Particles with non abelian charges
Fiorenzo Bastianelli, Roberto Bonezzi, Olindo Corradini, Emanuele Latini
aa r X i v : . [ h e p - t h ] S e p Particles with non abelian charges
Fiorenzo Bastianelli a , Roberto Bonezzi a , Olindo Corradini b , Emanuele Latini ca Dipartimento di Fisica e Astronomia, Universit`a di Bologna andINFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy b Centro de Estudios en F´ısica y Matem´aticas Basicas y Aplicadas,Universidad Aut´onoma de Chiapas, Ciudad Universitaria, Tuxtla Guti´errez 29050,Mexico c Institut f¨ur Mathematik, Universit¨at Z¨urich-Irchel,Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland
Abstract
Efficient methods for describing non abelian charges in worldline approachesto QFT are useful to simplify calculations and address structural properties, asfor example color/kinematics relations. Here we analyze in detail a method fortreating arbitrary non abelian charges. We use Grassmann variables to take intoaccount color degrees of freedom, which however are known to produce reduciblerepresentations of the color group. Then we couple them to a U(1) gauge fielddefined on the worldline, together with a Chern-Simons term, to achieve projectionon an irreducible representation. Upon gauge fixing there remains a modulus, anangle parametrizing the U(1) Wilson loop, whose dependence is taken into accountexactly in the propagator of the Grassmann variables. We test the method in simpleexamples, the scalar and spin 1/2 contribution to the gluon self energy, and suggestthat it might simplify the analysis of more involved amplitudes.
Dedicated to the Memory of V´ıctor Manuel Villanueva Sandoval
Introduction and outlook
The description of non abelian charges in worldline approaches has a long history. Yet, itseems useful to improve on existing methods to gain in efficiency and be able to addressin the worldline context properties like the color/kinematics relations, found in the studyof perturbative gauge and gravitational amplitudes [1].The classical limit of nonabelian charges was originally studied in [2], followed by theproposal of using Grassmann variables to treat them in first quantization [3, 4]. The use-fulness of Grassmann variables is that upon quantization they produce a finite dimensionalHilbert space that can be employed to describe both spin and color degrees of freedom.For the spin 1/2 particle the description of spin in terms of Grassmann variables [5, 6]allows to simplify in the electromagnetic coupling the spin factor introduced with a pathordering prescription by Feynman [7], so that the worldline description can combine intoone expression Feynman diagrams with different orderings of the external photons alonga line or loop, just as in the scalar case. This produces considerable simplifications for theorganization and calculation of the related amplitudes. However, the color Hilbert spacearising from the use of Grassmann variables is reducible. For that reason path orderedexponentials (i.e. Wilson lines) to take into account non abelian charges were often usedin the past, as for example in the worldline calculations of [8, 9, 10, 11, 12, 13, 14, 15].The quantization of Grassmann variables creates automatically the path ordering, but onemust devise ways to select the irreducible representation needed in specific applications.One way of projecting onto irreducible representations was proposed in [16] by consideringa sum over a discrete set of angles, that implements the required projection. This methodwas used for example in [17] and [18].Here we suggest a way of obtaining the projection by coupling the Grassmann variablesto a U(1) worldline gauge field with an additional Chern-Simon term, whose discretizedcoupling is tuned to obtain the required projection. This projection mechanism appearedin the worldline description of differential forms [19, 20, 21], defined by constrained modelswith worldline gauge fields making up an extended supergravity multiplet with a U(1)gauge field. Upon quantization the latter is seen to produce the projection needed toselect the degree of the differential form. This mechanism was then used in other worldlineapplications [13, 