aa r X i v : . [ h e p - t h ] N ov Gravity and Yang-Mills theory
Sudarshan Ananth
Indian Institute of Science Education and ResearchPune 411021, India
Abstract
Three of the four forces of Nature are described by quantum Yang-Mills theories with re-markable precision. The fourth force, gravity, is described classically by the Einstein-Hilberttheory. There appears to be an inherent incompatibility between quantum mechanics andthe Einstein-Hilbert theory which prevents us from developing a consistent quantum theoryof gravity. The Einstein-Hilbert theory is therefore believed to differ greatly from Yang-Mills theory (which does have a sensible quantum mechanical description). It is thereforevery surprising that these two theories actually share close perturbative ties. This articlefocuses on these ties between Yang-Mills theory and the Einstein-Hilbert theory. We discussthe origin of these ties and their implications for a quantum theory of gravity.ravity, although the oldest force known to man, is the force we understand the least. Theweakness of the gravitational force precludes the possibility of performing simple exper-iments that will teach us more about this omnipresent force. In addition, the fact thatthe gravitational constant κ is dimensionful seems to produce a plethora of problems whenattempting to unite quantum mechanics and general relativity.Apart from gravity there are three other fundamental forces in Nature: the electromagneticforce, the weak force and the strong force. These three forces are all described by quan-tum Yang-Mills theories and theoretical predictions have, with stunning accuracy, matchedexperimental observations. This leads us naturally back to gravity: to wonder why, unlikethe other three forces, it refuses to play well in the quantum mechanical playground. Ex-perimental checks on the general theory of relativity have proved very succesful and thissuggests that the correct quantum theory of gravity will prove to be an “extension” ofEinstein’s theory rather than a replacement.There are many reasons to believe that gravity is indeed very different from the otherthree forces. The Einstein-Hilbert Lagrangian, which describes gravity, differs greatly fromthe Yang-Mills Lagrangian suggesting that the theories they describe should also differdrastically.In this article we paint a picture that conveys quite the opposite impression - a picturewhere Yang-Mills and gravity seem to have far more in common than previously believed.We explore close ties between the theories that go beyond superficial on-shell relations.We begin with a quick review of why the two theories are expected to behave very differently.The Yang-Mills action reads S YM = − Z d x Tr ( F µν F µν ) , (1)with F µν = ∂ µ A ν − ∂ ν A µ + i g [ A µ , A ν ] . (2) A µ represents the gauge field and g the dimensionless Yang-Mills coupling constant. Grav-ity, on the other hand, is governed by the Einstein-Hilbert action S EH = 1 κ Z d x √− g R , (3)where R is the Ricci scalar, g the determinant of the metric and κ the dimensionful gravi-tational coupling constant. Significant differences between the two theories include1. Yang-Mills theory has only cubic and quartic interaction vertices while gravity involvesinfinitely many interaction vertices.2. The coupling constant g is dimensionless while κ has the dimensions of length.3. The trace in Yang-Mills, due to the gauge group, is absent in gravity.Each of these is a fairly substantial difference in its own right. Given these manifest differ-ences, close ties between the two theories are all the more surprising.1he first clear indications of a concrete connection between gravity and Yang-Mills the-ory arose from the Kawai-Lewellyn-Tye (KLT) relations [1]. These relations tells us thattree-level scattering amplitudes in pure gravity are the “square” of tree-level scatteringamplitudes in pure Yang-Mills theory. Thus, rather unexpectedly, graviton scattering isexpressible as a sum of products of pieces of non-abelian gauge theory scattering ampli-tudes [2]. Since scattering amplitudes contain information about the physics of a process,this points to a much stronger relationship between gravity and Yang-Mills theory than onemight anticipate from a cursory glance at the respective Lagrangians.The KLT relations are easy to write down but before doing so, we need to address thedifferences, between the two theories, listed above in points 2 and 3. To this end we define,for Yang-Mills, color-stripped partial amplitudes A tree n = g ( n − Tr( . . . ) × A tree n . (4) A tree n is a tree-level scattering process involving n gluons in pure Yang-Mills theory. Allinformation regarding the color structure is contained within the trace and A tree n now rep-resents the color-stripped partial amplitude. Similarly, in the case of gravity, we define M tree n = (cid:18) κ (cid:19) ( n − × M tree n , (5)where M tree n represents a tree-level gravity scattering process and κ = 32 πG N is the cou-pling in terms of the Newton constant. M tree n is now κ -independent and represents thecoupling-stripped gravity amplitude. We are now in a position to write down the KLTrelations M tree n ∝ (A tree n ) , (6)suggesting the possibility of a much broader and intriguing relation of the formGravity ∼ (Yang-Mills) × (Yang-Mills) . (7)Such a broad relationship however would require far more than simple tree-level connections.If the relationship is indeed deeper it ought to stem from similarities at the off-shell level:in particular, can we relate the two theories directly at the level of the Lagrangians? It turnsout that the answer to this question is at least partially a yes and the rest of this paperwill expand on this point.We start by observing that both the gauge field and the graviton field, in four dimensions,involve exactly two physical degrees of freedom . We make this degrees-of-freedom equalitymanifest in the Lagrangians by working in light-cone gauge where only physical helicitystates are present.The structure of the light-cone gauge Yang-Mills Lagrangian, in momentum space, is L YM ∼ ¯ A p µ A + g f ¯ AAA + g f ¯ A ¯ AA + g f ¯ A ¯ AAA , (8) These degrees of freedom correspond to the positive and negative helicity states under the SO (2) littlegroup in four dimensions. A , A are − , + helicity states of the gauge field, p µ stems from the d’Alembertian inthe kinetic term, g is the Yang-Mills coupling constant and f , f , f represent momentumcoefficients .It turns out, from explicit calculations, that tree-level scattering amplitudes constructedusing the ¯ AAA (helicity − + + ) vertex in (8) vanish [3]. So, when focusing on amplitudecalculations it makes sense to try and eliminate this vertex by means of a field redefinition.Such a canonical change-of-variables, for Yang-Mills theory, was found in [4,5]. The change-of-variables maps the first two terms of the interacting Yang-Mills Lagrangian in (8) into afree Lagrangian in new variables¯ A p µ A + g f ¯ AAA → ¯ B p µ B .
