Momentum Sum Rule Is Violated in The Operator Product Expansion in QCD At The High Energy Colliders
aa r X i v : . [ h e p - ph ] J un Momentum Sum Rule Is Violated in The Operator ProductExpansion in QCD At The High Energy Colliders
Gouranga C Nayak ∗ C. N. Yang Institute for Theoretical Physics,Stony Brook University, Stony Brook NY, 11794-3840 USA (Dated: September 26, 2018)
Abstract
To prove the momentum sum rule in the operator product expansion (OPE) in QCD at highenergy colliders it is assumed that < P | ˆ T ++ (0) | P > = 2( P + ) where | P > is the momentumeigenstate of the hadron H with momentum P µ and ˆ T ++ (0) is the ++ component of the gaugeinvariant color singlet energy-momentum tensor density operator ˆ T µν (0) of all the quarks plusantiquarks plus gluons inside the hadron H . However, in this paper, we show that this relation < P | ˆ T ++ (0) | P > = 2( P + ) is correct if ˆ T µν (0) is the energy-momentum tensor density operatorof the hadron but this relation < P | ˆ T ++ (0) | P > = 2( P + ) is not correct if ˆ T µν (0) is the gaugeinvariant color singlet energy-momentum tensor density operator of all the quarks plus antiquarksplus gluons inside the hadron. Hence we find that the momentum sum rule is violated in theoperator product expansion (OPE) in QCD at high energy colliders. PACS numbers: 11.30.-j, 11.30.Cp, 11.15.-q, 12.38.-t ∗ G. C. Nayak was affiliated with C. N. Yang Institute for Theoretical Physics in 2004-2007.
Typeset by REVTEX 1 . INTRODUCTION
After the discovery of the renormalization [1] and the asymptotic freedom [2, 3] in quan-tum chromodynamics (QCD) the prediction power of the perturbative QCD (pQCD) atshort distance is tested at various collider and fixed target experiments. For the incomingand outgoing hadrons at the high energy colliders the factorization theorem in QCD [4] hasplayed a major role along with the DGLAP evolution equation [5] to make prediction ofphysical observable by using pQCD calculation.One of the major problem remains to be solved in QCD is the non-perturbative QCDat log distance which will provide us information about hadron formation from quarks andgluons. Since the non-perturbative QCD is not solved yet the non-perturbative partondistribution function (PDF) inside the hadron and parton to hadron fragmentation function(FF) are extracted from various collider and fixed target experiments.The momentum sum rule in QCD plays an important role at high energy colliders. Inparticular it is widely used in various parton distribution function (PDF) sets such as CTEQ[6], GRV [7], MRST [8] etc. for the phenomenological prediction of quark and gluon distri-bution functions inside the hadron for practical use. The momentum sum rule [9, 10] statesthat the momentum of all the quarks plus antiquarks plus gluons inside the hadron at highenergy colliders is equal to the momentum of the hadron.In order to see the practical importance of the momentum sum rule in QCD at highenergy colliders consider the experimental measurement of the structure functions. Unlikequark structure function measurement it is not easy to measure gluon structure functionat high energy experiments. Since the gluon structure function is not easily experimentallymeasured one can use the momentum sum rule to determine the gluon structure functionfrom the experimentally measured quark structure function. This is one of the techniqueused in various PDF sets such as GRV [7], MRST [8] and CTEQ [6] etc. to determine thequark and gluon distribution functions inside the hadron. Hence the momentum sum rulein QCD at high energy colliders has been very useful for practical purpose.Since the momentum sum rule tells us that the momentum of all the quarks plus anti-quarks plus gluons inside the hadron is equal to the momentum of the hadron, the momentumsum rule can be derived from the conservation of momentum in QCD from the first principleby using gauge invariant Noether’s theorem. However, this is not straightforward like the1omentum conservation statement in QED because of confinement in QCD, a phenomenawhich is absent in QED. The main point is that the momentum sum rule in QCD at highenergy colliders equals the momentum of all the quarks plus antiquarks plus gluons insidethe hadron to the momentum of the