22, 23, 24], where a suitable projection was needed. After gauge fixing,there remains an integration over a modulus associated to the U(1) gauge field, an angle φ , that implements the projection in the amplitudes. In [16] it was indeed suggestedthat the discrete sum used there to achieve projection could be turned into a continuousintegration. In our model we see how that suggestion is explicitly realized. To proceedfurther, we find it useful to encode the coupling of the Grassmann variables to the modulus φ by using twisted boundary conditions. One-loop amplitudes are then obtained as usual1y computing worldline correlators of vertex operators suitably integrated over the moduli T and φ , the proper time and the U(1) modulus. Here we apply the mechanism to obtainprojection on a given representation of the color group for particles of spin 0 and 1/2,and as a test calculation compute their contribution to the gluon self energy. Gluons canalso be described in first quantization, as in [8, 12] or with the string inspired method of[13], and one could again employ in such a case the present description of non abeliancharges. It would be interesting to analyze how this description of color degrees of freedomperforms in calculating higher point one-loop amplitudes, and perhaps study the originof the color/kinematics relations in this context.A similar set up may be constructed using bosonic variables instead of fermionicones. The corresponding Hilbert space that gives rise to the color degrees of freedom isinfinite dimensional, containing all possible symmetric tensor products of the fundamentalrepresentation, but the coupling to the U(1) gauge field with a properly chosen Chern-Simons coupling can again select the finite subspace corresponding to the fundamentalrepresentation. The euclidean action of a scalar particle coupled to the photon is given by S [ x, e ; A ] = Z dτ (cid:16) e − ˙ x + 12 em − iqA µ ( x ) ˙ x µ (cid:17) (1)where x µ are the coordinates of the particle, e is the einbein, q is the charge of the particle,and A µ ( x ) is the background abelian gauge field. When inserted in the path integral Z DxDe
Vol(Gauge) e − S [ x,e ; A ] (2)one finds that the coupling gives rise to the Wilson line e iq R A µ dx µ . A way to generalizethis to non abelian fields A µ ( x ) = A aµ ( x ) T a , with T a hermitian generators of a simplegroup, is to use the path ordering prescription to guarantee gauge invariance P e ig R A µ dx µ . (3)The generators are taken in an arbitrary representation of the gauge group and describethe non abelian charge assigned to the particle, while g denotes the coupling constant.This procedure is correct and well-known, but it is also useful to introduce Grassmannvariables to create the Hilbert space associated to the color degrees of freedom and getrid of the path ordering prescription. The latter is generated by path integration over the2ew variables. The coupling part of the scalar particle action takes then the form∆ S NA = Z dτ (cid:18) ¯ c α ˙ c α − igA aµ ( x ) ˙ x µ ¯ c α ( T a ) αβ c β (cid:19) (4)which depend on the Grassmann variables c α and their complex conjugates ¯ c α . The gaugegroup generators ( T a ) αβ can be chosen in any desired representation, but for definitenesswe select SU( N ) as gauge group and choose the fundamental representation which hasdimension N . The Grassmann variables c α has thus an index in the fundamental repre-sentation, and ¯ c α in the complex conjugate one.The quantization of the Grassmann variables gives rise to fermionic creation and an-nihilation operators ˆ c α and ˆ c † α , satisfying the anticommutation relations { ˆ c α , ˆ c † β } = δ βα , { ˆ c α , ˆ c β } = 0 , { ˆ c † α , ˆ c † β } = 0 . (5)They are naturally represented by ˆ c † α ∼ ¯ c α and ˆ c α ∼ ∂∂ ¯ c α when acting on wave functionsof the form φ ( x, ¯ c ). The latter has a finite Taylor expansion on the Grassmann numbers¯ c α of the form φ ( x, ¯ c ) = φ ( x ) + φ α ( x )¯ c α + 12 φ αβ ( x )¯ c α ¯ c β + · · · + 1 N ! φ α ...