# Some remarks on Feynman rules for non-commutative gauge theories based on groups $Gneq U(N).pdf

arXiv:hep-th/0205286v1 28 May 2002May 2002HU Berlin-EP-02/21hep-th/0205286Some remarks on Feynman rules fornon-commutative gauge theories based ongroups G negationslash= U(N)Harald Dorn and Christoph Sieg 1Humboldt–Universit¨at zu Berlin, Institut f¨ur PhysikInvalidenstraße 110, D-10115 BerlinAbstractWe study for subgroups G ⊆ U(N) partial summations of the θ-expanded perturbationtheory. On diagrammatic level a summation procedure is established, which in the U(N)case delivers the full star-product induced rules. Thereby we uncover a cancellation mech-anism between certain diagrams, which is crucial in the U(N) case, but set out of work forG ⊂ U(N). In addition, an explicit proofis given that forG ⊂ U(N), G negationslash= U(M), M 2 vanish. In both cases from (10) we would get Feynman ruleswith a finite number of building blocks.For U(N) the equivalent representation (12) is due to a simple field redefinition of afree theory. Therefore, looking at the n-point functions of the, in terms of a,c,¯c, composite4operators A,C, ¯C (see (4),(6)) the summation of the perturbation theory with respect tos1[a,c,¯c]−ilogJ must yield the free field result guaranteed by (11).5On the other side for G ⊂ U(N) we cannot directly evaluate (11) and are forcedto work with (12). It will turn out to be useful to study both U(N) and G ⊂ U(N)in parallel. Since the result for U(N) is a priori known, one has some checks for thecalculations within the s1-perturbation theory.3 s1-perturbation theory for U(N) and G ⊂ U(N)In both cases our gauge fields take values in the Lie algebra of U(N). We writeAµ = ABµ TB (15)and use the following relations for the generators TA of the U(N) Lie algebra[TA,TB] = ifABC TC , {TA,TB} = dABC TC , tr(TATB) = 12δAB . (16)Then (4) and (6) implyAMµ = aMµ − 12θαβaPα∂βaQµ dMPQ + 14θαβaPα∂µaQβ dMPQ− 14θαβaPαaQβ aRµ dMPSfSQR + O(θ2) (17)andCM = cM + 14θαβ∂αcPaQβ dMPQ + O(θ2)¯CM = ¯cM . (18)In the case G ⊂ U(N), G negationslash= U(M), M 2.What changes if we switch from U(N) to G ⊂ U(N)? First of all, then we do notknow the answer in advance and have to rely only on s1-perturbation theory. Secondly, inthis perturbation theory the above cancellation mechanism is set out of work for externalpoints carrying a primed index, related to Lie algebra elements of U(N) not in the Liealgebra of G. Then according to (21) the external vertex to start with in the first diagramsof fig. 3 and fig. 4 is zero, i.e. the partner to cancel the second diagrams disappears. Thisobservation is a strong hint that for G ⊂ U(N) there remain non-vanishing connectedGreen functions 〈A(x1).A(xn)〉kin for all integer n. An explicit proof will be given inthe next section.4 Non-vanishing n-point Green functions generatedby logZkinGThe connected Green functionsGkin,M1.Mnn (x1,.,xn) =angbracketleftBigAM1(a(x1)).AMn(a(xn))angbracketrightBigkinare power series in θ and g. To prove their non-vanishing for generic θ and g it is sufficientto extract at least one non-zero contribution to Gkinn of some fixed order in θ and g.To find for our purpose the simplest tractable component of the Green function itturns out to be advantageous to restrict all of the group indices Mi to primed indicesthat do not correspond to generators of the Lie algebra of G. Then the Green functionsimplifies in first nontrivial order of the Seiberg-Witten map to:angbracketleftBig nproductdisplayi=1bracketleftbigA(2)m′i(a(xi)) + A(3)m′i(a(xi))bracketrightbigangbracketrightBigkin. (32)Here A(2), [A(3)] denote the ∝ θ part of the Seiberg-Witten map (17) with quadratic,[cubic] dependence on the ordinary field a. Thus the above function is Oparenleftbigθnparenrightbig. Focussingnow on the special contribution which is exactly ∝ θn, it is clear that in addition tothe external vertices further θ-dependence (e. g. higher order corrections to the Seiberg-Witten map) is not allowed. That means this special part of the connected Green functionis universal with respect to the θ-expansion of the constraint (4) where a ∈ G.9p1,µ1,m′1p2,µ2,m′2p3,µ3,m′3pn−1,µn−1,m′n−1pn,µn,m′n+ permFigure 5: graphs ∝ θng2n of the connected n-point Green functionThe special contribution to the Green function ∝ θn then consists of n to 32n, [32(n−1)+1] internal lines for n even, [odd]. Two or three of these originate from each of the npoints (external vertices). There are no further internal vertices present stemming fromthe interaction term s1[a,c,¯c] in (12) since this would increase the power in θ.In our normalization where the coupling constant g is absorbed into the fields eachpropagator enlarges the power of the diagram in g by g2. Thus for general coupling g it issufficient to check the non-vanishing of all connected diagrams with the same number ofpropagators. Here we choose the minimum case of n propagators where we can neglect allcontributions from A(3) in (32). Then it follows that the connected ∝ θng2n contributionsto the Green function are given by the type of diagrams shown in fig. 5.The total number of the diagrams can be determined as follows: The two lines start-ing at each point are distinguishable due to the derivative at one leg. To construct allconnected contributions we connect the first leg of the first external vertex to one of the2n−2 other legs that do not start at the same external point. The next one is connectedto one of the remaining 2n − 4 allowed legs, such that no disconnected subdiagram isproduced and so on. We thus have to add-up (2n − 2)!! = (n − 1)! 