Quantum chromodynamics through the geometry of M¨obiusstructures
John Mashford
School of Mathematics and Statistics University of Melbourne, Victoria 3010, Australia E-mail: [email protected]
September 13, 2021
Abstract
This paper describes a rigorous mathematical formulation providing a diver- gence free framework for QCD and the electroweak theory in curved space-time. The starting point of the theory is the notion of covariance which is interpreted as (4D) conformal covariance rather than the general (diffeomorphism) covari- ance of general relativity. Fock space which is a graded algebra composed of tensor valued function spaces associated with spaces of multiparticle bosonic or fermionic states is defined concretely. Algebra bundles whose typical fibers are algebras such as the Fock algebra are defined. Scattering processes are associ- ated with linear maps between fibers in such bundles. Such linear maps can be generated by linear operators between the multiparticle function spaces which intertwine with the actions of the group SU(3) associated with the strong in- teraction and a group locally isomorphic to SL(2, C)×U(1) which is associated arXiv:1809.04457v5 [physics.gen-ph] 10 Sep 2021 with the electroweak interaction and such intertwining, or covariant, operators may be generated by covariant integral kernels. It is shown how quark-quark scattering and gluon-gluon scattering are associated with such covariant ker- nels. The rest of the paper focuses on QCD vacuum polarization in order to compute and display the QCD running coupling. Through an application of the technique called spectral regularization the densities associated with the quark, gluon and ghost bubbles are computed without requiring renormalization and
1 hence the QCD vacuum polarization tensor is determined. It is found that the spectral QCD running coupling does not manifest a Landau pole and has the
property known as “freezing of αs” in the infrared.
Keywords: axiomatic QCD, conformal covariance, Fock space algebra, vacuum polarization, divergence-free, running coupling
Contents
1 Introduction 3
2 General objects transforming covariantly at a point x ∈ X where X denotes space-time 7
3 Covariant infinitesimal objects 10 3.1 Covariant infinitesimal objects for M¨obiusstructures ...... 10 3.2 Covariant infinitesimal objects for su(3) and M¨obiusstructures . . . . 12
4 The Fock space F = L∞ F of test functions for mul- k1,k2,k3,k4=0 (k1,k2,k3,k4) tiparticle states 13
5 Particle scattering processes and intertwining operators between
F and P 0 0 0 0 15 (k1,k2,k3,k4) (k1,k2,k3,k4)
6 Quark-quark scattering 18
7 Gluon-gluon scattering and the extended Fock space 20
8 QCD vacuum polarization 23 8.1 The quark fermion bubble ...... 23 8.2 The gluon bubble ...... 26 8.3 The ghost bubble ...... 32
9 The QCD running coupling on the basis of 1-loop vacuum polariza- tion 34 9.1 The contribution of vacuum polarization to the QCD running coupling 34 9.2 Computation of the QCD running coupling ...... 37
9.3 Determination of the bare QCD coupling constant gs ...... 40
2 10 Higher order calculations 42
11 Discussion 44
12 Conclusion 46
List of Figures
1 Feynman diagram for gluon-gluon scattering ...... 20 2 1-loop spectral QCD running coupling versus log energy (GeV) between 1 and 10000 GeV ...... 41 3 1-loop spectral QCD running coupling versus log energy (eV) between 1eV and 10 GeV ...... 42 4 2-loop spectral QCD running coupling versus log energy (GeV) between 1 and 10000 GeV ...... 45 5 2-loop spectral QCD running coupling versus log energy (eV) between 1eV and 10 GeV ...... 45
1 Introduction
Quantum chromodynamics (QCD) is the highly successful theory which organizes and explains the “particle zoo” made up of the baryons and the mesons, collectively known as hadrons, which experience the strong force. In QCD baryons and mesons are made up of collections of quarks, which are fermions with non-zero masses analogous to electrons in QED, and the strong force is mediated by gluons which are massless bosons analogous to photons in QED. One of the main problems met with in perturbative quantum chromodynamics
(pQCD) is the difficulty with setting the renormalization scale µr of the QCD running coupling such that fixed order predictions for physical observables can be obtained. In the usual method a single µr as the argument of the QCD running coupling is simply guessed at, and its value is then varied over an arbitrary range [1]. This procedure is ad hoc and contradicts renormalization scheme (RS) invariance. Two approaches to getting around this problem are the principle of maximum conformality (PMC) [1] and the Brodsky-Lepage-Mackenzie (BLM) approach [2]. The running of the coupling in QCD (and in QED) is an important phenomenon
3 [3, 4, 5] which needs to be taken into account in carrying out calculations in QCD
(e.g. [6]). The running fine structure constant in QCD αs can be determined ex- perimentally in a number of ways including deep inelastic scattering, τ decay and hadron-hadron scattering [4]. In addition it can be determined from lattice gauge theory computations. In all cases its determination involves theoretical QCD calcu- lations involving renormalization. In QCD the running coupling is not an observable
but observables are calculated from αs in the high energy domain where, due to asymptotic freedom [7, 8], pQCD is applicable, by expanding to nth order a series in
αs [9]. The methods for setting the renormalization scale such as the PMC and the BLM approach are all formulated in the context of the renormalization group (RG) ap- proach involving the RG equations [10, 11, 12]. Such calculations are neccessary when renormalization is carried out in order to circumvent divergences in non-tree level pQCD calculations. We have developed a method of regularization for quantum field theory (QFT) which we call spectral regularization and which is described in Refs. [13, 14, 15]. When using spectral regularization renormalization is not required and we may obtain
an expression for the QCD running coupling αs(τ) as a function of energy τ which does not require the setting of a renormalization scale µr. QCD is based on the non-abelian “gauge” group SU(3). Nobody really has ex- plained why the group SU(3) should be of such fundamental importance in describing the physical world at the subatomic level, though some not entirely convincing rea- sons have been suggested. The group SU(3) is described as an internal symmetry group because it is believed that it does not relate to space-time symmetry. Space- time symmetry in QFT is encapsulated in the Lorentz group and more generally in the Poincar´egroup. QED and QCD are formulated in Minkowski space which has the Poincar´egroup as the maximal symmetry group. More general space-time symmetries are conformal covariance, which is covariance with respect to conformal diffeomorphisms, and the “general covariance” of general relativity (GR), which is covariance with respect to arbitrary C∞ diffeomorphisms. GR is the currently accepted theory of gravity. QCD has certain general properties related to conformal symmetry [16, 17, 18, 19]. Gluodynamics has been considered in the context of curved conformally flat space- time [20]. In a previous paper [21] we have presented a theory in which space-time is modeled
4 as a (causal) locally conformally flat Lorentzian manifold so we are considering a subset of the set of space-times considered in GR. Locally conformally flat space- times form a wide class of space-times, including the well studied AdS space-time, and, in general, the class has a wide variety of geometric and topological properties. Actually we do not specify a metric, we only specify the causal, or interaction, structure. This results in any such structure being associated with an equivalence class of conformally equivalent metrics. Such objects are closely related to what are known as M¨obiusstructures. A M¨obiusstructure is a set X together with an atlas
A = {(Ui, φi)}i∈I where I is some index set and for all i ∈ I, Ui ⊂ X and φi : Ui → Vi 4 is a bijection of Ui onto some open subset Vi ⊂ R , such that ∀i, j ∈ I for which −1 ∞ Ui ∩ Uj 6= ∅, φi ◦ φj is a C conformal diffeomorphism. Throughout this paper we let X = (X, A) be a M¨obiusstructure where A is a maximal atlas. The category of M¨obiusstructures has the same logical status as the category of smooth manifolds which forms the starting point for GR except that the pseu- dogroup of transformations (Kobayashi and Nomizu, 1963 [22], p. 1) is the conformal pseudogroup Γ(1, 3) [21] rather than the pseudogroup of general coordinate trans- formations. Γ(1, 3) is made up of all C∞ conformal diffeomorphisms between open subsets of Minkowski space. In Ref. [21] we described how there is a natural principal bundle with structure group C(1, 3), the group of maximal conformal transformations in Minkowski space, associated with any M¨obiusstructure which is obtained by taking the unique element of C(1, 3) associated with each overlap conformal diffeomorphism as a (constant) C(1, 3) valued transition function. We will consider, in this paper, M¨obiusstruc- tures whose transition functions are generated by constant U(2, 2) valued transition functions. Consider G = U(2, 2) in its matrix representation defined with respect to the Hermitian form ! 0 1 g = 2 . (1) 12 0 We define the subgroup H ⊂ G by ( ! ) a b H = ∈ U(2, 2) : b = 0 , (2) c d
5 and the subgroup K ⊂ H by ( ! ) a 0 K = : a ∈ GL(2, C), |det(a)| = 1 . (3) 0 a†−1
K is a homomorphic image of H and is locally isomorphic to SL(2, C) × U(1). There is a natural reduction of structure group from G to H to a principal bundle, P say, with structure group H and a natural principal bundle which we call Q, with structure group K, can be obtained as a homomorphic image of the principal bundle P [21]. In Refs. [23, 24] we showed how QED and the electroweak theory can be formu- lated in terms of the principal bundle Q and K covariance. K acts on C4 and R4 in natural ways and the map κ 7→ (p 7→ κp) with p ∈ R4 is a homomorphism from K to the proper orthochronous Lorentz group O(1, 3)↑+. The fact that the symmetry group of the theory is the group K rather than the Poincar´egroup implies that the Coleman Mandula theorem which precludes the intermixing of “internal” and space-time symmetries may not apply. In the present paper we extend, and to a large extent complete, a formulation of a framework for particle physics theory from the electroweak interaction to elec- troweak + strong by extending the symmetry group from K to SU(3) × K using the relationships between K covariance, SU(3) covariance and U(2, 2) covariance. We will define a general Fock space which is an algebra formed as a direct sum of Fock spaces associated with multiparticle states which we define concretely. We will, as in Ref. [23], define scattering processes in terms of linear maps between fibers in bundles whose typical fibers are spaces related to the Fock spaces. Such maps can be generated by intertwining operators between the spaces. We show how quark-quark scattering and gluon-gluon scattering can be described by kernels which generate such intertwining operators. The paper then focuses on QCD vacuum polarization in order to determine and display the running coupling for QCD and its properties. We do not use renormal- ization but use spectral regularization to determine the vacuum polarization tensor.
6 2 General objects transforming covariantly at a point x ∈ X where X denotes space-time
Let, for any x ∈ X,Ix denote {i ∈ I : x ∈ Ui}. Let C(1, 3) denote the group of maximal C∞ conformal transformations in Minkowski space [21]. If U and V are open subsets of Minkowski space R4 and f : U → V is a C∞ conformal transfor- mation (diffeomorphism), let C(f) ∈ C(1, 3) denote the unique maximal conformal transformation g ∈ C(1, 3) such that f ⊂ g, i.e. f = g|U . ! a b Given a matrix A = ∈ U(2, 2) where a, b, c, d ∈ C2×2, let Ξ(A) denote c d the maximal conformal transformation defined in Minkowski space associated with A, i.e.
