University of Massachusetts Amherst ScholarWorks@UMass Amherst
Physics Department Faculty Publication Series Physics
2006 Graviton physics BR Holstein [email protected]
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Recommended Citation Holstein, BR, "Graviton physics" (2006). AMERICAN JOURNAL OF PHYSICS. 278. Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/278
This Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Physics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. arXiv:gr-qc/0607045v1 11 Jul 2006 om,adcosscin o aiu ecin r straigh are reactions methods. various scat helicity for graviton using sections evaluated that cross elec shown familiar and of is product forms, It a into factorized case. be can photon amplitudes familiar the with h neatoso rvtn ihmte r acltdin calculated are matter with gravitons of interactions The eateto Physics—LGRT of Department nvriyo Massachusetts of University rvtnPhysics Graviton mes,M 01003 MA Amherst, ar .Holstein R. Barry eray3 2008 3, February Abstract 1 tromagnetic tforwardly parallel tering 1 Introduction
The calculation of photon interactions with matter is a staple in an intro- ductory (or advanced) quantum mechanics course. Indeed the evaluation of the Compton scattering cross section is a standard exercise in relativistic quantum mechanics, since gauge invariance together with the masslessness of the photon allow the results to be presented in terms of simple analytic forms[1]. On the surface, a similar analysis should be applicable to the interactions of gravitons. Indeed, like photons, such particles are massless and subject to a gauge invariance, so that similar analytic results for graviton cross sections can be expected. Also, just as virtual photon exchange leads to a detailed understanding of electromagnetic interactions between charged systems, a careful treatment of virtual graviton exchange allows an understanding not just of Newtonian gravity, but also of spin-dependent phenomena associated with general relativity which are to be tested in the recently launched gravity probe B[2]. However, despite this obvious parallel, examination of quantum mechanics texts reveals that (with one exception[3]) the case of graviton interactions is not discussed in any detail. There are at least three reasons for this situation: i) the graviton is a spin-two particle, as opposed to the spin-one photon, so that the interaction forms are somewhat more complex, involving symmetric and traceless second rank tensors rather than simple Lorentz four-vectors; ii) there exist few experimental results with which to compare the theo- retical calculations; iii) as we will see later in some processes, in order to guarantee gauge invariance one must include, in addition to the usual Born and seagull diagrams, the contribution from a graviton pole term, involving a triple- graviton coupling. This vertex is a sixth rank tensor and contains a multitude of kinematic forms. However, recently, using powerful (string-based) techniques, which simplify conventional quantum field theory calculations, it has recently been demon- strated that the elastic scattering of gravitons from an elementary target of arbitrary spin must factorize[4], a feature that had been noted ten years pre- viously by Choi et al. based on gauge theory arguments[5]. This factorization
1 permits a relatively painless evaluation of the various graviton amplitudes. Below we show how this factorization comes about and we evaluate some rel- evant cross sections. Such calculations can be used as an interesting auxiliary topic within an advanced quantum mechanics course In the next section we review the simple electromagnetic case and develop the corresponding gravitational formalism. In section 3 we give the factor- ization results and calculate the relevant cross sections, and our results are summarized in a concluding section 4. Two appendices contain some of the formalism and calculational details.
2 Photon Interactions: a Lightning Review
