Towards Amplituhedron via One-Dimensional Theory
Maysam Yousefian∗ and Mehrdad Farhoudi†
Department of Physics, Shahid Beheshti University G.C., Evin, Tehran 19839, Iran (Dated: July 8, 2021) Inspired by the closed contour of momentum conservation in an interaction, we introduce an in- tegrable one-dimensional theory that underlies some integrable models such as the Kadomtsev- Petviashvili (KP)-hierarchy and the amplituhedron. In this regard, by defining the action and partition function of the presented theory, while introducing a perturbation, we obtain its scatter- ing matrix (S-matrix) with Grassmannian structure. This Grassmannian corresponds to a chord diagram, which specifies the closed contour of the one-dimensional theory. Then, we extract a solu- tion of the KP-hierarchy using the S-matrix of theory. Furthermore, we indicate that the volume of phase-space of the one-dimensional manifold of the theory is equal to a corresponding Grassmannian integral. Actually, without any use of supersymmetry, we obtain a sort of general structure in com- parison with the conventional Grassmannian integral and the resulted amplituhedron that is closely related to the Yang-Mills scattering amplitudes in four dimensions. The proposed theory is capable to express both the tree- and loop-levels amplituhedron (without employing hidden particles) and scattering amplitude in four dimensions in the twistor space as particular cases.
PACS numbers: 02.30.Ik; 05.45.Y v; 11.10. − z; 02.10.Kn Keywords: Amplituhedron; KP-Hierarchy; Integrable Models; Knot Theory
I. INTRODUCTION and also each material particle has mass. On the other hand, it has not been specified/discussed why the number One of the key points to understand particle physics is of dimensions should be exactly four in the amplituhed- the scattering amplitudes. On the other hand, the stand- ron and, in general, the scattering amplitude issues. ard method of calculating these scattering amplitudes In this work, we show that interesting integrable mod- is to apply the Feynman diagrams, which have many els can have a common foundation and being raised from complications [1–3]. These complications have always a single root. For this purpose, inspired by the fact that been a motivation for physicists to simplify their calcu- geometric representation of the momentum conservation lations. A specific example of such efforts is the Park- of participated particles in an interaction leads to the Taylor formula, which describes the interaction between formation of a closed contour, we introduce a theory gluons [4, 5]. in one dimension that underlies some integrable mod- As another example, in Ref. [6], it has been shown els in various dimensions. The important kinds of integ- that one can, with correct factorization, give a recurs- rable models that we are considering here are the KP- ive expression in terms of amplitudes with less external hierarchy [9], two-dimensional models [10–12], the amp- particles, in which a scattering amplitude is broken down lituhedron [7]. In the proposed theory, as there is only into simpler amplitudes. Also recently, in Ref. [7], by one dimension, in analogy with the closed contour resul- employing the recursive expression used in Ref. [6] and ted from the momentum conservation, there is only one the twistor theory, a geometric structure has been pro- point that evolves in itself, hence we refer to this theory posed that is coined amplituhedron. This structure en- as a point theory (PT). ables simplification of the computation of scattering amp- In the following, we first define the corresponding ac- litudes in some quantum field theories such as super- tion, and then its equation of state and path integral. arXiv:1910.05548v5 [hep-th] 8 Jul 2021 Yang-Mills theory. The amplituhedron challenges space- Also, we present a specific symmetry of the PT. Af- time locality and unitarity as chief ingredients of the terward, by defining S-matrix of the PT, we show the standard field theory and the Feynman diagrams. Hence, Grassmannian structure of the S-matrix of the PT, and due to spacetime challenges at the scale of quantum grav- in turn, we specify its closed contour. Thereupon, we ity [8] and since locality will have to be remedied if grav- define a hierarchy transformation by which, we repres- ity and quantum mechanics ought to coexist, the amp- ent some integrable models based on the PT. Thus, we lituhedron is of much interest. However, the presented express a vacuum solution in the PT as a solution of amplituhedron requires supersymmetry and introduces the KP-hierarchy. Accordingly, under these considera- the scattering amplitude of massless particles, neverthe- tions, we endeavor to move towards the amplituhedron less no trace of supersymmetry has been seen up to now, without employing supersymmetry. For this purpose, we indicate the tree and loop levels of the amplituhedron as special cases of the contour integrations of the volume of phase-space of the closed contour, wherein the volume ∗ M Yousefi[email protected] itself is as a Grassmannian integral. Finally, we discuss † [email protected] the physical meaning and importance of the linked closed 2
