Graviton self-energy from worldlines
GGraviton self-energy from worldlines
Fiorenzo Bastianelli and Roberto Bonezzi
Dipartimento di Fisica ed Astronomia, Universit`a di Bologna andINFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy
Abstract.
Worldline approaches, when available, often simplify and make more efficient thecalculation of various observables in quantum field theories. In this contribution we first reviewthe calculation of the graviton self-energy due to a loop of virtual particles of spin 0, 1/2 and1, all of which have a well-known worldline description. For the case of the graviton itself, anelegant worldline description is still missing, though one can still describe it by constructing aworldline representation of the differential operators that arise in the quadratic approximationof the Einstein-Hilbert action. We have recently analyzed the latter approach, and we use ithere to calculate the one-loop graviton self energy due to the graviton itself in this formalism.
1. Introduction
The worldline path integral formulation of quantum field theories provides an alternative efficientmethod for computing Feynman diagrams, especially in the one-loop approximation, see ref. [1]for a review. The worldline method was of particular interest to Victor, who has given variouscontributions to the subject. In particular, in [2] he contributed in analyzing the consequencesof the one-loop graviton-photon mixing in a constant electromagnetic background [3]. TheFeynman diagram describing the process is drawn in figure 1, and constitute a prime example
Figure 1.
One-loop correction to the graviton-photon mixing due to a virtual loop of an electronin a constant magnetic field.where the worldline approach has been used with great efficiency. The final result of that analysiswas that, even for the strong magnetic fields present in our universe, the one-loop contributionamounts to no more than a few percent of the tree level mixing. However, although it isnumerically only a small correction to the tree-level amplitude, unlike the tree-level amplitudethe one-loop conversion of photons into gravitons in a magnetic field leads to dichroism, andsurprisingly for the relevant range of parameters is even the largest standard model contributionto dichroism, as shown in [4]. a r X i v : . [ h e p - t h ] A p r kin to the diagram of the graviton-photon mixing, there are the graviton self-energydiagrams that take place in flat spacetime (without the need of additional background fields).Those that are due to virtual particles of spin 0, 1/2 and 1 may be depicted as in figure 2,and can be computed successfully with worldline methods. Indeed, a worldline description of Figure 2.
One-loop correction to the graviton self-energy due to particles of spin 0, 1/2 and 1circulating in the loop.particles of spin s ≤ Figure 3.
Graviton self-energy at one-loop due to a virtual graviton.quantization is still missing. One may try to use the O (4) spinning particle of refs. [5, 6], thatindeed describes the propagation of the graviton in flat space. The problem is that it is notknown how to couple that model to an external gravitational background, and the best onecould do so far is to couple it to AdS spaces [7] and to conformally flat spaces [8].Given this state of affairs, a less elegant but viable option is that of constructing a worldlinerepresentation of the differential operators that arise in the quadratic approximation of theEinstein-Hilbert action. We have recently analyzed the latter approach in [9], and we review ithere. Then, we use it to compute the one-loop graviton self-energy due to the graviton itselfwith worldline methods.