Before treating the case of gravitons it is useful to review the case of photon interactions, since this familiar formalism can be used as a bridge to our understanding of the gravitational case. We begin by generating the pho- ton interaction Lagrangian, which is accomplished by writing down the free matter Lagrangian together with the minimal substitution[6]
i∂ iD i∂ eA µ −→ µ ≡ µ − µ where e is the particle charge and Aµ is the photon field. As examples, we discuss below the case of a scalar field and a spin 1/2 field, since these are familiar to most readers. Thus, for example, the Lagrangian for a free charged Klein-Gordon field is known to be
= ∂ φ†∂µφ + m2φ†φ (1) L µ which becomes
=(∂ ieA )φ†(∂µ + ieAµ)φ + m2φ†φ (2) L µ − µ after the minimal substitution. The corresponding interaction Lagrangian can then be identified—
= ieA (∂µφ†φ φ†∂µφ)+ e2AµA φ†φ (3) Lint − µ − µ Similarly, for spin 1/2, the free Dirac Lagrangian
= ψ¯(i m)ψ (4) L 6∇ − 2 (a) (b) (c)
Figure 1: Diagrams relevant to Compton scattering. becomes = ψ¯(i e A m)ψ (5) L 6∇ − 6 − whereby the interaction Lagrangian is found to be = eψ¯ Aψ (6) L − 6 The single-photon vertices are then
= e(p + p )µ (7) f | em| i S=0 f i for spin zero and µ µ
∗ ∗ 1 ǫ (pi+ ki + m) ǫi ǫi(pf ki + m) ǫ Amp (S = )= e2u¯(p ) 6 f 6 6 6 6 6 − 6 6 f u(p ) Compton 2 f 2p k − 2p k i i i f i · · (11) Again, one can easily verify that this amplitude is gauge-invariant—
∗ ∗ 1 ǫ (p + k + m) k ki(pf ki + m) ǫ δAmp (S = ) = λe2u¯(p ) 6 f 6 i 6 i 6 i 6 6 − 6 6 f u(p ) Compton 2 f 2p k − 2p k i i · i f · i =u ¯(p ) ǫ∗ ǫ∗ u(p )=0 (12) f 6 f − 6 f i The corresponding cross sections can then be found via standard methods, as shown in many texts[7]. The results are usually presented in the labora- tory frame—pi =(m,~0)—wherein the incident and final photon energies are related by ω ω = i (13) f ωi 2 1 1+2 m sin 2 θ where θ is the scattering angle. For unpolarized scattering, we sum (average) over final (initial) spins via
(p + m) ǫ∗µǫν = ηµν , u(p,s) u¯(p,s) = 6 ij (14) λ λ − i j 2m λ s X X and the resulting cross sections are well known—
2 dσlab(S = 0) α ωf 2 1 2 = 2 ( ) (1 + cos θ) (15) dΩ m ωi 2 and
dσlab(S =1/2) α ωf 2 ωf ωi 2 = 2 ( ) [ + 1 + cos θ)] dΩ 2m ωi ωi ωf − 2 2 2 α ωf 1 2 ωi 2 1 ωi 4 1 = 2 2 [ (1 + cos θ)(1+2 sin θ)+2 2 sin θ] m ωi 2 m 2 m 2 (16)
4 2.1 Helicity Methods For use in the gravitational case it is useful to derive these results in an alternative fashion, using the so-called ”helicity formalism,” wherein one de- composes the amplitude in terms of components of definite helicity[8]. Here helicity is defined by the projection of the particle spin along its momentum direction. In the case of a photon moving along the z-direction, we choose states λi λi ǫ = (ˆx + iλiyˆ), λi = (17) i −√2 ± while for a photon moving in the direction
kˆf = sin θxˆ + cos θzˆ we use states
λf λf ǫ = (cos θxˆ + iλf yˆ sin θzˆ), λf = (18) f −√2 − ± Working in the center of mass frame we can then calculate the amplitude for transitions between states of definite helicity. Using
± ∗± p ǫ pf = ǫ pi = sin θ i · − f · ∓√2 1 1 ǫ∗± ǫ± = (1 + cos θ), ǫ∗± ǫ∓ = (1 cos θ) f · i −2 f · i −2 − we find for spin zero Compton scattering
p2 sin2 θ A = A = e2 1 + cos θ + ++ −− − p k i · f p2 sin2 θ A = A = e2 1 cos θ (19) +− −+ − − − p k i · f While these results can be found by direct calculation, the process can be simplified by realizing that under a parity transformation the momentum reverses but the spin stays the same. Thus the helicity reverses, so parity conservation assures the equality of Aa,b and A−a,−b while under time re- versal both spin and momentum change sign, as do initial and final states, guaranteeing that helicity amplitudes are symmetric—Aa,b = Ab,a.
5 Using the standard definitions
s =(p + k )2, t =(k k )2, u =(p k )2 i i i − f i − f it is easy to see from simple kinematical considerations that
