Quantum Chromodynamics: History and

Prospects Oberwölz, Austria,

September 3 - September 8, 2012

PUZZLE OF HIGH ENERGY PP-SCATTERING.

HELICITY CONSERVATION FROM PERTURBATIVE QCD

OR KINEMATIC HIERARCHY?

Mikhail P. Chavleishvili,

Laboratoryof Theoretical Physics, Joint Institute for Nuclear Research

end International University “Dubna”

Dubna, RUSSIA

Abstract We discuss pp elastic scattering at high energies and large fixed angle. This is the region of hard collisions where perturbative QCD must work. PQCD predicts "the helicity conservation" which gives a zero polarization and the value 1/3 for asymmetry parameter �� which are in contradiction with the experiment. We consider the general spin particle formalism based on symmetry properties, including requirements of angular momentum conservation in the t-channel. In such "a dynamic amplitude" approach obligatory kinematic factors arise in helicity amplitudes and consequently in expressions of all observable quantities. These spin structures give small parameters in the considered region. These parameters supress contributions of definite helicity amplitudes in observables. This "kinematic hierarchy" gives the nonzero polarisation and is closer to the esperimental value for �� than QCD.

PERPURBATIVE QCD AND HARD PROTON-PROTON SCATTERING There exists contradiction between the perturbative QCD and experiment. This is connected with proton-proton scattering at high energies and large fixed angles. This is just the region where PQCD must work. But one can say that "the naive PQCD" has some difficulty here. The point is that PQCD yields a "helicity conservation rule" [1-3] which gives in the lowest orders of perturbation theory a zero value for polarization and the value 1/3 for the asymmetry parameter ��. [1] S.J.Brodsky, G.P.Lapage, Phys. Rev., D24 (1981) 2848. [2] S.J.Brodsky, C.E.Carlson, H.J.Lipkin, Phys. Rev., D20. (1979) 2278 [3] G.R.Farrar, S.Gottlieb, D.Sivers, G.H.Thomas, Phys. Rev., D20 (1979) 202.

The use of perturbative QCD for this reaction is based on the following assumptions: --- Factorization property. The quark subprocess is separable. --- A simple connection between quark and proton helicities. The proton helicity is just the sum of quark helicities. --- Helicity does not change at the quark-gluon vertex in the asymptotical region. The result is the helicity conservation rule:

� � � � Where � means the helicity of a corresponding particle. The consequences of this rule are in contradiction with the experiment. It is difficult to expect for changing (and improving) the situation by calculating an enormous number of diagrams in higher order. (It seems That helicity conservation will remain in any finite order of aperturbative expansion.)

STRUCTUE OF HELICITY AMPLITUDES AND SOME RESULTS FOR BINARY PROCESSES [4] G.Cohen-Tannoudji, A.Morel, H.Navelet, Ann. of Physics, 46(1968) 239 [5] L.L.Wang, Phys. Rev., 142 (1966) 1187 [6] M.P.Chavleishvili, High Energy Spin Physics, Proceedings of the 9th International Symposium, Eds. K.-H.Althoff W.Meyer, Bonn, vol1 ,p.489 (Springer-Verlag, Berlin, 1991) [7] M.P.Chavleishvili, Polarization Dynamics in Nuclear and Particle Physics, Proceedings of the International Symposium, Trieste,1992 [8] M.P.Chavleishvili, Ludwig-Maximilian University Preprint LMU-02-93, M\"unchen, 1993 [9] M.P.Chavleishvili, Ludwig-Maximilian University Preprint LMU-03-93, M"unchen, 1993 [10] M.P.Chavleishvili, JINR Preprint, E2-92-385, Dubna, 1992 [11] M.P.Chavleishvili, High Energy Spin Physics, Proceedings of the 8th International Symposium, Ed.K.J.Heller, Minneapolis, 1988,vol 1 ,p.123. New York, 1989