hadron. Since we do not know how the hadron isformed from quarks plus antiquarks plus gluons due to the lack of our knowledge aboutconfinement in QCD which requires to solve non-perturbative QCD, we find that the proofof momentum sum rule in QCD from the first principle must include non-perturbative QCDat large distance where the hadron is formed.Recently by using the gauge invariant Noether’s theorem in QCD from the first principlewe have proved that the momentum sum rule in QCD is violated at high energy colliders [11]due to confinement in QCD which involves non-perturbative QCD at long distance. We havefound non-vanishing boundary surface term at large distance in QCD where confinementhappens because the potential energy at large distance r in QCD is an increasing functionof distance r [11].In the operator product expansion (OPE) in QCD the momentum sum rule is provedwhich does not need to use QCD at large distances [10]. To prove the momentum sum rulein the operator product expansion (OPE) in QCD it is assumed that [10] < P | ˆ T ++ (0) | P > = 2( P + ) (1)where the ˆ T ++ (0) is the ++ component of the gauge invariant color singlet energy-momentum tensor density operator ˆ T µν (0) of all the quarks plus antiquarks plus gluonsinside the hadron H and | P > is the (physical) momentum state of the hadron H withmomentum P µ normalized as < P ′ | P > = 2 P + δ ( P ′ + − P + ) δ (2) ( P ′ T − P T ) . (2)In terms of | ˜ P > which is normalized to unity we find from eq. (1) [12] < P | ˆ T ++ (0) | P > = 2 P + < ˜ P | Z d x ˆ T ++ ( x ) | ˜ P > = 2( P + ) . (3)However, in this paper, we show that this relation < P | ˆ T ++ (0) | P > = 2( P + ) in eq. (1)is correct if ˆ T µν (0) is the energy-momentum tensor density operator of the hadron but thisrelation < P | ˆ T ++ (0) | P > = 2( P + ) in eq. (1) is not correct if ˆ T µν (0) is the gauge invariantcolor singlet energy-momentum tensor density operator of all the quarks plus antiquarksplus gluons inside the hadron. 2his implies that the eq. (1) is an assumption in [10] but is not a proof based on the firstprinciple in QCD. Hence we find that the momentum sum rule is violated in the operatorproduct expansion (OPE) in QCD at high energy colliders.The paper is organized as follows. In section II we discuss the hadronic matrix elementof hadronic operator. In section III we discuss the hadronic matrix element of partonicoperator. In section IV we prove that the momentum sum rule is violated in the operatorproduct expansion (OPE) in QCD at high energy colliders. Section V contains conclusions. II. HADRONIC MATRIX ELEMENT OF HADRONIC OPERATOR
Consider a particle of mass M located at the space-time coordinate X µ ( t ) with velocity u µ = dX µ ( t ) dt , momentum P µ and energy P = E . In classical mechanics, similar to thecurrent density j µ ( x ), we find that the energy-momentum tensor density T µν ( x ) of thisparticle is given by T µν ( x ) = δ (3) ( ~x − ~X ( t )) P µ P ν P . (4)For free particle we find from eq. (4) ∂ µ T µν ( x ) = 0 (5)which gives conserved energy-momentum Z d xT µ ( x ) = P µ . (6)From eq. (4) we find the energy-momentum tensor Z d xT µν ( x ) = 1 P P µ P ν . (7)In quantum mechanics the momentum eigenvalue equation is given byˆ P µ | P > = P µ | P > (8)which gives < P | ˆ P µ | P > = P µ Z d x (9)where | P > is the momentum eigenstate with the normalization < P ′ | P > = δ (3) ( ~P ′ − ~P ) . (10)3ence similar to eq. (1) if we use the momentum density operator ˆ P µ (0) then we find fromeq. (9) < P | ˆ P µ (0) | P > = P µ (11)Extending the conserved momentum equation in classical mechanics from eq. (6) to quantummechanics we find from eqs. (11) [similar to eq. (3)] < P | ˆ P µ (0) | P > = < P | ˆ T µ (0) | P > = < ˜ P | Z d x ˆ T µ ( x ) | ˜ P > = P µ (12)which gives the conserved momentum P µ where ˆ T µν ( x ) is the energy-momentum tensordensity operator and | ˜ P > is normalized to unity.In [10] the normalization in the momentum eigenstate | P > of the hadron in eq. (2)differs from the normalization in the momentum eigenstate | P > in eq. (10) by a factor of √ P . Hence using the momentum eigenstate | P > of hadron of [10] from eq. (2) we findfrom eq. (12) that < P | ˆ T µ (0) | P > = 2 P < ˜ P | Z d xT µ ( x ) | ˜ P > = 2 P P µ (13)where | ˜ P > is normalized to unity. Eq. (13) in the light-cone coordinate system gives < P | ˆ T ++ (0) | P > = 2 P + < ˜ P | Z d x ˆ T ++ ( x ) | ˜ P > = 2 ( P + ) (14)which reproduces eq. (1) which agrees with [10]. III. HADRONIC MATRIX ELEMENT OF PARTONIC OPERATOR