contour. In this way, due to the twistor theory [15], each point (number) in the subspace M determines a line in the null projective twistor space, in which each twistor, say $, has II. FOUNDATION OF POINT THEORY coordinates $ ≡ ((ζ1, ζ2), (ζ3, ζ4)) ≡ (λ, µ). Also, one can specify a point in the null projective twistor space by It is convenient to work in the natural units (i.e., c = two points or a null line in the M. Second, we determine 1 = ~) with an action on a real 1-dimensional compact n different values of the parameter z, as different numbers Hausdorff manifold, say C as a vacuum, namely (points) zi ∈ M for i = 1, ··· , n, which fix the manifold C (see, e.g., Fig. 1). Now, due to 1 1 S = dS(z) ≡ − ψ(z)ϕ(z)dz, (1) κ ˛ κ ˛ ( ( C C λizi = µi λizi = µi 1 and (3) where ψ(z) and ϕ(z) are two fermionic fields over the λizi+1 = µi λi+1zi = µi+1, manifold, κ is a constant and z is the parameter of the manifold C. we choose the constraint such a way that Each real 1-dimensional compact Hausdorff manifold ||(z − z )||2 ≡ (z − z )(z − z )? = 0. (4) (e.g., knots and links), generally, has its own global topo- i i+1 i i+1 i i+1 logical properties. For instance, one of these features has been expressed in the Fary-Milnor theorem [13], which (using a specified local scalar curvature at any point on the curve) explains whether or not a 1-dimensional manifold is a knot. However, global topological prop- erties of compact manifolds cannot be determined by local coordinates. For this reason, non-local or higher- dimensional coordinates, e.g. complex numbers, can be used instead of real local coordinates to express the global properties. Therefore, if the corresponding manifold is a circle, as real projective line, then its corresponding parameter will at least be a unit complex number. In the case that the manifold C is an unknot and unlinked disconnected com- pact manifold (e.g., several circles), then the parameter z can at least be a complex number. Also, it is known that a knot is an embedding of a circle into 3-dimensional Euc- lidean space R3 [14] and R3 is a subspace of 1-dimensional biquaternion2 space. Now, if the manifold C is a knotted and/or a linked one, then its parameter z will be a mem- ber of a special subspace (say, M, which corresponds to 3 Figure 1. A 1-dimensional manifold in M. Minkowski space wherein R ⊂ M) of biquaternion space having3 Q∗ = Q?, with an additional constraint on the M. An additional constraint is required because the sub- As mentioned earlier, the 1-dimensional manifold cor- space M corresponds to 4-dimensional Minkowski space responds to the closed contour resulted from the mo- that has one dimension more than R3. mentum conservation of participated particles in an in- Before we determine such a constraint, first we write teraction. In this regard, the closed contour can be con- each number of the subspace M (say, Q) in the unitary sidered as a knot, and hence, the topological properties matrix representation in terms of the Euler parameters, of the knot can be related to some of the interaction namely properties. On the other hand, the closed contour can be a link. However in this research, we explain the basis a0 + ia3 ia1 + a2 Q = . (2) of the PT and its relation with the scattering amplitude ia1 − a2 a0 − ia3 and amplituhedron, and will examine the characteristics of the states as knot and link in another research. Nev- ertheless, in the following, we briefly discuss the physical 1Actually, these two fields are not independent, as it will be shown interpretation of the states as linked closed contour. through the variation process. The variation of action (1) gives two trivial solutions 2A biquaternion number Q is a kind of number that is defined as n P3 o Q := a0 + aiei | {ei, ej } = −2 δij ; a0, ai ∈ C , where ei’s δ∂ S(z) δ∂ S(z) i=1 z = 0 = z ⇒ ψ(z) = 0 = ϕ(z), (5) are quaternion basis. δϕ(z) δψ(z) 3The sign ? on Q? is the quaternion conjugate and is defined as ? P3 ∗ Q ≡ a0 − i=1 aiei, whereas Q is the complex conjugate of Q, as the Euler-Lagrange equations. However, there is also a ∗ ∗ P3 ∗ i.e. Q = a0 + i=1 ai ei. non-trivial solution that, without loss of generality about 3