On the basis of general space-time and crossing symmetry general analitic structure for amplitudes describing spin-particle binary reactions are considered. Using knowledge about kinematic structure of helicity amplitudes in dynamic amplitude approach we can get:

- dispersion relations for each individual helicity amplitudes, describing any elastic processes; - low-energy theorems (involving reactions with photon, graviton and gravitino); - sum rules (including Drell-Hern-Gerasimov SR);

-model-independent sum--rulle type inequalities for observable quantities and -some asymptotic relations between polarization parameters. ============

In the framework of the general spin formalism based on the symmetry properties ("dynamic amplitude" approach) obligatory kinematic factors arise in the expressions of observables. These spin structures for high energies give a small parameter that orders the contributions of helicity amplitudes to observables. Such a "kinematic hierarchy " predicts for pp elastic scattering at high energies and a large fixed angle (90o) a simple connection between asymmetry parameters and even numerical values for them.

SYMMETRY AND SPIN Symmetry means harmony, beauty, order. In physics, symmetry has three levels: 1. Coordinate systems, frames (spherical system, inertial systems); 2. Variables. (For example, for binary processes we have two independent variables, energy and angle, or invariant variables -- s and t); 3. Functions. Amplitudes.

Observables and helicity amplitudes The formalism can be used as the general basis for model building with guaranteed fulfillment of symmetry and conservation laws. This method is interesting for studying general characteristics of particle reaction theory and it also suits for exploring concrete processes. Based on the symmetry principles we can separate the kinematics and dynamics of the binary processes. In this approach we can define the dynamic amplitudes which are free of kinematic singularities, have the same dimensions and are connected with observable quantities very simply. At extremely high energies with fixed angles observables are expressed via dynamic amplitudes with definite kinematical factors. These kinematical factors in the considered region in fact are small parameters in different powers. This gives us something like perturbation theory: the hierarchy of contributions in observables ordered (sorted) by small parameter. In the first approximation such "a kinematic hierarchy" gives us definite relations between observables. For proton-proton scattering we will have relations between asymmetry parameters and even numerical values for them. For the asymmetry parameter

�� it seems that its observed value just goes up to the obtained asymptotic number.

BINARY REACTIONS We will consider a general binary reaction with particles of any spin � , masses � and helicities � a( � �, �) + b( � �, �) → c( � �, �) + d( � �, �). Spins of massive particles have 2s+1 projection, so the total number of helicity (or other) amplitudes for scattering of massive particles with definite spins is N=(2 �+1)(2 �+1)(2 �+1)(2 �+1). Discrete P, T and C symmetries decrease the number of independent amplitudes. Besides the conservation of the projection of angular momentum decreases the number of amplitudes for certain directions when the process has higher symmetry. For particles with a zero spin the binary process is described by one amplitude, A(s,t). This amplitude is decomposed in polynomials of both invariant variables. For particles with nonzero spin, the process (1) can bedescribed by Jacob and Wick helicity amplitudes (Ann of Physics, 7 (1959) 404.)

The helicity amplitudes have clear physical meaning, the same dimensions, observables (polarization, cross sections, asymmetries, etc.) are simply expressed via them. The differential cross section in a proper experiment when one measures the helicity of each particle is expressed in terms of helicity amplitudes in the following simple form: �σ�� � � � � ����;����

� For unpolarized reactions we have �σ��

����;����� (the sum runs over all helicities.)

Asymmetry parameters, such as P, ��, �� and �� in terms of the helicity amplitudes have the form

�� � �∗

� � Here m and n represent sets of helicity indices; �� . The sum is taken for all values of helicities.

