (1) (1) O O ]) def , 2 given in ( 1 Q 2 NδQ 3.8 I 2 2 (1) ) is finite in the limit O 1 ii ε 2 YM ii F NδJ NδJ 2 − g δJ ) have combined with the Q 2 g the one- and two-loop K R[ all divergence KR[ 2 2 YM YM ii β C.12 (1) en by φ ii 2 − g g ii F δ + 1 ), as follows from the loop ) ii F read ii Q − = = = UV divergence α to ( ). Taking the logarithm and Q constant ( ( ii F 3.8 = ii + ( Q grams. They generate products F F F 3.8 ii F Q Q Q   in ( Q non-1PI -function ε , F 1 + 4( st 2 β , dt Q , 2 2 , and they generate the contribution 2 (2) I − O ] + 2 (2) O ) into ( 1 def Z 2 − i , I Z 2 ] ε δ 1 γ δ 1 2 (1) O 2 I 3.23 ) . Moreover, in contrast to the + 6 δJ ) depending on 2 + 1 g sin − i times the I , ( γ – 13 – + 1PI i 1 2 K[K[ 1 F , O 2 γ 3.18 I Q , dt − i 2 , ii + ) and ( 2 γ sin 1 ii F ) is 2 (2) I O 2 + i Q ii Z 2 γ 3.10 ii F δ sin (1) , O 2 3.24 (16 sin Q +   i 1 Z 2 + I δ γ 2 1 non-def def st N ii 2 , , , 8 sin 2 2 2 1 2 of the product ii F = (1) (1) 4 (2) YM O O O 2 4 Q g ] − g Z 1 δ − 2 -pole in ( (1) I 16 sin φ 2 (2) O NδJ NδJ 1 ε + 2 2 = = K[ Z Nδ non-1PI , 2 2 ε 2 YM YM 1 δ N − g g (2) O dt ii 2 YM , 2 2 + 4 YM g 2 ii F Z g 2 − − (2) δ O ] = Q ) of the renormalization constant. The only deformation-de (1) 1 O 2 Z = I ). = = = 4 ] g Z δ 1 δ C.7 I F F F F 1PI , 3.17 = = 2 Q Q Q Q dt 2 , 2 K[ 2 O (2) O − Z The two-loop renormalization constant is given by insertin 2 Z 1 δ ln I given in ( guarantees that the anomalous dimension derived according corrections given respectively inexpanding ( it to second loop order, we obtain expansion ( diagrams involve the one-loop counter term The complete two-loop contribution to the renormalization The negative sum of the pole parts of the above diagrams is giv The coefficient of the The 1PI one-loop diagrams involving one-loop counter terms remaining diagrams such that the results depends on the over As discussed above, this result indeed contains the overall K[ have to consider alsoof one-particle one-loop reducible (non-1PI) counter dia terms which contribute to where the contributions from terms in ( JHEP09(2014)078 , i } γ 3 , , hout 2 (A.1) (3.26) (3.25) , and to 1 ] l ̺ -function φ β ∈ { k φ ) in Euclidean A α − i ] tr[ . , N ψ j γ ii c ¯ α φ 2 ii F i, j, k, l i µ 2 ˙ α Q ¯ ) φ D sin β ii D µ tr[ ˙ ii F + i α A ¯ γ ij lk Q ψ cussions. J.F. and M.W. calculation. c ∂ ( 2 ii i Q ̺ ii F + ction doubled spacetime in- 1 sin Q ) + ¯ N + i sion for ) 4 φ ermion and F-term-type scalar − g The deformation parameters i B α − ˙ ̺ β omotionsförderungsstipendium. α B µ γ 2 ] ¯ ψ l ψ 32 i g 2 i φ ¯ 2 φ ¯ φ )D j − i ˙ ¯ sin α A φ − ¯ and flavor indices φ ii labels the scheme and the α A ¯ k ψ + i ii F µ − i transforms under a scheme change as φ ψ } γ ̺ i γ 2 Q ¯ 2 BA φ , (D 2 ii 2 ) BA ], in particular the ones for raising, low- 1 i g ) ii F tr[ i † − ]. sin 41 † Q ρ ) sin 2 ij kl ρ ∈ { ) 26 ̺ + = 4 i ˜ µ Q γ 2 ˙ – 14 – + ( = 4 SYM theory α 2 A + (˜ O = 0 in the DR scheme. In a different scheme, the B α µ ] + α, Z l ∂ ψ sin N ̺ ˙ 4(1 + i ( α B -deformation with gauge group SU( φ 4 ln i ¯ φ -deformation and our notation and conventions. For k ψ α 4 g i 1 γ i φ 2 g reads γ φ j 8 32 α A ii ˙ ¯ ̺ α A φ − ψ i ii F ¯ − − ψ ∂ ¯ ) acquires a contribution which is proportional to φ depends on the chosen renormalization scheme, as discussed µν ii ii ∂Q i BA BA i 2 F tr[ ii F ii F ρ ρ ii O 3.25 Q Q ( (˜ lk ij µν ii F γ 2 2 -deformed ˆ Q Q F i g g YM YM β , spinor indices 4 1 γ , where the real parameter g g } ). In particular, ii − 3 = 4 = 4 2 YM − + + ii F , are summed. This is the only exception to the rule that throug g 2 Q 2 1.3 ∂ ̺ ∂g O , β } + tr 1 γ 2 ̺ 4 . At one loop, the coupling , εg , 0 3 D − , x 2 4 ii = , ∈ { d ii F 1 2 Q O Z γ is given in ( µ, ν ∈ { = . The result in the scheme = 0. It reads The gauge-fixed action of the The above result for ii ii ii S ii F ii F ̺ ii F → Q Q further details, we refer to our publication [ In this appendix, we present the A The action of Acknowledgments We thank Burkhardt Eden anddanken der Stijn Studienstiftung van des Tongeren Deutschen ein for Volkes für Pr useful dis where in the second line we have inserted the explicit expres only enter the coupling tensors of the Yukawa-type scalar-f β ering and contractions ofdices spinor indices. Note that in thisthis a paper Einstein’s summation convention never applies. β ε where we have adopted the conventions of [ A, B in the dimensional reduction (DR) scheme which we used in the Q space can be written as two-loop contribution in ( in appendix JHEP09(2014)078 . l , ]. i = + i 26 γ k (A.3) (A.8) (A.6) (A.5) (A.4) (A.2) (A.7) is de- ± , = B − i j lk q γ ij F = Q = i and 1 + ± j k i A . δ , φ with the index i l q q δ i . ]. This generates in ) can be introduced AB i ∧ − AB ˜ only for ρ 26 , Γ i    1.2 + ij i 2 φ − 1 Γ 2 i q γ 3 0 mber e γ ) e = φ − j l ] = A i ∗ δ δ 3 ) are given by 1 i k via 1 10 eynman rules for the quartic γ 2 B 4 0 δ ± γ δ φ BA i − + ij + i i d = ρ Γ 2 − 3 γ 1 ij lk 0 )-charge vectors 0 0 1 1 00 0 1 0 γ B i = (˜ φ reads [ 3 − δ , , . and Γ A 4 3    , q B . The coupling ( δ 2 B γ AB µ +2 ,Q + i ) 0 0 0 q ij lk = -product in the component expansion of i AB A , q ) γ ∧ † = ∗ ) are given by = ( Q 1 j k A ρ and C 2 q δ = (˜ q i l γ 1 1 1 2 2 2 i 2 4 α δ A.1 A AB i ψ +1 + + + ˜ ρ = i + ψ j l – 15 – = e 1 , , δ q ), the above action can be obtained by replacing γ 2 2 2 1 1 1 3 i α k B B δ ∧ ψ + − − ( , ∗ 1.2 is understood. In terms of these phase tensors, the i 4 1 i AB ψ i C q A ρ q 1 1 1 2 2 2 − 2 α T AB ) ∼ Γ = -deformation can be found in appendix B of our work [ ψ − + − = i i = 2 A ∗ γ e q ) ij kl +1 2 2 2 1 1 1 1 α + 3 ˜ Q i i = ( ψ i + − − iAB i BA Γ 4 ρ B , iǫ q 1 2 3 B B B j , k B q q q = ( δ = ∧ i l − i δ A γ 2 1 AB q ) i AB . = i − ρ † ij lk 4 ρ + ij ψ ( Γ Q i q e is a tree-level coupling tensor with nontrivial components ∧ j l -products of two component fields δ lk i ∗ i k ψ ij F δ q = 4 SYM theory before the auxiliary fields are integrated out [ Q = = The Feynman rules for the We define the antisymmetric phase tensors Γ The N lk ij 4 i ˆ Q Γ which have to vanish in the special case For the different fields, the components of the charge vectors fined as where where we trust that the reader will not confuse the complex nu Yukawa-type coupling tensors in the action ( where cyclic identification and for the anti-fields their signs are reversed. For the calculations in this paper, it is useful to alter the F the product of two fields by a non-commutative particular the double-trace coupling withby tensor redefining the couplings. Apart from the coupling ( where the antisymmetric product of the two ( They obey the conjugation relations The coupling tensors of the quartic scalar interactions rea JHEP09(2014)078 om ). In (B.3) (B.1) (B.2) we have A.1 , has scaling L , 1 . 