The Pauli Form Factor of Quark and Nontrivial Topological Structure of the QCD

TL;DR

This paper calculates the quark's electromagnetic Pauli form factor considering QCD vacuum instantons, revealing significant effects with implications for photon-hadron reactions and quark matter dynamics.

Contribution

It introduces a calculation of the quark Pauli form factor incorporating nontrivial topological QCD effects, highlighting their importance.

Findings

Instantons significantly influence the quark Pauli form factor.

Results impact understanding of photon-hadron interactions.

Findings relevant for quark matter in extreme conditions.

Abstract

We calculate the electromagnetic Pauli form factor of quark induced by the nontrivial topological fluctuations of QCD vacuum called instantons. It is shown that such contribution is significant. We discuss the possible implications of our result in the photon-hadron reactions and in the dynamics of quark-photon interactions in the dense/hot quark matter.

Equations

$${\Gamma }^{\mu }={\gamma }^{\mu }{F}_1(Q^2)+\frac{i{\sigma }^{\mu \nu }{q}_{\nu }}{2M_q}{F}_2(Q^2)$$

$$V_{\mu }^a(k_1^2,k_2^2,t^2)=\frac{i{g}_s{\sigma }^{\mu \nu }{q}_{\nu }}{2M_q}{F}_2(k_1^2,k_2^2,t^2)t^a,$$

$$F_2(k_1^2,{k_2^{\prime }}^2,t^2)={\mu }_a{F}_q(|k_1|\rho /2)F_q(|k_2|\rho /2)F_g(|t|\rho ),$$

$$F_q(z)$$

$$F_g(z)$$

$${\mu }_a=-\frac{3\pi (M_q{\rho }_c{)}^2}{4{\alpha }_s({\rho }_c)}.$$

$${{\mu }_a}^{\mathrm{MFA}}=-0.4{\mu }_a^{\mathrm{DP}}=-1.6$$

$$F_g(k_E^2)\approx F_q(k_E^2)\approx e^{-k_E^2/{\Lambda }^2},$$

$$i\mathcal{M}$$

$$=-i{e}_q\overline{u}(p^{\prime }){\Gamma }^{\mu }(p,p^{\prime })u(p),$$

$$N\equiv -i\overline{u}(p^{\prime }){\sigma }^{\alpha \rho }(/k^{\prime }+M_q){\gamma }^{\mu }(/k+M_q){\gamma }_{\rho }{t}_{\alpha }u(p).$$

$$F_2(q^2)=\text{tr}\left\{(/p+M_q){\Lambda }_{\rho }^{(2)}(p^{\prime },p)(/p^{\prime }+M_q){\Gamma }^{\rho }(p^{\prime },p)\right\},$$

$${\Lambda }_{\rho }^{(2)}(p^{\prime },p)\equiv \frac{M_q^2}{k^2(4M_q^2-k^2)}\left[{\gamma }_{\rho }+\frac{k^2+2M_q^2}{M_q(k^2-4M_q^2)}(p^{\prime }+p{)}_{\rho }\right].$$

$$\frac{1}{k^n}={\int }_0^{\infty }d\alpha \frac{{\alpha }^{n-1}}{(n-1)!}{e}^{-\alpha k},$$

$$F_2(Q^2)$$

$$\times \left\{\frac{3v_1+6v_2-7}{\Delta }-Q^2{v}_1{v}_2(v_1+2v_2-3)-M_q^2(v_1+v_2)((v_1+v_2)(v_1+2v_2)-2v_2)\right\},$$

$$\Delta$$

$$v_1$$

$$v_2$$

$$F_2^d(Q^2)=-\frac{1}{2}{F}_2^u(Q^2).$$

$$F_2(Q^2,M_q)=\frac{F_2(0,M_q)}{1+{\rho }_c{Q}^2/(4.7M_q)},$$

$${\mu }_a^u\approx \frac{2}{3}\left(-0.065+0.97(M_q{\rho }_c)\right).$$

$${\mu }_a^{e,u}$$

$$F_2(Q^2)\approx -\frac{1}{Q^2}\frac{e_q{C}_F{g}_s^2{\mu }_a}{4{\pi }^2}\sum _{n=0}^{\infty }\underset{0}{\overset{\infty }{\int }}d{k}^2\frac{k^{2(n+2)}}{n!(n+2)!}{M}_q^{2n}\times$$

$$\times [F_g{D}_g{]}_{n+1}\left(2[F_q{D}_q{]}_{n-1}-M_q^2\frac{k^2}{(n+3)}[F_q{D}_q{]}_{n+1}\right),$$

$$[F_q{D}_q{]}_{-1}=-\underset{k^2}{\overset{\infty }{\int }}d{l}^2{F}_q(l^2)D_q(l^2)$$