2n−1 diagrams. Allof them can be drawn like the one shown in fig. 5 by permuting the external momenta,Lorentz and group indices and the internal legs.To sum-up all diagrams it is convenient to define two classes of permutations: The firstincludes all permutations that interchange the two distinct legs at one or more externalvertices with the distribution of the external momenta, Lorentz and group indices heldfixed. The second contains all permutations which interchange the external quantitiessuch that this cannot be traced back to a permutation of the distinct lines at the externalvertices. We call its elements proper permutations in the following.In total 2n combinations exist, generated by interchanging the distinct legs when theexternal points are fixed. The proper permutations are the ones which are not identicalunder (anti)cyclic permutations. There are n! configurations of the external points andwith each one n−1, [n] others are identified under cyclic, [anticyclic] permutations, i. e.there are n!2n = (n−1)!2 proper permutations. This is consistent with the total number ofdiagrams.10The connected ∝ θng2n contributions to the momentum space Green function can thusbe cast into the following form:Gkin,m′1.m′nn µ1.µn (p1,.,pn)vextendsinglevextendsingle∝θng2n =summationdisplayperm{i1,.,in}(anti)cycl.pi1,µi1,m′i1pi2,µi2,m′i2pi3,µi3,m′i3pin−1,µin−1,m′in−1pin,µin,m′in((((( (( (((( (.(33)Here the brackets around the external vertices denote a sum over both configurationswhere the two legs are interchanged. These n sums are then multiplied, describing exactlythe 2n permutations of the distinct two legs at each vertex.The sum of the two permutations at one external vertex occurring n times in (33)readsp,µ,M +q,α,Ar,β,B= −i4 dabm′bracketleftBig2(θβγqγδαµ+θαγrγδβµ)+θαβ(qµ−rµ)bracketrightBig.Using this, the analytic expression for the ∝ θng2n part of the connected Green functionis given byGkin,m′1.m′nn µ1.µn (p1,.,pn)vextendsinglevextendsingle∝θng2n=summationdisplayperm{i1,.,in}(anti)cycl.g2n4nintegraldisplay dDk(2π)Dnproductdisplayr=1darar+1m′irbracketleftBig−2θαrγr(qr−1)γrgµirαr+1 + 2θ γrαr+1 (qr)γrδαrµir−θαrαr+1(qr−1 + qr)µirbracketrightBig 1q2r−1 ,(34)where summation over αr appearing twice in the sequence of multiplied square bracketsis understood. Thereby one has to identify an+1 = a1, αn+1 = α1, pin = −summationtextn−1r=1 pir. Theqr are defined byqr = qr(k,pi1,.,pir) = k +rsummationdisplays=1pis . (35)In appendix A we prove that this expression is indeed non-zero at least for even n andthe most symmetric non-trivial configuration of the external momenta, Lorentz and group11indices. This means that non-vanishing connected n point functions for arbitrary highn exist in the kinetic perturbation theory, leading to infinitely many building blocks inthe θ-summed case. In other words one needs infinitely many elements to formulateFeynman rules for the non-commutative G-gauge theory if one insists on keeping thenon-commutative U(N) vertices as components.Due to the fact that the expressions discussed above cannot be affected by higherorder corrections of (4) this statement is universal, i. e. independent of the power in θ towhich the constraint a ∈ G is implemented.5 The case with sources restricted to the Lie algebraof GUp to now we have looked for Feynman rules working with the original U(N) vertices andsources JM taking values in the full U(N) Lie algebra. This seemed to be natural sincein the enveloping algebra approach for G ⊂ U(N) the non-commutative gauge AM field,although constrained, carries indices M running over all generators of U(N).There is still another option to explore. First one can restrict the sources J, η, ¯ηin (9) by hand to take values in the Lie algebra of G only. Then instead of pulling outin (10) the complete interaction SI one separates only those parts of SI, which yieldvertices whose external legs carry lower case Latin indices referring to the Lie algebra ofG exclusively. The remaining parts of SI, generating vertices with at least one leg owninga primed index, are kept under the functional integral. The functional integration andthe constraint remain unchanged. We denote this splitting of SI bySI[A,C, ¯C] = Si[A,C, ¯C] + S′i[A,C, ¯C] (36)and the sources by hatted quantitieshatwideJa′ = hatwide¯ηa′ = hatwideηa′ = 0 . (37)ThenZG[hatwideJ,hatwide¯η,hatwideη] = eiSi[ δiδ hatwideJ , δiδhatwide¯η, δiδhatwideη] hatwideZG[hatwideJ,hatwide¯η,hatwideη] (38)andhatwideZG[hatwideJ,hatwide¯η,hatwideη] =integraldisplaya,c,¯c ∈GDA D ¯C DC ei(Skin[A,C, ¯C]+S′i[A,C,¯C]+AhatwideJ+hatwide¯ηC+ ¯C hatwideη)=integraldisplaya,c,¯c ∈GDa D¯c Dc J ei(Skin[a,c,¯c]+hatwides1[a,c,¯c]+A[a]hatwideJ+hatwide¯ηC[c,a]+¯chatwideη) , (39)where hatwides1[a,c,¯c] is defined bySkin[A[a],C[c,a],¯c] + S′i[A[a],C[c,a],¯c] = Skin[a,c,¯c] + hatwides1[a,c,¯c] . (40)12If nowlogparenleftBighatwideZG[hatwideJ,hatwide¯η,hatwideη]/hatwideZG[0]parenrightBig=summationdisplayninintegraldisplaydx1 .dxn hatwideJ(x1). hatwideJ(xn)〈A(x1).A(xn)〉kin +S′i + . , (41)e.g. for G = SO(N), in the spirit of (14) would generate only the free propagators, theSO(N) Feynman rules conjectured in ref.