Asymptocic Freedom of Gluons in Hamiltonian Dynamics

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by  María Gómez-Rocha

Few-Body Systems manuscript No.
(will be inserted by the editor)
Mar´ıa G´omez-Rocha · Stanis(cid:32)law D. G(cid:32)lazek
Asymptocic Freedom of Gluons in Hamiltonian
Dynamics
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Received: date / Accepted: date
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Abstract We derive asymptotic freedom of gluons in terms of the renormalized SU(3) Yang-Mills
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Hamiltonian in the Fock space. Namely, we use the renormalization group procedure for effective par-
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ticles(RGPEP)tocalculatethethree-gluoninteractionterminthefront-formYang-MillsHamiltonian
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using a perturbative expansion in powers of g up to third order. The resulting three-gluon vertex is a
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function of the scale parameter s that has an interpretation of the size of effective gluons. The corre-
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sponding Hamiltonian running coupling constant exhibits asymptotic freedom, and the corresponding

] Hamiltonian β-function coincides with the one obtained in an earlier calculation using a different
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generator.
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- Keywords Asymptotic freedom · Quantum chromodynamics · Renormalization group · Relativistic
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Hamiltonian dynamics
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1 Introduction
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v
1 The renormalization group procedure for effective particles (RGPEP) has been developed during the
0 lastyears[1;2;3]asanon-perturbativetoolforconstructingbound-statesinquantumchromodynamics
8 (QCD)[4].Itintroducestheconceptofeffectiveparticles,whichdifferfromthebareorcanonicalones,
7 byhavingsizes,correspondingtothemomentumscaleλ=1/s.Creationandannihilationofeffective
0 particles in the Fock space are described by the action of effective particle operators, a† and a , on
. s s
1 states built from the vacuum state |0(cid:105) using a†; bare particle operators, a† and a , appearing in the
s 0 0
0
canonical Hamiltonian, create and annihilate pointlike particles (with size s=0).
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We are interested in calculating the evolution of quark and gluon quantum states, describing their
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dynamics and studying their binding. In a single formulation, the sought effective Hamiltonian must
:
v provide a means for the constituent-like behavior of quarks and gluons in hadrons with the measured
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X quantum numbers, and also an explanation for the short-distance phenomena of weakly interacting
pointlike partons.
r
a In this work, we apply the RGPEP to interacting gluons in the absence of quarks. We will demon-
strate that the RGPEP passes the test of describing asymptotic freedom, which is a precondition for
any approach aiming at using QCD, especially for tackling nonperturbative issues, such as the ones
that emerge when one allows effective gluons to have masses [4].
PresentedbyMar´ıaGo´mez-RochaatLightCone2015,21-25September,2015,INFNFrascatiNationalLabo-
ratories.
Mar´ıa Go´mez-Rocha
ECT*, Strada delle Tabarelle, 286, I-38123 Villazzano Trento, Italy
E-mail: [email protected]
Stanisl(cid:32)aw D. G(cid:32)lazek
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
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We start from the regularized canonical Hamiltonian for quantum Yang-Mills field in the Fock
space, obtained from the corresponding Lagrangian density. We use the RGPEP to introduce effective
particles and calculate a family of effective Hamiltonians characterized by a scale or size parameter s.
These Fock-space Hamiltonians depend on the effective-particle size parameter in an asymptotically
free way: the coupling constant in a three-gluon interaction term vanishes with the inverse of ln(1/s).
In the following, we summarize the procedure, which is general and can be applied to any other
quantum field theory. For a more extended and detailed explanation, we refer the reader to Ref. [3].
2 Renormalization group procedure for effective particles
2.1 Initial Hamiltonian
We derive the canonical Hamiltonian for Yang-Mills theories from the Lagrangian density:
1
L=− trFµνF , (1)
2 µν
where Fµν = ∂µAν −∂νAµ +ig[Aµ,Aν], Aµ = Aaµta, [ta,tb] = ifabctc, which leads to the energy-
momentum tensor,
Tµν =−Faµα∂νAa +gµνFaαβFa /4 . (2)
α αβ
Wechoosethefront-form(FF)ofdynamics[5]whichconsistsofsettingthequantizationsurfaceonthe
hyperplane x+ =x0+x3 =0. Using the gauge A+ =0, the Lagrange equations lead to the condition
1 2
A− = 2∂⊥A⊥− ig[∂+A⊥,A⊥] , (3)
∂+ ∂+2
so that the only degrees of freedom are the fields A⊥.
Integration of T+− over the front x+ =0 leads to the FF energy of the constrained gluon field:
(cid:90)
1
P− = dx−d2x⊥T+−| . (4)
2 x+=0
Thisoperatorcontainsaseriesofproductsof2nd,3rdor4thpowersofthefieldAµ ortheirderivatives.
