Lecture 15: 2-Point Functions in P-space; Introduction to 3-Point Functions

8.821 Lecture 15: 2-Point Functions in P-space ; Introduction to 3-Point Functions 1 Introduction In today’s lecture we will continue to carry out the calculation of hOOi in momentum space. Along the way, we will deal with some issues that arise when we work with a real time coordinate (i.e. with a Lorentzian metric), which shows up for modes with k2 < 0. After a short discussion of which scaling dimensions of operators (namely Δ) are allowed from the bulk point of view, we will start our discussion on 3-point functions. The following material will be relevant: 1. d’Hoker-Freedman, hep-th/0201253, §8. 2. Klebanov-Witten, hep-th/9905104. 2 Some Real-time Issues We continue considering the equation, 0 = [z D+1 ∂z (z −D+1 ∂z ) − m2 L2 − z 2 k2 ]fk (z) (∗) from last lecture. If k2 > 0, that is spacelike (or Euclidean), the solution would be, fk (z) = Ak z D/2 Kν (kz) + AI z D/2 Iν (kz) where ν = Δ − D/2 = horizon as, p (D/2)2 + m2 L2 . Since each Bessel function shows definite behavior at the Iν (z) ∼ ekz Kν (z) ∼ e−kz we see that the regularity in the interior uniquely fixes fk and hence the bulk-to-boundary prop­ agator. Actually there is a theorem (the Graham-Lee theorem) exactly addressing this issue for 1 gravity fields which states that if you specify a Euclidean metric on the boundary of a Euclidean AdS (which in their formalism, would be a S D ) modulo conformal rescaling, the metric for the space inside of the S D , whose topology would be that of AdS, would be uniquely determined.1 A similar result holds for gauge fields. ~2 In contrary to this, in Lorentzian signature with timelike k2 , i.e. on-shell states with ω 2 > pk , there exist many normalizable solutions with the same leading z Δ− behavior. If we define q = ω 2 − ~k2 , K±ν (iqz) ∼ e±iqz (z → ∞) so these modes behave like oscillating modes near the poincare horizon. A better basis of writing the solution (one that is more convenient in dealing with this) is, fk (z) = A1 z D/2 Yν (qz) + A2 z D/2 Jν (qz) where the two terms behave at z → 0 as z Δ− (φ0 ) + z Δ+ and z Δ+ respectively. These are purely “normalizable.” The ambiguity of the propagator for k2 < 0, is equivalent to the ambiguity in the coefficient A2 . We see that now we can have different subleading terms in the propagator. But actually this is to be expected, as this corresponds to different choices of quantum states of the boundary QFT (Balasubramanian, Kraus, Lawrence and Trivedi, hep-th/9808017.) This is actually exactly the same statement as saying that adding different homogeneous solutions to propagators in a free QFT corresponds to the correlator of different states. As usual, the Wick rotation from (1) the Euclidean answer Kν (kz) gives the time-ordered Feynman propagator Hν (qz). One last thing we must deal with before proceeding is to define what we mean by a ‘normalizable’ mode, or solution, when we say that we have many normalizable solutions for k2 < 0 with a given scaling behavior. In Euclidean space, φ is normalizable P when S[φ] < ∞. This is because when we are thinking about the partition function Z[φ] = φ e−S[φ] , modes with boundary conditions which force S[φ] = ∞ would not contribute. In real time, we say φ is normalizable if E[φ] < ∞ where, Z Z √ µ ν D−1 E[φ] = d xdz hTµν [φ]n ξ = x0 =constant Σ √ dD−1 xdz hT 