[17] 7 would have been derived via partial sum-mation of the θ-expanded perturbation theory in the enveloping algebra approach.In the remaining part of this section we prove that this cannot happen. For thispurpose we consider 〈Am1(x1).Amn(xn)〉kin +S′i and look at it as a power series in g2and θ. To prove that it is not identically zero, it is sufficient to find a particular non-vanishing order in g2, θ. Let us concentrate on the lowest possible order in g2.At all xi the contributing diagrams in the hatwides1-perturbation theory have to start withat least one commutative gauge field propagator (29). This generates at least a factorg2n (One propagator at each xi corresponding to the lowest order of SW map.) Thediagrams have to be connected. To achieve this, with respect to power counting in g2, inthe most effective way one has to connect all the n legs in just one n-point vertex of thehatwides1-perturbation theory, ending up with a total g2-power of g2n−2.Now we search in addition for the lowest possible power in θ. The n-point verticesarise from expressing the non-commutative fields A either in the original non-commutativekinetic term or 3-point or 4-point interactions in S′i (see (36)) via (17) in terms of thecommutative field a. 8 Let us look for the most efficient way for simultaneously trading aminimal number of θ-factors combined with a maximal number of a-legs. Simple dimen-sional analysis shows that this is achieved by terms in the SW map (17) not containingderivatives, i.e. terms of the type (a)l(θ)l−12 . Then independent of the order of contribu-tion to the SW-map n-point vertices originating from the kinetic term, the original 3-pointor 4-point vertices behave like θn2−1, θn2−32 and θn2 −2, respectively. From this observationwe can conclude that for a given n within the lowest g2-power term the minimal numberof θ factors is exclusively realized by connecting the n-external legs in just one n-pointvertex generated by SW-mapping out of a 4-point interaction of S′i.In appendix B we prove for SO(3) that the corresponding contribution to the 8-pointfunction 〈Am1(x1).Am8(x8)〉kin +S′i is different from zero. This excludes the rules of [17].The more ambitious program to exclude rules based on the vertices in Si and anarbitrary but finite number of additional building blocks would require to show, similarto the previous section, that there is no n0 assuring vanishing connected n-point functionsfor n n0. Although we have practically no doubt concerning this conjecture, a rigorousproof is beyond our capabilities since for increasing n higher and higher orders of the7There have been given arguments [7] that their constraint is equivalent to requiring the image underthe inverse SW map to be in SO(N).8Note that the original 3-point or 4-point interactions by themselves areθ-dependent via the ⋆-product.But since we are searching for lowest order in θ this further θ-dependence can be disregarded.13SW-map contribute. This happens because in contrast to the proof in section 4 one isforced to look at Green functions with all external group indices referring to generators ofthe Lie algebra of G since no primed indices of the remaining generators spanning U(N)are probed.6 ConclusionsStarting from the enveloping algebra approach we have studied the issue of partial summa-tion of θ-expanded perturbation theory for subgroups G ⊂ U(N). The main motivationwas given by the search for some Feynman rules exhibiting UV/IR mixing similar to thewell known U(N) case. The original Feynman rules in the enveloping algebra approachcontain an infinite number of vertices. They are read off from the interactions in termsof the commutative gauge field aµ (and ghosts) taking values in the Lie algebra of G.Our aim was to decide, whether by some partial summation new rules related to theinteractions of the non-commutative gauge field Aµ (and ghosts) can be derived. Thenon-commutative fields take values in the Lie algebra of U(N), but are constrained to berelated to the commutative fields by the Seiberg-Witten map. Coming from the side ofθ-expanded perturbation theory the non-commutative fields are composites constructedout of the commutative fields.With our initial formula (10) we have decided to choose the vertices generated by theinteraction term SI in terms of Aµ,C, ¯C as part of the building blocks of the wanted Feyn-man rule