2 L , L − , i ), the required 1 O + 2 4 , i + i Lγ + Γ i i + A.5 e AC Lγ + Lγ i Γ . If cos i B i L δ e L Lγ cos O uate the contractions of A 2 i cos . Recall that Einstein’s ) J 1) δ L 4 − i − cos 1) − 3.1 al cases. Here, we split the f the coupling tensors that i − sions. rporated into the theory and iLγ i ions contribute to the tensor − ACi 4 ǫ − in a separate vertex. Γ ( i = 2( ns read iLγ e of four scalar fields in ( lk ]. L =1 4 = 2( 4 B ij F X C δ = 2 e 4 43 L i Q A A = 2 e B 4 , Γ L i δ δ 1 2 e L AC 42 1 − − 2 Γ + i Ai = = e + δ + ij 2 Γ − + ij ) ) can be added to the action regularized in ). Γ i 4 iL T CB T CB 4 ) ) − iL Γ i i i e e 1.5 ρ ρ – 16 – A.5 e ACi i i ( (˜ ǫ 4 3 3 ( 6= 6= =1 =1 A j j j j X X in ( AC AC − δ ) ) i i ). With these results as well as ( L = = − † † O =1 ρ ρ L L ( (˜ 4 ) ) A.7 4 =1 X ii ii ii ii X =1 =1 4 4 A A,C ˆ ˆ Q Q X X C C ) and then ( ] = ] = = = + ( + ( ] we have split these interactions into those originating fr L L A.8 ) and ( L L ) ) B B ) ) 26 T T A A is built from the remaining D-term interactions. Moreover, ) ) ij ji ij ji T T i i A.6 ˆ ˆ ) ) ρ ρ Q Q i i ij kl ( ( ρ ρ ˜ i i )(˜ )( Q i i 3 3 6= 6= )( )(˜ =1 =1 † † i i j j j j X X ρ ρ † † ρ ρ = = ( (˜ dimensions via a coupling to an external source L L tr[((˜ tr[(( ε ) ) ), while 2 ij ji ij ji ˆ ˆ Q Q − A.8 ( ( 3 3 =1 =1 j j Composite operators such as For the diagrams with a fermionic wrapping loop, we first eval For the scalar diagrams, we need the expressions X X in ( = 4 ij lk ˆ C Renormalization of compositeIn operators this appendix, we reviewhow how they composite are operators renormalized; are see inco e.g. the textbooks [ where we have first used ( B Tensor identities In this appendix,are we encountered explicitly in evaluate the the Feynman diagram combinations analysis o in section Q D traces are determined as two Yukawa-type coupling tensors. The resulting expressio summation convention does not apply in the following expres the F-term and D-terminteractions couplings according in to the supersymmetric thethis speci two case, single-trace the structures entire F-term and parts of the D-term interact kept the double-trace couplings with tensor structure scalar interactions: in [ where we have used ( JHEP09(2014)078 ams (C.8) (C.1) (C.7) (C.3) (C.6) (C.2) (C.4) (C.5) as 0 L, . O , L , as c O i anti-scalar fields ) . ) L i δJ L O L φ a ( i x , ) ( L δJ i L ¯ φ O x i rpart are then given in L ( = 1 + (1) φ i O ∆. The resulting term in and δ ... ¯ φ g ) i , is the negative sum of the 1PI − δJ 1 re quantities and second in φ introduce a renormalization , as well as 0 δ ibution can be interpreted in erm. The renormalized and ... 