Collaborators:
Rajen Kundu
Asmita Mukherjee
Dipankar Chakrabarti
Raghunath Ratabole
James P. Vary
Wei-Min Zhang

I. PROGRAM DESCRIPTION

With the recent and planned experiments on polarized and unpolarized deep inelastic scattering, the field of high energy hadron physics has entered a new era. For a proper understanding of new experimental data there is an urgent need to develop new theoretical tools which are based on physical intuition and which at the same time employ well-defined field theoretical calculational procedures. Towards this goal, for the past five years we have been developing and applying a new method of calculation of deep inelastic structure functions based on light front Hamiltonian field theory and Quantum Chromo Dynamics (QCD).

To provide an intuitive picture, we keep close contact with the original pre-QCD parton ideas due to Feynman. We are able to generalize this concept and introduce field theoretic partons as non-collinear, massive, yet on-mass-shell objects in interacting field theory. This is facilitated by the light front Hamiltonian description of composite systems in field theory which utilizes many body wave functions for the constituents. The basic features of light front field theory that makes it extremely suitable for the description of deep inelastic scattering phenomena based on a generalized parton picture are the following: (1) triviality of the light front vacuum (2) light front power counting which forces one to treat the longitudinal coordinate x^- and the transverse coordinate x^\perp differently (3) existence of constraints that makes it possible to describe relativistic fermions and gauge bosons with only two dynamical field components (4) special properties of light front boosts, the longitudinal boost being a simple scaling operation and the transverse boosts being simply Galilean boosts familiar from Newtonian mechanics.

Taking clue from Bjorken's original derivation of scaling, we make use of the Bjorken-Johnson-Low (BJL)expansion of scattering amplitudes specialized to light front kinematics and light front current algebra. Since the current commutators are evaluated in the interacting theory of QCD, we overcome some of the original shortcomings. The analysis directly leads to expressions for various quark distribution functions (which are related to structure functions in the leading logarithmic approximation) as Fourier Transforms of equal light front time correlation functions which involve bilocal vector and axial vector currents. Since the bilocality is only in the longitudinal direction we can immediately exploit the whole machinery of Fock space expansion techniques and bring in multi-parton wave functions. Since matrix elements relevant for deep inelastic scattering are measured at the scale Q, a major question is whether we can consistently carry out the renormalization program. We are able to achieve this using the tools of old fashioned perturbation theory, the elements of which are transition matrix elements and energy denominators. The use of light front gauge A^+ = 0 greatly simplifies matters. For example, the path ordered exponential involving the gauge field between the fermion field operators in the bilocal currents reduces to unity in this gauge. In our method, the physical picture is transparent at each stage of the calculation since one is using techniques that closely resembles those in nonrelativistic many body theory.

Since multi-parton wave functions carry both perturbative and nonperturbative information on the structure of hadrons, both aspects are treated in the same framework in our formalism and we are thus able to provide a unified picture. This is a unique feature of our program which is lacking in alternative methods currently in practice, namely, operator product expansion method and QCD factorization method.

II. RESEARCH HIGHLIGHTS

Our research in the area of high energy scattering can be broadly classified into four subgroups: (1) unpolarized scattering (2) polarized scattering (3) higher twist structure function and (4) formal theoretical investigations. In the following subsections we highlight the results from our investigations with references to published work where more details can be found.

A. Unpolarized Scattering

In our first published work [1] on the subject, we briefly discussed the emergence of parton model from field theory in the context of light front current algebra and canonical manipulations. We touched upon the shortcomings of the naive canonical picture, especially concerning renormalization issues. To illustrate the novel aspects of the renormalization problem in light front dynamics, the scaling behavior of different components of currents under light front power counting which leads to some apparent paradoxes was stressed.