α N ( x )¯ c α ... ¯ c α N (6)and contain wave functions transforming in all possible antisymmetric tensor products ofthe fundamental representation. Thus we see that the fermionic creation and annihilationoperators create a finite dimensional Hilbert space for the color degrees of freedom, whichis however reducible.To select the fundamental representation one must project to the sector with occupa-tion number one, so to isolate the wave function φ α ( x ) with an index in the fundamentalrepresentation. This can be achieved by coupling the variables c α and ¯ c α to a U(1) gaugefield a ( τ ) living on the worldline, with in addition a Chern-Simons term with quantizedcoupling s = n − N , where n is the occupation number and N is the dimension of thefundamental representation. The gauge field a acts as a Lagrange multiplier that imposesa constraint on physical states, and setting n = 1 selects precisely the sector with oc-cupation number one as possible physical states, corresponding to wave functions in thefundamental representation of the color group.Let us analyze this mechanism in detail. The free kinetic term of the Grassmannvariables that appears in (4) is modified by the coupling to the worldline U(1) gauge field a to S c ¯ c = Z dτ (cid:16) ¯ c α ( ∂ τ + ia ) c α − isa (cid:17) (7)where the last piece is the Chern-Simons term. The equations of motion of a produce theconstraint C ≡ ¯ c α c α − s = 0. Upon quantization the function C becomes the operatorˆ C ≡
12 (ˆ c † α ˆ c α − ˆ c α ˆ c † α ) − s ∼ ¯ c α ∂∂ ¯ c α − c and ¯ c , and used the quantized value of the Chern-Simons coupling s = 1 − N .Clearly, when acting on the generic wave function (6) the constraint ˆ Cφ ( x, ¯ c ) = 0 selectsthe wave function φ α ( x ). The conclusion is that one can compute quantum propertiesof a scalar particle coupled to non abelian gauge fields in an arbitrary representation byusing the above ingredients.In particular, the one-loop effective action induced by a scalar particle coupled to anon abelian gauge field is computed by path integrating on the circle S Γ[ A ] = Z S DxD ¯ cDcDeDa Vol(Gauge) e − S (9)where S = Z dτ (cid:18) e − ˙ x + 12 em + ¯ c α (cid:0) ∂ τ + ia (cid:1) c α − isa − igA aµ ( x ) ˙ x µ ¯ c α ( T a ) αβ c β (cid:19) . (10)The particle action has two local invariances, the standard reparametrization invarianceand the new U(1) local symmetry. One may gauge fix e ( τ ) = 2 T and a ( τ ) = φ , where T isthe standard Fock-Schwinger proper time and φ an angle describing U(1) gauge invariantconfigurations on the circle ( z = e i R dτa = e iφ is the Wilson loop). They make up modulithat must be integrated over in the path integral. Taking care of the related Faddeev-Popov determinants, one finds a final formula for the induced QFT effective action forthe non abelian field A aµ of the typeΓ[ A ] = − Z ∞ dTT e − m T Z π dφ π e isφ Z PBC Dx Z TBC D ¯ cDc e − S gf (11)with S gf = Z dτ (cid:18) T ˙ x + ¯ c α ˙ c α − ig A aµ ( x ) ˙ x µ ¯ c α ( T a ) αβ c β (cid:19) (12)where P BC denotes periodic boundary conditions, and
T BC twisted boundary conditionsspecified by c (1) = − e − iφ c (0), ¯ c (1) = − e iφ ¯ c (0). The latter take into account the couplingto the worldline U(1) gauge field, and arise after a field redefinition ( c ( τ ) → e iφτ c ( τ ) and itscomplex conjugate) that eliminates the U(1) coupling from the action while transferringit to the boundary conditions (for φ = 0 they reduce to ABC , antiperiodic boundaryconditions, as natural for fermionic variables). n -gluon amplitudes Having found the worldline representation for the one-loop effective action due to a scalarparticle, it is now easy to find a master formula of the Bern-Kosower type [25], though4ontaining only one-particle irreducible terms, for the corresponding contribution to theone-loop n -gluon amplitudes. This is obtained as usual by inserting in (11) a sum of planewaves A aµ ( x ) T a = n X i =1 ε µ ( k i ) e ik i · x T a i , (13)selecting the terms linear in each polarization ε µ ( k i ), and choosing the generators T a i inthe fundamental representation. One findsΓ scal ( k , ε , a ; ... ; k n , ε n , a n ) = − ( ig ) n Z ∞ dTT e − m T Z π dφ π e isφ × Z PBC Dx Z TBC D ¯ cDc e − S n Y i =1 V scal [ k i , ε i , a i ] (14)where the gluon vertex operator is given by V scal [ k, ε, a ] = ( T a ) αβ Z dτ ε · ˙ x ( τ ) ¯ c α ( τ ) c β ( τ ) e ik · x ( τ ) (15)and S denotes the quadratic part of the gauge fixed action (12). On the space of periodicfunctions x µ ( τ ) the kinetic operator − T d dτ is not invertible, due to constant zero modes.Thus, one may split the trajectories as x µ ( τ ) = x µ + y µ ( τ ), where y µ ( τ ) = X m =0 y µm e πimτ , x µ = Z dτ x µ ( τ ) , Z dτ y µ ( τ ) = 0 . (16)The integration measure factorizes as Dx = d D x Dy , and the integration over the zeromodes x µ produces the delta function for momentum conservation Z d D x n Y i =1 e ik i · x = (2 π ) D δ D ( k + k + ... + k n ) . (17)It is also convenient to define the normalized quantum averages h ... i with respect to thequadratic action as h f i = R DyD ¯ cDc f e − S R DyD ¯ cDc e − S , with S = Z dτ (cid:18) T ˙ y + ¯ c α ˙ c α (cid:19) , (18)while the free path integral normalization yields Z PBC Dy Z TBC D ¯ cDc e − S = (4 πT ) − D/ Det
TBC ( ∂ τ ) N = (4 πT ) − D/ (cid:18) φ (cid:19) N . (19)Also, from the action (18) one extracts the free worldline propagators h y µ ( τ ) y ν ( σ ) i = − T δ µν G ( τ − σ ) , h c α ( τ )¯ c β ( σ ) i = δ βα ∆( τ − σ ; φ ) (20)5here G ( τ − σ ) = | τ − σ | − ( τ − σ ) , ∆( τ − σ ; φ ) = 12 cos φ h e i φ θ ( τ − σ ) − e − i φ θ ( σ − τ ) i (21)with θ ( x ) the step function. A constant part in the bosonic propagator G ( τ − σ ) has beendropped, as it does not contribute in such calculations due to momentum conservation.At last, one may use an exponentiation trick by writing n Y i =1 ε i · ˙ y i = e P ni =1 ε i · ˙ y i (cid:12)(cid:12)(cid:12) lin ε ...ε n (22)with the shorthand notation y i = y ( τ i ), so that with all the previous definitions andmanipulations one may cast the non-abelian master formula for the n -gluon amplitude inthe following wayΓ scal ( k , ε , a ; ... ; k n , ε n , a n ) = − ( ig ) n (2 π ) D δ D ( k + ... + k n ) Z ∞ dTT e − m T (4 πT ) D/ Z π dφ π (cid:18) φ (cid:19) N e isφ × Z dτ ... Z dτ n exp ( T n X i,j =1 12 G ij k i · k j − i ˙ G ij ε i · k j + ¨ G ij ε i · ε j ) (cid:12)(cid:12)(cid:12)(cid:12) lin ε ...ε n × * n Y i =1 ( T a i ) α i β i ¯ c α i ( τ i ) c β i ( τ i ) + (23)where G ij = G ( τ i − τ j ), and dots stand for derivatives with respect to the first variable. Itis interesting to notice that the bosonic contributions yield exactly the same τ i integrandas in the abelian case: all the non-abelian features are captured by the extra fermionicaverage appearing in the last line, and by the modular integral over φ . The previous master formula can be tested by checking the scalar contribution to thegluon self-energy. From (23) one obtainsΓ scal ( k , ε , a ; k , ε , a ) = g (2 π ) D δ D ( k + k ) tr F ( T a T a ) Z π dφ π e isφ (cid:18) φ (cid:19) N − × ( ε · ε k − ε · k ε · k ) Z ∞ dTT e − m T (4 πT ) D/ T Z dτ Z dτ ˙ G e − T k G (24)6here we used ∆( τ − σ ; φ )∆( σ − τ ; φ ) = − (cid:0) φ (cid:1) − in the fermionic contractions,integrated by parts a term in the bosonic piece to get a manifestly transverse amplitude,and used momentum conservation to define k = k = − k . It can be casted in thefollowing formΓ scal ( k , ε , a ; k , ε , a ) = (2 π ) D δ D ( k + k ) C a a ε µ ε ν Π µν ( k ) (25)to isolate the color factor C ab = tr F ( T a T b ) Z π dφ π e isφ (cid:18) φ (cid:19) N − (26)and the abelian vacuum polarization tensor Π µν ( k ) = ( δ µν k − k µ k ν ) Π( k ) withΠ( k ) = g (4 π ) D/ Z ∞ dTT e − m T T − D Z du (1 − u ) e − T k u (1 − u ) . (27)The latter is obtained by using the translation symmetry on the circle to fix τ = 0, sothat denoting τ = u one has G ( u ) = u (1 − u ) and ˙ G ( u ) = 1 − u . For completeness, letus proceed further by using Z ∞ dtt e − at t z = a − z Γ( z ) (28)to perform the T integral and get it into a standard formΠ( k ) = g (4 π ) D/ Γ (cid:16) − D (cid:17) Z du (1 − u ) h m + u (1 − u ) k i D − (29)that is easily regulated in dimensional regularization with D = 4 − ε (and MS schemeconventions) to getΠ( k ) = 1 ε g π − g π Z du (1 − u ) ln M ( u, k ) µ + O ( ε ) (30)where M ( u, k ) = m + k u (1 − u ).In this simple example the effect of the color charge is to dress the abelian vac-uum polarization by the color factor C ab . In the fundamental representation one hastr F ( T a T b ) = δ ab . Using for simplicity the Wilson loop variable z = e iφ , and the specificvalue of the Chern-Simons coupling s = 1 − N , one finds from (26) C ab = 12 δ ab I C dz πi (1 + z ) N − z N − = 12 δ ab (31)where C is a contour of unit radius around the origin of the z complex plane. This is theexpected contribution for a particle in the fundamental representation of the color group.7t is also simple to consider other occupation numbers in the particle model to seehow the color charge corresponding to the various antisymmetric tensor products of thefundamental representation contributes to the self energy, and in particular to the one-loop beta function. One must set the Chern-Simons coupling to the value s = n − N ,with n = 0 , , ..., N , to select the other wave functions appearing in the expansion (6).One finds that the integral in (31) is changed to I C dz πi (1 + z ) N − z N − n = ( N − N − n − n − < n < N , while it vanishes for n = 0, as there is no residue, and for n = N , as thereis no pole. The latter two values are obviously correct, as the particle is colorless for suchcases. One may verify directly the color/anticolor duality, given by changing n → N − n .Of course, if one is interested in describing particles in other representations of thecolor group, one may just insert in (4) the generators in the chosen representation R , sayof dimension d R , take the variables c in that representation and ¯ c in its complex conjugate,and the previous analyses goes through just by substituting in the Chern-Simons couplingthe dimension N of the fundamental representation by d R . In this section we extend the previous construction to spin 1 / N = 1 local supersymmetry on the worldline [6]. The essential ingredient isthe supercharge Q , that gives rise to the Dirac equation as quantum constraint, while thehamiltonian H is completely determined by the supersymmetry algebra. For reducing toan irreducible representation of the color group we add an additional constraint J . Wepresent the massless case first, and then add a mass term by dimensional reduction.The massless model contains as degrees of freedom the particle coordinates and mo-menta x µ and p µ together with real Grassmann variables ψ µ that reproduce space-timespin degrees of freedom. The color structure of the particle is provided in exactly thesame way as before by adding the ¯ cc sector with its coupling to the U(1) gauge field a .In minkowskian time the phase space action takes the form S ph [ x, p, ψ, c, ¯ c, e, χ, a ; A ] = Z dτ h p µ ˙ x µ + i ψ µ ˙ ψ µ + i ¯ c α ˙ c α − e H − iχ Q − a J i (33)where H, Q and J are constraint functions that are set to vanish by the worldline gaugefields e, χ and a . The supercharge Q is defined by Q = ψ µ π µ = ψ µ (cid:0) p µ − g A aµ ( x ) ¯ c α ( T a ) αβ c β (cid:1) (34)8here π µ are gauge covariant momenta. It is easy to check that at the quantum level thecorresponding constraint produces the gauge covariant massless Dirac equation Q | Ψ i = 0 → D/ Ψ( x ) = 0 . The hamiltonian H is fixed by the supersymmetry algebra, that in terms of Poissonbrackets reads { Q, Q } = − i H → H = π + i g ψ µ ψ ν F aµν ¯ c α ( T a ) αβ c β (35)where the non-abelian field strength is given by F aµν = ∂ µ A aν − ∂ ν A aµ + g f abc A bµ A cν , with f abc the structure constants of the gauge group. At this stage the spinor wave functiontransforms in all possible tensor products of the fundamental representation, so that bytaking the U(1) current in the form J = ¯ c α c α − s , (36)that is the fermion number for the c and ¯ c variables modified by the Chern-Simons coupling s = n − N , it implements the required projection to the sector with occupation number n . This completes the construction at the hamiltonian level.We can now eliminate momenta from (33) by means of their equations of motion,perform a Wick rotation to euclidean time, and obtain the euclidean action S = Z dτ h e ( ˙ x µ − χψ µ ) + ψ µ ˙ ψ µ + ¯ c α ( ∂ τ + ia ) c α − isa − igA aµ ( x ) ˙ x µ ¯ c α ( T a ) αβ c β + ie g ψ µ ψ ν F aµν ( x ) ¯ c α ( T a ) αβ c β i , (37)which can be used in a path integral on a