2. The worldline formalism in gravitational backgrounds
The case of a virtual scalar particle contributing to the graviton self-energy is the simplest one,but contains already all the essential elements that enter in a worldline description of the process.It was treated in [10] to exemplify the worldline formalism in a gravitational background. Toreview that, one may start recalling that the action of a relativistic scalar particle in a spacetimeof D dimensions is proportional to the length of its worldline S [ x ] = − m (cid:90) ds (1)here ds = dτ (cid:112) − ˙ x µ ˙ x µ with x µ ( τ ) a generic worldline parametrized by τ and ˙ x µ = dx µ dτ . Thesquare root appearing in the action can be avoided if one uses an einbein e ( τ ) for the intrinsicgeometry of the worldline. The action takes the form [11] S [ x, e ] = (cid:90) dτ
12 ( e − ˙ x µ ˙ x µ − em ) (2)which has the advantage over (1) of having a smooth massless limit. This action is otherwiseessentially equivalent to (1), as the equation of motion of the einbein, e − ˙ x + m = 0, can besolved for m (cid:54) = 0 by taking a positive square root to give e = √− ˙ x m . The latter plugged backinto (2) produces the original action in (1). This model has a reparametrization invariance, thatmust be gauge-fixed upon quantization. One can show that: i) quantizing the model in flat spaceand choosing the topology of the worldline to be that of a segment gives the propagator of thefree Klein-Gordon field, ii) quantizing the model on a closed worldline gives instead its one-loopeffective action. The complication of the gauge fixing procedure, and the related emergence ofan integration over a modular parameter, can be short cut as the answer must be the same asthat obtained long ago by Schwinger with his proper-time representation of the propagator andone-loop effective action [12].Let us consider the one-loop effective action in a curved background with metric g µν . Weproceed with euclidean conventions obtained after a Wick rotation to euclidean times. Theclassical particle action (2), Wick rotated and coupled to the background metric g µν , is S [ x, e ] = (cid:90) dτ (cid:0) e − g µν ( x ) ˙ x µ ˙ x ν + e ( m + ξR ( x )) (cid:1) (3)where an arbitrary non-minimal coupling to the scalar curvature R , with coupling constant ξ ,has been naturally included (it has the same dimension of the mass term). Taking the worldlineto be a closed loop, i.e. having the topology of the circle S , and performing the path integralquantization with the related gauge-fixing procedure, one obtains the one-loop effective actionΓ[ g ] in terms of a standard Feynman path integral with an additional integration over the Fock-Schwinger proper time T Γ[ g ] = − (cid:90) ∞ dTT (cid:90) S D x e − S gf [ x ] (4)where S gf [ x ] = (cid:90) dτ (cid:18) T g µν ( x ) ˙ x µ ˙ x ν + T ( m + ξR ( x )) (cid:19) (5)is the action (3) evaluated in the gauge e ( τ ) = 2 T . The path integration is over functions on thecircle, i.e. periodic functions. Up to the normalization appropriate for a real (uncharged) scalarparticle, and up to the integration over the proper time T , one finds a path integral identical tothat of a non-relativistic particle (of mass M = T ) in a curved space (i.e. a one-dimensionalnonlinear sigma model) with additional scalar potential interactions.Thus, one is left with a path integration over the coordinates x µ . This is non-trivial asthe path integral for the nonlinear sigma model in (5) needs a regularization for making itwell-defined. The regularization must be introduced to fix certain ambiguities that arise in theperturbative calculation of the path integral. These ambiguities take the form of products ofdistributions, which are ill defined in the absence of a regularizing procedure. They are akinto the ordering ambiguities of the canonical quantization of the model. Such path integralsfor nonlinear sigma models have been used to evaluate trace anomalies for quantum fieldheories in 2, 4 and 6 dimensions [13, 14, 15], and in that context three different regularizationshave been analyzed and applied: mode regularization (MR) [13, 16, 17] time slicing (TS)[18, 19], and dimensional regularization (DR) [20]. The DR regularization with the