We studied [2] the matrix element of the transverse component of the bilocal vector current in the context of deep inelastic scattering. BJL limit of high energy amplitudes together with light front current algebra imply the same parton interpretation for its matrix element as that of the plus component. On the other hand, the transverse component depends explicitly on the gluon field operator in QCD, appears as ``twist three'' and hence its matrix element has no manifest parton interpretation. We performed calculations in light front time ordering perturbative QCD for a dressed quark target to order g^2 and demonstrateed that the matrix element of the transverse component of the bilocal vector current has the same parton interpretation as that of the plus component.

The systematic exploration of the deep inelastic structure functions of hadrons nonperturbatively in an inverse power expansion of the light front energy of the probe in the framework of light front QCD is presented in Refs. [3,4]. In Ref. [3] We arrived at the general expressions for various structure functions as the Fourier transform of matrix elements of different components of bilocal vector and axial vector currents on the light front in a straightforward manner. The complexities of the structure functions are mainly carried by the multi-parton wave functions of the hadrons, while, the bilocal currents have a dynamically dependent yet simple structure on the light front in this description. Further, the factorization theorem and the scale evolution of the structure functions are presented in this formalism by using old-fashioned light front time-ordered perturbation theory with multi-parton wave functions. Nonperturbative QCD dynamics underlying the structure functions can be explored in the same framework. We showed that once the nonperturbative multi-parton wave functions are known from low-energy light front QCD (for a summary of recent developments see [5]), a complete description of deep inelastic structure functions can be realized.

In Ref. [4] we showed how a perturbative analysis in the light front Hamiltonian formalism leads to the factorization scheme we have proposed recently. The analysis also showed that the scaling violations due to perturbative QCD corrections can be rather easily addressed in this framework by simply replacing the hadron target by dressed parton target and then carrying out a systematic expansion in the coupling constant ff s based on the perturbative QCD expansion of the dressed parton target. The tools employed for this calculation are those available from light front old-fashioned perturbation theory. We presented the complete set of calculations of unpolarized deep inelastic structure functions to order g^2 . We extracted the relevant splitting functions in all the cases. We explicitly verified all the sum rules to order g^2. Further we demonstrated the validity of approximations made in the derivation of the new factorization scheme. This is achieved with the help of detailed calculations of the evolution of structure function of a composite system carried out using multi-parton wavefunctions.

B. Polarized Scattering

We have investigated important issues in both longitudinally and transversely polarized structure functions.

For the transversely polarized structure function g_2 , if the twist three contributions are ignored, one gets an expression purely in terms of the longitudinally polarized structure function g_1. In order to examine the validity of this Wandzura-Wilczek relation for the polarized DIS structure function g_2, we have used [5] the light front time-ordering perturbative(p) QCD to calculate g_2 at order ff s on a quark target. In contrast to the folklore in pQCD, We found that the study of the transversely polarized structure function in pQCD is meaningful only if we begin with massive quarks. The result showed that the Wandzura-Wilczek relation for g_2 is strongly violated in pQCD.

In the case of longitudinally polarized scattering, the nucleon spin crisis has attracted lot of attention. The question of parton orbital angular momentum in deep inelastic scattering has arisen in the context of the helicity sum rule for the nucleon. We have addressed [6] several issues associated with orbital angular momentum relevant for leading twist polarized deep inelastic scattering. We have presented a detailed analysis of the light front helicity operator (generator of rotations in the transverse plane) in QCD. We explicitly showed that, the operator constructed from the manifestly gauge invariant, symmetric energy momentum tensor in QCD, in the gauge A^+ = 0, after the elimination of constraint variables, is equal to the naive canonical form of the light front helicity operator plus surface terms. Restricting to topologically trivial sector, we eliminated the residual gauge degrees of freedom and surface terms. Having constructed the gauge fixed light front helicity operator, we introduced quark and gluon orbital helicity distribution functions relevant for polarized deep inelastic scattering as Fourier transform of the forward hadron matrix elements of appropriate bilocal operators. The utility of these definitions was illustrated with the calculation of anomalous dimensions in perturbation theory. We explicitly verified the helicity sum rule for dressed quark and gluon targets in light front perturbation theory. In order to elucidate the issues that arise in the problem of angular momentum of a composite system, we also considered the internal orbital helicity of a composite system in an arbitrary reference frame and contrasted the results in the non-relativistic situation versus the light front (relativistic) case.