1 1 (s m2)2 1 ((s m2)2 + st) 2 (m4 su) 2 p2 = − , cos θ = − = − 4s 2 s m2 s m2 1− − 1 ( st) 2 sin θ = − (20) 2 (s m2) − We can write then[9] (s m2)2 + st A = A =2e2 − ++ −− (s m2)(u m2) − − m2t A = A =2e2 − (21) +− −+ (s m2)(u m2) − − (It is interesting that the general form of these amplitudes follows from simple kinematical constraints, as shown in ref. [10].) The cross section can now be written in terms of Lorentz invariants as dσ 1 1 = A 2 dt 16π(s m2)2 2 | ij| i,j=± − X (m4 su)2 + m4t2 = 4e4 − (22) 16π(s m2)4(u m2)2 − − and can be evaluated in any desired frame. In particular, in the laboratory frame we have
s m2 = 2mω , u m2 = 2mω − i − − f 1 1 m4 su = 4m2ω ω cos2 θ, m2t = 4m2ω ω sin2 θ (23) − i f 2 − i f 2 Since dt d 2ω2(1 cos θ) ω2 = i − = f (24) dΩ 2πd cos θ −1+ ωi (1 cos θ) π m − the laboratory cross section is found to have the form
2 2 dσlab dσ(S = 0) dt α ωf 4 1 4 1 (S =0)= = 2 2 (cos θ + sin θ) (25) dΩ dt dΩ m ωi 2 2
6 and, using the identity 1 1 1 cos4 θ + sin4 θ = (1 + cos2 θ) 2 2 2 Eq. 25 is seen to be identical to Eq. 15 derived by conventional means. A corresponding analysis can be performed for spin 1/2. Working again in the center of mass frame and using helicity states for both photons and spinors, one can calculate the various amplitudes. In this case it is conve- nient to define the photon as the ”target” particle, so that the corresponding polarization vectors are
λi λi ǫ = ( xˆ + iλiyˆ) i −√2 −
λf ∗ λf ǫ = (sin θxˆ + cos θzˆ iλf yˆ) (26) f −√2 − for initial (final) state helicity λi (λf ). Working in the center of mass, the corresponding helicity amplitudes can then be evaluated via ∗ ∗ ∗ ∗ 2ǫi pi ǫf 2ǫi pf ǫf ǫf ki ǫi ǫi ki ǫf Bsf λf ;siλi =u ¯(pf ,sf )[ · 6 · 6 + 6 6 6 6 6 6 ]u(pi,si) pi ki − pi kf pi ki − pi kf · · · · (27) Useful identities in this evaluation are
∗ pλiλf A + Σ λi(A + Σ) 1 ǫf ki ǫi = O ≡ 6 6 6 2 λi(A + Σ) (A + Σ) − − ∗ pλiλf A Σ λi(A Σ) 2 ǫi ki ǫf = − − − O ≡ 6 6 6 2 λi(A Σ) (A Σ) − − − p sin θλiλf 0 Υ ǫ p ǫ∗ = (28) O3 ≡ i · f 6 f 2 Υ 0 − where
A = λiλf + cos θ Σ = (cos θλ + λ )σ + λ sin θσ i sin θσ i f z i x − y Υ = cos θσ + sin θσ iλ σ (29) − x z − f y We have then E + m B = χ† −2sf p χ† sf λf ;siλi 2m f E+m f 1 1 χi 1 ( 2 + 3) 2sip (30) × s m2 O − u m2 O O χi − − E+m 7 and, after a straightforward (but tedious) exercise, one finds the amplitudes[11] 2 1 2√se p cos 2 θ 1 m 2 1 B 1 1 = B 1 1 = ( + sin θ) 2 1; 2 1 − 2 −1;− 2 −1 m2 u −m s 2 − 2 2e mp 2 1 1 B 1 1 = B 1 1 = B 1 1 = B 1 1 = sin θ cos θ 2 1; 2 −1 − 2 1;− 2 −1 2 −1; 2 1 − 2 −1;− 2 1 −(m2 u)√s 2 2 2 − 2√se p 3 1 B 1 1 = B 1 1 = − cos θ 2 −1; 2 −1 − 2 1;− 2 1 m(m2 u) 2 − 2 2e p 1 2 1 B 1 1 = B 1 1 = B 1 1 = B 1 1 = sin θ cos θ 2 1;− 2 1 − − 2 −1; 2 −1 − 2 1; 2 1 − 2 −1;− 2 −1 −(m2 u) 2 2 2 − 2e p 3 1 B 1 1 = B 1 1 = − sin θ 2 −1;− 2 1 − − 2 1; 2 −1 m2 u 2 −2 2 2e m p 3 1 B 1 1 = B 1 1 = sin θ (31) 2 1;− 2 −1 − − 2 −1; 2 1 −s(m2 u) 2 − Summing (averaging) over final (initial) spin 1/2 states we define spin-averaged photon helicity quantities 2p2e4s sin2 1 θ 1 1 m4 s B 2 = B 2 = 2 1 + cos4 θ + sin4 θ (1 2 ) | ++|av | −−|av m2(m2 u)2 2 2 s2 − m2 − 2p2e4 sin4 1 θ m2 1 m4 1 B 2 = B 2 = 2 2 cos2 θ +(1+ ) sin2 θ (32) | +−|av | −+|av (m2 u)2 s 2 s2 2 − which, in terms of invariants, have the form e4 B 2 = B 2 = (m4 su)(2(m4 su)+ t2) | ++|av | −−|av 2m2(u m2)2(s m2)2 − − − − e4t2(2m2 t) B 2 = B 2 = − (33) | +−|av | −+|av 2(s m2)2(u m2)2 − − The laboratory frame cross section can be determined as before (note that this expression differs from Eq. 25 by the factor 4m2, which is due to the different normalizations for fermion and boson states—m/E for fermions and 1/2E for bosons) dσ(S =1/2) 4m2ω2 = f [ B 2 + B 2 ] dΩ 16π2(s m2)2 | ++|av | +−|av 2 2 − 2 α ωf 1 2 ωi 2 1 ωi 4 1 = 2 2 [ (1 + cos θ)(1+2 sin θ)+2 2 sin θ] m ωi 2 m 2 m 2 (34)
8 which is seen to be identical to the previously form—Eq. 16. So far, all we have done is to derive the usual forms for Compton cross sections by non-traditional means. However, in the next section we shall see how the use of helicity methods allows the derivation of the corresponding graviton cross sections in an equally straightforward fashion.