correctcovariant counterterm was developed after the results of [21] and [22], which dealt with nonlinearsigma models in the infinite propagation-time limit. All these regularizations require differentcounterterms to produce the same physical results. The counterterms needed for nonlinear sigmamodels with N extended supersymmetry have been worked out in [23] for all the regularizationschemes mentioned above. They are needed for treating particles of spin N , and used for thatpurpose in [24]. For the present applications only the cases N = 0 , , D x = Dx (cid:89) ≤ τ< (cid:113) det g µν ( x ( τ )) , Dx = (cid:89) ≤ τ< d D x ( τ ) (6)and can be represented more conveniently by introducing bosonic a µ and fermionic b µ , c µ ghosts D x = Dx (cid:90) DaDbDc e − S gh [ x,a,b,c ] (7)where S gh [ x, a, b, c ] = (cid:90) dτ T g µν ( x )( a µ a ν + b µ c ν ) (8)so that all measures are translational invariant, a property useful for developing the perturbativeexpansion around a free gaussian theory. Upon regularization one may note that vertices arisingfrom the ghost action (8) help to cancel potential infinities, so that the counterterms of thevarious regularization schemes are finite. Details on the construction and applications of thesepath integrals in curved space may be found in the book [25].The path integral in (4) is on periodic functions (functions on S ), so that it naturallycomputes a trace in the Hilbert space of the particle (cid:90) S D x e − S gf [ x ] = Tr e − T ˆ H (9)where the quantum hamiltonian ˆ H arises from the canonical quantization of S gf , and isrepresented by the Klein-Gordon operator ( − (cid:3) + m + ξR ). This way one recognizes (4) asthe Schwinger formula for the euclidean one-loop effective action Γ[ g ]Γ[ g ] = 12 Tr log( − (cid:3) + m + ξR ) = − (cid:90) ∞ dTT Tr e − T ( − (cid:3) + m + ξR ) (10)obtained by quantizing the action of a real Klein-Gordon field φ coupled to gravity S [ φ, g ] = (cid:90) d D x √ g
12 ( g µν ∂ µ φ∂ ν φ + m φ + ξRφ ) (11)through a field theoretical path integral ( e − Γ[ g ] = (cid:82) D φ e − S [ φ,g ] = Det − ( − (cid:3) + m + ξR )). . The graviton self-energy due to particles of spin s ≤ h µν = g µν − δ µν .For definiteness, we adopt the DR scheme which requires the covariant counterterm∆ S DR [ x ] = − (cid:90) dτ T R ( x ) . (12)Collecting all terms, the formula for the gravitational effective action induced by a spin 0 particlein the worldline representation is given byΓ[ g ] = − (cid:90) ∞ dTT (cid:90) DxDaDbDc e − S (13)with S = (cid:90) dτ (cid:18) T g µν ( ˙ x µ ˙ x ν + a µ a ν + b µ c ν ) + T ( m + ¯ ξR ) (cid:19) (14)where ¯ ξ = ξ − includes the DR counterterm. This path integral representation may beexpanded to second order in the metric fluctuation h µν = g µν − δ µν to obtain the contributionto the graviton self-energy. One obtains directly its Fourier transform in momentum space bysubstituting in (14) g µν = δ µν + h µν with the fluctuations given by the sum of two plane waves h µν ( x ) = (cid:88) i =1 (cid:15) ( i ) µν e ik i · x (15)and picking up the terms linear in each polarization (cid:15) ( i ) µν . We denote the resulting contributionto the self-energy by (2 π ) D δ D ( k + k )Γ ( k ,k ) , anticipating momentum conservation.The detailed calculation may be found in [10], though the salient points are the followingones. One considers first the case with ¯ ξ = 0. From expanding the metric in the kinetic termone finds vertex operators for the emission (or absorption) of one graviton of the form V [ (cid:15), k ] = (cid:90) dτ T (cid:15) µν ( ˙ x µ ˙ x ν + a µ a ν + b µ c ν ) e ik · x (16)and the self-energy is obtained by calculating the correlation functions of two such operatorsin the free gaussian theory with g µν = δ µν . The zero mode (i.e. the constant part) of thequantum variables x µ ( τ ) can be separated from the path integral, and its integration producesimmediately momentum conservation. Subtleties for factoring out the zero mode in curved spacehave been discussed extensively in [26], however those issues are not crucial and can be avoidedhere as the calculation at the end is performed in the flat space limit. Thus, the remaining pathintegration can be carried out using Wick contractions to obtainΓ ( k, − k ) = −