the constant of integration, can set to be where S is the Seifert surface that corresponds to the closed curve C containing the point z = 0, S0 is the in- δ∂zS(z) = 0 =⇒ ψ(z)ϕ(z) = 1. (6) finitesimal closed neighborhood of z = 0 in the Seifert surface, and C0 is the boundary of S0. As ∂z? ∂z ln z on This solution is the continuity equation in one dimen- the surface S − S0 is zero, its integration over the sur- sion wherein ψ(z)ϕ(z) is the conserved current within it. face S −S0 vanishes. Therefore, the only remaining term Considering the non-trivial case, we choose an arbitrary of relation (12), as a contour integration over a general solution closed contour (e.g., knotted or linked closed contour), is
(6) the last term. That is, we have [20] ψ(z) ≡ eS(z) ==⇒ ϕ(z) = e−S(z), (7) z−1dz = z−1dz = 2πi , (13) ˛ ˛ 0 which, after quantization, it will be clear that is a good C C0 choice. Also, as there is no derivative of the field in the where, by i , we mean a specific choice of i that, without Lagrangian, the field momentum is zero. Thus, the en- 0 loss of generality, we choose it to be i = e . ergy density is 0 1 Accordingly, by employing (1), (9) and (13), the cor- h(z) = ψ(z)ϕ(z), (8) responding path integral of the PT can be written as P ψpϕp which, due to Eq. (6), is a constant.4 Also, for later on Z = DψDϕ e p , (14) ˆ uses, we expand ψ(z) and ϕ(z) as5
P −(p+1/2) where we have set the constant κ in action (1) equal to ψ(z) = ψpz , Q Q p∈ +1/2 2πe1, Dψ = dψp and Dϕ = dϕp. Then, Z p∈ +1/2 p∈ +1/2 P −(p+1/2) Z Z ϕ(z) = ϕpz , (9) using the path integral (14) and the Berezin integral [21], p∈ +1/2 Z we can calculate the many-point correlation function. Al- where ψp and ϕp are coefficients of the expansion. ternatively, to calculate the many-point correlation func- At this stage, similar to complex numbers, we extend tion, we can use the canonical quantization of ψp and ϕp. the Cauchy integral theorem to any biquaternion number In this way, we have the Clifford algebra with fermionic ˆ z ∈ M. In this regard, for each real number l 6= 1, we operators ψp andϕ ˆp that satisfy the anti-commutators have ˆ ˆ ˆ {ψp, ψq} = 0, {ϕˆp, ϕˆq} = 0 and {ψp, ϕˆq} = δp,−q, l6=1 z−ldz = −(l − 1)−1 d(z−l+1) = 0. (10) (15) ˛C ˛C where p and q ∈ Z + 1/2. To give a representation of this algebra, we define the vacuum state as When l = 1, it is known that one can write the point z ˆ with arbitrary parameters x, y ∈ R as ψp|0C >= 0 (p > 0), ϕˆp|0C >= 0 (p > 0), ˆ < 0C|ψp = 0 (p < 0), < 0C|ϕˆp = 0 (p < 0). (16) z = x + yn, (11) Using quantum forms of expansions (9), the quantum P3 2 form of the Lagrangian is where n = i=1 aiei with n = 1. In this case, us- ing the Stokes-Cartan theorem [17], the Wick rotation ˆ X ˆ −l method [18, 19], and transformations i = −hn and ∂zS(z) = − Sl−1z , (17) 2 y¯ = hy, wherein h = −1, then z = x +y ¯i. Hence, l∈Z we get where
−1 ? ˆ X ˆ z dz = dzdz ∂ ? ∂ ln z Sl−1 = : ψpϕˆl−p−1 :, (18) ˛ ˆ z z C S (12) p∈Z+1/2 ? −1 = dzdz ∂ ? ∂ ln z + z dz, ˆ z z ˛ and it is compatible [16] with the quantum forms of solu- S−S0 C0 tions (7), namely
ψˆ(z) =: eSˆ(z) : andϕ ˆ(z) =: e−Sˆ(z) : . (19) 4 As the fields ψ and ϕ have no momentum, and hence no dynam- ˆ ics, the one dimensional manifold of the PT is classically without In the above relations, :a ˆb : is the normal ordered form length. ofa ˆˆb. 5In the continuation of this section and a few later sections, we em- ploy the same formulation mentioned in Refs. [9, 16]. However, the lack of the standard structure of a field theory (such as Lagrangian Symmetry of Point Theory: According to solu- and field equations) is the main difference. Besides, Refs. [9, 16] tions (19), the field ψˆ(z) is equal to its partition func- have been formulated only in real one dimension. tion. Thus, due to the quantization process, it means 4
ˆ that the theory will not change after quantization. This Each S(k) is equivalent to the simultaneous interaction 0 0 symmetry is a special feature of the PT that results the of k ordered points of n specific points (z1, ··· , zn) within 6 ˆ uniformity of physics in large and small scales. For in- C. Also, each S(k) has a Grassmannian structure with a stance, the result of such a symmetry makes the physics Grassmannian G(k, n), say C, as of a statistical set of leptons and quarks being similar to 0 0 physics of each one of those.7 C1(z1) ··· C1(zn) . .. . C ≡ . . . , (25) C (z0 ) ··· C (z0 ) III. S-MATRIX AND GRASSMANNIAN k 1 k n STRUCTURE ˆ in a way that < 0C|S(k)|0C > being equal to the sum of minors of the Grassmannian C. To express the S-matrix, as an evolution operator for On the other hand, each minor of the Grassmannian C initial state |iC > and final state |fC > while using the is the probability of simultaneous interaction of k specific free Hamiltonian and relations (1), (10), (13) and (17), 0 0 ordered points of (z1, ··· , zn) within C. Also, due to we obtain the conditional probability in the probability theory, the 0 0 0 − 1 ψˆ(z)ϕ ˆ(z)dz Sˆ probability that each point za of (z1, ··· , zn) interacts == . Sif C ¸ C C C with the manifold in the α’th order is (20) In other words, using relations (15) and (18), one equi- c1a ··· c1,a+α−1 valently gets . .. . det . . . 