3 Gravitation
The theory of graviton interactions can be developed in direct analogy to that of electromagnetism. Some of the details are given in Appendix A. Here we shall be content with a brief outline. Just as the electromagnetic interaction can be written in terms of the µ coupling of a vector current jµ to the vector potential A with a coupling constant given by the charge e
= ej aµ (35) Lint − µ the gravitational interaction can be described in terms of the coupling of the µν energy-momentum tensor Tµν to the gravitational field h with a coupling constant κ 1 = κT hµν (36) Lint −2 µν Here the field tensor is defined in terms of the metric via
gµν = ηµν + κhµν (37) while κ is defined in terms of Newton’s constant via κ2 = 32πG. The energy- momentum tensor is defined in terms of the free matter Lagrangian via
2 δ√ g T = − Lint (38) µν √ g δgµν − where 1 √ g = detg = exp trlogg (39) − − 2 is the square root of the determinantp of the metric. This prescription yields the forms
T = ∂ φ†∂ φ + ∂ φ†∂ φ g (∂ φ†∂µφ m2φ†φ) (40) µν µ ν ν µ − µν µ − 9 for a scalar field and 1 1 i T = ψ¯[ γ i←→ + γ i←→ g ( ←→ m)]ψ (41) µν 4 µ ∇ ν 4 ν ∇ µ − µν 2 6 ∇ − for spin 1/2, where we have defined ψi¯ ←→ ψ ψi¯ ψ (i ψ¯)ψ (42) ∇ µ ≡ ∇µ − ∇µ The matrix elements of Tµν can now be read off as
= p p + p p η (p p m2) (43) f | µν | i S=0 fµ iν fν iµ − µν f · i − and 1 1
10 3.1 Graviton Photoproduction Before dealing with our ultimate goal, which is the treatment of graviton Compton scattering, we first warm up with a simpler process—that of gravi- ton photoproduction
γ + s g + s, γ + f g + f → → The relevant diagrams are shown in Figure 2 and include the Born dia- grams accompanied by a seagull and by the photon pole. The existence of a seagull is required by the feature that the energy-momentum tensor is momentum-dependent and therefore yields a contact interactions when the minimal substitution is made, yielding the amplitudes
∗ ∗ 2ǫf ǫiǫf (pf + pi) S =0
= H ǫ∗ ǫ T αβ (S) (53) f f f f | | i i i × fα iβ Compton where H is the factor ∗ ∗ κ ǫ pf kf pi ǫ pikf pf H = f · · − f · · (54) 4e k k i · f ∗ αβ and ǫfαǫiβTCompton(S) is the Compton scattering amplitude for particles of spin S calculated in the previous section. The gauge invariance of Eq. 53
11 (a) (b)
(c) (d)
Figure 2: Diagrams relevant to graviton photoproduction. is obvious, since it follows directly from the gauge invariance already shown for the corresponding photon amplitudes together with that of the factor H under ǫ ǫ + λk . f → f f Also we note that in Eq. 53 the factorization condition mentioned in the introduction is made manifest, and consequently the corresponding cross sections can be obtained trivially. In principle, one can use conventional techniques, but this is somewhat challenging in view of the tensor structure of the graviton polarization vector. However, factorization means that he- licity amplitudes for graviton photoproduction are simple products of the corresponding photon amplitudes times the universal factor H, and the cross sections are then given by the simple photon forms times the universal factor 2 ∗ ∗ H . In the CM frame we have ǫf pi = ǫf ki and the factor H assumes the form · − ·
1 ∗ 2 4 2 κ ǫ kikf pf κ p sin θ s m κ m st H = f · · = − = − (55) | | 4e| ki kf | 4e √2 t 2e 2t · − − In the lab frame this takes the form
2 2 2 1 2 κ m cos 2 θ Hlab = 2 2 1 (56) | | 8e sin 2 θ