(Ξ(A))(M) = (aM + b)(cM + d)−1, for M ∈ u(2) such that det(cM + d) 6= 0, (4) where Minkowski space has been identified with u(2) in the usual way [21]. Ξ is a homomorphism from U(2, 2) to C(1, 3). We will suppose in the present paper that we are given constant transition func- tions
gij(x) = gij ∈ G = U(2, 2) for x ∈ Ui ∩ Uj where Ui ∩ Uj 6= ∅, (5) such that −1 C(φi ◦ φj ) = Ξ(gij) for Ui ∩ Uj 6= ∅. (6)
The collection {gij : Ui ∩ Uj 6= ∅} defines a principal bundle over X with typical fiber G = U(2, 2) and, being transition functions, they satisfy the cocycle condition
gij(x) = gik(x)gkj(x), ∀x ∈ X, i, j, k ∈ Ix, (7) i.e. for all i, j, k ∈ I such that Ui ∩ Uj ∩ Uk 6= ∅
gij = gikgkj. (8)
Suppose that we have an action (g, v) 7→ gv of G on some space V (e.g. a vector space or an algebra). Then there is a natural (associated) fiber bundle with typical
fiber isomorphic to V whose fiber at a point x ∈ X consists of all maps v : Ix → V
7 which satisfy
vi = gijvj, ∀i, j ∈ Ix. (9)
We will call such maps (V valued) covariant objects (at x ∈ X). This is the “physics” definition of the associated fiber bundle as being made up of fibers such that the fiber at a point x ∈ X consists of all objects v which take a value
vi ∈ V in a coordinate system φi about x such that the values transform covariantly under a change of coordinate system. It can be shown that the “physics” definition of the associated fiber bundle is equivalent to the “math” definition of the associated fiber bundle.
Let (Ur, φr) ∈ A be a reference coordinate system about a point x ∈ X. Then we
have, for all i, j ∈ Ix −1 −1 gij = girgrj = girgjr = aiaj , (10) where
ai = gir ∈ G, ∀i ∈ Ix. (11)
Define T : G × G → G by −1 Tgh = gh . (12)
Then
gij = Taiaj ∀i, j ∈ Ix. (13)
Now a covariant object at x is a map v : Ix → V such that
vi = gijvj, ∀i, j ∈ Ix.
∼ Given such an object define a map v : G → V as follows. Set
∼ v(ai) = vi, ∀i ∈ Ix. (14)
−1 Suppose that ai = aj. Then gir = gjr. Hence 14 = girgjr = girgrj = gij. Therefore ∼ vi = gijvj = vj. Therefore v is a well defined map from {ai : i ∈ Ix} to V . But, ∼ since A is a maximal atlas, as i ranges over Ix ai ranges over G. Therefore v is a well defined function from G to V .
Now let g, h ∈ G. Choose i, j ∈ I such that g = ai, h = aj. Then
∼ ∼ ∼ ∼ v(g) = v(ai) = vi = gijvj = Taiaj vj = Tghv(aj) = Tghv(h).
8 Thus, associated with any covariant object v : Ix → V at x, there is an object ∼ ∼ Φ(v) = v which is a map v : G → V which satisfies
∼ ∼ v(g) = Tghv(h), ∀g, h ∈ G. (15)
Conversely, suppose that we have an object w which is a map w : G → V which satisfies
w(g) = Tghw(h), ∀g, h ∈ G. (16)
Define Ψ(w): Ix → V by
Ψ(w)i = w(ai). (17)
Then
Ψ(w)i = w(ai) = Taiaj w(aj) = Taiaj Ψ(w)j = gijΨ(w)j, ∀i, j ∈ Ix. (18) Thus Ψ(w) is a covariant object at x. It is straightforward to verify that ΨΦ = I and ΦΨ = I. Thus we have constructed a natural identification of the space of covariant objects at x with the set of maps w : G → V which satisfy Eq. 16. We may, without fear of confusion, call maps v : G → V which satisfy
v(g) = Tghv(h), ∀g, h ∈ G, (19)
covariant objects. Now consider what happens if we had chosen a different reference coordinate
system about x, φr0 say. Then we have
0 ai = gir0 = girgrr0 = aic (20)
where
c = grr0 . (21)
0 −1 Therefore if Tgh = gh then
0 0 −1 −1 T 0 0 = T = (aic)(ajc) = aia = Ta a . (22) aiaj (aic)(aj c) j i j
Thus the transition function Tgh is independent of the reference coordinate system chosen. An important way that a covariant V valued object can arise is if we have a
9 reference element vref ∈ V . Define
v(g) = gvref, ∀g ∈ G. (23)
Then
−1 −1 Tghv(h) = (gh )(hvref) = (gh h)vref = gvref = v(g), ∀g, h ∈ G, (24) and so v is a covariant object. Conversely, we will show that any covariant object arises in this way. To this effect let v : G → V be a covariant object. Then
−1 v(g) = Tghv(h) = gh v(h), ∀g, h ∈ G. (25)
Set vref = v(14). Then
−1 v(g) = g14 v(14) = gvref, ∀g ∈ G, as required.
3 Covariant infinitesimal objects
3.1 Covariant infinitesimal objects for M¨obiusstructures
We have defined the function T : G × G → G by
−1 Tgh = gh .
Differentiating this equation at (g, h) = (14, 14) results in the infinitesimal transition function which we may, without fear of confusion, denote by the same symbol T , defined by T : g × g → g with
TAB = A − B, (26) where g = u(2, 2) is the Lie algebra of G. Note that T satisfies the infinitesimal cocycle condition
TAC = TAB + TBC . (27)
10 We will now define what we mean by Av for any A ∈ g and v ∈ V . Well we know what gv means for any g ∈ G and v ∈ V . This is defined by the given action,
α : G × V → V say, of G on V . If we differentiate g 7→ α(g, v) at g = 14 for v ∈ V we
get a map (∂1α)(14, v): g → V . We have assumed that V has a smooth structure, g 7→ α(g, v) is smooth for all v ∈ V and that V is a vector space so that the tangent ∗ space Tv V can be identified with V for any v ∈ V . Thus we may define the product Av where A ∈ g by
Av = ((∂1α)(14, v))A, (28) as the natural product from g × V to V . Now suppose that v : G → V is a covariant object and suppose that v is smooth. Since v maps G to V we have that
(Dv)(14): g → V. (29)
We may, without fear of confusion, use the same symbol v for the map from G to V and the map from g to V which it induces, i.e.
v(A) = ((Dv)(14))(A) for A ∈ g, (30)
We will call any map from g to V a V valued infinitesimal object for g. With v a covariant object we have that
−1 v(g) = Tghv(h) = gh v(h), ∀g, h ∈ G. (31)
Differentiating the LHS and RHS of this equation with respect to (g, h) at (14, 14) and evaluating it at (A, B) ∈ g × g implies
v(A) = TABv(14) + v(B) = (A − B)v(14) + v(B), ∀A, B ∈ g. (32)
We will, given a reference element vref ∈ V , call any map v : g → V which satisfies
v(A) = TABvref + v(B), ∀A, B ∈ g, (33) a (V valued) covariant infinitesimal object for g with respect to the reference element vref. We have shown that a covariant object v : G → V induces a covariant infinitesimal
11 object for g with respect to the reference element vref = v(14). Let V denote the space of covariant infinitesimal objects for g with respect to vref v and let w ∈ V . Then for all v ∈ V ref vref vref
(v − w)(A) = v(A) − w(A) = TABvref + v(B) − (TABvref + w(B)) = (v − w)(B), ∀A, B ∈ g.
Thus v − w is an invariant. Conversely if v is an invariant then v + w ∈ V . vref Therefore V = w + I, (34) vref where I denotes the space of invariants.
3.2 Covariant infinitesimal objects for su(3) and M¨obiusstruc- tures
There is a natural imbedding Φ1 of u(2, 1) in u(2, 2) defined by
a11 a12 a13 0 a11 a12 a13 a21 a22 a23 0 Φ1( a a a ) = . (35) 21 22 23 a a a 0 a a a 31 32 33 31 32 33 0 0 0 0
Φ1 is a Lie algebra homomorphism. There is also a natural linear imbedding Φ2 of su(3) into u(2, 1) defined by
Φ2(A) = Ag1, (36) where g1 = diag(1, 1, −1). Let Φ = Φ1Φ2.
Conversely, there is a natural projection π1 of u(2, 2) onto u(2, 1) defined by
a a a a 11 12 13 14 a a a a a a a 11 12 13 21 22 23 24 π1( ) = a21 a22 a23 , (37) a31 a32 a33 a34 a31 a32 a33 a41 a42 a43 a44
and a natural projection π2 of u(2, 1) onto su(3) defined by
π2(A) = π3(Ag1), (38)
12 where π3 : u(3) → su(3) is defined by
1 π (A) = A − tr(A). (39) 3 3
Let π = π2π1. Then πΦ = I|su(3). It can be shown [25] that the tangent bundle TP to the principal bundle P with structure group H associated with any M¨obiusstructure has the structure of a bundle ∗ ∼ of Lie algebras isomorphic to g. For each p ∈ P the space Tp P = g is invariant under change of coordinate system (conformal diffeomorphism). In particular the subspace identified with su(3) is invariant. (By a similar argument a subspace which can be identified with su(2) is invariant.) Thus there is associated with any M¨obiusstructure a bundle of Lie algebras iso- morphic to su(3).
4 The Fock space F = L∞ F of test k1,k2,k3,k4=0 (k1,k2,k3,k4) functions for multiparticle states
We are interested in covariant objects for the groups K and SU(3). Such objects are associated with representations, or actions, of SU(3)×K. We will now consider some such actions of significance for QFT.
For k1, k2, k3, k4 ∈ {1, 2,...} define F(k1,k2,k3,k4) to be the space of all tensor valued C∞ functions u : {1, 2, 3}k1 × {0, 1, 2, 3}k2 × {0, 1, 2, 3}k3 × (R4)k4 → C which are
Schwartz in their continuous arguments (see Ref. [23]). If k1 = 0, k2 = 0, k3 = 0 or
k4 = 0 we omit the corresponding argument from the definition of F(k1,k2,k3,k4) and we
define F(0,0,0,0) = C. F(k1,k2,k3,k4) is a carrier space for a representation of SU(3) × K and is associated with the space of multiparticle states.
ιµα 4 k4 Elements of F(k1,k2,k3,k4) are tensor valued functions u :(R ) → C where ι ∈ {1, 2, 3}k1 , µ ∈ {0, 1, 2, 3}k2 , α ∈ {0, 1, 2, 3}k3 . The indices ι correspond to quark color polarizations, the indices µ are Lorentz indices while the indices α correspond to K polarizations. We call the Lorentz indices bosonic because the photon propagator is a kernel with Lorentz indices defined by the K covariant measure [23] Z Z ηµν + Dµν(Γ) = Dµν(q) dq = (− 2 ) dq = iπηµνΩ0 (Γ). (40) Γ Γ q + i
13 Also the W boson propagator is a kernel with Lorentz indices defined by the K covariant measure [23]
Z Z −ηµν + m−1qµqν Dµν (Γ) = Dµν (q) dq = W dq (41) W,mW W,mW 2 2 Γ Γ q − mW + i Z µν −1 µ ν = iπ (η − mW q q )ΩmW (dq). (42) Γ
Similarly for the Z boson. ± ± Here, for any m ≥ 0, Ωm denotes the standard Lorentz invariant measure Ωm : B(R4) → [0, ∞] on the future/past mass m mass shell for which
* Z Z * * ± dp ψ(p)Ωm(dp) = ψ(±ωm(p), p) * , (43) ωm(p)
* *2 2 1 4 in which ωm(p) = (m + p ) 2 , for any measurable ψ : R → C for which the integral + on the RHS of the equation exists and Ωm = Ωm, for any m > 0.