circle S to compute the QFT effective actioninduced by a quark loopΓ[ A ] = Z S DxDψD ¯ cDcDeDχDa Vol(Gauge) e − S . (38)The gauge fixing procedure goes along as in the scalar case, together with the gaugefixing of the local supersymmetry obtained by requiring χ = 0 . One may also introducea mass term, without breaking the supersymmetry algebra, by dimensional reduction ofthe massless theory in one dimension higher. The net effect is that, in the gauge chosen,one only finds an additional term e − m T . Thus, the gauge fixed path integral reduces toΓ[ A ] = Z ∞ dTT e − m T Z π dφ π e isφ Z PBC Dx Z ABC Dψ Z TBC D ¯ cDc e − S gf (39) The gravitino is antiperiodic and does not develop a modulus, and the associated Faddeev-Popovdeterminant is trivial. S gf = Z dτ h T ˙ x + ψ µ ˙ ψ µ + ¯ c α ˙ c α + ig A aµ ( x ) ˙ x µ ¯ c α ( T a ) αβ c β − iT g ψ µ ψ ν F aµν ( x ) ¯ c α ( T a ) αβ c β i . (40)The overall normalization of the effective action is fixed by comparing with QFT results,and includes the sign for the loop of a fermionic particle. In order to compute n -point amplitudes from (39) we proceed in the usual way by spe-cializing the background gauge field A aµ to a sum of plane waves. A complication in thespinor case is due to the non-abelian field strength appearing in (40), that yields twokind of vertices, a first one with one gluon and a second one with two gluons. The fullone-gluon vertex is given by V spin [ k, ε, a ] = ( T a ) αβ Z dτ h ε · ˙ x ( τ ) − iT k · ψ ( τ ) ε · ψ ( τ ) i ¯ c α ( τ ) c β ( τ ) e ik · x ( τ ) , (41)and the two-gluon contact vertex by V [ k , ε , a ; k , ε , b ] = − iT f abc ( T c ) αβ Z dτ ε · ψ ( τ ) ε · ψ ( τ ) ¯ c α ( τ ) c β ( τ ) e i ( k + k ) · x ( τ ) . (42)Again, we only present the explicit calculation of the two-point function, i.e. thespinor contribution to the gluon self-energy. The perturbative calculation is standard, asreviewed in the previous section but with the addition of the ψψ propagator for antiperi-odic fermions that reads h ψ µ ( τ ) ψ ν ( σ ) i = δ µν S ( τ − σ ) S ( τ − σ ) = ǫ ( τ − σ ) (43)where ǫ ( x ) is the sign function obeying ǫ (0) = 0. The free path integral normalizationgains an extra 2 D/ factor coming from the ψ fermions, and as before we use the symbol h ... i to denote the normalized quantum average with respect to the quadratic action. Onecan thus write the gluon self-energy asΓ spin ( k , ε , a ; k , ε , a ) = 2 D/ − Z ∞ dTT e − m T (4 πT ) D/ Z π dφ π e isφ (cid:18) φ (cid:19) N × * ( ig ) Y i =1 V spin [ k i , ε i , a i ] + g V [ k , ε , a ; k , ε , a ] + . (44)10y looking at the contact vertex (42), it is immediate to see that the c ¯ c contraction in(44) yields tr F T c = 0, and it is therefore irrelevant for the two-point amplitude. The com-putation of the other piece h V spin [ k , ε , a ] V spin [ k , ε , a ] i proceeds in the very same wayas for the scalar one: the c ¯ c contractions give the same color factor, also one can employthe standard techniques to deal with the bosonic part, and the ψψ Wick contractions arestraightforward. Hence, as for the scalar case, we write (44) asΓ spin ( k , ε , a ; k , ε , a ) = (2 π ) D δ D ( k + k ) C a a ε µ ε ν Π µν ( k ) , (45)where the color factor C ab is the same as in the scalar case, eq. (26), while the vacuumpolarization Π µν ( k ) = ( δ µν k − k µ k ν ) Π( k ) is now given byΠ( k ) = − D/ − g (4 π ) D/ Z ∞ dTT e − m T (4 πT ) D/ T Z dτ Z dτ e − k T G (cid:16) ˙ G − S (cid:17) . (46)One can fix τ = 0 and τ = u , giving S ( u ) = , and performing the T integral producesΠ( k ) = 2 D/ g (4 π ) D/ Γ (cid:16) − D (cid:17) Z du u (1 − u ) h m + u (1 − u ) k i D − (47)that is again easily regulated in dimensional regularization with D = 4 − ε toΠ( k ) = 1 ε g π − g π Z du u (1 − u ) ln M ( u, k ) µ + O ( ε ) (48)where M ( u, k ) = m + k u (1 − u ). Acknowledgments
We wish to thank Christian Schubert for useful comments on the manuscript. The work ofFB was supported in part by the MIUR-PRIN contract 2009-KHZKRX. The work of OCwas partly supported by the UCMEXUS-CONACYT grant CN-12-564. EL acknowledgespartial support of SNF Grant No. 200020-131813/1.
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