$$[F_q{D}_q{]}_0=F_q(k^2)D_q(k^2)$$

$$[F_q{D}_q{]}_{+1}=-\left(\frac{d}{d{k}^2}\right)(F_q(k^2)D_q(k^2))$$

$$...$$

$$[F_q{D}_q{]}_{+n}={\left(-\frac{d}{d{k}^2}\right)}^n(F_q(k^2)D_q(k^2)),$$

$$\displaystyle F_{2}(Q^{2})\approx\frac{1}{Q^{2}}\frac{e_{q}C_{F}g_{s}^{2}\mu_{a}}{4\pi^{2}}\biggl{\{}\int\limits_{0}^{\infty}dk^{2}\frac{e^{-k^{2}/\Lambda^{2}}}{k^{2}+M_{q}^{2}}$$

$$\times \left[2{\Lambda }^2-e^{-k^2/{\Lambda }^2}(k^2+2{\Lambda }^2)\right]$$

$$+\sum _{n=1}^{\infty }\underset{0}{\overset{\infty }{\int }}d{k}^2\frac{k^{2(n+2)}{M}_q^{2n}}{(n-1)!(n+2)!}$$

$$\displaystyle\quad\times\left(\frac{2}{n}(F_{g}D_{g})_{n+1}(F_{q}D_{q})_{n-1}-(F_{g}D_{g})_{n}(F_{q}D_{q})_{n}\right)\biggr{\}}$$

$$F_2(Q^2)$$

Full text

quarks, form factor, instanton, quantum chromodynamics

I Introduction

Nowadays the study of electromagnetic structure of the elementary particles is one of the hottest topics in the Standard Model (SM). One well known puzzle is the experimental value of the muon anomalous magnetic moment which shows the significant deviation from the SM prediction (see recent reviews Lindner:2016bgg ; Dorokhov:2016knu ). Electromagnetic probes of the hadrons give very important information about the structure of the strong interaction Punjabi:2015bba . Various models have been developed which enable us to study electromagnetic properties of hadrons in terms of form factors Altarelli:1995 ; Petronzio:2003 ; Scopetta:2003et ; Gutsche:2014zua ; Kochelev:2015pqd ; Arrington:2006zm ; Faccioli:2002jd ; Faccioli:2002wr ; Faccioli:2001qg ; Blotz:1996ad . In the past decades the significant progress has been made, especially regarding the study of the relation between generalized parton distribution functions (GPD) and electromagnetic form factors of hadrons Ji:1996 ; Radyushkin:1996nd . The quark form factors carry the information about internal structure of the constituent quark and provide the bridge between partonic picture of the hadrons and their constituent structure Kochelev:2015pqd ; kochelev1 ; Kochelev:2003 ; Dorokhov:2004fb ; Petronzio:2003 ; Simula:2003 ; Roberts:2007ji .

In this paper we consider a new nonperturbative contribution to electromagnetic Pauli form factor (EPFF) of quark arisen from instanton induced quark-gluon vertex. The instanton is the well-known solution of QCD equation of motion in the Euclidian space-time which has nonzero topological charge. It was shown that instantons play a very important role in hadron physics (see the reviews Schafer:1998 ; Diakonov:2002 ; Kochelev:2005xn ). In particular, the instantons leads to the spontaneous chiral symmetry breaking (SCSB) in strong interaction which is not only one of the main sources for the observed hadron masses but also leads to the various anomalies observed in the spin-dependent cross sections kochelev1 ; Ostrovsky:2004pd ; Cherednikov:2006zn ; Hoyer:2005ev ; Kochelev:2013zoa ; Qian:2015wyq ; Kochelev:2015pqd . One of the cornerstones of the instanton-based theory of the spin effects in the strong interaction is the instanton-induced anomalous chromomagnetic quark-gluon interaction introduced in kochelev1 . The strength of this interaction is determined by the dynamical mass of the quark in the instanton vacuum Diakonov:2002 ; kochelev2 which is directly related to the phenomenon of the SCSB. The first attempt to estimate the effect of instantons to EPFF was made in Kochelev:2003 where the so-called instanton’s perturbative theory was used. This approach was developed in the papers ringwald1 ; ringwald to obtain the effect of the small size of the instantons to the Deep-Inelastic Scattering (DIS) at large transfer momentum $Q^{2}=-q^{2}$. However, the final result for their contribution to DIS at the large $Q^{2}$ was found to be very small. The same conclusion is also valid for the contribution of the small instantons to the large $Q^{2}$ asymptotic of the EPFF of quark obtained in Kochelev:2003 . Here, we will use another way to calculate the instanton contribution to EPFF. This approach is based on the effective quark-gluon vertex induced by instantons and allows to obtain the prediction for EPFF in the wide interval of the $Q^{2}$ including even very important case of the real photon, $Q^{2}=0$.