The energy momentum tensor T+− can be written as
T+− =H +H +H +H , (5)
A2 A3 A4 [∂AA]2
where [6; 7; 8]
1
H =− A⊥a(∂⊥)2A⊥a , (6)
A2 2
H =gi∂ Aa[Aα,Aβ]a , (7)
A3 α β
1
H =− g2[A ,A ]a[Aα,Aβ]a , (8)
A4 4 α β
1 1
H = g2[i∂+A⊥,A⊥]a [i∂+A⊥,A⊥]a . (9)
[∂AA]2 2 (i∂+)2
ThequantumcanonicalHamiltonianisobtainedbyreplacingthefieldAµbythequantumfieldoperator
(cid:88)(cid:90) (cid:104) (cid:105)
Aˆµ = [k] tcεµ a e−ikx+tcεµ∗a† eikx , (10)
kσ kσc kσ kσc
x+=0
σc
where [k]=θ(k+)dk+d2k⊥/(16π3k+), the polarization four-vector is defined as εµ =(ε+ =0,ε− =
kσ kσ kσ
2k⊥ε⊥/k+,ε⊥) and the indices σ and c denote spin and color quantum numbers, respectively. The
σ σ
creation and annihilation operators satisfy the commutation relations
(cid:104) (cid:105) (cid:104) (cid:105)
a ,a† =k+δ˜(k−k(cid:48)) δσσ(cid:48)δcc(cid:48) , [a ,a ] = a† ,a† = 0 , (11)
kσc k(cid:48)σ(cid:48)c(cid:48) kσc k(cid:48)σ(cid:48)c(cid:48) kσc k(cid:48)σ(cid:48)c(cid:48)
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with δ˜(p)=16π3δ(p+)δ(p1)δ(p2).
The canonical Hamiltonian is divergent and needs regularization. At every interaction term, every
creation and annihilation operator in the canonical Hamiltonian is multiplied by a regulating factor1
r (κ⊥,x)=exp(−κ⊥/∆)xδθ(x−(cid:15)) , (12)
∆δ
where x is the relative momentum fraction x = p+/P+, κ is the relative transverse momentum,
p/P
κ = p⊥−xP⊥, and P is the total momentum in the term one considers. The regulating function
p/P
preventstheinteractiontermsfromactingifthechangeoftransversemomentumbetweengluonswere
to exceed ∆, or if the change of longitudinal momentum fraction x were to be smaller than δ.
2.2 Derivation of the effective Hamiltonian
The RGPEP transforms bare, or point-like creation and annihilation operators into effective ones [1].
Effective particle operators of size s=t1/4 are related to bare ones by certain unitary transformation
a =U a U† . (13)
t t 0 t
The fact that the Hamiltonian operator cannot be affected by this change requires,
H (a )=H (a ) , (14)
t t 0 0
which is equivalent to writing:
H (a )=U†H (a )U . (15)
t 0 t 0 0 t
Differentiating both sides of (15) leads to the RGPEP equation:
H(cid:48)(a )=[G (a ),H (a )] , (16)
t 0 t 0 t 0
(cid:16) (cid:17)
where G = −U†U(cid:48), and therefore, U = T exp −(cid:82)tdτG . T denotes ordering in τ. The RGPEP
t t t t 0 τ
equation (16) is the engine of this procedure. It governs the evolution of effective particles with the
scaleparametert.Itencodestherelationbetweenpointlikequantumgluonsappearinginthecanonical
Hamiltonian and the effective, or constituent ones referred to by effective phenomenological models
describing bound states.
We choose the generator to be the commutator G = [H ,H ],2 where H is the non-interacting
t f Pt f
term of the Hamiltonian and H is defined in terms of H .
Pt t
H is a series of normal-ordered products of creation and annihilation operators,
t
∞
(cid:88) (cid:88)
H (a )= c (i ,...,i ) a† ···a . (17)
t 0 t 1 n 0i1 0in
n=2 i1,i2,...,in
H differs from H by the vertex total +-momentum factor,
Pt t
∞ (cid:32) n (cid:33)2
(cid:88) (cid:88) 1(cid:88)
H (a )= c (i ,...,i ) p+ a† ···a . (18)
Pt 0 t 1 n 2 ik 0i1 0in
n=2 i1,i2,...,in k=1
Theinitialconditionforthedifferentialequation(16)isgivenbytheregularizedcanonicalHamiltonian
given in Section 2.1 plus counterterms. More precisely, the initial condition is given by the physical
fact that at very small distances or very high energies, the regularized canonical Hamiltonian must be
recovered and any regularization dependence must be removed.
We solve the RGPEP equation (16) for the effective Hamiltonians using an expansion in powers of
the coupling constant g up to third order and we focus our studies on the structure of the three-gluon
term [1; 3].
1 Other regulating functions are available [3]. Finite dependence of the effective Hamiltonian on the small-x
regularizationmaybethoughttoberelatedtothevacuumstateproblem,thephenomenaofsymmetrybreaking
and confinement [4].
2 Other generators are also allowed but may lead to more complicated expressions [9]. Our choice is similar
to Wegner’s [10].
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3 The three-gluon vertex
The third-order effective Hamiltonian expansion have the following structure:
H =H +H +H +H +H +H +H +H +H .(19)
t 11,0,t 11,g2,t 21,g,t 12,g,t 31,g2,t 13,g2,t 22,g2,t 21,g3,t 12,g3,t
Thefirstandsecondsubscriptsindicatethenumberofcreationandannihilationoperators,respectively.