These equations are quite simple, but helicity amplitudes contain kinematic singularities and the conservation laws are not guaranteed to fulfil (and these laws are not fulfilled automatically), thus kinematics and dynamics are not separated. For proton-proton scattering we have five independent amplitudes. The standard choise of them is the following:

� ��,��; ��,��

� �

�,��;���,��

�

� �

�,���; ��,��

�

� �

�,���;��

�,��

� �

�,��; ��,���

For the forward scattering two of them � � and for the backward scattering also two of them � � must be zero. This is a consequence of the symmetry properties and angular momentum projection conservation in the s-channel center of mass system. For three asymmetry parameters (initial state correlation parameters) have [9]: C.Bourrely, E.Leader, J.Soffer, Phys. Reports, 59 (1980) 96.

= Re[ f1 f2* - f3 f4

* -2 [׀

= Re[ f1 f2* + f3 f4

*]

= - { + +

Kinematic singularities Helicity amplitudes have kinematic singularities. When one considers spin-zero particle scattering, by definition, one has no kinematic singularities connected with spin. There exist several different but in fact equivalent definitions of "kinematic singularities": --- One can decompose the helicity amplitudes, using the Lorentz invariance, Dirac equation (for spin-1/2 particles) and energy-momentum conservation, in terms of invariant amplitudes (when the matrix structure is fixed by the Lorentz invariance and other general principles). The singularities of coefficient functions of the invariant amplitudes are kinematic singularities. --- Helicity states are defined from states at rest by boosting and rotating them. These boosts and rotations are realized by definite functions which have singularities and they are precisely the kinematic ones.

--- The definition of kinematic singularities in the perturbation theory language means to search for singular factors which do not vary from diagram to diagram for any given process and disappear in the case of spin-zero particles. --- Kinematic singularities can be manifested by the fact that the generalization of the partial-wave decomposition in the Legandre polynomials of the scattering amplitude in the spin zero case to spin-particle scattering (based on the symmetry arguments) is the decomposition of the helicity amplitudes in the Wigner rotation functions. But these functions are not polynomials, they contain singularities, just the kinematic singularities of the helicity amplitudes. So, helicity amplitudes have the kinematic singularities and we face the problem to find and separate them.

Conservation laws and dispersion amplitudes We can consider the problem of separating the kinematic singularities using symmetries and consequences of conservation laws. Let us consider the reaction in the s-channel described by the helicity amplitudes

����;���� �

and introduce the quantities

� � and

µ � �. Their meaning will be clear if we recall that two particles in the center of mass system are moving in opposite directions and thu and µ are projections of the total spin (or total angular momentum) on the directions of motion before and after collision.

Owing to the conservation of the projection of total angular momentum the amplitudes in the forward direction,

� must vanish in all the cases except for µ Analogously, for backward scattering,

� π the amplitudes must vanish for the same reasons in all the cases except for µ Two questions arise: Can the helicity amplitudes be parametrized so as to satisfy the above conditions automatically both in s- and t-channals ? Can kinematic singularities of helicity amplitudes be found and separated in a simple way? The answer is "Yes" .

Wigner rotation functions It is convenient to expand helicity amplitudes into a set of the Wigner rotation functions. In this decomposition the scattering amplitude in the c.m. system of the s-channel, is splitted into two parts; one part is defined by the symmetry properties and enters into the Wigner functions (that makes the conservation laws of the angular momentum valid,) and the other part has a dynamic nature. This decomposition is physically distinguished because the basis functions of the decomposition are eigenfunctions of the total angular momentum (which is a conservative quantity). This is the kinematic part in the decomposition, while dynamics is contained in coefficient functions.

Crossing and dynamic amplitudes During crossover from one channel to another in the case of helicity amplitudes, analytic continuation from a physical region from one channel to a physical region of another channel should be accompanied by transition from the c.m.system of the s-channel to the c.m.system of the t-channel with the use of the Lorentz complex transformation. The latter results in requantization of spins; and the connection between amplitudes is expressed in terms of the Wigner functions. Expressing s-channel dispersion amplitudes in terms of t-channel dispersion amplitudes we obtain the connection between the amplitudes having kinematic singularities in s with the amplitudes free from them, with definite coefficients containing the wanted singularities in s. In this way we can determine dynamic amplitudes. Using crossing relations we can separate separate kinematical part from helicity amplitudes defining the Dynamic amplitudes For any binary reactions with arbitrary masses and spins. {It may be strange. Any physicist study definite process !!!}

Elastic scattering For elastic scattering of equal mass spin-particles we can determine dynamic amplitudes by the following equation:

����;����[��µ]

�[��µ]

�

���� �

����;����

Where K = ��(��)∑ �

�.