12 and the graviton photoproduction cross sections are found to be dσ 1 = Gα cos2 θ dΩ 2 ωf 2 2 1 2 1 2 1 ( ω ) [ctn 2 θ cos 2 θ + sin 2 θ] S =0 i 2 ωf 3 2 1 2 1 2 1 2ωi 4 1 4 1 ωi 2 1 × ( ) [(ctn θ cos θ + sin θ)+ (cos θ + sin θ)+2 2 sin θ] S =1/2 ( ωi 2 2 2 m 2 2 m 2 (57) The form of the latter has previously been given by Voronov.[12]
3.2 Graviton Compton Scattering Finally, we can proceed to our primary goal, which is the calculation of gravi- ton Compton scattering. In order to produce a gauge invariant scattering amplitude in this case we require four separate contributions, as shown in Figure 3. Two of these diagrams are Born terms and can be written down straightforwardly. However, there are also seagull terms both for spin 0 and for spin 1/2, whose forms can be found in Appendix A. For the scalar case we have κ 2
= 2ǫ∗ ǫ (ǫ p ǫ∗ p + ǫ p ǫ∗ p ) f f f f | | i i i i seagull 2 − f · i i · i f · f i · f f · i 1 (ǫ∗ ǫ )2k k (58) − 2 f · i i · f while in the case of spin 1/2 κ 2
= u¯(p ) f f f f | | i i i i seagull 2 f 3 ǫ∗ ǫ (ǫ ǫ∗ (p + p )+ ǫ∗ ǫ (p + p )) × 16 f · i 6 i f · i f 6 f i · i f i + ǫ∗ ǫ ǫρσηλγ γ (ǫ ǫ∗ k ǫ ǫ k ) u(p ) 16 f · i λ 5 iη fσ fρ − fη iσ iρ i (59) Despite the complex form of the various contributions, the final form, which results (after considerable algebra-cf. Appendix B) upon summation of the various components, is remarkably simple: ǫ ǫ M αβ;γδǫ ǫ = F ǫ∗ ǫ T µν (S = 0) ǫ∗ ǫ T αβ (S) fα fβ grav iγ iδ × fµ iν Compton × fα iβ Compton (60)
13 (a) (b)
(c) (d)
Figure 3: Diagrams relevant for gravitational Compton scattering. where F is the universal factor κ2 p k p k F = i · i i · f (61) 8e4 k k i · f and the Compton amplitudes are those calculated in section 2. The gauge in- variance of this form is again obvious from the already-demonstrated gauge invariance of the photon amplitudes and this is the factorized form guar- anteed by general arguments[5]. Again, it is in principle possible but very challenging to evaluate the cross section by standard means, but the result follows directly by the use of helicity methods. From the form of Eq. 60 it is clear that the helicity amplitudes for graviton scattering have the simple form of a product of corresponding helicity amplitudes for spinless and spin S Compton scattering. That is, for graviton scattering from a spinless target we have C 2 = C 2 = F 2 A 4 | ++| | −−| | ++| C 2 = C 2 = F 2 A 4 (62) | +−| | −+| | +−| while for scattering from a spin 1/2 target, we find for the target-spin aver- aged helicity amplitudes D 2 = C 2 = F 2 A 2 B 2 | ++|av | −−|av | ++| | ++|av C 2 = C 2 = F 2 A 2 B 2 (63) | +−|av | −+|av | +−| | +−|av 14 Here the factor F has the form κ2 (s m2)(u m2) F = − − (64) 8e4 t whose laboratory frame value is
κ2m2 1 Flab = 4 2 1 (65) 8e sin 2 θ The corresponding laboratory cross sections are found then to be
dσ ω2 1 lab (S =0) = f F 2 ( C 2 + C 2) dΩ π 16π(s m2)2 | ++| | +−| − ω 1 1 1 = G2m2( f )2[ctn4 θ cos4 θ + sin4 θ] (66) ωi 2 2 2 for a spinless target and
dσ 1 ω2 1 lab (S = ) = f F 2 ( D 2 + D 2 ) dΩ 2 π 16π(s m2)2 | ++|av | +−|av − ω 1 1 1 = G2m2( f )3[ctn4 θ cos4 θ + sin4 θ) ωi 2 2 2 ω 1 1 1 ω2 1 1 + 2 i (ctn2 θ cos6 θ + sin6 θ)+2 i (cos6 θ + sin6 θ)] m 2 2 2 m2 2 2 (67) for a spin 1/2 target. The latter form agrees with that given by Voronov[12]. We have given the results for unpolarized scattering from an unpolarized target, but having the form of the helicity amplitudes means that we can also produce cross sections involving polarized photons or gravitons. That is, however, a subject for a different time and a different paper.