4 x L + 2 , L pansion of the above equation ( D L of the respective contribution. ) L O i O , s of the bare ones . . L 1 ) + ¯ O O i φ Z L x ) ) J δJ 2 L i φ ( . − i i O x ( n constants as L φ − 1 φ ( ¯ ) L φ L ( O i 0 ) − L J (1) Z L O O φ x , Z ( ( O hO δJ 0 1PI L 2 L i L L , i = L φ O L, L O − φ i L Z hO O J O (1) Z φ x J O Z L δ 1PI 1PI D O , , 2 1PI L ) = = , d 1 1 L L Z i 0 L – 17 – − − , O O L − φ O Z ( O Z Z i i 0 ,J J Z ) = (1) (2) φ φ i L, Z = = i 0 δ δ = φ ) = φ c a ( . The counter term O 2 2 i 0 L L = i 1 2 i L ν L 0 φ i ) ) , p O ( − L φ − − L L L O J 0 O x Z x O L L Z L, ( ( Z J (1) (2) O O 0 has to have scaling dimension 0 = O i, i, i L ¯ 0 δJ δJ ¯ φ φ , times the sum of the divergences of the 1PI self-energy diagr φ O L , = = J 2 O 1 ...... p i L L ) ) φ 1 (1) (2) 1 O O δ x J is are given in terms of the counter terms x x Z Z i D ( ( L δ δ φ 0 0 d δ O i, i, J ¯ ¯ φ φ with momentum Z ) = 1 + ) Z i x x i = ( ( φ φ 0 0 L Z and L, L, O i φ hO δS hO Z We consider Green functions that involve the operator Instead of renormalizing the sources, we can alternatively . The bare connected Green function and its amputated counte i ¯ terms of the renormalized ones as φ bare quantities are related via respective renormalizatio where the explicit expressionterms is of given renormalized first quantities in and terms a of the respective ba counter t where the superscript in parenthesisThe denotes products the of loop one-loop order counterterms terms of in non-1PI the diagrams. two-loop contr The counter term the action then reads Inserting the counter terms, the firstare two given terms by in the loop ex where dimension ∆, the source divergences of the 1PI diagrams involving one operator for the field constant that expresses the renormalized operators in term We make contact with the source renormalization by demandin which immediately yields JHEP09(2014)078 . ) g L ii i β ii F O φ , Q C.8 Z Z (C.9) β ), are (C.12) (C.10) (C.11) , ln 1.2 and that imension µ d such that . If d µ ε = 0 ε of ( 2 µ c a i ii ) = ii F L i Q x ams which renor- φ ( i . It guarantees that ¯ φ , , γ YM utated Green function, ) ... g L larization of the UV di- g ε ) . This scale is introduced tions on the rhs. of ( O β 1 µ . s which involve self-energy µ rticular, it specifies which ormalization constant x Z RGEs) for the renormalized ( + s pendence of the anomalous = L i is independent of , as well as ln ¯ ove Green functions are gauge O φ 0 λ ether with the UV divergences , π εg 0 ) µ ). Z 4 ( d √ g defined as x ε d ( YM ln µ g µ L = C.12 − = g hO ii g = ii F ε i ∂ 0, which has to be taken at the end. If µ L φ O ∂Q → Lγ ii ∂ ε ∂g ii F ± must not contain higher poles in g Q , γ L β β – 18 – L µ O α O + − γ d d Z µ + ∂ ∂ ∂g ∂µ = ) is, however, gauge invariant and thus independent of has to contain also higher-order poles in µ ∂ εg ∂α 0, ln L δ C.6 O = , δ + → = Z , one obtains 0 . ). ii ii µ g L ε dimensions. The bare Green functions have to be independent g α ε d is not renormalized in the theories we consider in this paper ii F ii F O d µ ε ∂ µ γ d 2 g Q µ 3.25 d ∂Q = − µ ii . Inserting this result into the definition of the anomalous d 0 ii F g 0 = εg = Q = 4 β − ii D ii F + = Q given in ( g 2 ∂ ∂g β O g γ , β β and hence also g µ ), one finds + d ) are determined by the same