SU(3) × K acts on F(k1,k2,k3,k4) in a natural way according to
ι0µ0α0 ι0 µ0 α0 ιµα −1 ((U, κ)u) (p) = U ιΛ µκ αu (Λ p), (44) where Λ is the Lorentz transformation corresponding to κ and, for quark color index 0 vectors ι0 = (i0 , . . . , i0 ), ι = (i , . . . , i ) ∈ {1, 2, 3}k1 and for any U ∈ SU(3), U ι 1 k1 1 k1 ι denotes the tensor
ι0 i0 i0 ι0 U = U 1 ··· U k1 = (U ⊗ · · · ⊗ U) (k times), (45) ι i1 ik1 ι 1
0 0 α µ ↑+ 4 k4 and κ α is defined similarly, Λ µ is defined similarly and O(1, 3) acts on (R ) according to
↑+ 4 k4 Λp = (Λp1,..., Λpk4 ), ∀Λ ∈ O(1, 3) , p = (p1, . . . , pk4 ) ∈ (R ) . (46)
For economy of notation in this paper we use the same symbol for an operator which acts on R4 and the operator which it induces which acts on (R4)k4 .
There is a natural Hermitian product on F(k1,k2,k3,k4) given by Z ιµα ∗ τνβ (u, v) = διτ ηµνgαβu (p) v (p) dp, (47)
14 where διτ = (δ ⊗ · · · ⊗ δ)ιτ (k1 times), similarly for ηµν and gαβ in which δ is the Kronecker delta, η = diag(1, −1, −1, −1) is the Minkowski space metric tensor and g = γ0 is the standard Hermitian form for C4 given by Eq. 1 (where γ0 is the
zeroth Dirac gamma matrix in the Weyl representation). This makes F(k1,k2,k3,k4) into a generalized Hilbert space for each k1, k2, k3, k4 ∈ {0, 1, 2,...} where, by a generalized Hilbert space, we mean a complex vector space together with a Hermitian form where the Hermitian form is not neccessarily positive definite. The Hermitian
form ( , ) is invariant under the action of SU(3) × K on F(k1,k2,k3,k4) and therefore the representation ((U, κ), u) 7→ (U, κ)u is a unitary representation of SU(3) × K on
F(k1,k2,k3,k4). Define ∞ M F = F(k1,k2,k3,k4). (48)
k1,k2,k3,k4=0 F forms an algebra under the operation (u, v) 7→ u⊗v of tensor multiplication where,
for u ∈ F(k1,k2,k3,k4), v ∈ F(l1,l2,l3,l4), u ⊗ v is given by
(u ⊗ v)(ιτ)(µν)(αβ)((pq)) = uιµα(p)vτνβ(q),
where (wz) denotes concatenation of vectors w and z.
5 Particle scattering processes and intertwining
operators between F and P 0 0 0 0 (k1,k2,k3,k4) (k1,k2,k3,k4)
The Fock space F(k1,k2,k3,k4) is associated with the space of multiparticle states. We
can define a more general space C(k1,k2,k3,k4) to be the same as F(k1,k2,k3,k4) but with the condition of rapid decrease relaxed. It is natural to define a scattering process at x ∈ X with < in| particles of type 0 0 0 0 (k1, k2, k3, k4) and < out| particles of type (k1, k2, k3, k4) to be a linear map from
a space associated with multiparticle states at x of type (k1, k2, k3, k4) to a space 0 0 0 0 associated with multiparticle states at x of type (k1, k2, k3, k4). Thus we define a scattering process to be a linear map from the fiber at x of the vector bundle with
typical fiber F(k1,k2,k3,k4) to the fiber at x of the vector bundle with typical fiber C 0 0 0 0 . (k1,k2,k3,k4) It can be shown (see Ref. [23]) that such mappings can be generated by linear
intertwining maps from F to C 0 0 0 0 , i.e. operators which commute with (k1,k2,k3,k4) (k1,k2,k3,k4)
15 the action of SU(3) × K.
One can define a class of operators M : F → P 0 0 0 0 using integral (k1,k2,k3,k4) (k1,k2,k3,k4) k +k0 k +k0 k +k0 4 k +k0 kernels M : {1, 2, 3} 1 1 × {0, 1, 2, 3} 2 2 × {0, 1, 2, 3} 3 3 × (R ) 4 4 → C which are polynomially bounded and C∞ in their continuous arguments, according to Z ι0µ0α0 0 ι0µ0α0 0 ιµα M(u) (p ) = M ιµα(p , p)u (p) dp. (49)
(For economy of notation we use the same symbol for the kernel and the operator that it induces.) Here P denotes the space of polynomially bounded elements of C. Note that P ⊂ F ∗ , the dual space to F . (k1,k2,k3,k4) (k1,k2,k3,k4) (k1,k2,k3,k4) We have Z ι0µ0α0 0 ι0µ0α0 0 ιµα (M((U, κ)u)) (p ) = M ιµα(p , p)((U, κ)u) (p) dp Z ι0µ0α0 0 ι µ α τνβ −1 = M ιµα(p , p)U τ Λ νκ βu (Λ p) dp.
Also
ι0µ0α0 0 ι0 µ0 α0 ιµα −1 0 ((U, κ)(M(u))) (p ) = U ιΛ µκ α(M(u)) (Λ p ) Z ι0 µ0 α0 ιµα −1 0 τνβ = U ιΛ µκ α M τνβ(Λ p , p)u (p) dp Z ι0 µ0 α0 ιµα −1 0 −1 τνβ −1 = U ιΛ µκ α M τνβ(Λ p , Λ p)u (Λ p) dp, where we have used the Lorentz invariance of the Lebesgue measure. Therefore
ι µ α ι0µ0α0 0 ι0 µ0 α0 M is an intertwining operator ⇔ U τ Λ νκ βM ιµα(p , p) = U ιΛ µκ α ιµα −1 0 −1 M τνβ(Λ p , Λ p), 4 k 0 4 k0 ∀U ∈ SU(3), κ ∈ K, p ∈ (R ) 4 , p ∈ (R ) 4 ι µ α ι0µ0α0 0 ι0 µ0 α0 ⇔ U τ Λ νκ βM ιµα(Λp , Λp) = U ιΛ µκ α ιµα 0 M τνβ(p , p), 4 k 0 4 k0 ∀U ∈ SU(3), κ ∈ K, p ∈ (R ) 4 , p ∈ (R ) 4 , where the equality must hold for all values of the free indices. ι0µ0α0 One may consider a more general class of kernels M where M = (M ιµα) in 0 0 0 0 k1+k k2+k k3+k 4 k 4 k4 which M : {1, 2, 3} 1 × {0, 1, 2, 3} 2 × {0, 1, 2, 3} 3 × (R ) 4 × B0(R ) → C
16 4 4 4 where B0(R ) denotes {Γ ∈ B(R ) : Γ is relatively compact} in which B(R ) is the 4 Borel algebra of R . Let K 0 0 0 0 denote the space of such kernels which (k1,k1,k2,k2,k3,k3,k4,k4) are polynomially bounded C∞ functions of each of their continuous arguments with the other arguments fixed and are Lorentz invariant tempered Borel measures as a function of each of their set arguments with their other arguments fixed. Elements of
K 0 0 0 0 act on elements of the Fock space F according to (k1,k1,k2,k2,k3,k3,k4,k4) (k1,k2,k3,k4) Z ι0µ0α0 0 ι0µ0α0 0 ιµα M(u) (p ) = M ιµα(p , dp)u (p). (50)
Let
I 0 0 0 0 = {M ∈ K 0 0 0 0 : M is an intertwining operator}. (k1,k1,k2,k2,k3,k3,k4,k4) (k1,k1,k2,k2,k3,k3,k4,k4) (51) Then, by an argument analogous to the one used above, one can show that for all
M ∈ K 0 0 0 0 (k1,k1,k2,k2,k3,k3,k4,k4)
ι µ α ι0µ0α0 0 ι0 µ0 α0 ιµα 0 M ∈ I 0 0 0 0 ⇔ U Λ κ M (Λp , ΛΓ) = U Λ κ M (p , Γ), (k1,k1,k2,k2,k3,k3,k4,k4) τ ν β ιµα ι µ α τνβ 0 4 k4 0 4 k ∀U ∈ SU(3), κ ∈ K, Γ ∈ B0(R ) , p ∈ (R ) 4 ,
(1) (2) We may consider spaces I 0 and I 0 0 0 defined by (k1,k1) (k2,k2,k3,k3,k4,k4)
(1) 0 0 0 k1+k1 ι ι ι ι M ∈ I 0 ⇔ M : {1, 2, 3} → C,U τ M ι = U ιM τ (k1,k1) ∀U ∈ SU(3), and
(2) 0 0 0 k2+k2 k3+k3 4 k4 4 k4 M ∈ I 0 0 0 ⇔ M : {0, 1, 2, 3} × {0, 1, 2, 3} × (R ) × B0(R ) → C, (k2,k2,k3,k3,k4,k4) µ α µ0α0 0 µ0 α0 µα 0 Λ νκ βM µα(Λp , ΛΓ) = Λ µκ αM νβ(p , Γ), 0 4 k4 0 4 k ∀κ ∈ K, Γ ∈ B0(R ) , p ∈ (R ) 4 .
(2) I 0 0 0 coincides with the space of intertwining kernels considered in Refs. (k2,k2,k3,k3,k4,k4) [23, 24] and so may be described as forming part of the electroweak algebra. Since the exponential map exp : su(3) → SU(3) is surjective, at the infinitesimal level, one
17 (1) has the following equivalent definition of the space I 0 (k1,k1)
(1) 0 0 0 k1+k1 ι ι ι ι M ∈ I 0 ⇔ M : {1, 2, 3} → C,A τ M ι = A ιM τ (k1,k1) ∀A ∈ su(3).
Since the kernel algebra considered in Refs. [23, 24] is a subalgebra of the algebra (2) generated by the spaces I 0 0 0 all the results of Refs. [23, 24] apply to the (k2,k2,k3,k3,k4,k4) work of this paper, in particular, the work of this paper covers QED and the weak interaction. Furthermore because the kernels have quark color indices, the work of this paper concerns the strong force, i.e. QCD. This will be the subject of the rest of the paper.
6 Quark-quark scattering
The Feynman amplitude associated with the tree level Feynman diagram for quark- quark (ud → ud) scattering is given by (see (Schwartz, [26], p. 512), and use Feynman- ’t Hooft gauge)
0 a µ 0 b ν iM = u(p1)igsTi0iγ u(p1)iEabµν(q)v(p2)igsTj0jγ v(p2), (52)
a 1 a a 8 where T = 2 λ for {λ }a=1 the Gell-Mann matrices, gs is the strong coupling con- stant, δ η E (q) = − ab µν , (53) abµν q2 + i 0 µ 3 is the gluon propagator, q = p2 − p2 is the momentum transfer, {γ }µ=0 are the Dirac gamma matrices, η = diag(1, −1, −1, −1) is the Minkowski metric tensor and u and v are Dirac spinors. Thus we define
0 0 2 a a 0 0 1 M 0 0 0 0 (p , p , p , p ) = g T 0 T 0 M 0 0 (p , p , p , p ) , (54) i j ijα β αβ 1 2 1 2 s i i j j 0α β αβ 1 2 1 2 q2 + i where
0 0 0 0 µ 0 0 ν M0α0β0αβ(p1, p2, p1, p2) = ηµνu(u, α , p1)γ u(u, α, p1)u(d, β , p2)γ u(d, β, p2), (55)
−1 u(type, α, p) = mtype(p/ + mtype)eα, (56)
18 4 for all quark types type ∈ {u, u, d, d, . . .}, α ∈ {0, 1}, p ∈ R in which eα is the αth basis vector for C4 in the standard basis, i.e.