4 Summary
While the subject of photon interactions with charged particles is a standard one in any quantum mechanics course, the same is not true for that of gravi- ton interactions with masses despite the obvious parallels between these two topics. The origin of this disparity lies with the complications associated with
15 the tensor structure of gravity and the inherent nonlinearity of gravitational theory. We have argued above that this need not be the case. Indeed in an earlier work we showed how the parallel between the exchange of virtual gravitons and photons could be used in order to understand the phenomena of geodetic and Lense-Thirring precession in terms of the the related spin-orbit and spin-spin interactions in quantum electrodynamics[2]. In the present pa- per, we have shown how the treatment of graviton scattering processes can benefit from use of this analogy. Of course, such amplitudes are inherently more complex, in that they must involve tensor polarization vectors and the addition of somewhat complex photon or graviton pole diagrams. However, it is remarkable that when all effects are added together, the resulting am- plitudes factorize into simple products of photon amplitudes times kinematic factors. Using helicity methods, this factorization property then allows the relatively elementary calculation of cross sections since they involve simple products of the already known photon amplitudes times kinematical factors. It is hoped that this remarkable result will allow introduction of graviton re- actions into the quantum mechanics cirriculum in at least perhaps a special topics presentation. In any case, the simplicity associated with this result means that graviton interactions can be considered a topic which is no longer only associated with advanced research papers. Appendix A: Gravitational Formalism Here we present some of the basics of gravitational field theory. Details can be found in various references[13, 14]. The full gravitational action is given by 1 S = d4x√ g R + (68) g − 16πG Lm Z where m is the Lagrange density for matter and R is the scalar curvature. VariationL of Eq. 68 via g η + κh µν → µν µν yields the Einstein equation 1 R g R = 8πGT (69) µν − 2 µν − µν where the energy-momentum tensor Tµν is given by 2 ∂ T = (√ g ) (70) µν √ g ∂gµν − Lm − 16 We work in the weak field limit, with an expansion in powers of the gravita- tional coupling G
g η + κh(1) + . . . µν ≡ µν µν gµν = ηµν κh(1)µν + κ2h(1)µλh(1) ν + . . . (71) − λ where here the superscript indicates the number of powers of G which appear and indices are understood to be raised or lowered by ηµν . We shall also need the determinant which is given by 1 1 √ g = exp trlog g =1+ κh(1) + . . . (72) − 2 2 The corresponding curvatures are given by
κ λ λ R(1) = ∂ ∂ h(1) + ∂ ∂λh(1) ∂ ∂ h(1) ∂ ∂ h(1) µν 2 µ ν λ µν − µ λ ν − ν λ µ R(1) = ηµνhR(1) = κ 2h(1) ∂ ∂ h(1)µν i (73) µν − µ ν In order to define the graviton propagator, we must make a gauge choice µν λ and we shall work in harmonic (or deDonder) gauge—g Γµν = 0—which requires, to first order in the field expansion, 1 0= ∂β h(1) ∂ h(1) (74) βα − 2 α Using these results, the Einstein equation reads, in lowest order, 1 1 1 2h(1) η 2h(1) ∂ ∂βh(1) ∂ h(1) ∂ ∂βh(1) ∂ h(1) = 16πGT matt µν −2 µν − µ βν − 2 ν − ν βµ − 2 µ − µν (75) which, using the gauge condition Eq. 74, can be written as 1 2 h(1) η h(1) = 16πGT matt (76) µν − 2 µν − µν or in the equivalent form 1 2h(1) = 16πG T matt η T matt (77) µν − µν − 2 µν Gravitational Interactions: Spin 0
17 The coupling to matter via one-graviton and two-graviton vertices can be found by expanding the spin zero matter Lagrangian