Let us denote the momentum eigenstate of the gauge invariant color singlet momentumoperator ˆ p µi of the parton i = q, ¯ q, g by | p i > with the eigenvalue equationˆ p µi | p i > = p µi | p i > (15)where p µi is the gauge invariant color singlet momentum of the parton i . Note that themomentum of the hadron in eq. (8) is denoted by capital P µ whereas the momentum of theparton i in eq. (15) is denoted by small p µi .Hence we find from eqs. (8) and (15) thatˆ p µi | P > = P µ | P > (16)4here ˆ p µi is the gauge invariant color singlet momentum operator of the parton i , the | P > is the (physical) momentum eigenstate of the hadron H and P µ is the momentum of thehadron H . The eq. (16) is true because momentum operator ˆ p µi of the parton i is not physicalbecause we have not directly experimentally observed quarks and gluons whereas | P > and P µ in eq. (16) are physical because P µ and | P > are the momentum and momentumeigenstate of the physical hadron.The gauge invariant color singlet momentum operator ˆ p µi of a parton i obtained from firstprinciple by using gauge invariant Noether’s theorem in QCD is given byˆ p µi = Z d x ˆ T µi ( x ) (17)where ˆ T µνi ( x ) is the gauge invariant color singlet energy-momentum tensor density of theparton i in QCD [13].It is important to mention here the important difference between the gauge invariantcolor singlet momentum operator ˆ p µi and momentum eigenstate | p i > of the parton i ineq. (15) and the momentum operator ˆ P µ and momentum eigenstate | P > of hadron in eq.(8). The momentum operator ˆ P µ and momentum eigenstate | P > of hadron in eq. (8) arephysical and hence the momentum is conserved, i. e. , d < P | ˆ P µ (0) | P >dt = d < P | ˆ T µ (0) | P >dt = d < ˜ P | R d x ˆ T µ ( x ) | ˜ P >dt = 0 . (18)Hence it is important to remember that the momentum operator ˆ P µ of the hadron is aphysically acceptable momentum operator because it is conserved, i. e. , it satisfies eq. (18).However, the gauge invariant momentum operator ˆ p µi and momentum eigenstate | p i > of the parton i in eq. (15) are not physical because we have not directly experimentallyobserved quarks and gluons. Hence it is not surprising that the momentum of the parton isnot physical and hence is not conserved i. e. , d < P | ˆ¯ p µi (0) | P >dt = d < P | ˆ T µi (0) | P >dt = d < ˜ P | R d x ˆ T µi ( x ) | ˜ P >dt = 0 (19)where ˆ¯ p µi represents the momentum density operator of the parton i .Note that even if the momentum operator ˆ p µi of the parton i is color singlet and gaugeinvariant but the momentum < P | ˆ¯ p µi (0) | P > of the parton i inside the hadron H is not aphysical observable because we have not directly experimentally observed quarks and gluonseven if the momentum eigenstate | P > of the hadron H in eq. (19) is physical. Therefore it5s important to remember that the momentum operator ˆ p µi of the parton is not a physicallyacceptable momentum operator because it is not conserved, i. e. , it satisfies eq. (19). Asmentioned above, since we have not directly experimentally observed quarks and gluons, theeigenstate | p i > and the eigenvalue equation for the parton i in eq. (15) does not correspondto any physical situation. We have just considered the eq. (15) as an illustration purposeto emphasize the validity of eq. (16).From eqs. (16) and (17) we find < P | ˆ T µi (0) | P > = < ˜ P | Z d x ˆ T µi ( x ) | ˜ P > = P µ . (20)If we use the extra normalization factor √ P in the (physical) momentum eigenstate | P > of the hadron H as used in eq. (2) [10] then we find from eq. (20) < P | ˆ T µi (0) | P > = 2 P < ˜ P | Z d x ˆ T µi ( x ) | ˜ P > = 2 P P µ (21)where | ˜ P > is normalized to unity. Eq. (21) in the light-cone coordinate system gives < P | ˆ T ++ i (0) | P > = 2 P + < ˜ P | Z d x ˆ T ++ i ( x ) | ˜ P > = 2( P + ) . (22) IV. MOMENTUM SUM RULE IS VIOLATED IN THE OPERATOR PRODUCTEXPANSION IN QCD AT HIGH ENERGY COLLIDERS