00 [φ] where Σ is a given spatial slice, h is the induced metric on that slice, nµ is a normal unit vector to Σ and ξ µ is a timelike killing vector. TAB is defined as, δ 2 SBulk [φ] TAB [φ] = √ g δgAB 3 Bulk to Boundary Propagator in Position Space We return to considering spacelike k in this section. The normalized solution at z = ǫ is given by the condition fk (z = ǫ) = 1. This means that the δ function in x space is given at z = ǫ, so the 1 There is a subtlety here as the specified metric on the boundary sphere needs to be in some finite region around the round metric. Witten provides some conjectural understanding of this region in terms of the conformal coupling of the N = 4 scalars to the boundary metric on page 11 of hep-th/9802150. 2 bulk to boundary propagator would be different from the one we have obtained in the last lecture. And indeed the position space green’s function obtained, fk (z) = z D/2 Kν (kz) ǫD/2 Kν (kǫ) is different from KW itten obtained previously. The general position space solution can be obtained as, Z [φ0 ] φ (x) = dD ke−ikx fk (z)φ0 (k, ǫ) so the action would be, Z √ η S[φ] = − dD x γφn · ∂φ 2 Z Z Z η D D d x d k1 dD k2 e−(k1 +k2 )x φ0 (k1 , ǫ)φ0 (k2 , ǫ)z −D fk1 (z)z∂z fk2 (z) =− 2 Z η dD kφ0 (k, ǫ)φ0 (−k, ǫ)Fǫ (k) =− 2 and therefore, hO(k1 )O(k2 )iǫc = − δ δ S = (2π)D δD (k1 + k2 )ηFǫ (k1 ) δφ0 (k1 ) δφ0 (k2 ) We note that if you don’t like functional derivatives, you may see this by calculating,   ∂2 W [φ0 (x) = λ1 eik1 x + λ2 eik2 x ] |λ1 =λ2 =0 hO(k1 )O(k2 )iǫc = ∂λ1 ∂λ2 Now Fǫ (k) could be calculated to be, Fǫ (k) = z −D f−k (z)z∂z fk (z)|z=ǫ + (k ↔ −k) = 2ǫ−D+1 ∂z z D/2 Kν (kz) ǫD/2 Kν (ǫz) ! |z=ǫ The near boundary behavior of Kν for integer ν is, Kν (u) =u−ν (a0 + a1 u2 + a2 u4 + · · · ) +uν ln u(b0 + b1 u2 + b2 u4 + · · · ) where the coefficients of the series ai , bi depend on ν. For non-integer ν, there would be no (ln u) multiplied in the second line. ! (kz)−ν+D/2 (a0 + · · · ) + (kz)ν+D/2 ln kz(b0 + · · · ) −D+1 Fǫ (k) = 2ǫ ∂z |z=epsilon (kǫ)−ν+D/2 (a0 + · · · ) + (kǫ)ν+D/2 ln kǫ(b0 + · · · )     2b0 D −D 2 2 4 4 2ν 2 = 2ǫ − ν(1 + c2 (ǫ k ) + c4 (ǫ k ) + · · · ) + ν (ǫk) ln(ǫk)(1 + d2 (ǫk) + · · · ) 2 a0 = (I) + (II) where (I) and (II) denote the first and second group of terms of the previous line. 3 (I) is a Laurent series in ǫ with coefficients which are positive powers of k (i.e. analytic in k at k = 0.) These are contact terms, i.e. short distance ‘goo’ that we can subtract off. We can see this by writing, Z D dD ke−ikx (ǫk)2m ǫ−D = ǫ2m−D �m x δ (x) for m > 0. The ǫ2m−D factor reinforces the notion that ǫ, which is an IR cutoff in AdS is a UV cutoff for the QFT. The interesting bit of F(k) which gives the x1 6= x2 behavior is non-analytic at k = 0. (II) = −η · 2ν · b0 2ν k ln(kǫ) · ǫ2ν−D (1 + O(ǫ2 )), a0 b0 (−1)ν−1 = 2ν for ν ∈ Z a0 2 νΓ(ν)2 The claim is that the fourier transformation of the leading term of (II) is given by, Z 2νΓ(Δ+ ) 1 2Δ− dD ke−ikx (II) = D/2 ǫ π Γ(Δ+ − D/2) x2Δ+ We note that for ν 6∈ Z obtaining the previous result is a lot more transparent as hO(k)O(−k)i ∼ k2ν 1 and hence in position space this would be ∼ x2Δ since there are no logs concerned. Δ− as before the operation, Also, since ǫ2ν−D = ǫΔ− if we let φ0 (k, ǫ) = φRen 0 (k)ǫ δ δ = ǫ−Δ− Ren δφ0 (k, ǫ) δφ0 (k) removes this factor. We also see that for ǫ → 0, the O(ǫ2 ) terms vanish. One last thing we must touch upon is the difference between the prefactors obtained in this lecture and the last, namely, ν hOOi = hOOiW itten Δ We claim that the l.h.s. is the correct correlator to obtain. This is discussed in d’Hoker-Freedman, hep-th/0201253; we’ll come back to this when we talk about three-point functions. 