The third subscript labels the order in powers of g. Finally, the last label indicates the dependence on
the scale parameter t.
The initial condition at t=0 has the form:
H =H +H +H +H +H +H +H +H +H (,20)
0 11,0,0 11,g2,0 21,g,0 12,g,0 31,g2,0 13,g2,0 22,g2,0 21,g3,0 12,g3,0
and consists of the regularized canonical Hamiltonian plus counterterms. The latter are calculated in
such a way that H remains finite when ∆ → ∞. It is not possible to remove the small-x cutoff δ at
t
thesepoint.However,thisdependencewillbeofinterestinhigher-ordercalculations,sincethesmall-x
phenomena are thought to be related to the vacuum-state behavior. The last step in the RGPEP is to
replace bare creation and annihilation operators by effective ones.
The three-gluon vertex and the running of the Hamiltonian coupling are encoded in the sum of
first- and third-order terms:
H =(H +H )+(H +H ) . (21)
(1+3),t 21,g,t 12,g,t 21,g3,t 12,g3,t
The third-order solution requires the knowledge of the first and second-order solutions. The sum of all
these contributions has the form3 [3] (see Fig. 1):
(cid:88)(cid:90) (cid:104) (cid:105)
H = [123] δ˜(k +k −k ) f Y˜ (x ,κ⊥,σ)a† a† a +Y˜ (x ,κ⊥,σ) a† a a
(1+3),t 1 2 3 12t 21t 1 12 1t 2t 3t 12t 1 12 3t 2t 1t
123
(22)
where f12t =e−(k1+k2)4t is a form factor and Y˜21t(x1,κ⊥12,σ) is the object of our study. We define the
Hamiltoniancouplingconstantg asthecoefficientinfrontofthecanonicalcolor,spinandmomentum
t
dependentfactorY123(x1,κ⊥12,σ)=ifc1c2c3[ε∗1ε∗2·ε3κ⊥12−ε∗1ε3·ε∗2κ⊥12x21/3 −ε∗2ε3·ε∗1κ⊥12x11/3]inthelimit
κ⊥ →0, for some value of x denoted by x . So,
12 1 0
lim Y˜(x ,κ⊥,σ)= lim (cid:2)c (x ,κ⊥)Y (x ,κ⊥,σ)+g3T˜ (x ,κ⊥,σ)(cid:3). (23)
t 1 12 t 1 12 123 1 12 3finite 1 12
κ⊥→0 κ⊥→0
12 12
where T˜ (x ,κ⊥,σ) is a finite part contained in the counterterm and does not contribute to the
3finite 1 12
running coupling,
lim c (x ,κ⊥)=g+g3 lim (cid:2)c (x ,κ⊥)−c (x ,κ⊥)(cid:3) . (24)
κ⊥→0 t 1 12 κ⊥→0 3t 1 12 3t0 1 12
12 12
And assuming some value for g at some small t , g =g ,
0 0 t0 0
g ≡c (x ) = g +g3[c (x )−c (x )] . (25)
t t 1 0 0 3t 1 3t0 1
We introduce now the momentum scale parameter λ=t−1/4. This yields,
g3 λ
g =g − 0 N 11 ln . (26)
λ 0 48π2 c λ
0
Differentiation of the latter with respect to λ leads to
d 11N
λ g =β g3 , with β = − c . (27)
dλ λ 0 λ 0 48π2
This result equals the asymptotic freedom result in Refs. [11; 12], when one identifies λ with the
momentum scale of external gluon lines in Feynman diagrams. Our result also coincides with the
expression obtained in [9], where an analogous calculation were performed using a different generator.
3 The subscripts 1,2,3 refer to the gluon lines indicated in Fig. 1. So, e.g. κ⊥ =x κ⊥ −x κ⊥ .
12 2/3 1/3 1/3 2/3
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Fig. 1 Graphical representation of terms contributing to the effective three-gluon vertex (third-order expan-
sion) [3]. Thin internal lines correspond to intermediate bare gluons and thick external lines correspond to
the creation and annihilation operators that appear in the three-gluon FF Hamiltonian interaction term for
effective gluons of size s. Dashed lines with transverse bars represent the combined contributions of terms (8)
and (9). The black dots indicate counterterms.
4 Summary and conclusion
We have applied the RGPEP to the quantum SU(3) Yang-Mills theory and extracted the running
coupling from the three-gluon-vertex term in the third-order effective Hamiltonian. The result turns
out to be independent of the choice of the generator, as it coincides with the one obtained in an
analogous calculation performed in [9], using a different generator. The present generator, however,
leads to simpler equations than the older one, which is desired and needed for our forthcoming forth-
order calculations, required for any attempt at description of physical systems using QCD [4]. The
obtained running coupling is of the form that is familiar from other formalism and renormalization
schemes and passes the test of producing asymptotic freedom, which any method aiming at solving
QCD must past.
Acknowledgements Part of this work was supported by the Austrian Science Fund (FWF) under project
No. P25121-N27. Fig. 1 was produced with JaxoDraw [13].
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