diagrams: these are the 1PI diagr d YM vanishes in the limit g µ ∂ and hence the effective planar coupling constant ∂µ A renormalization scheme defines a prescription for the regu The renormalization constants and renormalized Green func ii . This condition leads to renormalization group equations ( C.6 C.10 = µ ii F µ g Q YM β does not vanish, however, ln g cancellations of all poles occur among both terms in ( in a relation for the bare Yang-Mills coupling constant can be determined exactly. Using that the bare coupling in ( it obeys the relation The above result must be finite in the limit Since respectively, and the renormalization group functions are where the upper and lower sign holds for the connected and amp which yields in ( β In this appendix,dimension we discuss the renormalization-scheme de D Renormalization-scheme dependence Green functions. They are given by depend on the renormalization scale given by the ’t Hooft mas dimensionless in malize the amputated Green functioncorrections and of the the non-amputated non-1PI propagators. diagram dependent, While the the combination ab in ( the gauge-fixing parameter of The UV divergence of the connected Green function and the ren vergences and their absorptionfinite contributions into are the absorbed counter into the terms. counter terms In tog pa JHEP09(2014)078 - g µ i is of γ 2 and ii O ) as (D.4) (D.1) (D.3) (D.2) ̺ ii F ii γ dimen- Q ng con- ii F 1.3 18 ε Q of the 2 2 al subtrac- g and ii − ̺ ̺ ii F g = 4 Q 2 = 4 given in ( (1) O γ ii is not renormalized. D ). The latter can be ii F is present. g a description for handling Q ε . is obtained by applying β 3.24 ) µ , and then introducing 6 ̺ minimal subtraction (MS) g µ was determined in the DR ( 0 yields the relation ). In the DR scheme, ibutions sult in terms of the couplings In the dimensional reduction . ant ( . O ii ) → ) ii F ory, since D.2 + 4 4 ε g Q g -function 2 ( ( β (2) O in ( O O γ DR) scheme, the finite combination ε ii 1 + + 4 ̺ ii F 2 ii DR scheme, are members of a family of Q ) depends on the renormalization scheme + (1) ii F O at ’t Hooft mass g . in the respective counter terms yields the and the running coupling γ Q ii is obtained as ̺ ̺ ̺ 2 ̺ β ε ii F 2 ̺ g ) and the ̺ 3.25 g 2 ̺ Q g = β = e by – 19 – − 2 2 2 3.25 ) itself is scheme independent, since the replacement only ε ̺ O of the theory regularized in g ii g 2 g = 0) at ’t Hooft mass ii F Z 1.3 0 when no prefactor with poles in ε ), one finds that the difference of the logarithms in is obtained by inserting the couplings . Two schemes of this family are related via a change ̺ + Q ̺ ln 2 → 2 given in ( ̺ O D.2 = ε (1) O − 2 Z γ 2 ii O . Then reexpressing the result in terms of the couplings ε ̺ O γ ̺ ii F ̺ 1 2 given in ( can be expressed as expansions of those in the DR scheme. Z , the coupling Q taken from ( ii g ̺ = ) and ( ε ln . ii F − Both, the DR and the i 2 2 ̺ µ Q γ β O − D.1 2 = e Z 17 ̺ , which induces a change of the renormalized fields and coupli ]. µ g sin ln µ ε ̺ 37 = + i µ γ ̺ ], only the poles in is also absorbed, in analogy to the widely used modified minim -function 2 µ , the result for ln β 44 dimensions. π ̺ ]. Replacing the coupling