PDF) and Fragmentation Function (FF)

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By using the spectroscopic method involving the electronic hydrogen (the ordinary hydrogen atom consisting of proton and electron) the charge radius of the proton is measured to be 0.8768 ×10−15 meter. Similarly by using the electron-proton scattering method the charge radius of the proton is measured to be 0.8775 ×10−15 meter which is consistent with the spectroscopic method. The CODATA-2014 world average value of the charge radius of the proton by using the electrons, i. e., by using the above two methods, is 0.8751 ×10−15 meter [1]. However, the muonic hydrogen (hydrogen atom consisting of proton and muon) experi- ment in the year 2010 found that the charge radius of the proton is 0.84087 ×10−15 meter [2, 3]. The CODATA-2018 value of the charge radius of the proton is 0.8414 ×10−15 meter [4]. This disagreement between various experiments about the value of the charge radius of the proton is known as the proton radius puzzle which remains an unsolved problem in science. It is well known that the proton is not a point particle but it is a composite particle consisting of quarks and gluons which are the fundamental particles of the nature. The up 2e quark has the fractional electric charge 3 and the down quark has fractional electric charge e − 3 where e is the magnitude of the charge of the electron. Hence the charge radius of the proton depends on the charge distribution (the form factor) of the partons inside the proton.

The electric charge radius RP of the proton P is given by [5]

2 2 6 dGE(Q ) 2 RP = − 2 |Q =0 (1) GE(0) dQ

2 2 where GE(Q ) is the electric form factor and Q is the momentum transfer square of the virtual photon in the lepton-proton scattering. The interaction between the quarks and gluons inside the proton is described by the quantum chromodynamics (QCD) [6] which is a fundamental theory of the nature. The short distance partonic cross section can be calculated by using the perturbative QCD (pQCD) due to the asymptotic freedom in QCD [7]. Using the factorization theorem in QCD [8–10] the hadronic cross section can be calculated from the partonic cross section at the high energy colliders by using the experimentally extracted parton distribution function (PDF) and fragmentation function (FF).

1 The formation of the proton from the quarks and gluons is a long distance phenomenon in QCD. Due to the asymptotic freedom the QCD coupling becomes large at the large distance where the pQCD is not applicable. Hence the formation of the proton from the quarks and gluons cannot be studied by using the pQCD. The non-perturbative QCD is necessary to study the formation of the proton from the quarks and gluons. However, the analytical solution of the non-perturbative QCD is not known yet because of the presence of the cubic and quartic power of the gluon fields in the QCD lagrangian inside the path integration in the generating functional in QCD [see section II for details]. The path integration in QCD can be performed numerically in the Euclidean time by using the lattice QCD method. Hence the lattice QCD provides the first principle method to study the formation of the proton from the quarks and gluons. 2 Since the electric form factor GE(Q ) of the partons inside the proton in eq. (1) is a non-perturbative quantity in QCD it cannot be calculated by using the perturbative QCD (pQCD) method but it can be calculated by using the lattice QCD method. Recently we have formulated the lattice QCD method to study the proton formation from the quarks and gluons [11] and to study the proton spin crisis [12] and to study the proton decay [13] by implementing the non-zero boundary surface term in QCD which arises due to the confinement of quarks and gluons inside the finite size proton [14]. In this paper we extend this to formulate the lattice QCD method to study the proton radius puzzle. We derive the non-perturbative formula of the charge radius of the proton from the first principle in QCD which can be calculated by using the lattice QCD method. The paper is organized as follows. In section II we describe the lattice QCD method to study the proton formation from quarks and gluons by implementing the non-zero boundary surface term in QCD due to confinement. In section III we formulate the lattice QCD method to study the proton radius puzzle and derive the non-perturbative formula of the charge radius of the proton from the first principle in QCD which can be calculated by using the lattice QCD method. Section IV contains conclusions.

2 II. FORMATION OF PROTON FROM QUARKS AND GLUONS USING LAT- TICE QCD METHOD

The partonic operator for the proton (P ) formation is given by

T OP (x)= ǫklnUk (x)Cγ5Dl(x)Un(x) (2) where Uk(x) is the up quark ﬁeld and Dk(x) is the down quark ﬁeld, C is the charge conjugation operator and k,l,n =1, 2, 3 are the color indices. In the path integral formulation of QCD the vacuum expectation value of the non- † ′ ′′ perturbative partonic correlation function of the type < 0|OP (x )OP (x )|0 > is given by 1 δBs < 0|O† (x′)O (0)|0 >= [dA][dU¯][dU][dD¯][dD] × O† (x′)O (0) × det[ f ] P P Z[0] Z P P δωb 1 1 ×exp[i d4x[− F s (x)F σδs(x) − [Bs(x)]2 + U¯ (x)[δkn(i6 ∂ − m )+ gT s A/s(x)]U (x) Z 4 σλ 2α f k U kn n ¯ kn s s +Dk(x)[δ (i6 ∂ − mD)+ gTknA/ (x)]Dn(x)]] (3)

s s where Bf (x) is the gauge ﬁxing term with color index s = 1, ..., 8, the Aσ(x) is the gluon

