Free BMN Correlators With More Stringy Modes
UUSTC-ICTS/PCFT-21-04
Free BMN Correlators With More Stringy Modes
Bao-ning Du ∗ , Min-xin Huang † Interdisciplinary Center for Theoretical Study,University of Science and Technology of China, Hefei, Anhui 230026, ChinaPeng Huanwu Center for Fundamental Theory,Hefei, Anhui 230026, China
Abstract
In the type IIB maximally supersymmetric pp-wave background, stringyexcited modes are described by BMN (Berenstein-Madalcena-Nastase) opera-tors in the dual N = 4 super-Yang-Mills theory. In this paper, we continuethe studies of higher genus free BMN correlators with more stringy modes,mostly focusing on the case of genus one and four stringy modes in differenttransverse directions. Surprisingly, we find that the non-negativity of torustwo-point functions, which is a consequence of a previously proposed probabil-ity interpretation and has been verified in the cases with two and three stringymodes, is no longer true for the case of four or more stringy modes. Neverthe-less, the factorization formula, which is also a proposed holographic dictionaryrelating the torus two-point function to a string diagram calculation, is stillvalid. We also check the correspondence of planar three-point functions withGreen-Schwarz string vertex with many string modes. We discuss some issuesin the case of multiple stringy modes in the same transverse direction. Ourcalculations provide some new perspectives on pp-wave holography. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] J a n ontents A.1 S contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26A.2 S contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.3 S contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 References 34
The AdS/CFT correspondence [1, 2, 3] is a deep idea which relates two seeminglytotally different theories, namely string theory or supergravity on AdS backgroundand the N = 4 SU ( N ) super-Yang-Mills theory. Although the correspondence hasfound flourishing applications in many topics, the precise quantitative tests of theholographic dictionary are mostly restricted to supersymmetry protected quantitiesin the supergravity approximation, such as the spectrum and correlation functionsof BPS operators. Without an alternative effective method to handle string theoryin the deeply stringy regime, a common perspective is to simply take the super-Yang-Mills theory as a non-perturbative definition of AdS string theory at any finitecoupling and energy scale, assumed to be valid unless otherwise convincingly explicitlycontradicted. 1 particularly interesting avenue for progress in the precise tests of the holographiccorrespondence in the stringy regime is to take a Penrose limit [4] of the type IIB AdS × S background. The geometry becomes a pp-wave background [5] with alsomaximal supersymmetry ds = − dx + dx − − µ ( (cid:126)r + (cid:126)y )( dx + ) + d(cid:126)r + d(cid:126)y , (1.1)where x + , x − are light cone coordinates, (cid:126)r, (cid:126)y are 4-vectors, and the parameter µ isproportional to spacetime curvature as well as the Ramond-Ramond flux F +1234 = F +5678 ∼ µ . The free string spectrum can be solved in the light cone gauge usingGreen-Schwarz formalism similar to the flat space [6]. Berenstein, Maldacena andNastase (BMN) proposed the holographic dual operators in the gauge theory for thestringy states, a type of near-BPS operators known as the BMN operators, and it wasshown that the free string spectrum is reproduced by the planar conformal dimensionsof these BMN operators [7]. On the field theory side, one takes a large R-charge limit,previously considered in the context of giant gravitons, or D-branes in the AdS spacein [8, 9, 10, 11], and also in many subsequent literature e.g. [12, 13, 14, 15]. Thecalculations on the field theory side are perturbative in the large R-charge limit, sothe original strong-weak AdS/CFT duality becomes precisely testable in this setting.The Penrose limit provides a new twist to the holography story. In the celebratedAdS/CFT holographic dictionary in [3], the CFT lives at the boundary of a bulk AdSspace and its local operators couple to the boundary configurations of the AdS bulkfields. However, although the pp-wave background (1.1) comes from a Penrose limit ofthe AdS space, the geometry is rather different. As such, it is not clear how to directlyapply the standard AdS holographic dictionary, particularly in the situations withfinite string interactions. Our approach in some previous papers [16, 17, 18, 19, 20]is to consider another corner of the parameter space in the BMN limit, focusing onthe free gauge theory. In this case, the string theory side becomes infinitely curved µ ∼ ∞ , and strings are effectively infinitely long and tensionless, but can still havefinite string interactions. Most interestingly, since the string spectrum is completelydegenerate, the tensionless string can jump from one excited state to another withoutenergy cost through a quantum unitary transition. It turns out that in this case theeffective string coupling constant should be identified with a finite genus countingparameter g := J N , where J is the large R-charge and scales like J ∼ √ N ∼ ∞ inthe BMN limit. Some higher genus BMN correlators were first computed in [21, 22].Since the full fledged holographic dictionary is no longer available in the pp-wavebackground, our pragmatic approach is to try to compute the physical quantities onboth sides of the correspondence and find potential non-trivial agreements. In thissense, a mismatch with naive expectation is not necessarily a contradiction of theholographic principle. Instead, one should focus on finding aspects where the cal-culations from both side do match, and try to give physical derivations or proofs of2uch mathematical coincidence. Besides the free string spectrum originally consid-ered in [7], some more tests of the pp-wave holography are immediately clear. Forexample, the free planar three-point functions of BMN operator should correspondto the Green-Schwarz light cone string field cubic vertex [23, 24] in the infinitelycurved pp-wave background [16, 25]. In the papers [17, 18], we further proposed afactorization formula, where the free higher genus BMN correlators are holographi-cally related to string loop diagram calculations by pasting together the cubic stringvertices without propagator. More recently, we propose a probability interpretationof the BMN two-point functions [19]. This also provides yet another interesting newentry of the pp-wave holographic dictionary that the BMN two-point function doesnot naively correspond to a quantum transition amplitude on the string theory side,but rather to its norm square. A consequence of the probability interpretation is thenon-negativity of BMN two-point functions, which can be demonstrated for BMNoperators with two stringy modes at any genus, or three stringy modes at genus one.In this paper, we further test the non-negativity conjecture for BMN operators withfour and five stringy modes at genus one. Surprisingly, it turns out this is no longervalid. Of course, as mentioned earlier, this is not necessarily a contradiction of holo-graphic principle according to our philosophy, but rather provide a new perspectiveon the limitation of our probability interpretation.Motivated by the results, we further check the factorization formula for the caseof four stringy modes and confirm that it is still valid. We also check that thecorrespondence of planar