18 1(4 π ) D (cid:90) ∞ dTT D e − m T (cid:0) R I + R I − T k ( R I − R I ) + 4 T k R I (cid:1) (17)where k = k = − k and R i = (cid:15) (1) µν R µναβi (cid:15) (2) αβ with R µναβ = δ µν δ αβ , R µναβ = δ µα δ νβ + δ µβ δ να R µναβ = 1 k ( δ µα k ν k β + δ να k µ k β + δ µβ k ν k α + δ νβ k µ k α ) R µναβ = 1 k ( δ µν k α k β + δ αβ k µ k ν ) , R µναβ = 1 k k µ k ν k α k β (18)hile the integrals in the correlation functions are calculated in DR to the values I = (cid:90) dτ e − T k ( τ − τ ) , I = 14 T k − I , I = 18 − T k (1 − I ) I = 12 T k (1 − I ) , I = 18 T k − T k (1 − I ) . (19)The calculation is valid for arbitrary spacetime dimension D , which may be extended to complexvalues in view of spacetime renormalization. Carrying out the proper time integral one finds(4 π ) D Γ ( p, − p ) = −
18 Γ( − D ) (cid:16) ( K ) D ( R + R − R − R + 3 R ) − ( m ) D (2 R − R − R + 3 R ) (cid:17) −
132 Γ(1 − D ) k ( m ) D − ( R − R + 2 R ) (20)where we have used the definition( K ) x = (cid:90) dτ ( m + k ( τ − τ )) x . (21)Additional terms ∆Γ ( k, − k ) are present in the case with ¯ ξ (cid:54) = 0. First, there are corrections tothe vertex operator for the emission of a single graviton arising from expanding to the linearorder the scalar curvature in (14). Secondly, there is a vertex operator for the emission of twogravitons obtained by expanding the scalar curvature to second order. Their inclusion producesthe following additional contribution to (20)(4 π ) D ∆Γ ( k, − k ) = − ¯ ξ − D ) k (cid:16) ( m ) D − (2 R + R − R − R + 4 R ) − K ) D − ( R − R + R ) (cid:17) − ¯ ξ − D ) k ( K ) D − ( R − R + R ) . (22)This gives the final regulated (but not renormalized) self-energy contribution from spin 0.The final two-point function can be written in a more compact form using the tensors S = R − R + R , S = R − R + 2 R (23)which satisfy k µ S µναβ = k µ S µναβ = 0 (when the polarizations are stripped off), and make iteasier to check the gravitational Ward identities. For arbitrary ¯ ξ it readsΓ (0) ( k, − k ) = − Γ( − D )8(4 π ) D (cid:16) ( m ) D ( R − R ) + (( K ) D − ( m ) D )( S + S ) (cid:17) − Γ(1 − D )32(4 π ) D (cid:104) k ( m ) D − S + 4 ¯ ξk (cid:16) ( m ) D − (2 S + S ) − K ) D − S (cid:17)(cid:105) − Γ(2 − D )2(4 π ) D ¯ ξ k ( K ) D − S . (24)The value ¯ ξ = − ( ξ = 0) describes the self-energy from a scalar with minimal coupling. Aconformally coupled scalar needs instead the value ¯ ξ = − D ) (i.e. ξ = ( D − D − ) together with = 0. Finally, the value ¯ ξ = 0 ( ξ = ) allows for the simplest computation in the worldlineformalism as vertex operators may contain one graviton only.For a Dirac particle of spin 1/2 one proceeds in a similar way, using the spinning particle of[11]. From the worldline point of view this amounts to supersymmetrize the previous result for¯ ξ = 0, and one finds [27]Γ ( ) ( k, − k ) = 2 D π ) D (cid:104) Γ( − D ) (cid:16) ( m ) D ( R − R − S − S ) + ( K ) D ( S + S ) (cid:17) + 14 Γ(1 − D ) k ( K ) D − S (cid:21) . (25)Similarly, for a spin 1 particle one may use the N=2 extended spinning particle model. Ithas been used in [28] (see also [29]) to find the contribution from massless and massive p -formsin arbitrary dimensions. In D = 4 the only new result with respect to the previous ones (up todualities) is that of a particle of spin 1, whose contribution readsΓ (1) ( k, − k ) = N dof Γ (0) ( k, − k ) − π ) D ( S − S ) × (cid:20) Γ(1 − D ) k (cid:16) K ) D − − ( m ) D − (cid:17) + 12 Γ(2 − D ) k ( K ) D − (cid:21) (26)where Γ ( k, − k ) is the two-point function due to a minimally coupled scalar ( ¯ ξ = − , i.e. ξ = 0),while N dof = 2 for a massless spin 1 particle and N dof = 3 for a massive one.These worldline results agrees with those computed with standard Feynman rules, though itmay be noticed how the worldline computation produces simpler and more compact expressions.Early calculations of the graviton self-energy may be found in [30], [31], [32], where one finds thecontributions due to a massive scalar with minimal coupling, a massless fermion, and the photon,respectively. In [33] one finds a calculation of the graviton self-energy due to the graviton itself,which is the most tricky one to obtain with worldline methods.