1 1 √ g = √ g D φgµνD φ m2φ2 (78) − Lm − 2 µ ν − 2 via 1 √ g (0) = (∂ φ∂µφ m2φ2) − Lm 2 µ − κ 1 √ g (1) = h(1)µν ∂ φ∂ φ η (∂ φ∂αφ m2φ2) − Lm − 2 µ ν − 2 µν α − κ2 1 √ g (2) = h(1)µλh(1)ν h(1)h(1)µν ∂ φ∂ φ − Lm 2 λ − 2 µ ν κ2 1 h(1)αβ h(1) h(1)2 (∂αφ∂ φ m2φ2) (79) − 8 αβ − 2 α − The one- and two-graviton vertices are then respectively iκ τ (p,p′) = − p p′ + p′ p η (p p′ m2) αβ 2 α β α β − αβ · − ′ 2 ξ ρ ′σ ′ρ σ ταβ,γδ(p,p ) = iκ I αβ,ρξI σ,γδ p p + p p 1 (η I + η I ) p′ρpσ − 2 αβ ρσ,γδ γδ ρσ,αβ 1 1 I η η p p′ m2 (80) − 2 αβ,γδ − 2 αβ γδ · − where we have defined 1 I = (η η + η η ) αβ;γδ 2 αγ βδ αδ βγ
18 µν We also require the triple graviton vertex ταβ,γδ(k, q) whose form is
iκ 1 3 τ µν (k, q) = (I η η ) kµkν +(k q)µ(k q)ν + qµqν ηµν q2 αβ,γδ 2 αβ,γδ − 2 αβ γδ − − − 2 λσ, µν, λσ, µν, λµ, σν, σν, λµ, + 2qλqσ I αβI γδ + I γδI αβ I αβI γδ I αβI γδ µ λν, λν, −ν λµ, − λµ, + [qλq (ηαβI γδ + ηγδI αβ)+ qλq (ηαβI γδ + ηγδI αβ) 2 µν, µν, µν λ σ q (ηαβI γδ + ηγδI αβ) η q q (ηαβIγδ,λσ + ηγδIαβ,λσ)] − λ σν, µ − σµ, ν + [2q (I αβIγδ,λσ(k q) + I αβIγδ,λσ(k q) σν, µ σµ,− ν − I γδIαβ,λσk I γδIαβ,λσk ) − 2 σµ, ν− ν σµ, µν λ ρσ, ρσ, + q (I αβIγδ,σ + Iαβ,σ I γδ)+ η q qσ(Iαβ,λρI γδ + Iγδ,λρI αβ)] 1 + [(k2 +(k q)2) Iσµ, I ν + Iσν, I µ ηµν P − αβ γδ,σ αβ γδ,σ − 2 αβ,γδ (k2η Iµν, +(k q)2η Iµν, )] (81) − γδ αβ − αβ γδ Gravitational Interactions: Spin 1/2
For the case of spin 1/2 we require some additional formalism in order to extract the gravitational couplings. In this case the matter Lagrangian reads
√e = √eψ¯(iγae µD m)ψ (82) Lm a µ − µ and involves the vierbein ea which links global coordinates with those in a locally flat space[15, 16]. The vierbein is in some sense the “square root” of the metric tensor gµν and satisfies the relations
a b a e µe νηab = gµν, e µeaν = gµν aµ a aµ ν µν e ebµ = δb , e ea = g (83)
The covariant derivative is defined via i D ψ = ∂ ψ + σabω (84) µ µ 4 µab where 1 1 ω = e ν (∂ e ∂ e ) e ν(∂ e ∂ e ) µab 2 a µ bν − ν bµ − 2 b µ aν − ν aµ 1 + e ρe σ(∂ e ∂ e )e c (85) 2 a b σ cρ − ρ cσ µ 19 The connection with the metric tensor can be made via the expansion
a a (1)a e µ = δµ + κcµ + . . . (86)
The inverse of this matrix is
e µ = δµ κc(1)µ + κ2c(1)µc(1)b + . . . (87) a a − a b a and we find (1) (1) gµν = ηµν + κcµν + κcνµ + . . . (88) For our purposes we shall use only the symmetric component of the c- matrices, since these are physical and can be connected to the metric tensor. We find then 1 1 c(1) (c(1) + c(1))= h(1) µν → 2 µν νµ 2 µν We have κ det e =1+ κc + . . . =1+ h + . . . 2 and, using these forms, the matter Lagrangian has the expansion i √e (0) = ψ¯( γαδµ←→ m)ψ Lm 2 α ∇ µ − κ κ i √e (1) = h(1)αβ ψiγ¯ ←→ ψ h(1)ψ¯( ←→ m)ψ Lm − 2 α ∇ β − 2 2 6 ∇ − κ2 κ2 √e (2) = h(1)h(1)αβ ψiγ¯ γ ←→ ψ + (h(1))2ψiγ¯ γ ←→ ψ Lm 8 αβ ∇ λ 16 ∇ γ κ2 3κ2 h(1)ψiγ¯ αh λ←→ ψ + h(1)h(1)αµψiγ¯ δ←→ ψ − 8 α ∇ λ 16 δα ∇ µ κ2 κ2 + h(1)h(1)αβ ψmψ¯ (h(1))2ψmψ¯ 4 αβ − 8 iκ2 + h(1)(∂ h(1)ν ∂ h(1)ν )ǫαβδǫψγ¯ γ ψ (89) 16 δν β α − α β ǫ 5
20 The corresponding one- and two-graviton vertices are found then to be
iκ 1 1 1 τ (p,p′) = − (γ (p + p′) + γ (p + p′) ) η ( (p+ p′) m) αβ 2 4 α β β α − 2 αβ 2 6 6 − 1 1 τ (p,p′) = iκ2 ( (p+ p′) m)P αβ,γδ −2 2 6 6 − αβ,γδ 1 [η (γ (p + p′) + γ (p + p′) ) − 16 αβ γ δ δ γ ′ ′ + ηγδ(γα(p + p )β + γβ(p + p )α)] 3 + (p + p′)ǫγξ(I Iφ + I Iφ ) 16 ξφ,αβ ǫ,γδ ξφ,γδ ǫ,αβ i + ǫρσηλγ γ (I νI k′ I νI k ) (90) 16 λ 5 αβ,η γδ,σν ρ − γδ,η αβ,σν ρ Appendix B: Graviton Scattering Amplitudes
In this section we summarize the independent contributions to the various graviton scattering amplitudes which must be added in order to produce the complete amplitudes quoted in the text. We leave it to the (perspicacious) reader to perform the appropriate additions and to verify the factorized forms shown earlier.