ﬁeld with Lorentz index σ = 0, 1, 2, 3, the α is the gauge ﬁxing parameter, mU is the mass

of the up quark, mD is the mass of the down quark and Z[0] is the generating functional in QCD given by

δBs 1 1 Z[0] = [dA][dU¯][dU][dD¯][dD] × det[ f ] × exp[i d4x[− F s (x)F σδs(x) − [Bs(x)]2 Z δωb Z 4 σλ 2α f ¯ kn s s ¯ kn s s +Uk(x)[δ (i6 ∂ − mU )+ gTknA/ (x)]Un(x)+ Dk(x)[δ (i6 ∂ − mD)+ gTknA/ (x)]Dn(x)]] (4)

with

s s s scd c d Fσλ(x)= ∂σAλ(x) − ∂λAσ(x)+ gf Aσ(x)Aλ(x). (5)

In eq. (3) we do not have ghost ﬁelds as we directly work with the ghost determinant s δBf det[ δωb ]. The time evolution of the partonic operator in the Heisenberg representation is given by

−iHt iHt OP (t, ~r)= e OP (0, ~r)e (6) where the QCD Hamiltonian of the partons is given by H.

3 The complete set of proton energy-momentum eigenstates is given by

|Hl′′ >< Hl′′ | =1. (7) Xl′′ Using eqs. (6) and (7) in (3) we ﬁnd in the Euclidean time

† 2 − dt E ′′ (t) < 0|O (t, ~r)OP (0)|0 >= | | e l (8) P R X~r Xl′′ where dt is an indeﬁnite integration. In the large time limit we neglect the higher energy R level contribution to ﬁnd

† 2 − dt E(t) [ < 0|O (t, ~r)OP (0)|0 >]t→∞ = |

| e (9) P R X~r where E(t) is the energy of all the partons inside the proton in its ground state and |P > is the energy-momentum eigenstate of the proton in its ground state.

Due to non-vanishing boundary surface term EBS(t) in QCD due to the confinement of partons inside the finite size proton we find [14]

EP = E(t)+ EBS (t) (10)

where EP is the energy of the proton, E(t) is energy of all the partons inside the proton and

EBS(t) is the non-vanishing boundary surface term in QCD given by

′ † ′′ ′′ 3 ′ i0 ′ ′ dEBS(t ) ~r′′ < 0|OP (t , ~r )[ q,q,g¯ d r ∂iT (t , ~r )]OP (0)|0 > = [ ]t′′→∞. (11) ′ P P † R′′ ′′ dt ~r′′ < 0|OP (t , ~r )OP (0)|0 > P In eq. (11) the energy-momentum tensor density T σλ(x) of the partons in QCD is given by σλ σλ σµs λs g σ′µ′s λ′s σ ln λ s λs T (x)= F (x)F (x)+ F (x)F ′ (x)++U¯ (x)γ [δ i∂ − igT A (x)]U (x) µ 4 µ l ln n ¯ σ ln λ s λs +Dl(x)γ [δ i∂ − igTlnA (x)]Dn(x). (12)

Using eqs. (10) and (11) in (9) we ﬁnd

† ′ ′ ′ 2 −MP t ~r < 0|OP (t , ~r )OP (0)|0 > ′ |

| e = [ P† ′′ ′′ ′ 3 ′ i0 ′ ′ ]t →∞ ′′ <0|O (~r ,t )[ dt d r ∂iT (t ,~r )]OP (0)|0> ′ ~r P q,q,g¯ dt [ † ]t′′→∞ P P<0|O R(~r′′,t′′R)O (0)|0> eR ~r′′ P P P (13) which can be calculated by using the lattice QCD method where MP is the mass of the proton and dt′ is indeﬁnite integration. R Eq. (13) is the non-perturbative formula to study the proton formation from quarks and gluons using the lattice QCD method by implementing non-zero boundary surface term in QCD due to conﬁnement.