three-point functions with Green-Schwarz string vertex isrobust in the case of many string modes. Our mixed test results for this case shallmotivate potential physical explanations which might shed new light on the stillmysterious holographic principle.In some potentially related interesting recent developments, Gaberdiel and Gopaku-mar et al study string theory on a AdS background, dual to a symmetric productCFT [26, 27, 28], with ideas dating back to some early papers e.g. [29]. Althoughthe technical details are rather different, there appears to be some common featureswith our works that the strings are tensionless and the dual CFT is free. To ourknowledge, in various special situations where the higher genus string amplitudes canbe systematically computed, our setting by far most resembles the usual critical su-perstring theory on flat spacetime, with of course still certain notable simplificationsthat in our case there is no continuous light cone or transverse momentum due to theinfinite curvature and Ramond-Ramond flux in the background.The paper is organized as the followings. In Sec. 2 we review some notationsand previous results, with an emphasis on the real and symmetric properties of thetwo-point functions. In Sec. 3 we calculate the torus two-point functions of BMNoperators with four string modes with the notations of some standard integrals. Wealso compute the case five string modes for the generic situations of mode numbers3ith no degeneracy. In both cases we discover that they are not alway non-negative.In Sec. 4 we perform the one-loop string calculations and confirm that the factoriza-tion formula for the case of four string modes is still valid. In Sec. 5 we check thecorrespondence of planar three-point functions with Green-Schwarz string vertex withmany string modes. In Sec. 6 we consider the situations of multiple string modes inthe same transverse direction. We conclude with some discussions in Sec. 7. Let us first introduce some notations for the higher genus two-point functions, andreview some previous results. The integral formula is naively complex and we performa more careful analysis of its reality property. The string vacuum state in the pp-wave geometry is described by a dual vacuum BMN operator with large R-charge O J = Tr( Z J ), where Z = √ ( φ + iφ ) is a complex scalar field in the SU ( N ) adjointrepresentation, constructed from two of the six real scalar fields in the N = 4 SU ( N )super-Yang-Mills theory. We take the BMN limit J ∼ √ N ∼ ∞ with g := J N finite,and focus on free gauge theory. As in the previous papers, our notation omits theuniversal spacetime factors in the correlators.The stringy states with bosonic excited modes in the eight transverse directionsare constructed by inserting the four remaining real scalars φ I and four covariantderivatives D I where I = 1 , , , Z ’s with phases. For example,the BMN operators up to four scalar oscillator modes are the followings O J = 1 √ J N J T rZ J , O J = 1 √ N J +1 T r ( φ I Z J ) ,O J − m,m = 1 √ J N J +2 J − (cid:88) l =0 e πimlJ T r ( φ I Z l φ I Z J − l ) .O J ( m ,m ,m ) = 1 √ N J +3 J J (cid:88) l ,l =0 e πim l J e πim l J Tr( φ I Z l φ I Z l − l φ I Z J − l ) .O J ( m ,m ,m ,m ) = 1 √ N J +4 J J (cid:88) l ,l ,l =0 e πim l J e πim l J e πim l J Tr( φ Z l φ Z l − l φ Z l − l φ Z J − l ) . (2.1)Here one can use the cyclicity of the trace to move one scalar to the starting po-sition for convenience, the mode numbers (cid:80) i m i = 0 in the case of three and fourmodes. The operators are properly normalized to be orthonormal at the genus zeroor planar level. The convention is that the first operator O J corresponds to theclosed string vacuum state, and the positive and negative modes in the other opera-tors represent the left and right moving stringy excited modes, while the zero modes4re supergravity modes representing discretized momenta in the corresponding tra-verse direction. The construction ensures only operators satisfying closed string levelmatch conditioning are non-vanishing. As a consequence, the stringy excited stateshave at least two oscillator modes with opposite signs. Analogously, we can add morestringy modes and denote the properly normalized BMN operator O Jm ,m , ··· ,m k withthe closed string level matching condition (cid:80) ki =1 m k = 0. Unless otherwise specified,we use this notation to denote k different string modes.The free two-point functions at higher genus h ≥ Z ’s up to n ≤ h segments and Wick contracted according to a permutationof (1 , , · · · , n ). We only consider cyclically inequivalent permutations where no twoneighboring numbers are consecutive. The contributions of such Feynman diagramsof genus h are proportional to J n N h . So the dominant contributions come from those ofthe maximal number of segments n = 4 h and we can neglect the other cases n < h which are suppressed in the large R-charge limit. Furthermore, in the BMN limit,the contributions are proportional to J h N h = g h , confirming the finite parameter g as the genus counting parameter therefore the effective string coupling constant withour restriction to free gauge theory. We should note that a generic permutation of(1 , , · · · , h ) can give Feynman diagram with genus higher than h . A useful ruleto select genus h permutations is to generate them by string diagrams with h loops[18]. It is known that there are (4 h − h +1 such genus h permutations [30]. For example,at genus one there is only one such permutation, and can be generated by a one-loop string process (1234) → (12)(34) → (2143). The field theory torus diagram isdepicted in Figure 1. Please note that we denote the genus as h because the usualsymbol g has been used as the effective string coupling.Once the string of Z ’s is Wick contracted with ¯ Z ’s, we can add scalar insertionsand contract them along the lines of Z ’s to preserve the genus of the Feynman dia-gram. In the BMN limit each scalar insertion gives an integral with the correspondingphases. For example, the free torus two-point function can be written as (cid:104) ¯ O J ( m ,m , ··· ,m k ) O J ( n ,n , ··· ,n k ) (cid:105) torus = g (cid:90) dx dx dx dx δ ( x + x + x + x − × k (cid:89) i =1 ( (cid:90) x + e πin i ( x + x ) (cid:90) x + x x + e πin i ( x − x ) (cid:90) − x x + x + e − πin i ( x + x ) (cid:90) − x ) dy i e πi ( n i − m i ) y i = g (cid:90) dx dx dx dx δ ( x + x + x + x − (cid:90) x dy k e πi ( n k − m k ) y k × k − (cid:89) i =1 ( (cid:90) x + e πin i ( x + x ) (cid:90) x + x x + e πin i ( x − x ) (cid:90) − x x + x + e − πin i ( x + x ) (cid:90) − x ) dy i e πi ( n i − m i ) y i , (2.2)where in the second equality we use the cyclicity to put the one string mode into5 r( Z J ) 4. The contribution is I (2 , , , , , ( n − m , , − m , n , m − n , n − m + n − m ) . (3.11)4. The variables 0 < y < y < y < x < x + x + x < y < 1. There are also3! · y and permutations ofindices 2 , , 4. The contribution is I (2 , , , , ( m , n , , n + n − m , m + m − n ) . (3.12)11. The variables 0 < y < y < x < y < y < x + x . There are 3 · · y and exchange of indices between3 , , 2. The contribution is I (2 , , , , ( − m − m , − n − n , , − n − m , m + n ) . (3.13)6. The variables 0 < y < y < x < x + x < y < y < x + x + x . There arealso 3 · · y and exchange ofindices between 3 , , 2. The contribution is I (2 , , , , , ( n + n − m − m , , n + n , − m − m , n − m , − n + m ) . (3.14)7. The variables 0 < y < y < x < x + x + x < y < y < 1. There arealso 3 · · y and exchange ofindices between 3 , , 2. The contribution is I (2 , , , , ( n + n , m + m , , m + n , − n − m ) . (3.15)8. The variables 0 < y < y < x < y < x + x < y < x + x + x . There arealso 3 · · y and exchange ofindices between 3 , , 2. The contribution is I ( m , n , − m , − n , m + n , − m + n , m − n , m + n + m + n ) . (3.16)9. The variables 0 < y < y < x < y < x + x < x + x + x < y < 1. Thereare also 3 · · y and exchangeof indices between 3 , , 2. The contribution is I ( m , n , − m , − n , , m − n , n − m , n − m + n − m ) . (3.17)10. The variables 0 < y < y < x < x + x < y < x + x + x < y < 1. Thereare also 3 · · y and exchangeof indices between 3 , , 2. The contribution is I ( m , n , − m , − n , − m − n , m − n , n − m , m + m + n + n ) . (3.18)11. The variables 0 < y < x < y < y < y < x + x . There are 3! = 6 similarintegrals by counting the permutations of indices 1 , , 3. The contribution is I (2 , , , , ( m , n , , n + n − m , m + m − n ) . (3.19)12. The variables 0 < y < x < x + x < y < y < y < x + x + x . Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I (2 , , , , , ( n − m , , n , − m , − n + m , n + n − m − m ) . (3.20)123. The variables 0 < y < x < x + x + x < y < y < y < 1. There arealso 3! = 6 similar integrals by counting the permutations of indices 1 , , 3. Thecontribution is I (2 , , , , ( − m , − n , , − m − m + n , − n + m − n ) . (3.21)14. The variables 0 < y < x < y < y < x + x < y < x + x + x . Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I ( m , n , − m , − n , , m − n , n − m , m + m + n + n ) . (3.22)15. The variables 0 < y < x < y < y < x + x < x + x + x < y < 1. Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I ( m , n , , m + m , n + n , − n − m , m + m + n ) . (3.23)16. The variables 0 < y < x < y < x + x < y < y < x + x + x . Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I ( m , n , , m + m , n + n , − n − m , m + m + n , n + n + m ) . (3.24)17. The variables 0 < y < x < x + x < y < y < x + x + x < y < 1. Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I ( − m , − n , , m + m , n + n , n + m , m + m − n , n + n − m ) . (3.25)18. The variables 0 < y < x < y < x + x < x + x + x < y < y < 1. Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I ( − m , − n , , m + m , n + n , m + n , n + n − m , m + m − n ) . (3.26)19. The variables 0 < y < x < x + x < y < x + x + x < y < y < 1. Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I ( m , n , − m , − n , , n − m , m − n , m + m + n + n ) . (3.27)130. The variables 0 < y < x < y < x + x < y < x + x + x < y < 1. Thereare also 3! = 6 similar integrals by counting the permutations of indices 1 , , I ( m , n , − m , − n , m + m − n , m + m + n , n + n + m , n + n − m ) . (3.28)The 20 cases of integrals can be organized into 10 types of integrals, so that thetotal contribution to can be more succinctly written as (cid:104) ¯ O J ( m ,m ,m ,m ) O J ( n ,n ,n ,n ) (cid:105) torus = g (cid:88) ( i,j,k,l ) [ I (1 , , , ( n i − m i , n i − m i + n j − m j , − n k + m k , I (2 , , , , ( − m i , − n i , , − m i + n j − m j , − n i − n k + m k )+ I (2 , , , , ( m i , n i , , m i − n j + m j , n i + n k − m k )+ I (2 , , , , ( m i + m j , n i + n j , , m i + n j , − m k − n l )+ I (2 , , , , , ( n i − m i , , n i , − m i , − n j + m j , n i − m i + n k − m k )+ I ( m i , n i , − m j , − n j , , m i − n j , n i − m j , n i − m j + n k − m k )+ I ( m i , n i , − m j , − n j , − m j − n j , m i − n j , n i − m j , m i + n i + m k + n l )+ I ( m i , n i , − m j , − n j , m i + n i , m i − n j , n i − m j , m i + n i + m k + n l )+ g (cid:88) ( i,j ) ↔ ( k,l ) I (2 , , , , , ( n i + n j − m i − m j , , n i + n j , − m i − m j , n i − m i , − n k + m k )+ g (cid:88) ( i,j,k ) I ( m i , n i , − m j , − n j , m i + n i + m k , m i + n i + n k , m i + m k − n j , n i + n k − m j ) . ≡ g (cid:88) k =1 I k (3.29)We provide some explanations of the notations. For later convenience we denotethe 10 type of integrals by I k , k = 1 , , · · · , 10, according to the order as written inthe above equation, which should not be confused with labels of transverse spacedirection in the pp-wave geometry. The first 8 types of integrals are summed overthe 24 permutations ( i, j, k, l ) of 1234. The notation ( i, j ) ↔ ( k, l ) in I denotes wesum only once if two permutations are related by exchanging ( i, j ) ↔ ( k, l ). This canbe achieved e.g. by specifying 1 ∈ { i, k } in the permutations. The last integral I is summed over the 6 permutations ( i, j, k ) of 123. Of the 10 types of integrals, the I comes from case 1 in the above enumeration, the I from combining cases 2 and13, the I from combining cases 4 and 11, the I from combing cases 5 and 7, the I from combing cases 3 and 12, the I from combining cases 9, 14 and 19, the I fromcombing cases 10, 15 and 16, the I from combining cases 8, 17 and 18, the I fromcase 6, the I from case 20. Although the last two integrals I , I are not summed14ver the full 24 permutations of 1234, it is easy to show they are also permutationsymmetric using the closed string level matching conditions and the invariance of thestandard integral under a shift of all arguments by an integer.We can perform the calculations using a compute program. The calculations arestraightforward for a given set of mode numbers. An expression for the generic casewhere there is no further degeneracy in the arguments in (3.29) can be obtained but itis too long to write down here. We can check some special cases. For example, whentwo modes m = n = 0, this reduced to the case of three string modes consideredin [18]. Another special case is when m i = 0 and n i (cid:54) = 0, then the result identicallyvanishes, consistent with the conservation of discrete momentum in the transversedirection.The total contribution (3.29) is always real, although each individual integral canbe complex. Computing the results for some random mode numbers, we find thatthe result can be either positive or negative. We can provide some potentially helpfulempirical observations about the signs of the torus two-point functions. However fornow there seems no particular strong motivation to warrant a thoroughly rigorousanalysis. In the followings we assume all m i , n i , i = 1 , , , m i = n i , i = 1 , 2, then the torustwo-point functions are most likely positive. There may be some exceptions.For example, in the case ( m i , n i ) = ( − , − , ( − , − , (1 , , (19 , , i =1 , , , 4, the torus two-point function is negative. If all mode numbers are thesame, i.e. m i = n i , i = 1 , , , 4, then we have not found an example of negativetorus two-point function.2. For m i (cid:54) = n i , i = 1 , , , 4, the sign of torus two-point function is most likelythe same as (cid:81) i =1 ( m i − n i ). There are also some exceptions. For example, inthe case ( m i , n i ) = (8 , , (2 , − , ( − , − , ( − , , i = 1 , , , 4, the torus two-point function is positive. This phenomenon can be explained from the previousmethod in the case of five modes in (3.6). In the case of four modes we need tonow pick up the real parts in the integrals (3.7). Only two terms give the lastintegral with completely degenerate arguments. So the result can be roughlywritten as (cid:104) ¯ O J ( m ,m ,m ,m ) O J ( n ,n ,n ,n ) (cid:105) torus = g (2 π ) (cid:81) i =1 ( m i − n i ) ( 13 + · · · ) , (3.30)where the · · · denotes some correction terms which are inverse squares of non-zero integers from mode numbers and also suppressed by a factor of 2 π , sotheir absolute