sin ε ] for a complete definition in the context of QCD, including also 2 4 26 ]. In a modified dimensional reduction ( g 46 − and neglecting terms which vanish in the limit , + ln 4 36 32 ii 45 E = 4 − ii F γ MS) scheme [ D Q = The effective planar coupling constant The anomalous dimension − The one-loop See [ 2 in 18 17 = (2) O 5 this new scheme into thein above the expression. DR Expressing scheme the re via ( the subtractions of the DR scheme ( determined from the logarithm of the renormalization const sions are absorbed intoscheme the [ counter terms, like in the famous From the relation γ generates terms that vanish in the limit the two schemes is finite and given by rewritten in terms of the individual one- and two-loop contr because of the finite redefinition of the coupling schemes labeled by a free parameter ̺ stants. In particular, the theory in the scheme of the ’t Hooft mass This relation holds to all orders in planar perturbation the In the scheme γ deformation in the scheme via the relation (DR) scheme [ and are hence subtracted from the regularized expressions. tion ( renormalization in the scheme and The one-loop renormalization of the running coupling scheme in [ JHEP09(2014)078 ) ), ), l D.2 ]. D.4 (D.6) (D.5) C.12 t ] ommons , SPIRE via ( IN (1974) 461 . Since this , ii ][ ̺ ) ii F (1996) 53 6 (1998) 253 Q g B 72 ( global symmetry (2005) 069 2 ]. O ]. 05 + B 376 U(1) hep-th/9503121 ii . [ e DR scheme originates SPIRE e scheme × ii F SPIRE ii IN Q ]. IN JHEP ii F ][ d two loops. The individual , redited. Nucl. Phys. Q hep-th/0505071 ][ in the two schemes are hence ̺ β [ U(1) , 2 er-loop contributions to ( g ̺ β 2 2 SPIRE Phys. Lett. (1995) 95 ) leads to the same result, which imme- , the resulting term is a two-loop g , ) with respect to IN − ii 2 ii ii F ][ ii F 2 − Q O (2005) 3 D.4 m originates from the finite redefinition ( 2 γ β g, Q (2) B 447 O ( 2 Gauge theory correlators from noncritical = γ Adv. Theor. Math. Phys. O , 2 hep-th/0502086 γ = [ ̺ O hep-th/9802109 B 723 [ Z ) = 2 (2) ii ̺ O ln – 20 – -function ̺ ii F β Nucl. Phys. ,Q hep-th/9711200 ii [ ̺ , ii F g ( ∂ (2005) 033 , γ 2 (1998) 105 Nucl. Phys. ̺ O ∂Q Deforming field theories with 2 Exactly marginal operators and duality in four-dimensiona , γ 05 limit of superconformal field theories and supergravity ii (1) O ), which permits any use, distribution and reproduction in ii F γ ]. ]. N Q = β B 428 Wrapping interactions and the genus expansion of the 2-poin (1999) 1113 JHEP − 2 (1) , . ̺ SPIRE SPIRE O ii 38 ∂ γ ∂g ii F IN IN Q CC-BY 4.0 ][ ][ The large- εg This article is distributed under the terms of the Creative C Phys. Lett. A planar diagram theory for strong interactions , = Anti-de Sitter space and holography 2 Lax pair for strings in Lunin-Maldacena background ̺ O ]. ]. ]. Divergencies in a field theory on quantum space γ supersymmetric gauge theory 19 = 1 SPIRE SPIRE SPIRE IN IN IN hep-th/9802150 hep-th/0503201 function of composite operators and their gravity duals [ [ [ [ Int. J. Theor. Phys. string theory [ N Note that a direct identification of 19 [1] S. Frolov, [7] G. ’t Hooft, [8] C. Sieg and A. Torrielli, [9] T. Filk, [5] O. Lunin and J.M. Maldacena, [6] R.G. Leigh