1 ψˆ(z)ϕ ˇ(z)dz 0 − 1 ψˆ(z)ϕ ˆ(z)dz 0 cαa ··· cα,a+α−1 :e κ C : ψˆ(z ):e κ C := e ψˆ(z ), ¸ ¸ c¯αa ≡ , (26) (21) c1a ··· c1,a+α−2 where e is the Euler number. That is, we have a trivial . .. . det . . . case wherein the initial and final states are the same. c ··· c However, adding a perturbation to the free Hamilton res- α−1,a α−1,a+α−2 ults a general nontrivial expression for the S-matrix in the where these two determinants are the probability of sim- PT, wherein the initial state evolves. In this respect, we ultaneous interaction of (α − 1) ordered points and the define a perturbed Hamiltonian as the interaction of su- same (α−1) ordered points plus one extra point of n spe- 0 0 0 perposition of n specific points of C with the other points, cific points (z1, ··· , zn) within C, and each cαa ≡ Cα(za) namely is the element of the matrix C. Thus, due to the number of minors of the Grassman- ψˆ(z)ϕ ˆ(z) → ψˆ(z)ϕ ˆ(z)+ nian C (that is equal to the combination Cn) and given (22) k ι C(z)ψˆ(z)ϕˆ(z0 ) + ··· +ϕ ˆ(z0 ), that the sum of the probabilities of all possible states 1 n of the simultaneous interaction of k ordered points of n 0 0 where ι is a constant and C(z) for a = 1, ··· , n is the specific points (z1, ··· , zn) within C is equal to one, by probability function of interaction of ψˆ(z) with one or employing the Wick theorem [16] more points of C. Accordingly, we express the S-matrix ˆ ˆ 0 0 < 0C|ψ(z1) ··· ψ(zn)ϕ ˆ(z ) ··· ϕˆ(z )|0C >= operator in terms of the Dyson series as 1 n ˆ 0 det < 0C|ψ(zi)ϕ ˆ(zj)|0C > , (27) n k " n # X( ι )k Y X ˆ = κ : C (z )ψˆ(z )ϕ ˆ(z0 ) : dz and relations (9) and (13) that result S k! ˛ α α α a α k=1 α=1 C a=1 n 1 ˆ 0 X ˆ <0C| ψ(z)ϕ ˆ(z )dz|0C >= 1, (28) ≡ S(k), (23) κ ˛C k=1 we obtain wherein the constant ι is determined in a way that n Ck k! ˆ X ˆ <0C|S(k)|0C >= Mh = 1. (29) < 0C|S|0C >= 1, (24) ιk h=1 and given that the product of two identical Fermion fields Furthermore, the minors M of the Grassmannian C, ˆ h is zero, then S(k) for k > n vanishes. as probability of simultaneous interaction of k specific 0 0 points of (z1, ··· , zn) within C, are, in general, posit- ive numbers. Therefore, the Grassmannian C is a pos- itive Grassmannian. In addition, in Ref. [24], it has 6Also, in Ref [22], we have shown that, on macro scales, elastic waves in asymmetric environments behave analogous to QED. been shown that each non-negative Grassmannian cor- 7In Ref. [23], we have proposed a way toward a quantum gravity responds to a chord diagram. In this way, as a chord dia- based on a ‘third kind’ of quantization approach, in which that gram is a closed contour, each non-negative Grassman- similarity has been explained. ˆ nian, and in turn each S(k), specifies a closed contour C. 5
Moreover, since a Grassmannian G(k, n) is a space that to a diffeomorphism as parameterizes all k-dimensional linear subspaces of the n-dimensional vector space, if we mod out it by the gen- H(x) = − ξ(z0, x)dz0ψ(z0)ϕ(z0)/(2πe ) eral linear group GL(k), there will be no change in it. ˛ 1 C (33) Then, dividing relation (29) by one of Mh or a constant, 00 0 00 0 00 makes no change in C. Eventually, it is obvious that the = − dz ψ (z )ϕ (z )/(2πe1), ˛C0 S-matrix in the PT has a Grassmannian structure. where dz00 ≡ ξ(z0, x)dz0, ψ0(z00)ϕ0(z00) = ψ(z0)ϕ(z0) and C0 is a diffeomorphism of C. Hence, after some calcula- IV. HIERARCHY TRANSFORMATION AND tions [16], transformation (31) reads KP-HIERARCHY ( ψˆ(z) → e−ξ(z,x)ψˆ(z) ≡ ψˆ(z, x) (34) To investigate the integrability and solitonic proper- ϕˆ(z) → eξ(z,x)ϕˆ(z) ≡ ϕˆ(z, x). ties of the PT, we inquire relation between the PT and the KP-hierarchy. The KP-hierarchy is based on math- ˆ ematical foundation in terms of a Grassmannian vari- Now, we define S(k,x) as ety [9, 16, 24]. It is supplemented with a number of k " n # ( ι )k classical and non-classical reductions that contain many ˆ κ Y X ˆ 0 S(k,x)≡ : Cα(zα)ψ(zα, x)ϕ ˆ(za) :dzα. (35) integrable equations and solitonic models [9]. Indeed, the k! ˛C KP-hierarchy is a hierarchy of nonlinear partial differen- α=1 a=1 tial equations in a function, say τ(x), of infinite number Since, the function τ(x), as a solution of the KP- of variables xl for l = 1, ··· , ∞ as hierarchy coordin- hierarchy, is define as ates. As the KP-hierarchy is an integrable system, the plausible (physical) meaning of the hierarchy coordinates k! τ(x) = <0 |ˆ |0 >= Wr(f , ··· , f ), (36) can be related to the infinite numbers of possible con- ιk C S(k,x) C 1 k served quantities that any KP-hierarchy soliton solution can contain [25, 26]. Such a τ(x) is a function that should where Wr(f1, ··· , fk) is the Wronskian of functions fα satisfy the bilinear identity [9] for α = 1, ··· , k as each fα is