values are most likely small comparing to .15 latexit sha1_base64="LaiPe5Cph4m7/l9DxW2hBXqhn8M=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Kokoeix6EU8t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dlZW19Y3Ngtbxe2d3b390sFhU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj26nfekKleSwfzDhBP6IDyUPOqLFS/b5XKrsVdwayTLyclCFHrVf66vZjlkYoDRNU647nJsbPqDKcCZwUu6nGhLIRHWDHUkkj1H42O3RCTq3SJ2GsbElDZurviYxGWo+jwHZG1Az1ojcV//M6qQmv/YzLJDUo2XxRmApiYjL9mvS5QmbE2BLKFLe3EjakijJjsynaELzFl5dJ87ziXVbc+kW5epPHUYBjOIEz8OAKqnAHNWgAA4RneIU359F5cd6dj3nripPPHMEfOJ8/odOM0g== J Since we have now discovered a new phenomenon in the case of more than three stringmodes that the BMN torus two-point functions can be negative, it is worthwhile totest the other proposals for the holographic dictionary, in particular the factorizationformulas in [17, 18]. This also serves as a check of the somewhat complicated calcu-lations in the previous Section 3. In this section we focus on the case of four stringmodes. First we shall calculate the relevant free planar three-point functions, which corre-spond to the string vertices. There are 3 ways to distribute the 4 scalar insertions asthe long string is cut into two short strings with J ≡ xJ and J ≡ (1 − x ) J numberof Z ’s (0 < x < ¯ O Jm ,m ,m ,m O J k ,k ,k ,k O J (cid:105) = g √ J (1 − x ) x (cid:90) x (cid:89) i =1 dy i e − πi ( m i − kix ) y i = g √ J x (1 − x ) (cid:89) i =1 sin( πm i x ) π ( m i x − k i ) , (cid:104) ¯ O Jm ,m ,m ,m O J k ,k ,k O J (cid:105) = g √ J x [ (cid:90) x (cid:89) i =1 dy i e − πi ( m i − kix ) y i ][ (cid:90) x dy e − πim y ] = − gx √ J [ (cid:89) i =1 sin( πm i x ) π ( m i x − k i ) ] sin( πm x ) πm , (cid:104) ¯ O Jm ,m ,m ,m O J − k,k O J − l,l (cid:105) = g [ (cid:82) x dy dy e − πi ( m + kx ) y e − πi ( m − kx ) y ][ (cid:82) x dy dy e − πi ( m y + l ( y − x )1 − x ) e − πi ( m y − l ( y − x )1 − x ) ] (cid:112) J x (1 − x )= g √ J [ x (1 − x )] (cid:81) i =1 sin( πm i x ) π ( m x + k )( m x − k )( m (1 − x ) + l )( m (1 − x ) − l ) . (4.1)For the simplicity of notation, we do not label the specific string modes in the oper-ators with the implicit understanding that the string modes appearing in the sameorder in ¯ O and O operators are the same. We note that the integral formulas are validfor any mode numbers, but the integrated results in the above equation may not bevalid in some special cases where the denominator vanishes, e.g. some m i = k i = 0.In those cases one needs to do the integral separately. As in the cases with less stringymodes, the three-point functions are always suppressed by a factor √ J , so they arevanishing or “virtual” in the BMN limit J ∼ ∞ . There are also 3 string one-loop processes corresponding to the torus two-point func-tions, depicted in Figure 3. In the BMN limit, the sum over the operator lengthbecomes integral (cid:80) J − J =1 = J (cid:82) dx . We denote the contributions S , S , S as thefollowings S = J (cid:90) dx (cid:88) (cid:80) i k i =0 (cid:104) ¯ O Jm ,m ,m ,m O J k ,k ,k ,k O J (cid:105)(cid:104) ¯ O J k ,k ,k ,k ¯ O J O Jn ,n ,n ,n (cid:105) ,S = J (cid:90) dx (cid:88) i =1 (cid:88) (cid:80) i k i =0 (cid:104) ¯ O Jm i ,m i ,m i ,m i O J k ,k ,k O J (cid:105)(cid:104) ¯ O J k ,k ,k ¯ O J O Jn i ,n i ,n i ,n i (cid:105) ,S = J (cid:90) dx (cid:88) i =2 + ∞ (cid:88) k,l = −∞ (cid:104) ¯ O Jm ,m i ,m i ,m i O J − k,k O J − l,l (cid:105)(cid:104) ¯ O J − k,k ¯ O J − l,l O Jn ,n i ,n i ,n i (cid:105) , (4.2)where ( i , i , i , i ) in S is a cyclic permutation of (1234) and ( i , i , i ) in S is acyclic permutation of (234). 17 latexit sha1_base64="EnIDaY1IqI+42zgQ11t/Q7qlowo=">AAACAHicbVDLSgMxFM3UV62vURcu3ASLUEHKTK3osuhG3FjBPqAdh0yatqFJZkgyQhlm46+4caGIWz/DnX9j2s5CqwcOHM65l+SeIGJUacf5snILi0vLK/nVwtr6xuaWvb3TVGEsMWngkIWyHSBFGBWkoalmpB1JgnjASCsYXU7y1gORiobiTo8j4nE0ELRPMdLG8u29m/trPylx3z3mfsXwxLB6lPp20Sk7U8C/ws1EEWSo+/ZntxfimBOhMUNKdVwn0l6CpKaYkbTQjRWJEB6hAekYKRAnykumB6Tw0Dg92A+lodBw6v7cSBBXaswDM8mRHqr5bGL+l3Vi3T/3EiqiWBOBZw/1YwZ1CCdtwB6VBGs2NgJhSc1fIR4iibA2nRVMCe78yX9Fs1J2T8vObbVYu8jqyIN9cABKwAVnoAauQB00AAYpeAIv4NV6tJ6tN+t9Npqzsp1d8AvWxzfwlJSy O J ( m ,m ,m ,m ) 2. The bosonic operator a I † m ( r ) creates the r ’s string statein the I ’s transverse direction with BMN mode number m . The Neumann matrixencodes the string interactions. Its element N , m,n = 0 for any m, n , and it has asymmetry N r,sm,n = N s,rn,m . Since the number of string modes in the 3rd long string isthe sum of those of the two r = 1 , N ,rm,n type of matrix elements. This corresponds to the calculations of free planar BMNthree-point functions where the string modes are contracted between a long stringand the two short strings.The Neumann matrix elements were computed in the pp-wave background [25] andbecomes much simplified in the infinite curvature limit [16]. We denote the light conewidth of the two short strings as x and 1 − x , corresponding to the relative lengths ofthe two short operators in the free planar three-point function. The relevant matrixelements in the infinite curvature limit are N , , = √ x, N , , = √ − x,N , m,n = √ x sin( πmx ) π ( mx − n ) , N , m,n = − √ − x sin( πmx ) π [ m (1 − x ) − n ] , for ( m, n ) (cid:54) = (0 , , (5.2)We should note that we use a different convention for the basis of the bosonic creationoperators from the literature [25, 16]. Due to the different conventions, there arealso some sign differences in the Neumann matrix elements with the literature. Thecurrent convention is most convenient from the field theory perspective.Suppose three BMN operators O , O , O correspond to three string states | (cid:105) , | (cid:105) , | (cid:105) ,then the planar three-point functions are related to the string vertex (cid:104) ¯ O O O (cid:105)(cid:104) ¯ O J O xJ O (1 − x ) J (cid:105) = (cid:104) |(cid:104) |(cid:104) | V (cid:105)(cid:104) | V (cid:105) , (5.3)where | (cid:105) is the string vacuum state, and the the normalization factor of BMN vacuumcorrelator is simply (cid:104) ¯ O J O xJ O (1 − x ) J (cid:105) = g (cid:112) x (1 − x ) √ J . (5.4)The right hand side of (5.3) can be computed by expanding the bosonic operator(5.1) to appropriate order and extract the relevant Neumann matrix elements. Forexample, for the case of four string modes, the BMN operators correspond to thestring states as O Jm ,m ,m ,m ⇐⇒ a I † m (3) a I † m (3) a I † m (3) a I † m (3) | (cid:105) ,O J k ,k ,k ,k ⇐⇒ a I † k (1) a I † k (1) a I † k (1) a I † k (1) | (cid:105) ,O J k ,k ,k ⇐⇒ a I † k (1) a I † k (1) a I † k (1) | (cid:105) ,O J − k,k ⇐⇒ a I †− k (1) a I † k (1) | (cid:105) , O J − l,l ⇐⇒ a I †− l (2) a I † l (2) | (cid:105) ,O J ⇐⇒ | (cid:105) , O J ⇐⇒ a I † | (cid:105) . (5.5)20e can expand the exponential operator (5.1) to 4th order and compute planar three-point functions with the usual commutation relation of creation and annihilationoperators. The only non-vanishing contributions come from the 4th order whichprovides the same numbers of creation operators as those of the annihilation operators.The results are (cid:104) ¯ O Jm ,m ,m ,m O J k ,k ,k ,k O J (cid:105) = g (cid:112) x (1 − x ) √ J (cid:89) i =1 N , m i ,k i , (cid:104) ¯ O Jm ,m ,m ,m O J k ,k ,k O J (cid:105) = g (cid:112) x (1 − x ) √ J ( (cid:89) i =1 N , m i ,k i ) N , m , , (cid:104) ¯ O Jm ,m ,m ,m O J − k,k O J − l,l (cid:105) = g (cid:112) x (1 − x ) √ J N , m , − k N , m ,k N , m , − l N , m ,l . (5.6)This agrees with the field theory results (4.1) using the Neumann matrix elementsin (5.2). One can also