(eα)β = δαβ, ∀α, β ∈ {0, 1, 2, 3}, (57)
mu, md are the masses of the up quark and down quark respectively and md = md. M can be written as
0 0 0 0 Mi0j0ijα0β0αβ(p1, p2, p1, p2) = M1i0j0ijM2α0β0αβ(p1, p2, p1, p2), (58)
where 2 a a M1i0j0ij = gs Ti0iTj0j, (59) and 0 0 0 0 1 M 0 0 (p , p , p , p ) = M 0 0 (p , p , p , p ) . (60) 2α β αβ 1 2 1 2 0α β αβ 1 2 1 2 q2 + i
(2) From Ref. [23] it follows that M2 ∈ I(0,0,2,2,2,2). We will now show that M1 ∈ (1) I(2,2). We want to show that
i j i0j0 i0 j0 ij A kA lM1 ij = A iA jM1 kl, ∀A ∈ su(3). (61)
a Using the Fierz identity for the su(3) generators Ti0i we have that
i0j0 2 a a 1 2 1 M = g T 0 T 0 = g (δ 0 δ 0 − δ 0 δ 0 ). (62) 1 ij s i i j j 2 s i j ij 3 i i j j Let A ∈ su(3). Then, for Eq. 61 we have
1 2 1 i j 1 2 j0 i0 1 i0 j0 LHS = g (δ 0 δ 0 − δ 0 δ 0 )A A = g (A A − A A ), 2 s i j ij 3 i i j j k l 2 s k l 3 k l 1 1 0 0 1 0 0 1 0 0 RHS = g2(δ δ − δ δ )Ai Aj = g2(Ai Aj − Ai Aj ). 2 s il kj 3 ik jl i j 2 s l k 3 k l Thus LHS = RHS and the required result follows. (1) (2) Therefore M ∈ I(2,2) ⊗ I(0,0,2,2,2,2) and so defines a covariant object.
19 Figure 1: Feynman diagram for gluon-gluon scattering
7 Gluon-gluon scattering and the extended Fock space
The Feynman diagram for tree level gluon-gluon gg → gg scattering is shown in Figure 1. This diagram may form part of a larger diagram in which the Lorentz indices on the external lines may be associated with with gluon polarization vector functions or else gluon propagators. Applying the Feynman rules, the Feynman amplitude for this process is written as
0 0 0 0 a1c1a1 µ1ν1 0 µ1 ν1µ1 µ1 µ1µ1 0 ν1 M =gsf [η (−p1 + k) + η (−k − p1) + η (p1 + p1) ]
Ec1c2ν1ν2 (k) 0 0 0 0 a2c2a2 µ2ν2 0 µ2 ν2µ2 µ2 µ2µ2 0 ν2 gsf [η (−p2 − k) + η (k − p2) + η (p2 + p2) ],
where Eabµν is the gluon propagator. Thus we have
0 0 0 0 a1a2a1a2µ1µ2µ1µ2 0 0 M (p1, p2, p1, p2) (63) 0 0 0 0 µ µ1ν1 µ µ2ν2 2 a1c1a1 a2c2a2 1 0 2 0 = gs f f M1 (p1, p1)M2 (p2, p2)Ec1c2ν1ν2 (k),
20 where
0 µ µ1ν1 0 0 0 1 0 µ1ν1 0 µ1 ν1µ1 µ1 µ1µ1 0 ν1 M1 (p1, p1) = η (−p1 + k) + η (−k − p1) + η (p1 + p1) ,
0 µ µ2ν2 0 0 0 2 0 µ2ν2 0 µ2 ν2µ2 µ2 µ2µ2 0 ν2 M2 (p2, p2) = η (−p2 − k) + η (k − p2) + η (p2 + p2) .
We have
0 µ µ1ν1 0 0 1 0 µ1ν1 µ1 0 ρ1 ν1µ1 µ1 ρ2 M1 (κp1, κp1) =η Λ ρ1 (−p1 + k) + η Λ ρ2 (−k − p1) + 0 µ1µ1 ν1 0 ρ3 η Λ ρ3 (p1 + p1) 0 µ1 ν1 σ1σ2 µ1 0 ρ1 =Λ σ1 Λ σ2 η Λ ρ1 (−p1 + k) 0 ν1 µ1 α1α2 µ1 ρ2 + Λ α1 Λ α2 η Λ ρ2 (−k − p1) 0 µ1 µ1 β1β2 ν1 0 ρ3 + Λ β1 Λ β2 η Λ ρ3 (p1 + p1) 0 µ1 µ1 ν1 σ1σ2 0 ρ1 ρ1σ2 σ1 =Λ σ1 Λ ρ1 Λ σ2 [η (−p1 + k) + η (−k − p1) +
σ1ρ1 0 σ2 η (p1 + p1) ] 0 µ1 µ1 ν1 σ1ρ1σ2 0 =Λ σ1 Λ ρ1 Λ σ2 M (p1, p1).
(2) Therefore M1 ∈ I(0,3,0,0,2,2) and so is covariant. Similarly M2 is covariant. Since the kernels now have gluonic color indices we extend the basic Fock space defined in Section 4. The gluonic color indices are intrinsically associated with the Lie algebra su(3) rather than with the Lie group SU(3). We define the gluonic Fock space to be the real vector space
⊗kg F(g,kg) = su(3) = su(3) ⊗ · · · ⊗ su(3) (kg times), (64)
0 and the set K(g,kg,kg) of gluonic operators to be the set of linear maps from F(g,kg) to
0 F(g,kg). We define the general Fock space to be composed of spaces of the form
F(kg,k1,k2,k3,k4) = F(g,kg)F(k1,k2,k3,k4), (65) and the general kernel algebra to be composed of spaces of the form
K 0 0 0 0 0 = K 0 K 0 0 0 0 . (66) (kg,kg,k1,k1,k2,k2,k3,k3,k4,k4) (g,kq,kq) (k1,k1,k2,k2,k3,k3,k4,k4)
0 We now want to define a notion of covariant operators in K(g,kg,kg). Elements M
21 ⊗k ⊗k0 0 g g of K(g,kg,kg) are, in general, arbitrary linear maps from su(3) to su(3) and do not have a distinguished or natural status. However there is a class of distinguished operators of the form
⊗kg MA(u) = [A, u], for u ∈ su(3) , (67) where A ∈ su(3)⊗kg and we define the bracket of elements of su(3)⊗kg in such a way that the bracket commutes with the operation of forming a tensor product, i.e.
[A1 ⊗ · · · ⊗ An,B1 ⊗ · · · Bn] = [A1,B1] ⊗ · · · ⊗ [Akg ,Bkg ], (68)
⊗kg (however (su(3) , [. , . ]) does not form a Lie algebra). An operator MA can operate with elements of su(3)⊗kg through the element A. Therefore we define
BMA = M[B,A], MAB = M[A,B]. (69)
We will define an operator MA to be covariant if
⊗kg BMA = MAB, ∀B ∈ su(3) . (70)
In the case when kg = 1 MA is covariant if and only if A lies in the trivial center of su(3).
Now consider the case that we are now interested in, that is the case of K(g,2,2).
Then, for A = A1 ⊗ A2 and B = B1 ⊗ B2
BMA = M[B,A] = M[B1⊗B2,A1⊗A2] = M[B1,A1]⊗[B2,A2] = M[A1,B1]⊗[A2,B2]
M[A1⊗A2,B1⊗B2] = M[A,B] = MAB,
(note that two minus signs have cancelled at the end of the first line). Therefore since decomposable elements of su(3) ⊗ su(3) span su(3) ⊗ su(3) and {A ∈ su(3) ⊗
su(3) : MA is covariant } is a subspace of su(3) ⊗ su(3) we may conclude that MA is covariant for any A ∈ su(3) ⊗ su(3). In particular, when A = I is the identity matrix ∼ 8×8 in su(3) ⊗ su(3) = R , M = MI is covariant in which case one can easily verify thet M is given by 0 0 0 0 a1a2 ca1a1 ca2a2 M a1a2 = f f . (71) Therefore the kernel given by Eq. 63 describing gluon-gluon scattering is coveriant.
22 8 QCD vacuum polarization
In this section we compute the vacuum polarization tensor for QCD using spectral regularization in order to compute the running coupling by computing an equivalent potential function using the Born approximation. We take the view that the QCD coupling is determined by insertions into the gluon propagator associated with the tree level quark-quark scattering graph, and that other Feynman diagrams such as the quark self-energy and the vertex correction are not relevant to determining the coupling but are relevant to determining the effect of the coupling on a scattered quark. In standard QFT using dimensional regularization and renormalization the bare charge is written as (Schwartz, 2014 [26], p. 526)
Z1 4−d g0 = gR √ µ 2 , (72) Z2 Z3 where gR is the (finite) renormalized charge, µ is the (finite) subtraction point and
Z1,Z2,Z3 are the (infinite!?) renormalizations. In QED one has the “theorem” to the effect that Z1 = Z2, and it therefore follows that in QED the coupling only depends on the vacuum polarization. It is often stated that since in QCD “Z1 6= Z2” the coupling depends on the quark self energy and the vertex correction as well. However, with spectral regularization [13, 14, 15], everything is finite, one does
not need the mysterious quantities Z1,Z2,Z3 and, we believe, the running coupling depends only on the gluon insertions, i.e. on the vacuum polarization.