This is achieved by the following canonical redefinition of the gauge field [4, 5] A → B + b B + b B + . . . + b n − B n , ¯ A → ¯ B + c B ¯ B + c B ¯ B + . . . + c n − B n − ¯ B , where b n , c n are functions of the momenta and ¯ B, B represent the “shifted” gauge field.This canonical change-of-variables results in a new form for the Yang-Mills Lagrangian L ′ YM ∼ ¯ B p µ B + g F ¯ B ¯ BB + g F ¯ B ¯ BBB + . . . + g n − F n − ¯ B ¯ BB n , (9)where the F n are the mometum coefficients for the new interaction vertices. The price paidfor eliminating the offending term from (8) is the appearance of infinitely many interactionvertices exactly like in gravity . Note that all interaction vertices in (9) involve exactly twonegative helicity fields .The advantage of the new form of the Yang-Mills Lagrangian is that certain classes ofscattering amplitudes (referred to as MHV) become trivial to compute. For example, theamplitude for A ( −− ++) is obtained trivially by taking the coefficient F from (9) and puttingit on-shell. To compute the same process starting from (8), one would have to deal withnot just the quartic vertex but also contact diagrams arising from combinations of the twocubic vertices.We now have a “close to on-shell physics” form for the Yang-Mills Lagrangian where ampli-tude structures are manifest. Since amplitudes in Yang-Mills are related to those in gravity,it seems natural to look for a similar amplitude-friendly form for the gravity Lagrangian -such a gravity Lagrangian ought to have manifest similarities to (9).We start with the momentum-space Einstein-Hilbert Lagrangian in light-cone gauge [6] L EH ∼ ¯ h p µ h + κ l ¯ hhh + κ l ¯ h ¯ hh + κ l ¯ h ¯ hhh + . . . (10)where ¯ h , h are − , + helicity states of the graviton and l , l , l , . . . represent momentumcoefficients. Notice that this structure, which is similar in some ways to (8), reveals thekey difference highlighted earlier: that Yang-Mills theory involves only a finite numberof interaction vertices (making it renormalizable) while gravity involves infinitely many Integrals over momenta and δ -functions are not shown here explicitly. The apparent disparity in the Lagrangian between + and − is an artifact of the convention-choice [3]. h → C + q C + q C + . . . + q n − C n , ¯ h → ¯ C + r C ¯ C + r C ¯ C + . . . + r n − C n − ¯ C , where q n , r n represent functions of the momenta and ¯ C, C the “shifted” graviton. Thisredefinition sucessfully eliminates the “bad” vertex resulting in the new form L ′ EH ∼ ¯ C p µ C + κ L ¯ C ¯ CC + κ L ¯ C ¯ CCC + . . . + κ n − L n − ¯ C · · · ¯ C C · · · C + . . . (11)where the L n are the momentum coefficients for the new interaction vertices.We are now in a position to compare the two structures in (9) and (11). Up to secondorder in the coupling constants we find that gravity does indeed behave like the “square”of Yang-Mills. For instance L ∼ ( F ) × ( F ) . (12)This is a much stronger statement than a KLT relation since it relates off-shell coefficientsin a Lagrangian as opposed to on-shell amplitudes. Unfortunately, beyond second order inthe coupling new problems crop up. This happens because gravity vertices in (11) involvevarying numbers of fields of both helicities, unlike Yang-Mills where all vertices in (9)involve exactly two negative helicity fields. A complete understanding of the relationshipin (7) could prove invaluable for a detailed finiteness-analysis [8] of gravity or supergravity.* * *In this article, we have reviewed two equivalent forms of the Yang-Mills Lagrangian. Thefirst has a finite number of interaction vertices while the second involves infinitely manysuch vertices. We have also described two equivalent light-cone Lagrangians for gravity,both involving infinitely many vertices. We conclude by asking whether there exists a thirdequivalent form of the gravity Lagrangian that involves only a finite number of interactionvertices . If yes, what implications will such a form of the Lagrangian have for the ultra-violet properties of gravity? Acknowledgments : This work was supported by a Ramanujan Fellowship from the De-partment of Science and Technology (DST), Government of India and by the Max PlanckSociety and DST through the Max Planck Partner Group in Quantum Field Theory. The answer to this question is most likely a No. However, where the search for a quantum theory ofgravity is concerned, concrete avenues of progress are rare and this approach, even if unsuccesful, is certainto teach us more about the UV structure of gravity - this makes it worth pursuing to its logical conclusion. eferences [1] H. Kawai, D.C. Lewellen and S.H.H. Tye, Nucl. Phys.
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