check the case of three string modes previously computed in[18] and various degenerate cases. Of course, as we mentioned, the planar three-pointfunctions are vanishing in the BMN limit J ∼ ∞ , but their ratios with the vacuumcorrelator are finite and meaningfully related to the Neumann matrix elements.It is not difficult to see that the Neumann matrix elements in (5.2) simply cor-respond to the integrations of the positions of relevant string mode with phases inthe BMN operators with proper normalization, using the closed string level match-ing condition in the long 3rd string to cancel out an overall phase. We infer thatalthough the physical setting has at most eight string modes of distinct directions,the mathematical structure of the holographic dictionary (5.3) is quite robust andsurvives even in a hypothetical situation with any number of different string modes,i.e. not just valid for I = 1 , , · · · Since we have now studied BMN operators with many string modes, it is appropriateto consider the situation of multiple modes in the same transverse direction. To ourknowledge, this situation has not been much discussed in the literature. Naively, thecorresponding BMN operators can be similarly constructed, using the same scalarfield (or covariant derivative) going through the string of Z ’s with multiple sums withphases, with possibly a different normalization discussed below.21or simplicity we consider BMN operators with only the 4 scalar field insertions.First we introduce some notations, if multiple string modes correspond to the samedirection, we use a square bracket to enclose the mode numbers. For example, theBMN operator with two identical scalar fields is denoted O J [ − m,m ] , and the BMNoperator with three string modes where two of them have the same direction is denoted O J ([ m ,m ] ,m ) . The closed string level matching condition is still the same that allmode numbers should sum to zero. Since the scalar fields in the square bracket areexchangeable, e.g. the operators O J ([ m ,m ] ,m ) and O J ([ m ,m ] ,m ) are the same, we canchoose to order the mode numbers in the square bracket, e.g. in a non-decreasingorder.However, this brings a subtle issue. We recall that the chiral primary operatorswith lowest dimension in a short multiplet of N = 4 super-Yang-Mills theory areconstructed by the 6 real scalars in the SO (6) symmetric traceless representation, seee.g. the review [31, 32]. They are BPS operators whose conformal dimensions areprotected by supersymmetry. When a real scalar appears multiple times, an operatormay no longer be chiral primary. For example, the operator Tr(( φ I ) ), known as theKonishi operator, is not a chiral primary operator, since it is not traceless in the SO (6). The conformal dimension of this operator would grow at least as ( g Y M N ) .On the other hand, the BMN vacuum operator Tr( Z J ) is a chiral primary operatorsince a power of the complex scalar Z is automatically traceless in the SO (6).In the original calculations of planar anomalous conformal dimensions of the BMNoperator O J − m,m [7], one used the fact that for m = 0, the operator O J , is a chiralprimary operator whose conformal dimension is not corrected by gauge interactions.So one only needs to compute the mode number m -dependent part which is pertur-bative in an effective gauge coupling constant λ (cid:48) ≡ g Y M NJ , a small parameter in theBMN limit. In this sense the BMN operators of distinct scalar field insertions withnon-zero modes are “near BPS” operators. As mentioned, if we put two identical realscalars into the string of Z ’s, the zero mode operator, namely O J [0 , , is no longer achiral primary operator. There may be large (field theory) quantum corrections tothe m -independent part of its conformal dimension. So in this case the calculations ofplanar conformal dimension is no longer reliable. We are not aware a simple naturalfix which also matches the expectations from the string theory side.In any case, we may hope by restricting ourselves to free gauge theory, this issuewith large quantum gauge corrections does not cause problems. We shall retest ourearlier results for the cases involving BMN operators with multiple identical scalarfields. We find that the comparison with Green-Schwarz light cone string field cubicvertex [16] and factorization formula [17, 18] still go through smoothly. However, theprobability interpretation [19] begins to encounter an issue in the case of three scalarfield insertions with two of them identical.22irst we consider the case of two identical scalar fields. The BMN operators are O J [ − m,m ] = 1 √ J N J +2 J − (cid:88) l =0 e πimlJ T r ( φ I Z l φ I Z J − l ) , m > ,O J [0 , = 1 √ J N J +2 J − (cid:88) l =0 T r ( φ I Z l φ I Z J − l ) , (6.1)where φ I is any one of 4 remaining real scalars. We only need to consider m ≥ m gives the same operator. The zero mode has an extra normalizationfactor √ a I † | (cid:105) , a I † | (cid:105) , a I †− m (3) a I † m (3) | (cid:105) .We have an extra contribution if the directions are the same I = I , namely, (cid:104) | a I † a I † a I − m (3) a I m (3) | V (cid:105)(cid:104) | V (cid:105) = (cid:40) N , − m, N , m, , I (cid:54) = I N , − m, N , m, + N , m, N , − m, , I = I . (6.2)The extra contribution for I = I also appears in the extra contraction for identicalscalar fields in the field theory calculations. So the comparison of BMN three-pointfunctions with cubic string vertex is still valid in the case of multiple modes in thesame direction.The factorization formula also works in this case. We note that with the extracontraction due to identical scalars, for m, n (cid:54) = 0 we have the formula (cid:104) ¯ O J [ − m,m ] O J O J (cid:105) = 2 (cid:104) ¯ O J − m,m O J O J (cid:105) , (cid:104) ¯ O J [ − m,m ] O J [ − n,n ] O J (cid:105) = (cid:104) ¯ O J − m,m O J − n,n O J (cid:105) + (cid:104) ¯ O J − m,m O J n, − n O J (cid:105) , (cid:104) ¯ O J [ − m,m ] O J [ − n,n ] (cid:105) torus = (cid:104) ¯ O J − m,m O J − n,n (cid:105) torus + (cid:104) ¯ O J − m,m O Jn, − n (cid:105) torus , (6.3)where J = J + J and the three-point functions without label are planar. Using thethe fact (cid:104) ¯ O J − m,m O J O J (cid:105) = (cid:104) ¯ O Jm, − m O J O J (cid:105) , (cid:104) ¯ O J − m,m O J − n,n O J (cid:105) = (cid:104) ¯ O Jm, − m O J n, − n O J (cid:105) and the factorization formula for the case of two different modes [18, 19], we canwrite the analogous formula for the current case2 (cid:104) ¯ O J [ − m,m ] O J [ − n,n ] (cid:105) torus = J − (cid:88) J =1 ∞ (cid:88) k =0 (cid:104) ¯ O J [ − m,m ] O J [ − k,k ] O J (cid:105)(cid:104) ¯ O J [ − k,k ] ¯ O J O J [ − n,n ] (cid:105) + [ J ] (cid:88) J =1 (cid:104) ¯ O J [ − m,m ] O J O J (cid:105)(cid:104) ¯ O J ¯ O J O J [ − n,n ] (cid:105) . (6.4)We note that the difference is that we only need to sum over k ≥ J ≤ J since the scalars in the two operators O J and O J are the same.The formula for case of m, n = 0 is much simpler and also works in this case, takinginto account the normalization in (6.1). 