4. Graviton self-energy due to spin 2 loop
In this section we are going to review the model developed in [9] to produce one-loop quantumgravity amplitudes in four dimensions, and present the computation of the graviton self-energyinduced by a graviton loop. As mentioned in the introduction, a worldline description ofgravitons is not easy to achieve. The O (4) extended spinning particle indeed describes thefree propagation of a spin two particle in conformally flat spacetimes, but its coupling to generalcurved backgrounds is prevented by obstructions in the constraint algebra defining the model[8]. A viable alternative, though less elegant, to reproduce one-loop graviton contributions hasbeen developed in [9]: its starting point is the quadratic expansion, in background field method,of the Einstein-Hilbert action. After fixing the quantum gauge symmetry in Fock-De Dondergauge, the one-loop effective action is given in terms of functional traces asΓ[ g ] = Γ TT + Γ S − V , (27)where the three pieces represent contributions from traceless tensor fluctuations, scalar tracefluctuations and vector ghosts, respectively. Each contribution to the effective action is givenby a functional trace of the form Γ = Tr ln[Π (cid:3) + R ], where Π is a tensor projector for the givenrepresentation and R stands for curvature couplings. By means of the Schwinger exponentiationof logarithms with the proper time, each functional trace in (27) can be represented as a quantummechanical partition function, as reviewed for the scalar in section 2. At this stage this is theell-known heat kernel method developed by DeWitt for obtaining the one-loop effective actionfor quantum gravity [34]. The goal of [9] was to engineer suitable worldline actions able toreproduce the various contributions of (27) as their partition functions, namelyΓ TT = − (cid:90) ∞ dTT e − m T (cid:90) π dφ π e iφ (cid:90) P D x (cid:90) A D ¯ ψDψ e − S TT [ x, ¯ ψ,ψ ; φ ] , Γ V = − (cid:90) ∞ dTT e − m T (cid:90) π dφ π e iφ (cid:90) P D x (cid:90) A D ¯ λDλ e − S V [ x, ¯ λ,λ ; φ ] , Γ S = − (cid:90) ∞ dTT e − m T (cid:90) P D x e − S S [ x ] , (28)where the modular integrals consist of integration over the proper time T and U (1) modulus φ ,with the latter ensuring projection onto the desired sectors of the particle Hilbert spaces , asexplained in detail in [9]. The subscripts A and P stand for (anti)-periodic boundary conditions,and the fictitious mass m is an infrared regulator that will be eventually set to zero. Theworldline actions appearing in (28) read S TT [ x, ¯ ψ, ψ ; φ ] = (cid:90) dτ (cid:104) T g µν ˙ x µ ˙ x ν + 12 ¯ ψ ab ( ∂ τ + iφ ) ψ ab + ω µab ˙ x µ ¯ ψ a · ψ b − T (cid:16) R abcd ψ ac ¯ ψ bd + R ab ψ a · ¯ ψ b + R (cid:17) (cid:105) ,S V [ x, ¯ λ, λ ; φ ] = (cid:90) dτ (cid:104) T g µν ˙ x µ ˙ x ν + ¯ λ a ( ∂ τ + iφ ) λ a + ω µab ˙ x µ λ a ¯ λ b − T (cid:16) R ab λ a ¯ λ b + R (cid:17) (cid:105) ,S S [ x ] = (cid:90) dτ (cid:104) T g µν ˙ x µ ˙ x ν − T R (cid:105) , (29)where the worldline fermions ψ ab ( τ ), ¯ ψ ab ( τ ) are symmetric traceless tensors in spacetime: ψ ab = ψ ba , ψ aa = 0, with a, b, .. being four dimensional flat Lorentz indices, while the fermions λ a ( τ ) and ¯ λ a ( τ ) are spacetime vectors. The above actions, being nonlinear sigma models, requireregularization, and the corresponding DR counterterms found in [9] are already included in (29).At this stage one can expand the background metric in plane waves around flat space in orderto compute contributions to the n -point functions in momentum space. For the self-energy onesets g µν ( x ) = δ µν + (cid:80) i =1 (cid:15) ( i ) µν e ik i · x and keeps the terms linear in (cid:15) (1) (cid:15) (2) . The calculation is verymuch akin to the one performed for the spin one loop, due to the extra angular integrals withrespect to spin one half and spin zero cases, and we shall present only the final result, that isgiven by