and M.J. Strassler, [2] J.M. Maldacena, [3] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, [4] E. Witten, derivative is multiplied by the one-loop Open Access. contribution. Accordingly, it was possible to discard high from the partial derivative of the one-loop term in ( where the term which is added to the anomalous dimension of th of the coupling constant References any medium, provided the original author(s) and source are c since they contribute toone- the and anomalous two-loop dimension contributions only torelated beyon the as anomalous dimension one obtains diately shows that the scheme dependence of the two-loop ter Attribution License ( Using this result to determine the anomalous dimension in th JHEP09(2014)078 ] ]. , , , , ] , = 4 = 4 SPIRE ] SYM N = 4 ]. IN N ]. Yang-Mills ][ , N = 4 ]. SYM theory , = 4 N SYM theory SPIRE SPIRE med hep-th/0510171 IN ]. N [ = 4 IN -deformed ][ super-Yang-Mills = 4 SPIRE ][ β N ns, orbifolds and IN arXiv:0811.4594 N [ SYM theory = 4 ][ SPIRE hep-th/0312218 ]. [ IN N tor of ][ = 4 ]. super Yang-Mills theory (2006) 288 , arXiv:0806.2103 [ ]. N SPIRE ]. IN ]. -deformed = 4 ]. i Twisted Bethe equations from a (2009) 034 ][ SPIRE γ hep-th/0506150 N ]. SPIRE B 736 (2004) 023 [ IN -deformed Single impurity operators at critical Finite-size effects in the 08 SPIRE IN arXiv:1312.2959 β [ ][ SPIRE IN 09 Twisting the mirror TBA ][ SPIRE (2008) 057 -deformed IN ][ arXiv:1012.3998 IN ]. [ β SPIRE ]. ]. ][ ][ 08 hep-th/0407029 IN JHEP [ , ][ JHEP Wrapping interactions and a new source of (2005) 023 duality cascade from a deformation of , SPIRE Nucl. Phys. Two-point correlators in the SPIRE SPIRE , (2014) 150 IN Operators with large R charge in SYM 10 JHEP IN IN – 21 – ][ = 1 , arXiv:1010.3229 Y-system and [ 07 ][ ][ (2012) 375 N = 4 hep-th/0205066 Non-conformality of The complete one-loop dilatation operator of planar real (2004) 034 [ 99 SYM arXiv:0902.1427 JHEP N [ An arXiv:1008.3351 , 12 [ JHEP Comments on the arXiv:1012.3982 , [ = 4 arXiv:1006.5438 [ hep-th/0307015 [ (2011) 027 N ]. Beauty and the twist: the Bethe ansatz for twisted (2002) 31 JHEP 02 , -deformed (2009) 130 β hep-th/0506128 arXiv:1009.4118 hep-th/0505187 [ [ [ (2012) 3 301 SPIRE (2004) 3 IN 04 SYM theory [ 99 (2011) 045014 JHEP Lett. Math. Phys. Review of AdS/CFT integrability: an overview -deformed , , β (2011) 015402 = 4 Wrapping in maximally supersymmetric and marginally defor B 676 JHEP N Review of AdS/CFT integrability. Chapter IV.2: deformatio On spin chains and field theories The complete one loop dilatation operator of D 84 , (2005) 042 (2011) 025 (2005) 039 A 44 ]. ]. ]. Superspace computation of the three-loop dilatation opera Annals Phys. 11 02 08 , SPIRE SPIRE SPIRE -deformed IN IN IN open boundaries SUSY Yang-Mills theory wrapping order in the Yang-Mills theory arXiv:1308.4420 β JHEP [ corrections to the spin-chain/string duality [ superconformal Phys. Rev. J. Phys. JHEP twisted S-matrix Lett. Math. Phys. JHEP Nucl. Phys. [ SYM at the next-to-leading order [22] F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, [23] J. Gunnesson, [13] N. Beisert et al., [27] J. Fokken, C. Sieg and M. Wilhelm, [28] S. Penati, A. Santambrogio