23he calculations with more stringy modes are similar based on the experience. Sowe conclude that as long as the comparison with cubic string vertex and the factor-ization formula are valid in the case of many different string modes, then identifyingsome of the modes as the same shall not cause problems.Now we consider the probability interpretation for two and three string modes.Identifying some modes to the same apparently does not change the non-negativity ofthe correlators. So we only need to consider the normalization relation. For the casetwo string modes in the same direction we still also have the normalization relationsimilar as (2.6) ∞ (cid:88) n =0 (cid:104) ¯ O J [ − m,m ] O J [ − n,n ]) (cid:105) h = (4 h − h + 1)(4 h )! g h , (6.5)where we now only need to sum over non-negative integer k . The formula is valid forboth m = 0 and m > O J ([ m ,m ] ,m ) = c √ N J +3 J J (cid:88) l ,l =0 e πim l J e πim l J Tr( φ Z l φ Z l − l φ Z J − l ) . (6.6)Comparing to the case of three different modes (2.1), we add a normalization constantwhich is c = 1 if m < m and c = √ if m = m , so that the operators areorthonormal at the planar level. Again we compute the sum over one set of modenumbers. Suppose m < m , (cid:88) n ≤ n (cid:104) ¯ O J ([ m ,m ] ,m ) O J ([ n ,n ] ,n ) (cid:105) h = (cid:88) n (cid:54) = n (cid:104) ¯ O J ( m ,m ,m ) O J ( n ,n ,n ) (cid:105) h + √ (cid:88) n (cid:104) ¯ O J ( m ,m ,m ) O J ( n,n,n ) (cid:105) h . (6.7)Unlike the case of two string modes, the second term does not generally vanish. Sobecause of the √ The SO (8) rotational symmetry of the transverse directions in the pp-wave back-ground (1.1) is broken by the Ramond-Ramond flux into SO (4) × SO (4), where thebosonic string modes are described differently by covariant derivatives and scalar fieldinsertions in the dual CFT. As such, it is reasonable to expect our proposed entriesof pp-wave holographic dictionary, e.g. (2.10, 4.5, 5.3), to face some challenges withmore than four distinct string modes as the infinite Ramond-Ramond flux in our set-ting shall separate the two types of string modes. However, it is rather surprising that24ven for the case of four string modes, the torus two-point function can be negative,so the probability interpretation may no longer valid. Of course, since the two-pointfunction is always real and symmetric, the arguments in [19] are still valid that itcan not be naively identified with a quantum transition amplitude on the string the-ory side, which would then violate fundamental principle of unitarity. It would beinteresting to provide a reasonable explanation, or improve the proposed holographicdictionary (2.10) to include this case of four string modes.On the other hand, we confirm that the factorization formulas e.g. (4.5) are stillvalid for the case of four string modes, while the comparison with cubic string vertex(5.3) is seen to be straightforwardly applied to any hypothetical number of stringmodes, not even restricted by the eight dimensions of transverse directions in thepp-wave background.We also discuss the situation with multiple string modes in the same direction. Inthis case the BMN operators are no longer “near-BPS”, and there are potentially largequantum corrections on the field theory side if one turns on the gauge coupling. Wecheck that the mathematical structures in the factorization formula and comparisonwith cubic string vertex, e.g. (4.5, 5.3), are robust and remain valid in this situationas we stay in free gauge theory. However, the proposed probability interpretation(2.10) again seems rather fragile and further breaks down in the case of three stringmodes because of a problem with normalization, though it still holds up in the caseof two string modes due to the decoupling of the zero mode with non-zero modes.It is interesting to further explore aspects of the pp-wave holographic dictionary.For example, in the case of three string modes, the non-negativity of torus two-point functions can be shown by explicit calculations, where there are numerousdegenerate cases to deal with separately. One may ask whether there is a universalformalism which can deal with all cases regardless of mode number degeneracy andmay also generalize to higher genus h ≥ 2. It is also interesting to check whether thefactorization formulas are still valid in the case of more than four string modes orfurther in a hypothetical situation of any number of (different) string modes. Withouta significant improvement of mathematical tools, the calculations are much morecomplicated. In any case, it seems worthwhile to push forward with the laboriousendeavor for the purpose of a better understanding of pp-wave holography.As mentioned in [20], the probability interpretation of two-point function impliesthe string perturbation series is convergent. In this sense, the holographic higher genuscalculations are not asymptotic perturbative expansions as familiar in most examplesof quantum theories, but may in principle provide exact complete string amplitudesvalid for any string coupling. If no new non-perturbative effect is discovered in thefuture, then perhaps we have luckily found a rare example of perturbatively completestring theory, at least for the case of two string modes and very likely also for the caseof three distinct string modes pending more tests of non-negativity at higher genus25 ≥ 2. In the cases of four or more string modes, the torus two-point functions are nolonger always non-negative. One can nevertheless similarly follow the method in [20]to give an upper bound on the higher genus two-point functions and show that thegenus expansions remain convergent. For small string coupling and two different setsof mode numbers, the torus contribution is dominant, so the total two-point functioncould certainly be negative and is no longer a probability distribution although theycan be still similarly normalized to sum to unity. It would be desirable to betterunderstand the physical meaning of the two-point function on the string theory sideof the correspondence in this situation. Acknowledgments We thank Jun-Hao Li, Jian-xin Lu, Gao-fu Ren, Pei-xuan Zeng for helpful discus-sions. This work was supported in parts by the national Natural Science Foundationof China (Grants No.11675167, No.11947301 and No.12047502). A Some calculational details of the one-loop stringintegrals We will convert the formulas for one-loop string diagrams (4.3) into the 10 types ofintegrals in (3.29). In the calculations, some cases are simply related to others bya transformation ( m i , n i ) → ( − n i , − m i ). It is helpful to first list the action of thetransformation on the integrals I i invariant , i = 1 , , , , , ,I ↔ I , I ↔ I . (A.1)We discuss the dissection of the multi-dimensional integral domain in many cases,and introduce some positive variables z ’s and z (cid:48) ’s such that they sum to one. A.1 S contribution We assume integral variables y (cid:48) > y . The other case is related by switching y (cid:48) ↔ y and the transformation ( m i , n i ) → ( − n i , − m i ). We have 0 < y i + y (cid:48) − y < x . Inthis case first we write 1 − x = (cid:82) − x dz dz δ ( z + z − (1 − x )). Then the variables z , z do not appear in the exponent. There is always an argument 0 with at leastmultiplicity two in the standard integral. We define z = x − y (cid:48) , z = y and discussvarious cases.1. x < y i + y (cid:48) − y < x, i = 1 , , 3. The delta function constrains fix y (cid:48) i = y i + y (cid:48) − y − x, i = 1 , , 3. Without loss of generality we assume y > y > y .26e change integration variables z i = x − y i , i = 1 , , x = z + z + z + z , z = z (cid:48) + z , z = z (cid:48) + z . The integral is then (cid:90) dz dz (cid:48) dz (cid:48) [ (cid:89) i =4 dz i ] δ ( z + z (cid:48) + z (cid:48) + (cid:88) i =4 z i − × e πi [ m ( z + z )+ n ( z + z )+( n + n − m ) z (cid:48) +( m + m − n ) z (cid:48) ] = I (2 , , , , ( m , n , , m + m − n , n + n − m ) . (A.2)This is a I type integral.2. x < y i + y (cid:48) − y < x, i = 2 , 3, and 0 < y + y (cid:48) − y < x . The delta functionsconstrain y (cid:48) i = y i + y (cid:48) − y − x, i = 2 , 3, and y (cid:48) = y + y (cid:48) − y . Without lossof generality we assume y > y . We change variables z i = x − y i , i = 2 , , x = z + z + z + z , z = z (cid:48) + z . We have y < z + z and this further dividesinto two sub-cases.