4 III. LATTICE QCD METHOD TO STUDY CHARGE RADIUS OF THE PROTON

In the previous section we have formulated the lattice QCD method to study the formation of the proton from the quarks and gluons by implementing the non-zero boundary surface term in QCD due to the conﬁnement. In this section we will extend this to formulate the lattice QCD method to study the proton radius puzzle. We will derive the non-perturbative formula of the charge radius of the proton from the ﬁrst principle in QCD which can be calculated by using the lattice QCD method. This method is also used to study various quantities in QCD in vacuum [12, 13, 16] and in QCD in medium [17] to study the quark-gluon plasma at RHIC and LHC [18–20]. In the single (virtual) photon exchange the amplitude M for the electron-proton (eP ) elastic scattering process is given by

gλδ 4πα M = [¯u (k )γ u (k )][ieu¯ (p )Γ (p ,p )u (p )] = l J λ (14) q2 e f λ e i P f δ f i P i Q2 λ

µ µ µ where ki(kf ), pi(pf ) are the initial (ﬁnal) momentum of the electron, proton, q = pf − pi is the momentum of the virtual photon, ue,uP are the electron, proton spinors respectively, lµ,J µ are the leptonic, hadronic (electromagnetic) currents respectively and q2 = −Q2. The λ general form of the vertex function Γ (pf ,pi) satisfying the relativistic invariance and current conservation is given by

λδ λ λ 2 σ qδ 2 2 λ 1 λ J =u ¯P (pf )[γ F1(Q )+ F2(Q )]uP (pi)=

. (15) 2MP 3 3

2 The electric form factor GE(Q ) is given by [5]

2 2 2 Q 2 GE(Q )= F1(Q ) − 2 F2(Q ). (16) 4MP Using eqs. (6) and (7) in (3) we ﬁnd in the Euclidean time

~p·~r † 2 − dt E ′′ (~p,t) e < 0|O (t, ~r)OP (0)|0 >= | | e l (17) P R X~r Xl′′ where ~p is the momentum of the proton and dt is an indeﬁnite integration. In the large R time limit we neglect the higher energy level contribution in eq. (17) to ﬁnd

~p·~r † 2 − dt E(~p,t) [ e < 0|O (t, ~r)OP (0)|0 >]t→∞ = |

| e (18) P R X~r

5 where E(~p, t) is the energy of all the partons inside the proton of momentum ~p in its ground state and |P (~p) > is the energy-momentum eigenstate of the proton P of momentum ~p in its ground state. λ From eq. (15) the electromagnetic current operator jq (x) of the quarks inside the proton is given by

q 2 ¯ 1 ¯ jλ(x)= U(x)γλU(x) − D(x)γλD(x)= ef q¯f (x)γλq(x) (19) 3 3 Xf

where qf (x) and ef are the quark ﬁeld and the fractional electrical charge of the quark of the ﬂavor f respectively. In the path integral formulation of QCD the vacuum expectation value of the non- ′′ q ′ perturbative 3-point partonic correlation function of the type < 0|OP (x )j0 (x )OP (0)|0 > is given by

1 δBs < 0|O† (x′′)jq(x′)O (0)|0 >= [dA][dU¯][dU][dD¯][dD] × O† (x′′)jq(x′)O (0) × det[ f ] P 0 P Z[0] Z P 0 P δωb 1 1 ×exp[i d4x[− F s (x)F σδs(x) − [Bs(x)]2 + U¯ (x)[δkn(i6 ∂ − m )+ gT s A/s(x)]U (x) Z 4 σλ 2α f k U kn n ¯ kn s s +Dk(x)[δ (i6 ∂ − mD)+ gTknA/ (x)]Dn(x)]]. (20)

In this paper we assume that the initial proton is at rest, i. e., ~pi = 0. This means

~pf = ~q = ~p. Using eqs. (6) and (7) in (20) we ﬁnd in the Euclidean time

i~q·(~r′′−~r′) ′′ ′′ q ′ ′ e < 0|OP (t , ~r )j0(t , ~r )OP (0)|0 >= < 0|OP |Hn′′ (~p) > ~rX′′,~r′ nX′′,n′ ′′ ′′ ′ ′ ′ ′ q −[ dt E ′′ (~p,t )− dt E ′′ (~p,t )] − dt E ′ (t ) < Hn′ |OP |0 > e n n e n (21) 0 R R R

where ~p is the momentum of the ﬁnal state proton and ~q = ~p. In the limit t′′ >>> t′, t′ →∞ we ﬁnd by neglecting the higher energy level contributions

i~q·(~r′′−~r′) ′′ ′′ q ′ ′ [ e < 0|OP (t , ~r )j0 (t , ~r )OP (0)|0 >]t′′>>>t′, t′→∞ =< 0|OP |P (~p) > ~rX′′,~r′ q −[ dt′′E(~p,t′′)− dt′E(~p,t′)] − dt′E(t′)

e e (22) 0 R R R

where E(~p, t) is the energy of all the partons inside the proton of momentum ~p, the E(t) is the energy of all the partons inside the proton at rest, |P (~p) > is the energy-momentum eigenstate of the proton of momentum ~p and |P > is the energy-momentum eigenstate of the proton at rest.