0 0 0 −ξ(z ,x) −ξ(z ,x) eξ(k,x−x )τ x − (k)τ x0 + (k)dk ≡ 0, (30) fα ≡ cα1e 1 , ··· , cαne n . (37) ˛C At this stage, as in Refs. [16, 24, 27], it has been shown P∞ l where ξ(k, x) ≡ xlk , k ∈ M is an arbitrary 9 l=0 that the Wr(f1, ··· , fk) is a solution of the KP-hierarchy, 2 point in the manifold C, (k) ≡ 1, 1/k, 1/(2k ), ··· , thus we conclude that a vacuum solution in the PT is 0 and arbitrary variables x ≡ (x0, x1, x2, ··· ) and x ≡ equivalent to a solution of the KP-hierarchy. 0 0 0 0 (x0, x1, x2, ··· ) with xl, xl ∈ M. To show the relation between the PT and the KP- hierarchy, we employ a transformation, which we will V. PHASE-SPACE OF C refer to it as a hierarchy transformation (HT), in the form Consider a quantum system with a Hamiltonian op- ( erator Hˆ and its basis states. The time evolution of ψˆ(z) → eH(x)ψˆ(z)e−H(x) the probability of a state of the system, say p , in this HT: H(x) −H(x) (31) i ϕˆ(z) → e ϕˆ(z)e , Hamiltonian basis is
where ∂tpi = iHˆpi = iHipi (38)
0 0 H(x) ≡ ξ(z , x)dS(z )/(2πe1). (32) that leads to ˛C d ln pi = iHidt. (39) Given that the function ξ(k, x), for constant parameters x ≡ (x0, x1, x2, ··· ), is regular with respect to k, rela- For a one-dimensional quantum system, relation (39) is tion (32) means that the closed contour C in relation (1), proportional to the volume element of phase-space of the without changing the topological properties,8 is subjected
9Incidentally, in Refs. [10–12], it has been shown that the n-point 8 Via a smooth variation of x ≡ (x0, x1, x2, ··· ), the function ξ(k, x) correlation function of fields of a kind of string theory can be ob- can tend to one. In this way, C0 smoothly tends to C. tained via the KP-hierarchy solutions. 6
state pi. In this way, the volume element of phase-space Then, due to the property of the matrix C, this matrix of total system is proportional to z contains k independent column-vectors, which (within the first case) are also independent from 4 n-component Y d ln pi. (40) vector $a. Accordingly, we can choose a matrix, say Z, i consists of n row-vectors Za ≡ (ζa1, ··· , ζa4, za1, ..., zak), which contains (4+k) independent n-component column- Inspired by this explanation, due to definition (26) and vectors as a basis for the above (4+k)-plane. Such a basis as the Grassmannian C specifies the closed contour C, and the above constraint eliminate the gauge redundancy we (plausibly) assume that the volume element of phase- GL(k). In this case, we can assume a polygon in a (3+k)- space of the closed contour C is10 dimensional projective space whose n vertices are Za’s. k,n At this stage, we satisfy the elimination of the gauge Y d lnc ¯αa. (41) redundancy GL(k) and also the application of those two α,a=1 mentioned cases (44) by imposing the Dirac delta func- tions to the volume element of the phase-space as Then, using definition (26) and k×n d cαa 0 c1,b ··· c1,b+k−1 k×k k ×4 n δ (y − Ik×k)δ (y − 0k0×4). (46) . .. . Q Mb ≡ det . . . , (42) Mb b=1 ckb ··· ck,b+k−1
k×k expression (41) reads as Indeed, the Dirac delta functions δ (y − Ik×k) elim- inate the gauge redundancy GL(k) and the Dirac delta k×n k0×4 d cαa functions δ (y − 0k0×4) impose those two mentioned n . (43) Q cases of (44). Therefore, expression (46) indicates that Mb b=1 the volume of the phase-space is equal to a correspond- ing Grassmannian integral.12 In addition, due to defini- However, there are two points that must be considered tion Mb’s in (42), the volume element, and in turn the in order to define the volume element of the phase-space corresponding Grassmannian integral, is obviously cyclic- of C. First, as the Grassmannian C is invariant under invariant. general linear group GL(k), the volume element of the phase-space of it must be mod out by GL(k) ‘gauge’ redundancy. The second point is about the relation between the Grassmannian C and the coordinates za. VI. FOUR-DIMENSIONAL Given relation (3), twistors, say $a = (ζa1, ··· , ζa4), can TREE-AMPLITUHEDRON correspond to the lines between two points za and za−1 in analogy with the momentum of particles in the closed In this section, we endeavor to show that the volume of contour of momentum conservation. Accordingly, if we the phase-space of C is equivalent with the amplituhed- define11 ron in a four-dimensional spacetime. For this propose, first we indicate that the tree-amplituhedron in a four- cαa$a ≡ yα, (44) dimensional spacetime can be deduced from the first case two different cases will be possible. One case is that the of definition (44). Actually, in this case, the integration inner product of the 4 n-component vector $ to each of expression (46) becomes of the k n-component row-vectors of the C matrix being zero (i.e., all of yα = 0). The other case is that some of k×n d cαa k×(4+k) yα’s being zero. n δ (Y − Y0) ≡ An,k, (47) ˆ Q In the above first case, there would be a (4 + k)-plane. Mb Now, we employ some auxiliary parameters, say a matrix b=1 z made of elements zaα, as where Y is a [k × (4 + k)]-matrix generally defined as C · z ≡ y with constraint: y = Ik×k. (45) Y ≡ C · Z with constraint Y0 = (0k×4, Ik×k). Further- more, consider each cell decomposition (say T ) of the polygon with the n vertices Za’s, which has a few cells (say Γ’s), wherein any cell of it, as a particular example 10Given that each Grassmannian specifies a chord diagram and each chord diagram determines a one-dimensional manifold, hence the number of dimensions of the phase-space of the one-dimensional manifold is equal to the number of degrees of the Grassmannian freedom. 