(a) y < z . Then we define y = z , z = z (cid:48) + z . The delta function constrainis δ ( z + z + z (cid:48) + z + z (cid:48) + z − x ). The exponents is now e πi [( n + n )( z + z )+( m + m )( z + z )+( − n − m ) z (cid:48) +( m + n ) z (cid:48) ] . (A.3)The integral is I (2 , , , , ( m + m , n + n , , m + n , − n − m ), which isa I type integral.(b) z < y < z + z . Then we define z = y − z , z = z (cid:48) + z . The deltafunction constrain is δ ( z + z + z (cid:48) + z (cid:48) + z + z − x ). The exponents is e πi [( n + n )( z + z )+( m + m )( z (cid:48) + z )+( − n − m ) z (cid:48) +( m + n ) z ] . (A.4)The integral is I (2 , , , , ( m + m , n + n , , m + n , − n − m ), which isalso a I type integral.3. x < y + y (cid:48) − y < x , and 0 < y i + y (cid:48) − y < x, i = 1 , 2. The delta functionsconstrain y (cid:48) = y + y (cid:48) − y − x , and y (cid:48) i = y i + y (cid:48) − y , i = 1 , 2. Without loss ofgenerality we assume y < y . Define variables z = x − y , x = z + z + z + z .We have y i < z + z , i = 1 , y < y < z . Then we define y = z , z = y − y , z = z (cid:48) + z + z . Thedelta function constrain is δ ( z + z + z + z + z (cid:48) + z − x ). The exponentsis e πi [ − n ( z + z ) − m ( z + z )+( − n + m − n ) z +( − m + n − m ) z (cid:48) ] . (A.5)The integral is I (2 , , , , ( − m , − n , , − m + n − m , − n + m − n ), whichis a I type integral. 27b) y < z < y < z + z . Then we define y = z , z = y − z , z = z (cid:48) + z , z = z + z (cid:48) . The delta function constrain is δ ( z + z + z + z (cid:48) + z (cid:48) + z − x ).The exponents is e πi [ − n ( z + z ) − m ( z (cid:48) + z )+( − m + n − m ) z +( − n + m − n ) z (cid:48) ] . (A.6)The integral is I (2 , , , , ( − m , − n , , − m + n − m , − n + m − n ), whichis also a I type integral.(c) z < y < y < z + z . Then we define z = y − z , z = y − y , z = z + z + z (cid:48) . The delta function constrain is δ ( z + z + z + z (cid:48) + z + z − x ).The exponents is e πi [ − n ( z + z ) − m ( z (cid:48) + z )+( − m + n − m ) z +( − n + m − n ) z ] . (A.7)The integral is I (2 , , , , ( − m , − n , , − m + n − m , − n + m − n ), whichis also a I type integral.4. 0 < y i + y (cid:48) − y < x, i = 1 , , 3. The delta functions constrain y (cid:48) i = y i + y (cid:48) − y , i =1 , , 3. Without loss of generality we assume y < y < y . Define variables x = z + z + z . We have y i < z + z , i = 1 , , y < y < y < z . Then we define z = y , z = y − y , z = y − y , z = z (cid:48) + z + z + z . The delta function constrain is δ ( z + z + z + z + z (cid:48) + z − x ).The exponents is e πi [( m − n ) z +( n + n − m − m ) z +( − m + n ) z (cid:48) ] . (A.8)The integral is I (5 , , , (0 , m − n , − m + n , n + n − m − m ), which isa I type integral.(b) y < y < z < y < z + z . Then we define z = y , z = y − y , z = y − z , z = z (cid:48) + z + z , z = z (cid:48) + z . The delta function constrain is δ ( z + z + z + z (cid:48) + z (cid:48) + z − x ). The exponents is e πi [( m − n ) z +( − m + n ) z +( n + n − m − m ) z (cid:48) ] . (A.9)The integral is I (5 , , , (0 , m − n , − m + n , n + n − m − m ), which isalso a I type integral.(c) y < z < y < y < z + z . Then we define z = y , z = y − z , z = y − y , z = z (cid:48) + z , z = z (cid:48) + z + z . The delta function constrain is δ ( z + z + z + z (cid:48) + z (cid:48) + z − x ). The exponents is e πi [( m − n ) z (cid:48) +( − m + n ) z +( n + n − m − m ) z ] . (A.10)The integral is I (5 , , , (0 , m − n , − m + n , n + n − m − m ), which isalso a I type integral. 28d) z < y < y < y < z + z . Then we define z = y − z , z = y − y , z = y − y , z = z (cid:48) + z + z + z . The delta function constrain is δ ( z + z + z + z (cid:48) + z + z − x ). The exponents is e πi [( m − n ) z +( − m + n ) z +( n + n − m − m ) z ] . (A.11)The integral is I (5 , , , (0 , m − n , − m + n , n + n − m − m ), which isalso a I type integral.Summarizing the total contributions, taking into account various permutationsof indices, we find S = g (2 I + I + I + I ) . (A.12) A.2 S contribution We only need to consider the first expression for S in (4.3), and the others can besimply obtained by permutations of indices. First we consider the integrals of y , y (cid:48) .There are two cases1. y (cid:48) > y . We define variables z = y − x, z (cid:48) = y (cid:48) − y , z = 1 − y (cid:48) . There is adelta function constrain δ ( z + z (cid:48) + z + x − y , y (cid:48) variablesbecome e πi [(0) z + n z (cid:48) +( n − m ) z +( n − m ) x ] . (A.13)2. y (cid:48) < y . This is simply obtained from the above by switching n → − m , m →− n . delta function constrain δ ( z + z (cid:48) + z + x − 1) is the same. The exponentis now e πi [(0) z − m z (cid:48) +( n − m ) z +( n − m ) x ] . (A.14)Next we consider the integrals of y i , y (cid:48) i , i = 1 , , 3. We assume y (cid:48) > y , with theother cases obtained by the transformation (A.1). We have 0 < y i + y (cid:48) − y < x, i =1 , , . We define z (cid:48) = x − y , z = y and discuss various cases1. x < y i + y (cid:48) − y < x, i = 1 , 2. The delta functions constrain y (cid:48) i = y i + y (cid:48) − y − x, i = 1 , 2. Without loss of generality we assume y > y . Define variables z = z (cid:48) + z , x = z + z (cid:48) + z + z (cid:48) + z . Including the factor e πi ( n − m ) x , theexponents of y i , y (cid:48) i , i = 1 , , e πi [( n + n − m ) z +( − n − m − m ) z (cid:48) +( n + n ) z +( m + m ) z (cid:48) + m z ] . (A.15)There are two contributions. Combining with equation (A.13) we have an inte-gral I ( m , n , − m , − n , − m − n , m − n , n − m , m + n + m + n ) , (A.16)29hich is a I type integral, while combining with equation (A.14) we have anintegral I ( m , n , − m , − n , , m − n , n − m , n − m + n − m ) , (A.17)which is a I type integral.2. x < y + y (cid:48) − y < x, < y + y (cid:48) − y < x . The delta functions constrain y (cid:48) = y + y (cid:48) − y − x, y (cid:48) = y + y (cid:48) − y . Define variables z = x − y , x = z + z + z (cid:48) + z . We have y < z + z (cid:48) , and discuss two sub-cases(a) y < z . Define variables z = y , z = z + z (cid:48) . Including the factor e πi ( n − m ) x , the exponents of y i , y (cid:48) i , i = 1 , , e πi [ − n z +( m + n + n ) z (cid:48) +( − m − n ) z +( − m − m + n ) z (cid:48) +( − m − m ) z ] . (A.18)There are two contributions. Combining with equation (A.13) we have anintegral I ( m , n , − m , − n , m + n , m − n , n − m , m + n + m + n ) , (A.19)which is a I type integral, while combining with equation (A.14) we havean integral I ( m , n , − m , − n , , m − n , n − m , m + n + m + n ) , (A.20)which is a I type integral.(b) z < y < z + z (cid:48) . Define variables z = y − z , z (cid:48) = z + z (cid:48) . Includingthe factor e πi ( n − m ) x , the exponents of y i , y (cid:48) i , i = 1 , , e πi [( m + n + n ) z +( − m − m + n ) z (cid:48) +( − m − n ) z − n z +( − m − m ) z ] . (A.21)There are two contributions. Combining with equation (A.13) we have anintegral I ( m , n , − m , − n , m + n , m − n , n − m , m + n + m + n ) , (A.22)which is a I type integral, while combining with equation (A.14) we havean integral I ( m , n , − m , − n , , m − n , n − m , m + n + m + n ) , (A.23)which is a I type integral.3. 