Graviton Photoproduction: Spin 0
∗ 2 (ǫf pf ) ǫi pi Born a : Ampa =4eκ · · − 2pi ki ∗ · 2 (ǫf pi) ǫi pf Born b : Ampb = 4eκ · · − − 2pi kf ∗ ∗· Seagull : Ampc = 2eκǫf ǫiǫf (pi + pf ) −eκ · · γ pole : Amp = [ǫ∗ (p + p )(k k ǫ∗ ǫ ǫ∗ k ǫ k ) − d k k f · i f i · f f · i − f · i i · f i · f + ǫ∗ k (ǫ∗ ǫ k (p + p ) ǫ∗ k ǫ (p + p )] (91) f · i f · i i · i f − f · i i · i f
21 Graviton Photoproduction: Spin 1/2
∗ ǫf pf ∗ Born a : Ampa = eκ · u¯(pf )[ǫf (pi+ ki + m) ǫi]u(pi) − 2pi ki 6 6 6 6 ∗· ǫ pi Born b : Amp = eκ f · u¯(p )[ǫ (p k + m) ǫ∗ ]u(p ) − b − 2p k f 6 i 6 i− 6 f 6 f i i · f Seagull : Amp = eκu¯(p ) ǫ∗ u(p ) c − f 6 f i 1 γ pole : Amp = eκ u¯(p )[ǫ∗ (k k ǫ∗ ǫ ǫ∗ k ǫ k ) − d k k f 6 f i · f f · i − f · i i · f i · f + k ǫ∗ ǫ ǫ∗ k ǫ (ǫ∗ k )2]u(p ) (92) 6 f f · i f · i− 6 i f · i i
Graviton Scattering: Spin 0
2 ∗ 2 2 (ǫi pi) (ǫf pf ) Born a : Ampa =2κ · · − pi ki ∗ · 2 2 (ǫ pi) (ǫi pf ) Born b : Amp = 2κ2 f · · − b − p k i · f 1 Seagull : Amp = κ2 ǫ∗ ǫ (ǫ p ǫ∗ p + ǫ p ǫ∗ p ) k k (ǫ∗ ǫ )2 c f · i i · i f · f i · f f · i − 2 i · f f · i 4κ2 g pole : Amp = ǫ∗ p ǫ∗ p (ǫ (p p ))2 + ǫ p ǫ p (ǫ∗ (p + p ))2 − d k k f · f f · i i · i − f i · i i · f f · i f i · f + ǫ (p p )ǫ∗ (p p )(ǫ∗ p ǫ p + ǫ∗ p ǫ p ) i · i − f f · f − i f · f i · i f · i i · f ǫ∗ ǫ ǫ (p p )ǫ∗ (p p )(p p m2) − f · i i · i − f f · f − i i · f − + k k (ǫ∗ p ǫ p + ǫ∗ p ǫ p )+ ǫ (p p )(ǫ∗ p p k + ǫ∗ p p k ) i · f f · f i · i f · i i · f i · i − f f · f i · f f · i f · f + ǫ∗ (p p )(ǫ p p k + ǫ p p k ) f · f − i i · i f · i i · f i · i 1 + (ǫ∗ ǫ )2 p k p k + p k p k (p k p k + p k p k ) f · i i · i f · i i · f f · f − 2 i · i f · f i · f f · i 3 + k k (p p m2)2 (93) 2 i · f i · f −
22 Graviton Scattering: Spin 1/2
∗ 2 ǫf pf ǫi pi ∗ Born a : Ampa = κ · · u¯(pf )[ǫf (pi+ ki + m) ǫi]u(pi) − 8pi ki 6 6 6 6 ∗ · ǫ piǫi pf Born b : Amp = κ2 f · · u¯(p )[ǫ (p k + m) ǫ∗ ]u(p ) − b − 8p k f 6 i 6 i− 6 f 6 f i i · f 3 Seagull : Amp = κ2u¯(p ) ǫ∗ ǫ (ǫ ǫ∗ (p + p )+ ǫ∗ ǫ (p + p )) c f 16 f · i 6 i f · i f 6 f i · i f i + ǫ∗ ǫ ǫρσηλγ γ (ǫ ǫ∗ k ǫ∗ ǫ k ) u(p ) 8 f · i λ 5 iη fσ fρ − fη iσ iρ i κ2 g pole : Amp = u¯(p ) (ǫ ǫ∗ k + ǫ∗ ǫ k )(ǫ p ǫ∗ p ǫ p ǫ∗ p ) − d k k f 6 i f · i 6 f i · f i · i f · f − i · f f · i i · f (ǫ∗ ǫ ) k k (ǫ∗ ǫ k + ǫ ǫ∗ p ) − f · i i · f 6 f i · f 6 i f · i + k (ǫ∗ p ǫ p ǫ∗ p ǫ p )+ p k (ǫ ǫ∗ k + ǫ∗ ǫ k ) 6 i f · f i · i − f · i i · f i · i 6 i f · i 6 f i · f 1 + (ǫ∗ ǫ )2 k (p k k k ) u(p ) (94) f · i 6 i i · i − 2 i · f i
Acknowledgement
This work was supported in part by the National Science Foundation under award PHY02-44801. Thanks to Prof. A Faessler and the theoretical physics group at the University of T¨ubingen, where this work was completed, for hospitality.
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