11We use the Einstein summation rule, unless it is specified. 12For definition of the Grassmannian integral, see, e.g., Refs. [2, 28]. 7
with Grassmannian13 Moreover, relation (52), at k = 2 and n = 6 with choice (53), is structurally similar to relation (3.18) mentioned c ··· c 1,a(1) 1,a(4+k) in Ref. [31]. Incidentally, as the Grassmannian integral . .. . . . . , (48) (47) is cyclic-invariant, the resulted relation (52) from it is cyclic-invariant as well. ck,a ··· ck,a (1) (4+k) As the scattering amplitude is typically defined in is specified as the momentum twistor space, and the above relation does not also indicate the energy-momentum conserva- k+3 X tion, it is not suitable to calculate the scattering amp- Yα ≡ cα,a(1+i) Za(1+i) . (49) litude. Thus, by change of variables via the Fourier i=0 transformation of expression (47), similar to Ref. [32], we transform An,k into the momentum twistor space where In here, Yα is αth row-vector of the defined matrix Y but with (4 + k) components, wherein k × (n − k − 4) new results emerge. That is, we write components of the matrix C have been set to be zero. ˜ Hence, using (49), each of the corresponding minors (42) d2nµ eiλb∧µb A (λ , µ , z ) = ˆ a n,k a a aα and the related multiplication of differential elements for k×n any cell are d cαa 2k k×k 2k 2n ˜ n δ (cαbλb)δ (χ) d ραδ (λa − ρβcβa), ˆ Q ˆ < Y,Za , ··· ,Za > Mc (1+i) (4+i) c=1 Ma(i) = (50) < Za(1) ,Za(2) , ··· ,Za(4+k) > (54)
and where we have used the Dirac delta function expression δ2k(c µ ) = d2kρ exp (−iρ c ∧ µ ) with arbitrary 4+k 4+k αa a α β βa a Y d Yα dc = . (51) ρα ≡ (ρα1, ρα2´), χ ≡ y − Ik×k, λa ≡ (ζa1, ζa2), µa ≡ αa(i) < Z , ··· ,Z > ˜ i=1 a(1) a(4+k) (ζa3, ζa4) and λa ≡ (ϑa1, ϑa2) is the Fourier conjugate of µa. Then, in relation (54), we first integrate over cµa for, Now, by inserting relations (50) and (51) into expres- without loss of generality, µ = 1, 2 and afterward, over sion (47), the volume of the phase-space becomes ρα. Hence, after some manipulations, it reads
dk×(4+k)Y
cef ˜ ˜ δa−1,a,a+1 < λc, λe > δfb Qab ≡ , (57) 2 < λ˜ , λ˜ >< λ˜ , λ˜ > 13In fact for these particular examples, the components of the corres- a−1 a a a+1 ponding Grassmannian are determined by the Dirac delta functions cef in the Grassmannian integral. And, the rest of components of the where δa−1,a,a+1 is the generalized Kronecker delta, then C matrix in the Grassmannian integral are removed by contour abc ˜ ˜ ˜ the Schouten identity, i.e. < λa, λb > λc ≡ 0, implies integration. In other examples of cells/contour integration, the Q λ˜ = 0. Accordingly, if we replace c with c (δ − corresponding Grassmannian integral leads to other solutions that ab b αbˆ αaˆ ab ˜ ˜ can principally be obtained in similar process of these particular Qaa0 < λa0 , λb >) in expression (55), it will not change. examples. Of course, to avoid complicating the issue in this work, Also, we define [(k − 2) × n]-matrix G with elements gαaˆ we will present some other examples in further studies. as gαbˆ ≡ cαaˆ Qab, which, for fixed b, it implies the relation 8
between minors (42) and the minors of matrix G as particular choice (53). Indeed, in expression (61), if the variablesz ˜ are replaced by the multiplication of two ˜ ˜ ˜ ˜ aαˆ < λb, λb+1 > ··· < λb+k−2, λb+k−1 > fermionic variables (named bosonization [29]), one will Mb = Nb+1 , (58) < ρ1, ρ2 > arrive at the same common amplituhedron presented in Refs. [2, 7, 28, 32]. where Nb’s are independent minors of G as Nb = ˆ det[gα,bˆ +βˆ−1] with the minors indicesα ˆβ. Then, by em- ploying relation (58) and the gauge constraints cαaˆ zaµ = VII. FOUR-DIMENSIONAL 0, and definingµ ˜b as Qabµ˜b ≡ λa andz ˜bαˆ as Qabz˜bαˆ ≡ LOOP-AMPLITUHEDRON 14 zaαˆ into expression (55), after some manipulations, it becomes To get the loop-amplituhedron in a four-dimensional kˆ×n spacetime, we employ the second case of definition (44) MHV d gαaˆ 2kˆ ˜ 2kˆ kˆ×kˆ An n δ (gαbˆ λb)δ (gαeˆ µ˜e)δ (ˆχ ˆ), (59) and, without loss of generality, consider that (k − l) of ˆ Q αˆβ Nc yα’s are equal to zero. Also, we again assume that za ∈ c=1 M. Thus, we get a (4 + k − l)-plane, wherein we choose where (4+k−l) independent n-component column-vectors Za ≡ (ζ , ··· , ζ , z , ··· , z ) as a basis for it. Hence, a1 a4 a1 