II The contribution of anomalous quark-gluon interaction to the quark electromagnetic form factor

The general vertex for photon-quark interaction for on-shell quark is

[TABLE]

where $F_{1},F_{2}$ are electromagnetic Dirac and Pauli form factors, respectively, $M_{q}$ is the dynamical mass of the quark and $\sigma_{\mu\nu}=i(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu})/2$. The anomalous quark-gluon chromomagnetic (AQGC) vertex induced by the instantons can be written in the form  kochelev1 ; Diakonov:2002 ; kochelev2

[TABLE]

where $k_{1}^{2}$ and $k_{2}^{2}$ are the virtuality of the initial and final quarks, respectively, $t=k_{1}-k_{2}$ and the general case for non-zero virtualities of quarks and gluon is considered. The form factor $F_{2}(k_{1}^{2},k_{2}^{2},t^{2})$ suppresses the AQGC vertex at short distances when the respective virtualities are large. Within the instanton model it is explicitly related to the Fourier-transformed of both quark zero-mode and instanton field, which take the forms

[TABLE]

where

[TABLE]

$I_{\nu}(z)$, $K_{\nu}(z)$ are the modified Bessel functions, $\rho$ is the instanton size and $\mu_{a}=F_{2}(0,0,0)$ is the anomalous quark chromomagnetic moment (AQCM). Within the instanton liquid model Schafer:1998 , Diakonov:2002 , where all instantons have the same size $\rho_{c}\approx 1/3$ fm, AQCM is kochelev2 ; Diakonov:2002

[TABLE]

The first feature is that the strong coupling constant enters into the denominator showing a clear nonperturbative origin of AQCM. The second feature is the negative sign of AQCM. As we will see below, this sign of AQCM leads to the positive sign of the anomalous quark magnetic moment (AQMM). The value of AQCM strongly depends on the dynamical quark mass which is $M_{q}=170$ MeV in the mean field approximation (MFA) Schafer:1998 and $M_{q}=350$ MeV in the Diakonov-Petrov model (DP) Diakonov:2002 . Therefore, for the value of the strong coupling constant in the instanton model, $\alpha_{s}(\rho_{c})\approx 0.5$ and average size of instantons $\rho_{c}=1/600$ MeV*-1* Diakonov:2002 we get

[TABLE]

The contribution to the electromagnetic Pauli form factor coming from the AQGC vertex is obtained by the consideration of the diagrams presented in Fig. 1.

To perform analytical calculations the gaussian approximation for the form factors in Eq.(4)

[TABLE]

is used with $\Lambda=2/\rho_{c}$.

The contribution from Fig. 1(1(b)) is the same as that from Fig. 1(1(a)), hence the final result should be doubled. Therefore the total matrix element is

[TABLE]

where $C_{F}=\text{tr}(T^{a}T^{a})=\frac{4}{3}$ is the color factor and $e_{q}$ is the electric charge of the quark and

[TABLE]

One way to extract Pauli form factor $F_{2}(Q^{2})$ from $i\mathcal{M}$ is to rearrange the gamma matrices in Eq. (9) and find the term proportional to ${i\sigma^{\mu\nu}}/{2M_{q}}$. However, a simpler way is to use projector operator method Brodsky:1966mv ; Knecht:2001 , by making use of identity

[TABLE]

where $q^{2}=(p^{\prime}-p)^{2}\equiv-Q^{2}$ and

[TABLE]

By working in the Euclidean space-time,with the help of Feynman parametrization and identity

[TABLE]

we obtain

[TABLE]

where $\int d^{3}x\equiv 2\int_{0}^{1}dx_{1}\;dx_{2}\;dx_{3}\;\delta(1-x_{1}-x_{2}-x_{3})$ and

[TABLE]

III Numerical results

In our model the form factor $F_{2}(Q^{2})$ is proportional to the quark charge. Therefore, there is the relation between u- and d-quark form factors

[TABLE]

For simplicity, below only the result for u-quark case is presented in the figures. In Fig.2 the result of the calculation of electromagnetic form factor as the function of $Q^{2}$ is presented for two different masses of u-quark. Our numerical result can be fitted very well by the formula

[TABLE]

which can be useful for the applications. We would like to emphasize that positive sign of $F_{2}$ form factor for u-quark (see Fig.2), is fixed by the negative sign of the AQCM, Eq. (5). In the Fig.3 the dependency of the value of the magnetic moment of u-quark on the value of the its dynamical mass is shown. It’s behavior as a function of quark mass in the range between $80$ and $500$ MeV can be fitted very well with the linear function

[TABLE]

The results for $\mu_{a}^{e,u}=F_{2}^{u}(Q^{2}=0)$ at the two different values of the dynamical quark masses obtained in mean field approximation  Schafer:1998 and within the Diakonov-Petrov model Diakonov:2002 are

[TABLE]

Our value for the quark magnetic moment at $M_{q}=350$ MeV is in the qualitative agreement with the result of calculation within different approach based on the Dyson-Schwinger equation Chang:2010hb . However, we would like to emphasize that in this paper the $Q^{2}$ dependency of the EPFF is not considered. This $Q^{2}$ dependency in our model is presented in the Fig.2. One can mention its rather strong dependency on the virtuality of photon. In the model it is coming from the quark and gluon form factors presented in the Eq. (7).