0 < y i + y (cid:48) − y < x, i = 1 , 2. The delta functions constrain y (cid:48) i = y i + y (cid:48) − y , i = 1 , 2. Without loss of generality we assume y < y . Define variables x = z + z (cid:48) + z . We have y i < z + z (cid:48) , i = 1 , 2, and discuss three sub-cases30a) y < y < z . Define variables z = y , z = y − y , z = y + z (cid:48) . Includingthe factor e πi ( n − m ) x , the exponents of y i , y (cid:48) i , i = 1 , , e πi [(0) z +( m − n ) z +( n + n − m − m ) z (cid:48) +( n − m ) z (cid:48) − m z ] . (A.24)There are two contributions. Combining with equation (A.13) we have anintegral I (2 , , , , , ( n − m , , n , − m , m − n , n − m + n − m ) , (A.25)which is a I type integral, while combining with equation (A.14) we havean integral I (2 , , , , ( m , n , , n + n − m , m + m − n ) , (A.26)which is a I type integral.(b) y < z < y < z + z (cid:48) . Define variables z = y , z = y − z , z = z + z (cid:48) , z (cid:48) = z + z (cid:48) . Including the factor e πi ( n − m ) x , the exponents of y i , y (cid:48) i , i = 1 , , e πi [(0) z +( m − n ) z (cid:48) +( n + n − m − m ) z +( n − m ) z (cid:48) − m z ] . (A.27)There are two contributions. Combining with equation (A.13) we have anintegral I (2 , , , , , ( n − m , , n , − m , m − n , n − m + n − m ) , (A.28)which is a I type integral, while combining with equation (A.14) we havean integral I (2 , , , , ( m , n , , n + n − m , m + m − n ) , (A.29)which is a I type integral.(c) z < y < y < z + z (cid:48) . Define variables z = y − z , z = y − y , z (cid:48) = z + z + z (cid:48) . Including the factor e πi ( n − m ) x , the exponents of y i , y (cid:48) i , i = 1 , , e πi [( m − n ) z +( n + n − m − m ) z +( n − m ) z (cid:48) +(0) z − m z ] . (A.30)There are two contributions. Combining with equation (A.13) we have anintegral I (2 , , , , , ( n − m , , n , − m , m − n , n − m + n − m ) , (A.31)which is a I type integral, while combining with equation (A.14) we havean integral I (2 , , , , ( m , n , , n + n − m , m + m − n ) , (A.32)which is a I type integral. 31ummarizing the total contributions, taking into account various permutations ofindices, we find S = g ( I + I + 2 I + 2 I + I + I ) . (A.33) A.3 S contribution We only need to consider the first expression for S in (4.3), and the others canbe simply obtained by permutations of indices. First we consider the integrals of y i , y (cid:48) i , i = 3 , 4. We assume y (cid:48) > y , with the other cases obtained by the transforma-tion (A.1). Define variables z = y − x, z (cid:48) = 1 − y (cid:48) . We have x < y + y (cid:48) − y < − x .There are two cases with a subdivision into a total of three cases1. 1 < y + y (cid:48) − y < − x . The delta function constrain y (cid:48) = y + y (cid:48) − y − (1 − x ).We define variable y = 1 − z , − x = z + z + z (cid:48) + z . The exponents of y i , y (cid:48) i , i = 3 , e πi [ n z +( n − m − m ) z − m z (cid:48) +( n + n − m ) z +( n + n − m − m ) x ] . (A.34)2. x < y + y (cid:48) − y < 1. The delta function constrain y (cid:48) = y + y (cid:48) − y . We have y − x < z + z (cid:48) , which divides into two sub-cases(a) y − x < z . Define variables z = y − x, z = z + z (cid:48) . The exponents of y i , y (cid:48) i , i = 3 , e πi [( n + n − m − m ) z +( n − m ) z (cid:48) +(0) z (cid:48) +( n + n ) z +( n + n − m − m ) x ] . (A.35)(b) z < y − x < z + z (cid:48) . Define variables z = y − x − z , z (cid:48) = z + z (cid:48) . Theexponents of y i , y (cid:48) i , i = 3 , e πi [( n − m ) z +(0) z (cid:48) +( n + n − m − m ) z (cid:48) +( n + n ) z +( n + n − m − m ) x ] . (A.36)Next we consider the integrals of y i , y (cid:48) i , i = 1 , 2. We mainly consider y (cid:48) > y and the results for y (cid:48) < y can be simply obtained by transforming ( m i , n i ) → ( − n i , − m i ) , i = 1 , 2. We define z (cid:48) = x − y (cid:48) , z = y . We have 0 < y + y (cid:48) − y < x and discuss some cases1. x < y + y (cid:48) − y < x . The delta function constrain y (cid:48) = y + y (cid:48) − y − x . Define y = x − z , x = z + z + z (cid:48) + z . The exponents of y i , y (cid:48) i , i = 1 , e πi [ n z +( n − m − m ) z − m z (cid:48) +( n + n − m ) z ] . (A.37)There are three contributions. Combining with equation (A.34) we have anintegral I ( m , n , − m , − n , m + m + n , n + n + m , m + m − n , n + n − m ) , (A.38)32hich is a I type integral, combining with equation (A.35) we have an integral I ( m , n , − m , − n , − m − n , m − n , n − m , m + n + m + n ) , (A.39)which is a I type integral, and combining with equation (A.36) we have anintegral I ( m , n , − m , − n , − m − n , m − n , n − m , m + n + m + n ) , (A.40)which is also a I type integral.2. We transform equation (A.37) by ( m i , n i ) → ( − n i , − m i ) , i = 1 , 2, and get afactor e πi [ − m z +( − m + n + n ) z + n z (cid:48) +( − m − m + n ) z ] . (A.41)Again there are three contributions. Combining with equation (A.34) we havean integral I ( m , n , − m , − n , m + m + n , n + n + m , m + m − n , n + n − m ) , (A.42)which is a I type integral, combining with equation (A.35) we have an integral I ( m , n , − m , − n , − m − n , m − n , n − m , m + n + m + n ) , (A.43)which is a I type integral, and combining with equation (A.36) we have anintegral I ( m , n , − m , − n , − m − n , m − n , n − m , m + n + m + n ) , (A.44)which is also a I type integral.3. 0 < y + y (cid:48) − y < x . The delta function constrain y (cid:48) = y + y (cid:48) − y . We have y < z (cid:48) + z and discuss some sub-cases(a) 0 < y < z . Define z = y , z = z + z (cid:48) , x = z + z (cid:48) + z (cid:48) + z . Theexponents of y i , y (cid:48) i , i = 1 , e πi [( n + n − m − m ) z +( n − m ) z (cid:48) +(0) z (cid:48) +( n + n ) z ] . (A.45)There are three contributions. Combining with equation (A.34) we havean integral I ( m , n , − m , − n , − m − n , m − n , n − m , m + n + m + n ) , (A.46)33hich is a I type integral, combining with equation (A.35) we have anintegral I (2 , , , , , ( n + n − m − m , , n + n , − m − m , n − m , − n + m ) , (A.47)which is a I type integral, and combining with equation (A.36) we havean integral I (2 , , , , , ( n + n − m − m , , n + n , − m − m , n − m , − n + m ) , (A.48)which is also a I type integral.(b) We transform equation (A.45) by ( m i , n i ) → ( − n i , − m i ) , i = 1 , 2, and geta factor e πi [( n + n − m − m ) z +( n − m ) z (cid:48) +(0) z (cid:48) +( − m − m ) z ] . (A.49)Again there are three contributions. Combining with equation (A.34) wehave an integral I ( m , n , − m , − n , m + n , m − n , n − m , m + n + m + n ) , (A.50)which is a I type integral, combining with equation (A.35) we have anintegral I (2 , , , , ( m + m , n + n , , m + n , − m − n ) , (A.51)which is a I type integral, and combining with equation (A.36) we havean integral I (2 , , , , ( m + m , n + n , , m + n , − m − n ) , (A.52)which is also a I type integral.(c) z < y < z + z (cid:48) . Define variables z = y − z , z (cid:48) = z + z (cid:48) , x = z + z (cid:48) + z + z . The exponents of y i , y (cid:48) i , i = 1 , e πi [( n + n − m − m ) z +( n − m ) z +(0) z (cid:48) +( n + n ) z ] . (A.53)We notice this is just (A.45) with the index switch 1 ↔ 2. So we cansimply obtain the remaining results by index switching the last two sub-cases. 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