4 Allowed Scaling Behavior at the Boundary Now let’s think about which Δ are attainable in our setting. We’ve seen that in Euclidean space, φ is normalizable if S[φ] < ∞. This depends on the z → 0 behavior of φ. For Z √ SBulk = dD+1 x g(gAB ∂A φ∂B φ + m2 φ2 ) ǫ with √ g = z −D−1 , if we have some φ ∼ z Δ (1 + O(z 2 )) with Δ = Δ+ or Δ− , gzz (∂z φ)2 = (z∂z φ)2 ∼ Δ2 z 2Δ 4 and hence, gAB ∂A φ∂B φ + m2 φ2 ≃ Δ2 z 2Δ + k2 z 2Δ+2 + m2 z 2 = (Δ2 + m2 )z 2Δ (1 + O(z 2 )) in the limit z → 0. Since for Δ = Δ± , Δ2 + m2 = −DΔ = 6 0, Z SBulk [z Δ ] ∼ dzz −D−1 (−DΔ)z 2Δ (1 + O(z 2 )) ∝ ǫ 1 e2Δ−D 2Δ − D We emphasize that only the boundary behavior of φ is defined, and it is not assumed that it satisfies the equation of motion. We see that SBulk [z Δ ] < ∞ ⇔ Δ > D/2 This does not saturate the unitary bound. We note that the calculation of SBulk [z Δ ] can be done by looking at the boundary term, in which case we get, Z √ γφn · ∂z φ ∼ Δe2Δ−D z=ǫ which is off by the same factor in the coefficient mentioned at the end of the last section, namely Δ/ν. Now consider the alternative action by Klebanov-Witten which is, Z Z D+1 √ KW 2 SBulk = d x gφ(−� + m )φ = SBulk − ǫ ∂AdS √ γφn · ∂z φ For this action we see that, KW SBulk [φ Δ 2 ∼ z (1 + O(z ))] = ∼ Z Zǫ ǫ dzz −D−1 z Δ (1 + O(z 2 ))[(−Δ(Δ − D) + m2 )z Δ (1 + O(z 2 )) + k2 z 2Δ+2 ] dzz −D−1+2Δ+2 ∼ ǫ2Δ−D+2 < ∞ is equivalent to D−2 2 which is exactly the unitary bound. We see that in this case both Δ± give normalizable modes for ν ≤ 1. Note that it is actually Δ− that gives the lowest value in the unitray bound, that is when,   s  2 D D−2 D = + m 2 L2  Δ− =  − 2 2 2 2 Δ≥ m2 =1− D4 The coefficient of z Δ+ would be the source in this case. What we have here is a different boundary CFT with the same bulk action, which we have obtained by adding a boundary term to the action. 5 5 3-Point Functions Next we will talk about connected correlation functions of three or more operators. Unlike twopoint functions, such observables are sensitive to the details of the bulk interactions, and we need to make a choice. We will consider the three point functions of the scalar fields coming from the action Z 3 1 √ X dD+1 x g( (∂φi )2 + m2i φ2i + bφ1 φ2 φ3 ) S= 2 i=1 The arguments presented could be easily extended to n-point functions with n > 3. The equations of motion we get is, (� − m2i )φi (z, x) = bφj φk ǫijk We solve this perturbatively to obtain, Z i φ (z, x) = dD x1 K Δi (z, x; x1 )φi0 (x1 ) Z Z Z Δi ′ ′ D ijk D ′ ′√ +bǫ d x dz gG (z, x; z , x ) d x1 dD x2 K Δj (z ′ , x′ ; x1 )K Δk (z ′ , x′ ; x2 )φj0 (x1 )φk0 (x1 ) +O(b2 φ30 ) where GΔi (z, x; z ′ x′ ) is the bulk-to-bulk propagator given by the condition, 1 (� − m2i )GΔi (z, x; z ′ , x′ ) = √ δ(z − z ′ )δD (x − x′ ) g so that (� − for a source J. m2i ) √ Z gGJ = J The first and second terms would be obtained from the Witten diagrams, figure 1 and 2. A typical higher-order diagram would look something like figure 3. φ 0j (x 1) φ 0i (x 1) Ki (z,x) K G i j (z’,x’) K (z,x) Figure 1: Witten diagram 1 k φ k0 (x 2) Figure 2: Witten diagram 2 6 Figure 3: Witten diagram 3