summing the three contributions as dictated by (27)Γ (2) ( k, − k ) = − π ) D/ (cid:104) Γ( − D ) (cid:16) (cid:0) K (cid:1) D ( S + S ) + ( m ) D ( R − R − S − S ) (cid:17) + k Γ(1 − D ) (cid:16) (cid:0) K (cid:1) D − (4 S − S ) + ( m ) D − S (cid:17) + k Γ(2 − D ) (cid:16) (cid:0) K (cid:1) D − ( S − S ) (cid:17)(cid:105) , (30) This projection mechanism was observed in [29] and applied often in worldline applications, as in [35] forprojecting to some irreducible color degrees of freedom, and in [36, 37] to achieve the description of quantum( p, q )-forms on K¨ahler spaces. Since the vielbein and spin connection are present, one actually expands those in plane waves and, after thecalculation is performed, one goes back to the metric basis. here we have used the same conventions of the previous section for tensor structures. It ispossible now to remove the infrared regulator by sending m →
0. In this limit the factors of K yield Euler beta functions, sincelim m → (cid:0) K (cid:1) x = k x (cid:90) dτ [ τ (1 − τ )] x = k x B ( x + 1 , x + 1) , and we can write the dimensionally regulated (in spacetime) result asΓ (2) ( k, − k ) = − (cid:18) k π (cid:19) D (cid:104) Γ( − D ) B ( D + 1 , D + 1)( S + S )+ Γ(1 − D ) B ( D , D )(4 S − S ) + Γ(2 − D ) B ( D − , D − S − S ) (cid:105) . (31)In order to display the physical result in four dimensions we take D = 4 − ε , and use the massscale µ of dimensional regularization as in the MS scheme (note also that we have kept thegravitational coupling constant absorbed in the polarisation tensor). The final result readsΓ (2) ( k, − k ) = − k π (cid:20)(cid:18) ε − ln k µ (cid:19) (cid:18) S + 2330 S (cid:19) + 41150 S + 724225 S (cid:21) . (32)One can define linearized curvature invariants in momentum space by using the plane waves in(15) R µν ( k ) = R (lin) µν ( (cid:15) e ikx ) R µν (lin) ( (cid:15) e − ikx ) = 18 k (2 S + S ) R ( k ) = R (lin) ( (cid:15) e ikx ) R (lin) ( (cid:15) e − ikx ) = k S (33)so that we can rewrite (32) in a more suggestive form, that makes manifest the relation withthe effective action in configuration spaceΓ (2) ( k, − k ) = − π (cid:20)(cid:18) ε − ln k µ (cid:19) (cid:18) R µν ( k ) + 1120 R ( k ) (cid:19) + 41150 R µν ( k ) + 6011800 R ( k ) (cid:21) , (34)where one can easily recognize the well known logarithmic divergencies of Einstein gravity infour dimensions [38].The relevant log term at large k can be extracted also for the other particles in the loop.From eqs. (24), (25) and (26) one finds the following additional contribution to the gravitonself-energyΓ( k, − k ) = 18 π ln k µ (cid:34) R µν (cid:0) N + 6 N + 12 N (cid:1) + R (cid:16) (1 − ξ + 60 ξ ) N − N − N (cid:17)(cid:35) (35)where N , N and N , are the number of particles of spin 0, and 1, respectively. For simplicity,we have taken all scalars with the same non-minimal coupling ξ and considered the spin particles as Dirac fermions. These particles are all taken massless, as for large enough k themass can be disregarded, so that a massive spin 1 particle would count as a massless one plus aminimally coupled scalar.As a check on (35), one may note that the same expression multiplying ln k µ sits also inhe diverging part, multiplied by − ε as in (34). For conformal fields this expression givesthe conformal anomaly. Indeed, setting ξ = as appropriate for a conformal scalar in fourdimensions, one finds that the two tensor structures combine to form the square of the Weyltensor (one must take into account that the topological Euler density is a total derivative, andvanishes in the plane wave basis considered in (33), so that a term R µνλσ can be expressed infunction of R µν and R ). Thus one can read off the correct conformal anomaly coefficient, theone that depends on the Weyl tensor squared. Acknowledgments
We thank Christian Schubert for valuable comments on the manuscript.
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