IV The large $Q^{2}$ behavior of EPFF

The formula for expansion of the form factor at large $Q^{2}\gg M_{q}^{2}$ is

[TABLE]

where

[TABLE]

and the same formula for $[F_{g}D_{g}]_{i}$ with the corresponding changing of index $q\rightarrow g$. Using the expressions for the form factors in gluon and quark sector in Eq. (7) and $D_{g}(k^{2})=1/k^{2}$, $D_{q}(k^{2})=1/(k^{2}+M_{q}^{2})$ one can rewrite it in the form

[TABLE]

and show that the main contribution to EPFF is coming from the first term in brackets. Moreover, one can perform additional expansion over $M_{q}^{2}/\Lambda^{2}$. In this approximation and ignoring the second term in Eq. (21), one can write the leading orders in $M_{q}^{2}/\Lambda^{2}$ expansion in the form

[TABLE]

where $\gamma_{E}$ is the Euler’s constant. By using the relation $\Lambda\approx 2/\rho_{c}$, it can be rewritten as

[TABLE]

Therefore, at large $Q^{2}$ form factor behaves as $F_{2}(Q^{2})\sim 1/Q^{2}$.

V The low $Q^{2}$ behavior of EPFF

It can be shown that in the limit $Q^{2}\rightarrow 0$ the form factor is

[TABLE]

where

[TABLE]

The expansion given by Eq.24 describes the exact result very well in the region of small $M_{q}\rho_{c}<1$, Fig.3. One can see that $F_{2}(0)$ vanishes in the limit $M_{q}\rightarrow 0$. It means that this contribution to form factor is directly related to the phenomenon of SCSB.

VI Conclusion

In this paper, we calculate the quark electromagnetic form factor within the nonperturbative approach based on the instanton picture for the QCD vacuum. It is shown that anomalous quark-gluon chromomagnetic interaction induced by instantons leads to large magnetic moment of u- and d-quarks. Possible applications of our results are as follows. One of the tasks is to consider the influence of EPFF on the hadron electromagnetic form factors. We would like to mention that instanton contribution to the electromagnetic form factors of proton, neutron and pion were calculated using different versions of instanton model in Forkel:1994pf ; Faccioli:2002jd ; Faccioli:2002wr ; Faccioli:2001qg ; Blotz:1996ad in semi-classical approaches to the corresponding correlators. However, it would be interesting to study the electromagnetic properties of hadrons based on the constituent quark model with an effective quark-photon and quark-gluon vertices induced by instantons. In this way, one can take into consideration the confinement effects as well in spirit of the calculation of nucleon electromagnetic form factors carried out in Petronzio:2003 for constituent quarks with inner structure. It is evident that due to the existence of an additional scale in our model related to the instanton size $\rho_{c}\approx 1/3$ fm, one can expect the deviation of the $Q^{2}$ dependency of the hadron form factors from the quark-counting rule prediction Matveev:1972gb ; Brodsky:1973kr ; Brodsky:1974vy .

We should stress that EPFF leads to quark spin-flip. Therefore, it should make contribution to various spin-dependent photon-hadron cross sections, including polarized semi-inclusive DIS. Another possible application, in the line of Fayazbakhsh:2014mca , is the study of the influence of the non-zero value of the anomalous quark magnetic moment on the dynamics of Quark-Gluon Plasma (QGP) in the strong magnetic field. We would like to emphasize that our new type of photon-quark interaction is very sensitive to the topological structure of the QCD vacuum which might be drastically changed during the deconfinement transition Ilgenfritz . This phenomenon can lead to, for example, the suppression of direct photon production induced by our anomalous quark-photon vertex in the QGP.

VII Acknowledgments

We are grateful to Aleksander Dorokhov, Sergo Gerasimov and Michael Ivanov for the useful discussions. This work was partially supported by the National Natural Science Foundation of China (Grant No. 11575254 and 11175215), and by the Chinese Academy of Sciences visiting professorship for senior international scientists (Grant No. 2013T2J0011) and President’s international fellowship initiative (Grant No. 2017VMA0045).