8.1 The quark fermion bubble
The contribution of the quark fermion bubble to the QCD vacuum polarization is (c.f. Schwartz [26], p. 517)
Z dk 1 1 Π(q)abµν(p) = −g2Tr(T aT b) s (2π)4 (p − k)2 − m2 + i k2 − m2 + i Tr[γµ(k/ − p/ + m)γν(k/ + m)]. (73)
Therefore since 1 Tr(T aT b) = δab, (74) 2
23 we have 1 Π(q)abµν(p) = g2δabΠ(q)µν(p), (75) 2 s where Z dk 1 1 Π(q)µν(p) = − Tr[γµ(k/ − p/ + m)γν(k/ + m)]. (2π)4 (p − k)2 − m2 + i k2 − m2 + i
In line with the work of Refs. [23, 15] suppose (“pretend”) that Π(q)µν existed point- wise and then we can compute (formally) the measure associated with the “function”
Z 1 1 p 7→ Πµν(p) = − Tr[γµ(k/ − p/ + m)γν(k/ + m)] dk, (p − k)2 − m2 + i k2 − m2 + i
4 as follows. Let Γ ∈ B0(R ). Then Z Πµν(Γ) = Πµν(p) dp Γ Z µν = χΓ(p)Π (p) dp Z 1 1 = − χ (p) Γ (p − k)2 − m2 + i k2 − m2 + i Tr[γµ(k/ − p/ + m)γν(k/ + m)] dk dp Z 1 1 = − χ (p) Γ (p − k)2 − m2 + i k2 − m2 + i Tr[γµ(k/ − p/ + m)γν(k/ + m)] dp dk Z 1 1 = χ (p + k) Γ p2 − m2 + i k2 − m2 + i Tr[γµ(p/ − m)γν(k/ + m)] dp dk Z 2 µ ν = −π χΓ(p + k)Tr[γ (p/ − m)γ (k/ + m)] Ωm(dp)Ωm(dk),
where, for any set Γ, χΓ denotes the characteristic function of Γ defined by ( 1 if p ∈ Γ χΓ(p) = (76) 0 otherwise,
24 and we have used the ansatz [14, 23, 15]
1 → −πiΩ± (p). (77) p2 − m2 + i m
Therefore the measure associated with Π(q)µν is 1 Z Π(q)µν(Γ) = − χ (p + k)Tr[γµ(p/ − m)γν(k/ + m)] Ω (dp)Ω (dk). (78) 16π2 Γ m m
Π(q)µν is analogous to the QED vacuum polarization tensor and it can be shown [15, 23] that Π(q)µν is a tempered Borel measure on Minkowski space and is associated with a density Π(q)µν given by
Π(q)µν(q) = (q2ηµν − qµqν)π(q), (79) where ( 2 m3s−3Z(s)(3 + 2Z(s)2) if q2 ≥ 4m2, q0 > 0 π(q) = 3π (80) 0 otherwise,
1 s2 1 2 2 2 s = (q ) and Z(s) = ( 4m2 − 1) . It can be shown [14] that in the spacelike domain π is given by ( − 2 m3s−3Z(s)(3 + 2Z(s)2) if s ≥ 2m π(q) = 3π (81) 0 otherwise,
2 1 where s = (−q ) 2 . Therefore
1 Π(q)abµν(q) = − g2δabm3s−3Z(s)(3 + 2Z(s)2)(q2ηµν − qµqν), s ≥ 2m. (82) 3π s
Now there are 6 quark types with flavors fq = u, d, c, s, t, b and masses mq ∈
{m1, . . . , m6} say. Therefore the total quark bubble contribution to the hadronic vacuum polarization tensor is given by
6 (q,tot)abµν X (q)abµν Π (q) = Π (mk, q), (83) k=1
25 where
Π(q)abµν(m, q) = ( − 1 g2δabm3s−3Z(m, s)(3 + 2Z(m, s)2)(q2ηµν − qµqν) if s ≥ 2m 3π s (84) 0 otherwise,
in which 1 s2 2 Z(m, s) = − 1 , (85) 4m2
2 1 and s = (−q ) 2 . It is not neccessary to specify which quarks are “active” at any given energy. This is automatically worked out in Eq. 84 by the value determined by that equation corresponding to any given quark type at any given energy.
8.2 The gluon bubble
The contribution from the gluon bubble is written (c.f. Schwartz [26], 2014, p. 518)
1 Z dk 1 1 Π(g)abµν(p) = − g2 f acef bdf δcf δedN µν 2 s (2π)4 k2 + i (k − p)2 + i 3 Z dk 1 1 = g2δab N µν, (86) 2 s (2π)4 k2 + i (k − p)2 + i
where
µν µα ρ αρ µ ρµ α N (p, k) = [η (p + k) + η (p − 2k) + η (k − 2p) ]ηαβηρσ ×[ηνβ(p + k)σ − ηβσ(2k − p)ν − ησν(2p − k)β].
26 We compute (formally) the measure Πµν associated with the “function” p 7→ Πµν(p) = R 1 1 µν k2+i (k−p)2+i N (p, k) dk as follows. Z Πµν(Γ) = Πµν(p) dp Γ Z µν = χΓ(p)Π (p) dp Z 1 1 = χ (p) N µν(p, k) dk dp Γ k2 + i (k − p)2 + i Z 1 1 = χ (p) N µν(p, k) dp dk Γ k2 + i (k − p)2 + i Z 1 1 = χ (p + k) N µν(p + k, k) dp dk Γ k2 + i p2 + i Z 2 µν = −π χΓ(p + k)N (p + k, k)Ω0(dk)Ω0(dp),
+ µν 4 4 where Ω0 denotes the measure Ω0 . Define M : R × R → R by
M µν(p, k) = N µν(p + k, k). (87)
Then Π(g)abµν is given by
3 Z Π(g)abµν(Γ) = − g2δab χ (p + k)M µν(p, k)Ω (dk)Ω (dp), 32π2 s Γ 0 0
27 where
M µν(p, k) = N µν(p + k, k) µα ρ αρ µ ρµ α ηαβηρσ[η (p + 2k) + η (p − k) − η (2p + k) ] × [ηνβ(p + 2k)σ − ηβσ(k − p)ν − ησν(2p + k)β] µα ρ νβ σ µα ρ βσ ν = ηαβηρσ[η (p + 2k) η (p + 2k) − η (p + 2k) η (k − p) − ηµα(p + 2k)ρησν(2p + k)β + ηαρ(p − k)µηνβ(p + 2k)σ− ηαρ(p − k)µηβσ(k − p)ν − ηαρ(p − k)µησν(2p + k)β− ηρµ(2p + k)αηνβ(p + 2k)σ + ηρµ(2p + k)αηβσ(k − p)ν+ ηρµ(2p + k)αησν(2p + k)β] µα νβ ρ σ = ηαβηρση η (p + 2k) (p + 2k) µα βσ ρ ν − ηαβηρση η (p + 2k) (k − p) µα σν ρ β − ηαβηρση η (p + 2k) (2p + k) αρ νβ µ σ + ηαβηρση η (p − k) (p + 2k) αρ βσ µ ν − ηαβηρση η (p − k) (k − p) αρ σν µ β − ηαβηρση η (p − k) (2p + k) ρµ νβ α σ − ηαβηρση η (2p + k) (p + 2k) ρµ βσ α ν + ηαβηρση η (2p + k) (k − p) ρµ σν α β + ηαβηρση η (2p + k) (2p + k) = ηµν(p + 2k)2 − (p + 2k)µ(k − p)ν − (p + 2k)ν(2p + k)µ + (p − k)µ(p + 2k)ν − 4(p − k)µ(k − p)ν − (p − k)µ(2p + k)ν − (2p + k)ν(p + 2k)µ + (2p + k)µ(k − p)ν + ηµν(2p + k)2 = ηµν(5p2 + 5k2 + 8p.k) − 2pµpν − 2kµkν − 7pµkν − 7kµpν.
28 It is straightforward to show that Π(g)abµν is a well defined Borel measure on Minkowski space R4. Write Π(g)abµν(Γ) = Π(g)abΠ(g)µν(Γ), (88) where 3 Π(g)ab = − g2δab, (89) 32π2 s Z (g)µν µν Π (Γ) = χΓ(p + k)M (p, k)Ω0(dk)Ω0(dp). (90)
↑+ 4 Using the Lorentz invariance of Ω0, for any Λ ∈ O(1, 3) , Γ ∈ B0(R ) Z (g)µν µν Π (ΛΓ) = χΛΓ(p + k)M (p, k)Ω0(dk)Ω0(dp), Z −1 −1 µν = χΓ(Λ p + Λ k)M (p, k)Ω0(dk)Ω0(dp), Z µν = χΓ(p + k)M (Λp, Λk)Ω0(dk)Ω0(dp).
But we have
M µν(Λp, Λk) = ηµν(5(Λp)2 + 5(Λk)2 + 8(Λp).(Λk)) − 2(Λp)µ(Λp)ν− 2(Λk)µ(Λk)ν − 7(Λp)µ(Λk)ν − 7(Λk)µ(Λp)ν µ ν ρσ 2 2 ρ σ = Λ ρΛ σ(η (5p + 5k + 8p.k) − 2p p − 2kρkσ − 7pρkσ − 7kρpσ) µ ν ρσ = Λ ρΛ σM (p, k).
Therefore
(g)µν µ ν (g)ρσ ↑+ 4 Π (ΛΓ) = Λ ρΛ σΠ (Γ), ∀Λ ∈ O(1, 3) , Γ ∈ B0(R ), (91)
and so Πµν is Lorentz covariant in the sense defined in Ref. [15]. By a similar argument to that used in Ref. [14] the support of Π(g)µν satisfies supp(Π(g)µν) ⊂ {p ∈ R4 : p2 ≥ 0, p0 ≥ 0}. Thus Π(g)µν is causal. We will now show, using the spectral calculus, [13, 14, 15, 27] that Π(g)µν has a spectral representation
Z ∞ Z ∞ (g)µν 2 µν 0 µ ν 0 Π (Γ) = p η Ωm0 (Γ) σ1(dm ) + p p Ωm0 (Γ) σ2(dm ), (92) m0=0 m0=0
where the spectral measures σ1, σ2 are continuous functions. Let a, b ∈ R, 0 < a < b.