From the gauge invariant Noether’s theorem [13] in QCD we find < ˜ P | X i = q, ¯ q,g Z d x∂ µ ˆ T µνi ( x ) | ˜ P > = 0 (23)where | ˜ P > is the (physical) state of the hadron H normalized to unity, the P i = q, ¯ q,g representsthe sum of all the quarks plus antiquarks plus gluons inside the hadron H andˆ T νµg ( x ) = ˆ F νλc ( x ) ˆ F µcλ ( x ) + 14 g νµ ˆ F cλδ ( x ) ˆ F λδc ( x ) (24)is the gauge invariant energy-momentum tensor density operator of the gluon in QCD andˆ T νµq ( x ) = i ψ l ( x )[ γ ν ( δ lk −→ ∂ µ − igT clk ˆ Q µc ( x )) − γ ν ( δ lk ←− ∂ µ + igT clk ˆ Q µc ( x ))] ˆ ψ k ( x ) (25)is the gauge invariant energy-momentum tensor density operator of the quark in QCD. Ineqs. (24) and (25) the ˆ ψ i ( x ) is the quark field, ˆ Q aµ ( x ) is the gluon field and F cµδ ( x ) = ∂ µ ˆ Q cδ ( x ) − ∂ δ ˆ Q cµ ( x ) + gf cdb ˆ Q dδ ( x ) ˆ Q bµ ( x ) . (26)6rom eq. (23) we find < ˜ P | X i = q, ¯ q,g Z d x ˆ T µi ( x ) | ˜ P > = − < ˜ P | X i = q, ¯ q,g Z d x∂ k ˆ T kµi ( x )] | ˜ P > (27)where in the R d x integration the R dt integration is an indefinite integration and R d x integration is definite integration.Using eq. (17) in (27) we find < ˜ P | X i = q, ¯ q,g Z d x ˆ T µi ( x ) | ˜ P > = < P | X i = q, ¯ q,g ˆ¯ p µi (0) | P > = − < ˜ P | X i = q, ¯ q,g Z d x∂ k ˆ T kµi ( x )] | ˜ P > (28)where ˆ¯ p µi represents the momentum density operator of the parton i and | P > is the mo-mentum eigenstate of the hadron H with momentum P µ satisfying the normalization in eq.(10) and | ˜ P > is normalized to unity.Note that if the boundary surface term does not vanish in QCD, i. e. , if < ˜ P | X i = q, ¯ q,g Z d x∂ k ˆ T kµi ( x )] | ˜ P > = 0 (29)then we find < ˜ P | X i = q, ¯ q,g Z d x∂ k ˆ T kµi ( x )] | ˜ P > = time dependent (30)which gives from eq. (28) < ˜ P | X i = q, ¯ q,g Z d x ˆ T µi ( x ) | ˜ P > = < P | X i = q, ¯ q,g ˆ¯ p µi (0) | P > = time dependent . (31)Hence we find that if the boundary surface term does not vanish in QCD, i. e. , if the eq.(29) is satisfied then we find from eq. (31) that < P | ˆ T µ (0) | P > = < P | X i = q, ¯ q,g ˆ T µi (0) | P > = < ˜ P | X i = q, ¯ q,g Z d x ˆ T µi ( x ) | ˜ P > = < ˜ P | X i = q, ¯ q,g ˆ¯ p µi (0) | ˜ P > = P µ (32)If we use the extra normalization factor √ P in the (physical) momentum eigenstate | P > of the hadron H as used in eq. (2) [10] then we find from eq. (32) < P | ˆ T µ (0) | P > = < P | X i = q, ¯ q,g ˆ T µi (0) | P > = 2 P < ˜ P | X i = q, ¯ q,g Z d x ˆ T µi ( x ) | ˜ P > = < P | X i = q, ¯ q,g ˆ¯ p µi (0) | P > = 2 P P µ (33)7hich in the light-cone coordinate system gives < P | ˆ T ++ (0) | P > = < P | X i = q, ¯ q,g ˆ T ++ i (0) | P > = 2 P + < ˜ P | X i = q, ¯ q,g Z d x ˆ T ++ i ( x ) | ˜ P > = 2( P + ) . (34)Note that we have found non-vanishing boundary surface term in QCD as given by eq. (29)in [11]. This implies that by using < P | T ++ (0) | P > = 2 ( P + ) from eq. (34) in eq. (3.28)of [10] we find X i Z dξ ξ f i/H ( ξ ) = 12 ( P + ) < P | T ++ (0) | P > = 1 (35)which proves that the momentum sum rule is violated in the operator product expansion(OPE) in QCD at high energy colliders where f i/H ( ξ ) is the parton distribution function(PDF) of the parton i inside the hadron H and ξ is the longitudinal momentum fraction ofthe parton with respect to the hadron.Note that the momentum sum rule in QCD involving parton distribution function insidenuclei also plays an important role to determine the initial condition for the quark-gluonplasma formation at RHIC and LHC heavy-ion colliders [14–17].The relation < P | ˆ T ++ (0) | P > = 2( P + ) in eq. (1) in [10] was based on the assumptionthat the total momentum of all the quarks plus antiquarks plus gluons inside the hadron isconserved. However, since the quark and gluon are not physical, i. e. , since we have notdirectly experimentally observed quarks and gluons, the momentum of a parton inside thehadron is not conserved as shown in eq. (19). Only when the total momentum of all thequarks plus antiquarks plus gluons inside the hadron is conserved then it becomes equal tothe momentum of the hadron so that eqs. (18) and (1) are satisfied.Hence we find that the momentum sum rule is violated in the operator product expansion(OPE) in QCD at high energy colliders. The correct momentum sum rule in QCD at highenergy colliders is given by eq. (5) of [11]. V. CONCLUSIONS
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