C. Transverse Spin

We hve also addressed [7,8] various issues connected with transverse spin in light front QCD. First we took up the long standing problem of the construction of relativistic spin operators for a composite system in QCD. Exploiting the kinematical boost symmetry in light front theory, we showed that transverse spin operators for massless particles can be introduced in an arbitrary reference frame, in analogy with those for massive particles. The transverse spin operators in QCD, in $A^+ = 0$ gauge, expressed in terms of the dynamical variables are explicitly interaction dependent unlike the helicity operator which is interaction independent in the topologically trivial sector of light-front QCD. Although it cannot be separated into an orbital and a spin part, we have shown that there exists an interesting decomposition of the transverse spin operator. We discuss the physical relevance of such a decomposition. In light front QCD, the complete set of transverse spin operators were identified for the first time, which are responsible for the helicity flip of the nucleon. We established the direct connection between transverse spin in light front QCD and transverse polarized deep inelastic scattering. We discussed the theoretical and phenomenological implications of our results.

Next, we performed a one loop renormalization of the full transverse spin operator in light-front Hamiltonian perturbation theory for a dressed quark state. We explicitly showed that all the terms dependent on the center of mass momenta get canceled in the matrix element. The entire non-vanishing contribution comes from the fermion intrinsic -like part of the transverse spin operator as a result of cancellation between the gluonic intrinsic-like and the orbital-like part of the transverse spin operator. We compareed and contrasted the calculations of transverse spin and helicity of a dressed quark in perturbation theory. In polarized deep inelastic scattering off nucleon, one can measure two structure functions, the helicity structure function g_1 and the transverse structure function g_T. The measurement of g_1 has lead to the famous "proton spin crisis" problem. Various experimental programmes are being planned to measure the contribution of gluon helicity and quark and gluon orbital angular momentum to proton helicity. Very little attention is paid to g_T because of the interaction dependent operators that appear in the theoretical discription of this observable that makes parton picture not viable. Our work is the first one that pointed out and explored the connection between transverse spin of the proton and g_T that is very similar to the relation between helicity of the proton and g_1.

D. Twist four structure function

To resolve various outstanding issues associated with the twist four longitudinal structure function we have performed an analysis [9,10] based on the BJL expansion for the forward virtual photon-hadron Compton scattering amplitude and equal (light front) time current algebra. We have showed that the integral of twist four F_L is related to the expectation value of the fermionic part of the light front Hamiltonian density at fixed momentum transfer. Using the Fock space expansion for states and operators, we have evaluated the twist four longitudinal structure function for dressed quark and gluon targets in perturbation theory. The new relation, in addition to providing physical intuition on F_L, relates the quadratic and logarithmic divergences of F_L to mass corrections in light front QCD and hence provides a new pathway for the renormalization of the corresponding twist four operator. The mixing of quark and gluon operators in QCD naturally leads to a twist four longitudinal gluon structure function and to a new sum rule which is the first sum rule obtained for a twist four observable. The validity of the sum rule in a non-perturbative context is explicitly verified in two-dimensional QCD. We have presented numerical results for the F_2 and F_L structure functions for the meson in two-dimensional QCD in the one pair approximation. We have pointed out the relevance of our results for the problem of the partitioning of hadron mass in QCD.

D. Formal Theoretical Investigations

Old-fashioned perturbation theory which we employ in our calculations has not been traditionally pushed far beyond tree level. We have studied the utility of a kinematical symmetry, namely Galilean boost symmetry in loop calculations [12]. This is the first work to address this issue in light front field theory.