29 We have
gµν(a, b, ) = Π(g)µν(Γ(a, b, )) Z µν = χΓ(a,b,)(p + k)M (p, k)Ω0(dk)Ω0(dp)
* * Z * * * * * * * * µν dk dp ≈ χ(a,b)(|p| + |k|)χ * (p + k)M ((|p|, p), (|k|, k)) * * B( 0 ) |k| |p| * * Z * * * * * * * µν dk dp = χ(a,b)(|p| + |k|)χ * *(k)M ((|p|, p), (|k|, k)) p * * B( 0 )− |k| |p| * Z * * * * * µν dp 4 3 ≈ χ(a,b)(2|p|)M ((|p|, p), (|p|, −p)) * ( π ), |p|2 3
for all a, b ∈ R with 0 < a < b. Now
* * * a b χ (2|p|) = 1 ⇔ 2|p| ∈ (a, b) ⇔ |p| ∈ ( , ). (a,b) 2 2 Hence, using spherical polar coordinates,
b Z 2 Z π Z 2π µν µν ga (b) = M ((s, s sin(θ) cos(φ), s sin(θ) sin(φ), s cos(θ)), a s= 2 θ=0 φ=0 4 (s, −s sin(θ) cos(φ), −s sin(θ) sin(φ), −s cos(θ))) sin(θ) dφ dθ ds ( π). 3 Thus, using the Leibniz integral rule
Z π Z 2π µν0 µν b ga (b) = M ( (1, sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)), θ=0 φ=0 2 b 2 (1, − sin(θ) cos(φ), − sin(θ) sin(φ), − cos(θ))) sin(θ) dφ dθ ( π) 2 3 Z π Z 2π = b2 M µν((1, sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)), θ=0 φ=0 1 (1, − sin(θ) cos(φ), − sin(θ) sin(φ), − cos(θ))) sin(θ) dφ dθ ( π). 6 Therefore, since if p = (1, sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)) and k = (1, − sin(θ) cos(φ), * * − sin(θ) sin(φ), − cos(θ)) then p2 = k2 = 0, p.k = 2 and k = (k0, k) = (p0, −p), we
30 have, for all i ∈ {1, 2, 3}
M ii(p, k) = −5p2 − 5k2 − 8p.k − 2pipi − 2kiki − 7piki − 7kipi = −16 − 2(pi)2 − 2(pi)2 + 7(pi)2 + 7(pi)2 = −16 + 10(pi)2, and so
Z π Z 2π 110 1 2 2 2 ga (b) = πb (−16 + 10 sin (θ) cos (φ)) sin(θ) dφ dθ 6 θ=0 φ=0 1 4 = πb2(−16(4π) + 10( )π) 6 3 76 = − π2b2, 9 Z π Z 2π 220 1 2 2 2 ga (b) = πb (−16 + 10 sin (θ) sin (φ)) sin(θ) dφ dθ 6 θ=0 φ=0 1 4 = πb2(−16(4π) + 10( )π) 6 3 76 = − π2b2, 9 Z π Z 2π 330 1 2 2 ga (b) = πb (−16 + 10 cos (θ)) sin(θ) dφ dθ 6 θ=0 φ=0 1 2 = πb2(2π)(−16(2) + 10( )) 6 3 76 = − π2b2. 9 Thus ii0 jj0 ga (b) = ga (b), ∀i, j ∈ {1, 2, 3}, b > a, (93) and as it should from the general theory [27] and
76 gii0(b) = − π2b2, ∀i ∈ {1, 2, 3}, b > a, (94) a 9 From this theory (the spectral theory for Lorentz covariant tensor valued measures [23, 15, 27]) the measure Πgµν is given by
Z ∞ Z (g)µν µν 2 (g) 0 µ ν (g) 0 0 Π (Γ) = χΓ(q)(η q σ1 (m ) + q q σ2 (m )) Ωm0 (dq) dm , (95) m0=0 R4
31 where 3 1 σ(g)(b) = − gii0(b), i ∈ {1, 2, 3}, (96) 1 4π b a 3 1 σ(g)(b) = g000(b) − σ (b), (97) 2 4π b a 1
and is associated with a density Π(g)µν : {q ∈ R4 : q2 > 0, q0 > 0} → R given by
(g)µν µν (g) µ ν −1 (g) Π (q) = η sσ1 (s) + q q s σ2 (s),
2 0 2 1 for q > 0, q > 0 where s = (q ) 2 . From Eqns. 94 and 96 we have that
3 1 76 19π σ(g)(s) = (− )(− π2s2) = s, ∀s > 0. (98) 1 4π s 9 3
In the spacelike domain it can be shown [14] that Π(g)µν : {q ∈ R4 : q2 < 0} → R is given by (g)µν µν (g) µ ν −1 (g) Π (q) = −(η sσ1 (s) + q q s σ2 (s)). (99)
2 − 1 where s = (−q ) 2 . Hence, from Eq. 88
3 Π(g)abµν(q) = g2δab(ηµνsσ(g)(s) + qµqνs−1σ(g)(s)). (100) 32π2 s 1 2
8.3 The ghost bubble
The contribution of the ghost bubble is defined by the equation (Schwartz, 2014 [26], p. 520) Z dk 1 1 Π(gh)abµν(p) = g2 f cadf dbckµ(k − p)ν. (101) s (2π)4 (k − p)2 k2 Now f cadf dbc = −f acdf bcd = −3δab.
Therefore 3 Π(gh)abµν(p) = − g2δabΠ(gh)µν(p), (102) 16π4 s where Z 1 1 Π(gh)µν(p) = kµ(k − p)ν dk. (103) (k − p)2 k2
32 We (formally) compute the measure associated with Π(gh)µν as follows. Z (gh)µν (gh)µν Π (Γ) = χΓ(p)Π (p) dp Z 1 1 = χ (p) kµ(k − p)ν dk dp Γ (k − p)2 k2 Z 1 1 = χ (p) kµ(k − p)ν dp dk Γ (k − p)2 k2 Z 1 1 = − χ (p + k) kµpν dp dk Γ p2 k2 Z 2 µ ν = π χΓ(p + k)k p Ω0(dp)Ω0(dk).
It is straightforward to show that Π(gh)µν is a causal Lorentz covariant 2-tensor valued tempered Borel measure. We use the spectral calculus to compute the spectrum of Π(gh)µν as follows.
gµν(a, b, ) = Π(gh)µν(Γ(a, b, )) * * Z * * * * * * * * 2 µ ν dp dk ≈ π χ(a,b)(|p| + |k|)χ * (p + k)(|k|, k) (|p|, p) * * B( 0 ) |p| |k| * * Z * * * * * * * 2 µ ν dp dk = π χ(a,b)(|p| + |k|)χ * *(p)(|k|, k) (|p|, p) * * B( 0 )−k |p| |k| * Z * * * * * 2 µ ν dk 4 3 ≈ π χ(a,b)(2|k|)(|k|, k) (|k|, −k) ( π ). *2 3 k Therefore
* Z * 33 4 3 3 2 dk g (b) = − π χ(a,b)(2|k|)(k ) a 3 *2 k b Z 2 Z π Z 2π 4 3 2 2 1 2 = − π s cos (θ) 2 s sin(θ) dφ dθ ds. 3 a s s= 2 θ=0 φ=0
33 Thus, using the Leibniz integral rule
Z π 2 330 4 3 b 2 1 ga (b) = − π cos (θ) sin(θ) dθ (2π)( ) 3 θ=0 4 2 2 = − π4b2. 9
Hence the density for Π(gh)µν is
(gh)µν µν (gh) µ ν −1 (gh) Π (q) = η sσ1 (s) + q q s σ2 (s), (104)
2 1 where s = (q ) 2 in the timelike domain and
(gh)µν µν (gh) µ ν −1 (gh) Π (q) = −(η sσ1 (s) + q q s σ2 (s)), (105)
2 1 (gh) (gh) where s = (−q ) 2 in the spacelike domain for spectral functions σ1 , σ2 : (0, ∞) → R and 3 3 2 π3 σ(gh)(s) = − s−1g330(s) = (− s−1)(− π4s2) = s. (106) 1 4π a 4π 9 6 Therefore 3 Π(gh)abµν(p) = g2δab(ηµνsσ(gh)(s) + qµqνs−1σ(gh)(s)), (107) 16π4 s 1 2
2 1 s = (−q ) 2 .
9 The QCD running coupling on the basis of 1- loop vacuum polarization
9.1 The contribution of vacuum polarization to the QCD running coupling
We have computed the tensor densities associated with the quark fermion bubble, the gluon bubble and the ghost bubble. It is common (Schwartz [26], p. 517) to include a term also for the counterterm. This term is included in order for regulariza- tion/renormalization to work. In our work we do not need to carry out renormaliza- tion, we have used spectral regularization. Therefore we do not need counterterms. The contribution of vacuum polarization is given by the Feynman amplitude de-
34 fined by
((vp) 0 0 T 0 0 a ρ iM 0 0 0 0 (p1, p2, p1, p2) =(c 0 ⊗ u(u, p1, α1))(igsT ⊗ γ )(ci1 ⊗ u(u, p1, α1)) i1i2i1i2α1α2α1α2 i1 (vp)cdµν iEacρµ(q)iΠ (q)iEdbνσ(q) T 0 0 b σ (c 0 ⊗ u(d, p , α ))(igsT ⊗ γ )(ci ⊗ u(d, p2, α2)), i2 2 2 2 where Π(vp)abµν = Π(q,tot)abµν + Π(g)abµν + Π(gh)abµν, (108) is the total vacuum polarization tensor, u(type, p, α) is a Dirac spinor for a quark of type type corresponding to momentum p and K polarization α, Eabµν is the gluon 0 propagator, q = p2 − p2 is the momentum transfer and {ci}i=1,2,3 are the standard basis vectors for the quark color space (C3). Now
6 (q,tot)abµν X (q)abµν Π (q) = Π (mk, q), k=1 3 Π(g)abµν(q) = g2δab(ηµνsσ(g)(s) + qµqνs−1σ(g)(s)), 32π2 s 1 2 3 Π(gh)abµν(q) = g2δab(ηµνsσ(gh)(s) + qµqνs−1σ(gh)(s)), 16π4 s 1 2 where
Π(q)abµν(m, q) = ( − 1 g2δabm3s−3Z(m, s)(3 + 2Z(m, s)2)(q2ηµν − qµqν) if q2 ≤ −4m2 3π s (109) 0 otherwise, in which 1 s2 2 Z(m, s) = − 1 , (110) 4m2
2 1 with s = (−q ) 2 and we are considering q to be spacelike because the momentum transfer q is spacelike. Therefore using a well known conservation property of the Dirac spinors (Wein- berg, 2015 [28], p. 480) which implies that
0 0 µ u(u, p1, α1)qµγ u(u, p1, α1) = 0, (111)
35 we have, after contracting indices,
(vp) 0 0 2 a b M 0 0 0 0 (p1, p2, p1, p2) =gs δabT 0 T 0 i1i2i1i2α1α2α1α2 i1i1 i2i2 g2 s (π(q)(s) + π(g) + π(gh)) q2 0 0 M 0 0 (p , p , p , p ), 0α1α2α1α2 1 2 1 2 where
6 (q) X (q) π (s) = π (mk, q), (112) k=1 3 19π 19 π(g)(= π(g)(s)) = s−2( s)( s) = . (113) 32π2 3 32π 3 π3 1 π(gh)(= π(gh)(s)) = s−2( s)( s) = . (114) 16π4 6 32π in which ( − 1 m3s−3Z(m, s)(3 + 2Z(m, s)2) if s ≥ 2m π(q)(m, s) = 3π 0 otherwise,
0 0 M 0 0 (p , p , p , p ) = 0α1α2α1α2 1 2 1 2 0 0 ρ 0 0 σ u(u, p1, α1)γ u(u, p1, α1)ηρσu(d, p2, α2)γ u(d, p2, α2),
2 1 s = (−q ) 2 . In the non-relativistic (NR) approximation, if the Dirac gamma matrices are taken to be in the Dirac representation,
0 0 M 0 0 (p , p , p , p ) = δ 0 δ 0 . (115) 0α1α2α1α2 1 2 1 2 α1α1 α2α2
Therefore
(vp) 0 0 M 0 0 0 0 (p1, p2, p1, p2) i1i2i1i2α1α2α1α2 2 2 a b gs (q) (g) (gh) 0 0 = gs δabTi0 i Ti0 i (π (s) + π + π )δα α1 δα α2 . 1 1 2 2 q2 1 2
36 9.2 Computation of the QCD running coupling
The Feynman amplitude for quark-quark ud → ud scattering at 1-loop level, taking into account vacuum polarization, is
M = M(tree) + M(vp), (116) where
(tree) 0 0 T 0 0 a ρ iM 0 0 0 0 (p1, p2, p1, p2) =(c 0 ⊗ u(u, p1, α1))(igsT ⊗ γ )(ci1 ⊗ u(u, p1, α1)) i1i2i1i2α1α2α1α2 i1
iEabρσ(q) T 0 0 b σ (c 0 ⊗ u(d, p , α ))(igsT ⊗ γ )(ci ⊗ u(d, p2, α2)). i2 2 2 2
Thus