a,k−l 4 4 ˜ < ρ1, ρ2 > δ |λb >< λb| expression (47) changes as MHV An ≡ n . (60) Q ˜ ˜ k×n < λa, λa+1 > d cαa [k×(4+k−l)−4l] n δ (Y − Y0) ≡ An,k−l,l, (63) a=1 ˆ Q Mb This result is related to the corresponding one men- b=1 4 tioned in Ref. [2] except that < ρ1, ρ2 > has been re- placed by fermionic delta function. In fact, definition (60) where, here, the corresponding Y is a [k × (4 + k − l)]- can be considered as a sort of general of the one men- matrix with constraint tioned in Ref. [2], which with the particular choice (53), 0(k−l)×4 I(k−l)×(k−l) is equivalent to the one mentioned there. Eventually, in Y0 = . (64) analogous with expression (47), expression (59) actually 0l×4 0l×(k−l) reads To proceed, due to relations (50) and (51) for the C MHV ˜ An An,kˆ(λa, µ˜a, z˜aαˆ). (61) matrix, we need to increase the number of components of the corresponding vectors Za’s from (4+k −l) to (4+k). 0 By substituting the corresponding relations (50) and (51) Accordingly, we define soft row-vectors Zn+i’s, with (4 + into expression (61), it gives a simple contour integration k)-component for i = 1, ··· , l, and new (4+k)-component 0 that reads vectors Za’s, for each a, through their components as MHV ˜ ˜ 4 An
With appropriate contour integration, expression (67) for tion [33] of each pair of (Yk−l+1, ··· , Yk) with l/2 loops. MHV a few fixed values of n, k and l, becomes An An,kˆ−l. Moreover, for l as even numbers, four-dimensional As an example, for a simple contour integration of (66), loop-amplituhedron arises from the ‘entangled’ integra- using relations (50) and (51), with Grassmannian13
c1,a(1) ··· c1,a(4+k−l) c1,a(5+k−l) ··· c1,a(4+k) ...... ...... ck−l,a(1) ··· ck−l,a(4+k−l) ck−l,a(5+k−l) ··· ck−l,a(4+k) , (68) ck+1−l,a(1) ··· ck+1−l,a(4+k−l) ck+1−l,a(5+k−l) ··· ck+1−l,a(4+k) ...... ......
ck,a(1) ··· ck,a(4+k−l) ck,a(5+k−l) ··· ck,a(4+k)
into expression (66), it reads and after some manipulations, it becomes
dk×(4+k) 0
4 4 ˜ ˜ 3 MHV d y1d y2 < Z1, ··· , Z4 > A4 . (71) ˆ < Z˜2, Z˜3, Z˜4, Y1 >
Then, by using the ‘entangled’ contour of integration [33], result of (71) gives
4 4 ˜ ˜ 2 MHV d y1d y2 < Z1, ··· , Z4 > A4 , (72) ˆ vol[GL(2)] < Y1, Y2, Z˜1, Z˜2 >< Y1, Y2, Z˜2, Z˜3 >< Y1, Y2, Z˜3, Z˜4 >< Y1, Y2, Z˜4, Z˜1 >
that is equivalent to the one mentioned in Refs. [2, 33] VIII. DISCUSSION except that the variablesz ˜aαˆ’s must be replaced with the fermionic variables. Actually, as in the previous sec- tion, through the resulted expression (72), a sort of gen- Physical Description of Closed Contour Link eral structure in comparison with the conventional loop- Mode: In order to attain some view on the physical de- amplituhedron has been introduced, of which the com- scription of the closed contour link mode, let us give some mon amplituhedron presented in Refs. [2, 33] can be re- examples. Consider schematically the Feynman diagram garded as a particular choice (53). of the interaction of two particles in a massless theory as shown in Fig. 2. When the time interval between vertexes x and y is very short, the magnitude of the energy-momentum vector for the propagator is non-zero and hence, the propagator is a virtual particle. However, when the time interval between these points is very long, 10
the PT, geometrical difference of the left and right parts of Fig. 3 leads to different topology, and hence, physical properties. Since links and knots have topological properties, we expect a correspondence between these topological prop- erties, and in turn, the properties of the consisted struc- ture of particles and interactions among those. For ex- Figure 2. The schematic Feynman diagrams of the interaction ample, the carbon nanotubes, graphite and diamond are of two particles in a massless theory. all made of carbon but have completely different prop- erties. It is widely accepted that the difference in their properties is due to their different geometric structures the magnitude of the energy-momentum vector for the in the placements of atoms of carbon. As an another ex- propagator is zero and hence, the propagator is a real ample, let us mention the topological molecules [36], in particle. Therefore, for the first case, one needs to have which those atoms, that make up some of the molecules, one closed contour for real particles, while for the second may be the same but have completely different topolo- case, two closed contours for real particles are required. gical structures [37, 38]. Now consider another example in which five groups of particles interact with each other in a massless theory. If these five groups of particles move away from each PT Description of Loop Level Scattering Amp- other in such a way that the distances between those litude: From mathematical point of view, the result of increase very much, then the propagators among those the hidden particles method in describing