The motivation for our work is the following. Investigations have revealed a very complex structure for the coefficient functions accompanying the divergences for individual time (x^+ ) ordered diagrams in light front perturbation theory. No guidelines seem to be available to look for possible mistakes in the structure of these coefficient functions emerging at the end of a long and tedious calculation, in contrast to covariant field theory. Since, in light front field theory, transverse boost generator is a kinematical operator which acts just as the two-dimensional Galilean boost generator in non-relativistic dynamics, it may provide some constraint on the resulting structures. In this work we investigated the utility of Galilean symmetry beyond tree level in the context of coupling constant renormalization in light front QCD using the two-component formalism. We showed that for each x^+ ordered diagram separately, underlying transverse boost symmetry fixes relative signs of terms in the coefficient functions accompanying the diverging logarithms. We also summarized the results leading to coupling constant renormalization for the most general kinematics. This calculation also serves as an intermediate step in a full-fledged fourth order calculation of the dressed parton structure function.

A major obstacle to a newcomer who wishes to enter the research area of light front field theory is the lack of coherent introductory materials on the subject. To remedy this we have published a set of lecture notes [12]. Despite being introductory in nature, this paper does contain some original research results. They are (1) the utility of Galilean symmetry in the construction of tree level light front Hamiltonians from light front power counting and (2) the construction of the Hamiltonian in closed form for the massive Thirring model.

REFERENCES

[1] A. Harindranath, Parton Model from Field Theory: the Good, the Bad, and the Terrible, in Theory of Hadrons and Light Front QCD, St. D. G/lazek (ed.), World Scientific, Singapore (1995).
[2] A. Harindranath and Wei-Min Zhang, On the Matrix Element of the Transverse Component of the Bilocal Vector Current and its Parton Interpretation, Phys. Lett. B390, 359 (1997).
[3] A. Harindranath, Rajen Kundu, and Wei-Min Zhang, Deep Inelastic Structure Functions in Light Front QCD: A Unified Description of Perturbative and Nonperturbative Dynamics, Phys. Rev. D59, 094012 (1999).
[4] A. Harindranath, Rajen Kundu, and Wei-Min Zhang, Deep Inelastic Structure Functions in Light Front QCD: Radiative Corrections, Phys. Rev. D59, 094013 (1999).
[5] A. Harindranath and Wei-Min Zhang, Examination of Wandzura-Wilczek Relation for g_2 in Perturbative QCD, Phys. Lett. B408, 347 (1997).
[6] A. Harindranath and Rajen Kundu, Orbital Angular Momentum in Deep Inelastic Scattering, hep-ph/9802406, Phys. Rev. D 59, 116013 (1999).
[7] A. Harindranath, Asmita Mukherjee and Raghunath Ratabole, Transverse Spin in QCD and Transverse Polarized Deep Inelastic Scattering , B 476, 471 (2000).
[8]A. Harindranath, Asmita Mukherjee, Raghunath Ratabole, Transverse Spin in QCD: Radiative Corrections, Phys. Rev. D 63, 045006 (2001).
[9] A. Harindranath, Rajen Kundu, Asmita Mukherjee and James P. Vary, Sum Rule for the Twist Four Longitudinal Structure Function, Phys. Lett. B417, 361 (1998).
[10] A. Harindranath, Rajen Kundu, Asmita Mukherjee and James P. Vary, Twist Four Longitudinal Structure Function in Light Front QCD, Phys. Rev. D 58, 114022 (1998).
[11] A. Harindranath and Rajen Kundu, Utility of Galilean Symmetry in Light Front Perturbation Theory: A Nontrivial Example in QCD, Int. J. Mod. Phys. A 13, 4591 (1998).
[12] A. Harindranath, An Introduction to Light Front Dynamics for Pedestrians, first chapter of the book Light Front Quantization and Non Perturbative QCD, edited by James P. Vary and Frank Woelz, published and distributed by International Institute of Theoretical and Applied Physics, Ames, I.A., U.S.A. (1997), hep-ph/9612244.