(tree) 0 0 2 ab a b 1 0 0 M 0 0 0 0 (p , p , p , p ) = g δ T 0 T 0 M 0 0 (p , p , p , p ), i i i i α α α α 1 2 1 2 s i i1 i i2 0α α α1α2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 q2 1 2 and so in the NR approximation
(tree) 0 0 2 ab a b 1 M 0 0 0 0 (p , p , p , p ) = g δ T 0 T 0 δ 0 δ 0 . (117) i i i i α α α α 1 2 1 2 s i i1 i i2 α α1 α α2 1 2 1 2 1 2 1 2 1 2 q2 1 2
Therefore using the results of the last subsection the total Feynman amplitude for the process under consideration is given by
0 0 M 0 0 0 0 (p , p , p , p ) i1i2i1i2α1α2α1α2 1 2 1 2 2 ab a b 1 0 0 = gs δ Ti0 i Ti0 i δα α1 δα α2 + 1 1 2 2 q2 1 2 2 2 a b gs (q) (g) (gh) 0 0 gs δabTi0 i Ti0 i (π (s) + π + π )δα α2 δα α1 1 1 2 2 q2 2 1
2 a a 1 2 (q) (g) (gh) 0 0 = gs Ti0 i Ti0 i 1 + gs (π (s) + π + π ) δα α1 δα α2 . 1 1 2 2 q2 1 2
As is usual in QFT calculations we average over incoming quark color polarizations and sum over outgoing quark color polarizations, Thus, using the Fierz identity we
37 obtain a color factor of
3 8 1 X X a a Ti0 i Ti0 i = 9 1 1 2 2 0 0 a=1 i1,i2,i1,i2=1 3 1 X 1 1 (δ 0 δ 0 − δ 0 δ 0 ) i1i2 i1i2 i1i1 i2i2 9 0 0 2 3 i1,i2,i1,i2=1 1 1 1 1 = (9 − 9) = . 9 2 3 3 Hence we obtain
M 0 0 (q) = α1α2α1α2 2 1 gs 2 (q) (g) (gh) (1 + g (π (s) + π + π ) δα0 α δα0 α . 3 q2 s 1 1 2 2
* Another aspect of the NR approximation is that q0 is negligible compared to q . Therefore
M 0 0 (q) = α1α2α1α2 2 1 gs 2 (q) (g) (gh) − 1 + g (π (s) + π + π ) δ 0 δ 0 . *2 s α1α1 α2α2 3 q
We will now compare the scattering matrix associated with the Feynman amplitude that we have computed with that obtained using the Born approximation as in Ref. [15]. We do this so that we may compute an equivalent potential and hence effective coupling constant. In general the scattering matrix associated with the process described by a Feyn- man amplitude M is
0 4 X 0 X 0 S = Sa0ι0µ0α0aιµα(p , p) = (2π) δ( pi − pi)Ma0ι0µ0α0aιµα(p , p). (118) i i
Now, in the Born approximation, for two particle scattering, if polarizations are pre- served in NR scattering, the relationship between the scattering matrix for a process and the associated equivalent potential is up to a color factor given by [15]
Z * * 4 0 0 * −i q . r * S = −(2π) δ(p + p − p − p ) V (r )e dr δ 0 δ 0 . (119) Born 1 2 1 2 α1α1 α2α2
38 Hence in the Born approximation the potential associated with a process with Feyn- man amplitude M satisfies, up to a color factor,
Z * * * −i q . r * 0 0 V (r )e dr δ 0 δ 0 = −M 0 0 (p , p , p , p ). (120) α1α1 α2α2 α1α2α1α2 1 2 1 2
We will assume that the color factor for the process that we are currently consid- 1 ering has the value of 3 that we computed above. Therefore, for the process under consideration
* * 2 Z * q * g V (r )e−i . r dr = s 1 + g2 π(q)(s) + π(g) + π(gh) . *2 s q
Thus, taking the inverse Fourier transform (which can be shown to exist) of each side of this equation we obtain
* * * Z 1 q * V (r ) =(2π)−3g2 1 + g2 π(q)(s) + π(g) + π(gh) ei . r dq . s *2 s q
*2 (q) 2 1 The function π depends only on s = (−q ) 2 = q . Therefore, since the Fourier transform of a rotationally invariant function is rotationally invariant, V is rotation- ally invariant and we can write
* Z 1 q * V (r) = (2π)−3g2 1 + g2 π(q)(s) + π(g) + π(gh) ei .(0,0,r) dq s *2 s q Z ∞ Z π −2 2 2 (q) (g) (gh) irs cos(θ) = (2π) gs 1 + gs π (s) + π + π e sin(θ) dθ ds s=0 θ=0 Z ∞ −2 2 1 2 (q) (g) (gh) 1 irs −irs = (2π) gs 1 + gs π (s) + π + π (e − e ) ds, ir s=0 s Z ∞ 1 2 1 2 (q) (g) (gh) sin(rs) = 2 gs 1 + gs (π (s) + π + π ) ds 2π r s=0 s Z ∞ 1 2 1 2 (q) −1 (g) (gh) = 2 gs 1 + gs (π (r s) + π + π ) sinc(s) ds. 2π r s=0
Thus the running coupling constant g(r) at range r is given by
2 Z ∞ g(r) 1 2 1 2 (q) −1 (g) (gh) = 2 gs 1 + gs (π (r s) + π + π ) sinc(s) ds, (121) r 2π r s=0
39 and so Z ∞ 2 1 2 2 (q) −1 (g) (gh) g(r) = 2 gs 1 + gs (π (r s) + π + π ) sinc(s) ds. (122) 2π s=0
Therefore the running coupling g(τ) as a function of energy τ = r−1 is given by
Z ∞ 2 1 2 2 (q) (g) (gh) g(τ) = 2 gs 1 + gs (π (τs) + π + π ) sinc(s) ds. (123) 2π s=0
9.3 Determination of the bare QCD coupling constant gs
The running coupling depends on the units used for gs or in other words, one might say, on the magnitude of the (finite) bare coupling. In addition it depends on the quark masses through the dependence of π(q) on the quark masses. We will seek an 2 optimal value of gs by a least squares fit of the running coupling function g(τ) of the form given by Eq. 123 to experimental data. We will take the quark masses to be given constants. Therefore we have a parameter, that is gs, to vary when carrying out the optimization. However, in addition, we have another parameter to vary. This is because the QCD coupling constants obtained at various energies by means of experiment coupled with QFT calculations using regularization/renormalization are renormalization scheme (RS) dependent. The calculations are usually carried out using dimensional regular- ization and renormalization with the modified minimal subtraction (MS) scheme. If the renormalization scheme was any other scheme (e.g. minimal subtraction (MS)) then the running coupling graph would be translated in the direction of the αs axis. No renormalization scheme is canonical, the scheme used is a matter of convenience.
Therefore there is an additional parameter apart from gs involved in the optimiza- tion, that is the parameter defining a translation of the running coupling graph in the direction of the αs axis. We have carried out an optimal least squares fit of the running coupling function of the form of Eq. 123 to the experimental data used to generate the graph of Figure 9.3 of the 2018 Particle Data Group Review of Particle Physics [29] using the simulated annealing optimization method. The graph of the 1-loop spectral QCD running coupling for energies between 1 and 10000 GeV (with the τ-axis on a logarithmic scale) is shown in Figure 2. (The C++ code used to produce this figure can be found in Appendix 1.) The optimal values of the parameters involved in the optimization are
40 Figure 2: 1-loop spectral QCD running coupling versus log energy (GeV) between 1 and 10000 GeV
• gs = 4.66152
• translation= −2.68627
The figure shows that the optimized running coupling agrees well with the exper- imental data, i.e. when compared with Figure 9.3 of the 2018 Particle Data Group Review of Particle Physics [29]. Agreement may be improved by better determination of the quark masses and inclusion in the analysis of other Feynman diagrams in addition to the 1-loop vacuum polarization diagram. The graph of the 1-loop spectral QCD running coupling versus energy for energies in the infrared from 1 eV to 10 GeV (with the τ-axis on a logarithmic scale) is shown in Figure 3. It can be seen that the spectral QCD running coupling does not have a
Landau pole and has the property of “freezing of αs” [30].
41 Figure 3: 1-loop spectral QCD running coupling versus log energy (eV) between 1eV and 10 GeV 10 Higher order calculations
The Feynman amplitude for the insertion of two quark, gluon and ghost bubble blobs into the gluon propagator of the tree diagram is given by
(vp,2-loop) 0 0 iM 0 0 0 0 (p1, p2, p1, p2) = i1i2i1i2α1α2α1α2 T 0 0 a ρ (c 0 ⊗ u(u, p , α ))(igsT ⊗ γ )(ci ⊗ u(u, p1, α1)) i1 1 1 1
(vp)a1a2ρ1ρ2 (vp)a3a4ρ3ρ4 iEaa1ρρ1 (q)iΠ (q)iEa2a3ρ2ρ3 (q)iΠ (q)iEa4bρ4σ(q) 0 0 b σ (c 0 ⊗ u(d, p , α ))(ig T ⊗ γ )(c ⊗ u(d, p , α )). i2 2 2 s i2 2 2
42 Therefore
(vp,2-loop) 0 0 M 0 0 0 0 (p1, p2, p1, p2) = i1i2i1i2α1α2α1α2 2 a b 1 3 gs δabTi0 i Ti0 i ( ) 1 1 2 2 q2
2 2 ρ1ρ2 ρ1 ρ2 (tot) gs ηρρ1 (q η + q q )π (q)
2 2 ρ3ρ4 ρ3 ρ4 (tot) gs ηρ2ρ3 (q η + q q )π (q) 0 0 η M 0 0 0 0 (p , p , p , p ), ρ4σ 0,i1i2i1i2α1α2α1α2 1 2 1 2 where π(tot)(q) = π(q)(q) + π(g) + π(gh). (124)
Thus we need to compute the tensor
2 ρ1ρ2 ρ1 ρ2 2 ρ3ρ4 ρ3 ρ4 Xρσ = ηρρ1 (q η + q q )ηρ2ρ3 (q η + q q )ηρ4σ.
When expanded out Xρσ has four terms. These are
2 ρ1ρ2 2 ρ3ρ4 2 2 ηρρ1 q η ηρ2ρ3 q η ηρ4σ = (q ) ηρσ,
ρ1 ρ2 ρ3 ρ4 ρ3 2 ηρρ1 q q ηρ2ρ3 q q ηρ4σ = qρqρ3 q qσ = q qρqσ,
2 ρ1ρ2 ρ3 ρ4 2 ρ3 ρ4 2 ηρρ1 q η ηρ2ρ3 q q ηρ4σ = ηρρ3 q q q ηρ4σ = q qρqσ,
ρ1 ρ2 2 ρ3ρ4 ρ2 2 2 ηρρ1 q q ηρ2ρ3 q η ηρ4σ = qρq ηρ2σq = q qρqσ.