6 From eq. (9) we ﬁnd for the proton at rest

′ ′ 2 − dt′E(t′) [ < 0|OP (t , ~r )OP (0)|0 >]t′→∞ = |

| e . (23) R X~r′

Similarly from eq. (18) we ﬁnd for the proton with momentum ~p

i~p·~r′′ ′′ ′′ ′ [ e < 0|OP (~r , t − t )OP (0)|0 >]t′′>>>t′, t′→∞ X~r′′ 2 −[ dt′′E(~p,t′′)− dt′E(~p,t′)] = |

| e . (24) R R

From eqs. (22), (23) and (24) we ﬁnd

q 2 2

= |

| × |

For the initial proton at rest with energy-momentum eigenstate |P > and the ﬁnal proton of momentum ~p with energy-momentum eigenstate |P (~p) > we ﬁnd that the electric form 2 factor GE(Q ) of the partons inside the proton is related to the proton matrix element q q

of the quark electromagnetic current j0 via the relation [15]

2E (~p) 2 v P q GE(Q )= u

(26) uEP (~p)+ MP t 2 2 where EP (~p) is the energy of the proton of momentum ~p = ~q and Q = −q . From eq. (13) we ﬁnd for the proton at rest

′′ † ′′ ′′ MP t 2 ~r′′ < 0|OP (t , ~r )OP (0)|0 > ×e ′′ |

where MP is the mass of the proton. Similarly for the proton with momentum ~p we ﬁnd

′′ ′′ i~p·~r † ′′ ′′ t EP (~p) 2 ~r′′ e < 0|OP (t , ~r )OP (0)|0 > ×e ′′ |

where EP (~p) is the energy of the proton with momentum ~p.

7 Using eqs. (27), (28) and (25) in (26) we ﬁnd

2E (~p) 2 v P GE(Q )= u uEP (~p)+ MP t 1 2 ′′ ′′ i~p·~r † ′′ ′′ t EP (~p)  ~r′′ e < 0|OP (t , ~r )OP (0)|0 > ×e  ′′ × [ P i~p·~r′′′ † ′′′ ′′′ ′′ 3 ′′ i0 ′′ ′′ ]t →∞ ′′′ e <0|O (t ,~r )[ dt d r ∂iT (t ,~r )]OP (0)|0>  ′′ ~r P q,q,g¯   dt [ ′′′ † ]t′′′→∞   P ei~p·~r P<0|O R(t′′′,~r′′′R )O (0)|0>   eR ~r′′′ P P  P 1 2 ′′ † ′′ ′′ MP t  ~r′′ < 0|OP (t , ~r )OP (0)|0 > ×e  ′′ × [ P † ′′′ ′′′ ′′ 3 ′′ i0 ′′ ′′ ]t →∞ ′′′ <0|O (t ,~r )[ dt d r ∂iT (t ,~r )]OP (0)|0>  ′′ ~r P q,q,g¯   dt [ † ]t′′′→∞   P P<0|O R(t′′′,~r′′′R )O (0)|0>   eR ~r′′′ P P  i~qP·(~r′−~r′′′) ′ ′ q ′′′ ′′′ ~r′′′,~r′ e < 0|OP (t , ~r )j0(t , ~r )OP (0)|0 > ′ ′′′ ′′′ ×[ P ′′′ ′′′ i~p·~r′ ′ ′′′ ′ ]t >>>t , t →∞ [ ~r′′′ < 0|OP (t , ~r )OP (0)|0 >][ ~r′ e < 0|OP (t − t , ~r )OP (0)|0 >] P P (29) where the initial proton is at rest and the ﬁnal proton momentum is ~p = ~q with Q2 = −q2. 2 The eq. (29) is the non-perturbative formula of the electric form factor GE(Q ) of the proton derived from the ﬁrst principle in QCD which can be calculated by using the lattice

QCD method. The charge radius RP of the proton is obtained from this electric form factor 2 GE(Q ) in eq. (29) by using eq. (1).

IV. CONCLUSIONS

Recently there has been disagreement between various experiments about the value of the proton radius which is known as the proton radius puzzle. Since the proton is not a point particle the charge radius of the proton depends on the charge distribution (the form factor) of the partons inside the proton. Since this form factor is a non-perturbative quantity in QCD it cannot be calculated by using the perturbative QCD (pQCD) method but it can be calculated by using the lattice QCD method. In this paper we have formulated the lattice QCD method to study the charge radius of the proton. We have derived the non-perturbative formula of the charge radius of the proton from the ﬁrst principle in QCD

8 which can be calculated by using the lattice QCD method.

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