loop correc- will become real particles (see, e.g., Fig. 3). Each group tions (mentioned in Refs. [2, 33]) is equivalent to what we have mentioned in the previous section. In fact, in the hidden particles method, two hidden particles are considered for each loop, and their momentum twisters are integrated in a certain way. On the other hand, in the previous section, instead of integrating on the mo- mentum twisters of two hidden particles, the same num- ber of Dirac delta functions have been removed and in- tegrated on variables y1 and y2 in the same specific way. Physically, from the PT point of view (also from the point of view of on-shell physics [2, 33]), a loop means the Figure 3. The schematic Feynman diagrams of the interaction existence of a pair of entangled real particle-antiparticle of five groups of particles in a massless theory. whose energy-momentum vectors are opposite to each other. Thus, the existence of these pairs does not change of particles behaves like a composite particle and also the total energy-momentum, and therefore all possible interacts with other groups in a way that the outcome values for their energy-momentum must be considered. forms a closed contour of the momentum conservation. This state is schematically as a state in which two points For example, see Fig. 4, wherein the reason for linking of the one-dimensional manifold touch each other [2] (see, e.g., Fig. 5).
Figure 4. The closed contour of the momentum conservation of five groups of particles that move away from each other in such a way that the propagators among those become real particles, related to Fig. 3. these five closed contours in this figure is that these five Figure 5. Forming a loop in the closed contour of the mo- sets of particles are not independent of each other and mentum conservation. After the points z1 and z6 touch each interact with each other so that the result forms a closed other, the energy-momentum vectors p6 and p7 will be equal contour of the momentum conservation. In the stand- and in opposite directions [2]. ard field theory, there is no significant physical difference between the left and right parts of Fig. 3. However, in 11
IX. CONCLUSIONS amplituhedron. Thus, we have represented that the S- matrix of the PT, after hierarchy transformation, is a solution of the KP-hierarchy. Inspired by the fact that geometric representation of Furthermore, we have indicated that the volume of the momentum conservation of particles participating in phase-space of the one-dimensional manifold of the the- an interaction leads to the formation of a closed contour, ory is equivalent with the Grassmannian integral. In- we have introduced an integrable one-dimensional the- deed, contrary to Refs. [2, 7, 28, 33], without any use ory, i.e., the PT. This theory underlies some integrable of supersymmetry, we have obtained a sort of general models such as the KP-hierarchy and the amplituhedron. structure in comparison with the conventional Grassman- In this regard, we have defined the action and then the nian integral and the resulted amplituhedron, which is partition function of the PT. Afterward, we have shown closely related to the Yang-Mills scattering amplitudes in that the fields of the PT are equivalent with their parti- four dimensions. Accordingly, the common amplituhed- tion functions. Due to the quantization process, this issue ron presented in the literature can be derived from the means that the theory will not change after quantization. volume of phase-space of the proposed one-dimensional Therefore, this theory possesses a particular symmetry as manifold as a special contour integration of the Grass- a statistical symmetry. This symmetry implies that the mannian integral presented in this work, and being re- theory, in small-scales, is statistically equivalent to large- garded as a particular choice of it. Indeed, the proposed scales (such a characteristic is what we have benefited in theory is capable to express both the tree- and loop-levels another work [22]). amplituhedron (without employing hidden particles) in Also, by defining the S-matrix of the PT, we have indic- four dimensions in the twistor space. Finally, we have ated that the S-matrix has the Grassmannian structure. discussed the physical meaning and importance of the This Grassmannian corresponds to a chord diagram, linked closed contour. which specifies the closed contour of the PT. Thereupon, by introducing and using a transformation (i.e., the hier- archy transformation), we have shown that the PT under- ACKNOWLEDGEMENTS lies some important kinds of integrable models in differ- ent dimensions such as the types of soliton models derived We thank the Research Council of Shahid Beheshti from the KP-hierarchy, two-dimensional models and the University.
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