and D. Zanon, [24] D.Z. Freedman and U. Gürsoy, [25] T.J. Hollowood and S.P. Kumar, [26] J. Fokken, C. Sieg and M. Wilhelm, [20] D.J. Gross, A. Mikhailov and R. Roiban, [21] J. Ambjørn, R.A. Janik and C. Kristjansen, [18] F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, [19] C. Sieg, [16] G. Arutyunov, M. de Leeuw and S.J. van Tongeren, [17] C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, [14] K. Zoubos, [15] N. Gromov and F. Levkovich-Maslyuk, [11] N. Beisert, [12] R. Roiban, [10] N. Beisert and R. Roiban, JHEP09(2014)078 , , 5 ]. ]. S , × 5 SPIRE SPIRE IN [ IN oops and AdS , ][ , mensional (2006) 040 , splitting functions 02 , renormalization group, (1978) 3998 ]. al reduction JHEP , , super-Yang-Mills theory ]. D 18 SPIRE arXiv:1108.4914 , Clarendon Press, Oxford U.K. IN [ = 4 [ ]. ]. TBA, NLO correction Lüscher and ]. ]. N SPIRE IN Deep inelastic scattering beyond the SPIRE ]. ][ SPIRE Superspace or one thousand and one SPIRE IN ]. SPIRE Phys. Rev. IN IN , ][ space technique IN New approach to evaluation of multiloop (2011) 059 (1980) 417 ][ ][ ][ x SPIRE -deformed i 12 SPIRE 43 IN γ [ IN [ – 22 – ]. ]. JHEP ]. The spectral problem for strings on twisted Orbifolded Konishi from the mirror TBA , hep-ph/9506451 [ , Cambridge University Press, Cambridge U.K. (1984). The calculation of the two loop spin splitting functions SPIRE -deformed conformal Yang-Mills SPIRE IN arXiv:1201.1451 β (1985) 356 SPIRE [ [ hep-ph/9512218 arXiv:1103.5853 IN [ [ arXiv:0906.0499 [ IN [ [ ]. ]. hep-th/0108200 , Temperature quantization from the TBA equations Theor. Math. Phys. B 164 (1996) 637 , SPIRE SPIRE (1980) 345 (2012) 339 IN IN (2009) 60 [ (1973) 455 Method for computing renormalization group functions in di C 70 (1979) 193 (1996) 2023 Evaluation of a class of Feynman diagrams for all numbers of l ][ Quantum field theory and critical phenomena (2011) 325404 A rederivation of the spin dependent next-to-leading order Dimensional regularization and the renormalization group Amplitudes in the Renormalization. An introduction to renormalization, the Phys. Lett. B 61 B 174 B 860 B 84 B 679 D 54 , Supersymmetric dimensional regularization via dimension A 44 Z. Phys. The emergence of supersymmetry in ), x ( (1) ij hep-th/0512194 renormalization scheme dimensions Phys. Lett. P Phys. Rev. Nucl. Phys. lessons in supersymmetry leading order in asymptotically free gauge theories [ Feynman integrals: the Gegenbauer polynomial Nucl. Phys. arXiv:1311.7391 double wrapping in twisted AdS/CFT J. Phys. Phys. Lett. Nucl. Phys. (1996). and the operator product expansion [45] R. Mertig and W.L. van Neerven, [46] W. Vogelsang, [42] J. Zinn-Justin, [40] A.A. Vladimirov, [41] S.J. Gates, M.T. Grisaru, M. and Roˇcek W. Siegel, [38] V.V. Khoze, [39] K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, [35] D.J. Broadhurst, [36] G. ’t Hooft, [37] W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, [33] C. Ahn, Z. Bajnok, D. Bombardelli and R.I.[34] Nepomechie, M. de Leeuw and S.J. van Tongeren, [30] M. de Leeuw and S.J. van Tongeren, [31] S. Frolov, private communication. [32] Q. Jin, [29] S. Frolov and R. Suzuki, [43] J.C. Collins, [44] W. Siegel,