Therefore when sandwiched between the Dirac spinors, by the well known conser- 2 2 vation property (Eq. 111), the only term which survives is ηρσ(q ) . Thus, after evaluating the color factor, the Feynman amplitude is
(vp,2-loop) 1 2 1 2 ( ) 2 0 0 tot 0 0 0 0 Mα0 α0 α α (q) = gs (gs π (q)) M0,i i i1i2α α α1α2 (p1, p2, p1, p2). (125) 1 2 1 2 3 q2 1 2 1 2
By similar computations it can be shown, by induction, that the Feynman amplitude for the insertion of n quark, gluon and ghost bubble blobs into the gluon propagator of the tree level quark-quark scattering diagram is
(vp,n-loop) 1 2 1 2 ( ) n 0 0 tot 0 0 0 0 Mα0 α0 α α (q) = gs (gs π (q)) M0,i i i1i2α α α1α2 (p1, p2, p1, p2). (126) 1 2 1 2 3 q2 1 2 1 2
43 Arguing as in the previous section we see that the spectral QCD n-loop running coupling is given by
n 1 Z ∞ X g(τ)2 = g2 (g2π(tot)(τs))isinc(s) ds. (127) 2π2 s s s=0 i=0
For a 2-loop calculation we optimized using simulated annealing the fit of a function of the form
2 1 Z ∞ X g(g2, c, τ)2 = g2 (g2(π(tot)(τs) + c))isinc(s) ds. (128) s 2π2 s s s=0 i=0 to the data of the 2018 Particle Data Group Review of Particle Physics [29] and found the optimal values of the parameters to be
2 • gs = 12.654
• c = −0.0364263
The graph of the 2-loop spectral QCD running coupling for energies between 1 and 10000 GeV (with the τ-axis on a logarithmic scale) is shown in Figure 4. The graph of the 2-loop spectral QCD running coupling versus energy for energies in the infrared from 1 eV to 10 GeV (with the τ-axis on a logarithmic scale) is shown in Figure 5.
11 Discussion
We have written down an analytic expression Eq. 127 for the n-loop spectral QCD running coupling. It might be argued that we have not achieved anything because we are fitting the functional form given by this equation to the experimental data rather than predicting the experimental data. However there are two reasons why the procedure that we have adopted is useful. Firstly the QCD coupling constant is not an observable and, when its computation involves renormalization, it is RS dependent. The RS affects the vacuum polarization function by the addition of a constant. For example, with Pauli-Villars regularization the unrenormalized quark bubble vacuum polarization function is given by (Itzykson
44 Figure 4: 2-loop spectral QCD running coupling versus log energy (GeV) between 1 and 10000 GeV
Figure 5: 2-loop spectral QCD running coupling versus log energy (eV) between 1eV and 10 GeV
45 and Zuber, 1980 [31], p. 323)
( " 1 α Λ2 1 2m2 4m2 2 π(k2, m, Λ) = − − log( ) + + 2 1 + − 1 3π m2 3 k2 k2 1 #) 4m2 2 ×arcot − 1 − 1 , k2 where Λ is the fictitious mass parameter and therefore the counterterm involves the Λ2 factor − log( m2 ). For dimensional regularization the unrenormalized quark bubble vacuum polarization function is given by (Schwartz, 2014 [26], p. 309)
2 Z 1 −γE 2 2 gs 2 (4πe µ) π(p ) = − 2 dx x(1 − x) + log 2 2 , (129) 2π 0 m − p x(1 − x) where µ is the subtraction point. For MS RS scheme the counterterm involves the
1 1 −γE factor while for MS RS scheme the counterterm involves the factor +log(4πe ). In fact, for the so called Rδ RS [32] an arbitrary value δ is added. In any case the effect of a change in RS is the addition of a constant to π. Spectral regularization is finite at all stages and no renormalization is required, there are no counterterms. To compare our prediction with the experimental data which has been obtained using calculations based on the MS RS we need to add a constant to the vacuum polarization function. If all the calculations of the running fine structure constant at any given energy based on experiments were done using spectral regularization then no such constant would be required because renormalization would not be required. The second issue is the need to optimize the (finite) bare coupling constant. This is because this quantity is a constant of nature like the electron charge, and needs to be determined by experiment. We have determined it based on the 2018 Review of Particle Physics [29] data to have the value given above, which enables one to predict the behavior of the QCD running coupling at low energies and at high energies into the TeV region and beyond as given by Figures 2 to 5.
12 Conclusion
We have described a rigorous mathematical formulation providing a divergence-free framework for QCD and the theory of the electroweak interaction in curved space-
46 time. Our approach does not use variational principles as a starting point, the starting point is the notion of covariance which we interpret as 4D conformal covariance rather than the general (diffeomorphism) covariance of GR. We have described the general Fock space which is a graded algebra composed of tensor valued test functions for spaces of multiparticle states, where the particles can be fermions such as quarks and electrons or bosons such as gluons or photons. We consider algebra bundles whose typical fibers are the Fock spaces and related spaces. Scattering processes are associated with linear maps between the fibers of the associated vector bundles with these spaces as typical fibers. Such linear operators can be generated by intertwining operators between the spaces under consideration. We have shown how quark-quark scattering and gluon-gluon scattering are associated with kernels which generate such intertwining operators. The rest of the paper focuses on QCD vacuum polarization in order to compute the effective potential and hence the running coupling for QCD at different scales. Using spectral regularization we compute the densities associated with the quark bubble, the gluon bubble and the ghost bubble and hence the vacuum polarization tensor. It remains to be seen what else is needed to generate all the physics of the standard model. We have shown that the kernels that are associated with quark-quark and gluon-gluon scattering as well as electron-electron scattering and muon decay [23, 24] are covariant (i.e. intertwining). It remains to be verified that all possible scattering processes in the standard model are covariant. Covariance is a very strong condition, much stronger than simple Lorentz covariance.
If M1 and M2 are covariant kernels then so is α1M1 + α2M2 for any α1, α2 ∈ C.
Furthermore so is M1 ⊗ M2 so the covariant kernels form an algebra. An overall multiplicative factor does not matter since, by general principles of quantum theory, one is interested in rays in Hilbert space. But it is required to have a way to decide on valid relative weightings. We believe that a possible approach to selecting physically significant covariant kernels is through a cohomology algebra associated with the Fock space algebra bundle which can be defined in a fashion analogous to de Rham cohomology for smooth manifolds. The topology of M¨obiusstructures can, in general, be very complex. It is therefore possible that there could be a correspondence between the topology of space-time and the particle scattering processes that are occurring.
47 Acknowledgements
The author thanks G¨unther Dissertori and Siegfried Bethke for providing, and allow- ing the use of, the data for the experimental results for the QCD running coupling published in the 2018 Particle Data Group Review of Particle Physics.
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Appendix 1: C++ code to produce graph of run- ning coupling versus energy (Figure 2)
// QCD_running_coupling.cpp : This file contains the ’main’ function. // Program execution begins and ends there. //
50 #include
51 const double T_factor = 0.1; // factor by which temperature T is reduced after each epoch const int Epoch_size = 100; // SA epoch size double T = T_init; double pi_g = 19.0 / (32.0 * pi); // gluon bubble double pi_gh = 1.0 / (32.0 * pi); // ghost bubble double g_s_sq; double translation; double best_g_sq; double best_translation; double best_Delta; int main() { std::cout << "Hello World!\n"; std::ofstream outFile("file.txt"); std::ifstream inFile("data.txt");
inFile >> n_data; create_arrays(); make_quark_masses(); std::cout << "n_data = " << n_data << "\n"; int i; for (i = 0; i < n_data; i++) { inFile >> tau_data[i] >> alpha_data[i]; } for (i = 0; i < n_data; i++) tau_data[i] *= 1.0e9; initialize_parameters(); // initialize g_s_sq and trans optimize_parameters(); // find optimal values for g_s_sq and trans using SA std::cout << "optimal bare coupling = " << sqrt(g_s_sq) << " optimal translation = " << translation << "\n";
52 // compute running constant squared for (i = 0; i <= n; i++) { double tau = start + i * delta; double answer = compute_value(best_g_sq, best_translation, tau); outFile << (log(tau) - log(start))/log(10.0) << ’\t’ << answer/(4.0*pi) << "\n"; } return(0); } double compute_value(double g_sq, double trans, double tau) { double I = 0.0; int j; for (j = 1; j <= n_int; j++) { double s = j * delta_int; double factor = 1.0 + g_sq*(pi_q(s * tau)+pi_g+pi_gh); I += sin(s) * factor / s; } I *= delta_int; I *= g_sq / (2.0 * pi * pi); I += trans; return(I); } double compute_value_0(double tau) { double I = 0.0; int j; for (j = 1; j <= n_int; j++) { double s = j * delta_int;
53 double factor = pi_q(s * tau); I += sin(s) * factor / s; } I *= delta_int; return(I); } void create_arrays() { mass = new double[6]; tau_data = new double[n_data]; alpha_data = new double[n_data]; } void make_quark_masses() { mass[0] = 2.3e6; // up quark mass[1] = 4.8e6; // down quark mass[2] = 1275.0e6; // charm quark mass[3] = 95.0e6; // strange quark mass[4] = 173210.0e6; // top quark mass[5] = 4180.0e6; // bottom quark } double pi_q(double s) { // vacuum polarization associated with quark bubble double answer = 0.0; int k; for (k = 0; k < 6; k++) { m = mass[k]; if (s > (2 * m)) answer -= m * m * m * Z(s) * (3.0 + 2.0 * Z(s) * Z(s)) / (3.0 * pi * s * s * s); // fermion bubble
54 } return(answer); } double Z(double s) { if (s < 2.0 * m) { std::cout << "error in Z function\n"; exit(1); } double answer = sqrt(s * s / (4.0*m * m) - 1.0); return(answer); } void initialize_parameters() { double d_1 = compute_value_0(tau_data[0]); double d_2 = compute_value_0(tau_data[n_data-1]); double g_1_sq = 4.0 * pi * alpha_data[0]; double g_2_sq = 4.0 * pi * alpha_data[n_data - 1]; g_s_sq = sqrt(2.0*pi*pi*(g_2_sq - g_1_sq) / (d_2 - d_1)); translation = g_1_sq - (1.0/(2.0*pi*pi))*g_s_sq * (pi / 2.0 + g_s_sq * (d_1 + (pi / 2.0) * (pi_g + pi_gh))); } void optimize_parameters() // optimize g_s_sq and translation using simulated annealing { best_g_sq = g_s_sq; best_translation = translation; double log_g_s_sq = log(g_s_sq); double Delta = 0.0; // variable representing error of fit int j; for (j = 0; j < n_data; j++)
55 { double v = compute_value(g_s_sq, translation, tau_data[j]) - 4.0*pi*alpha_data[j]; Delta += v * v; } best_Delta = Delta; std::cout << "initial Delta = " << Delta << "\n"; // carry out simulated annealing int i; for (i = 0; i < n_iter; i++) { double new_Delta = 0.0; int sgn = make_sign(); double new_log_g_s_sq = log_g_s_sq + sgn * log_g_s_sq_step; double new_g_s_sq = exp(new_log_g_s_sq); sgn = make_sign(); double new_trans = translation + sgn * trans_step; for (j = 0; j < n_data; j++) { double v = compute_value(new_g_s_sq, new_trans, tau_data[j]) - 4.0*pi*alpha_data[j]; new_Delta += v * v; } if (new_Delta < Delta) { log_g_s_sq = new_log_g_s_sq; g_s_sq = new_g_s_sq; translation = new_trans; Delta = new_Delta; } else if(exp(-(new_Delta-Delta)/T)>my_rand()) { log_g_s_sq = new_log_g_s_sq; g_s_sq = new_g_s_sq; translation = new_trans;
56 Delta = new_Delta; } if (new_Delta < best_Delta) { best_g_sq = new_g_s_sq; best_translation = new_trans; best_Delta = new_Delta; } if(i\%Epoch_size==0) T *= T_factor; } std::cout << "final Delta = " << Delta << "\n"; } int make_sign() { int sgn = 2.0 * my_rand(); if (sgn == 0) sgn -= 1; return(sgn); } double my_rand() { return(((double)rand() + 1.0) / (RAND_MAX + 2.0)); }
57