As the connection between the helicity and dynamic amplitudes is one-to-one, every helicity amplitude for elastic scattering is expressed in terms of one dynamic amplitude. Hence it follows that all attractive features of the helicity amplitudes, clear physical meaning, simple relations with observables, and equal dimensions, are also inherent in the dynamic amplitudes. The formalism of dynamic amplitudes is simple for low spins and remains such also for higher spins: the formalism is simple for any spins.

The differential cross section with fixed helicities is expressed only via one dynamic amplitude with the kinematical factors which contain all kinematic singularities (the same cross section is expressed in terms of all N invariant amplitudes). Kinematic factors ensure that conservation laws are fulfilled automatically. In this case we have the simple separation of kinematics and dynamics in the description of spin particle elastic scattering. The dynamic amplitudes have a clear physical meaning and the same dimensions. Analogous simple equations connect other observables and dynamic amplitudes. The summation in these formulae runs over all helicities, but because of "kinematic hierarchy" at high energies we can dramatically simplify these expressions separating the dominant contributions and reducing the number of terms.

Kinematic hierarchy and proton-proton scattering At high energies it was often assumed that spin effects are died out, and consequently, the helicity amplitudes do not depend on spin. However, this cannot be assumed directly. Simplifications like that are not correct, as the obligatory kinematic conditions are not taken into account. One cannot neglect the kinematic factors or consider them to be all equal. In that case the helicity amplitudes could possess kinematic singularities, and the angular momentum would not be conserved, say, for forward scattering "forbidden" amplitudes would not vanish. Besides the experiment gives that at high energies the spin effects are considerable. In studying the binary processes at fixed scattering angles and high energies it is convenient to represent kinematic factors in the definition of dynamic amplitudes as functions of the scattering angle in the c.m. system and invariant variable s. Kinematic factors expressed in terms of and s and are factorizable, and we can write

����;���� ����;���� ����;���� ����;����

����;�����√�

�(����;����).

For elastic scattering at

∞ we get the small kinematic factor

�(����;����)

For different values of helicities

��� � � � � ���. In observables, some of contributions of amplitudes are kinematically increased (such amplitudes will give leading contributions) whereas others are suppressed (and can be neglected in the first approximation). So we have the "kinematic hierarchy" -- the helicity amplitudes are divided into classes giving the leading contribution, the first corrections, second correct ons, and so on. The kinematic hierarchy gives also definite relations between various observables.

Nucleon-nucleon scattering For nucleon-nucleon scattering we have five independent amplitudes. The connection between helicity and dynamic amplitudes at fixed angles and asymptotic s can be expressed as follows: �(s,t)= �

√��

�(s,t), �(s,t)= m

√s2D2(s,t),

f3(s,t)= cos2 �2

D3(s,t),

f4(s,t)= sin2 �2

D4(s,t), f5(s,t)= √s

2mD5(s,t).

In the high-energy large-fixed-angle region m√s

1 and the helicity amplitudes are splitted into three classes in the order of smallness determined by the kinematic factors. So we obtain

12,12; 12,�1

2 1

2,�12; 12,�1

2 1

2,�12;�1

2,12 1

2,12; 12,12 1

2,12;�12,12

.

here "a b" means that the contribution of b is suppressed relative to the contribution of a in the observables. For proton-proton scattering at =900 we have from s-u crossing symmetry that

12,12; 12,�1

2900 =0,

12,�1

2; 12,�12

900 =- 12,�1

2;�12,12

900

Taking into account the dominating amplitudes we get for asymmetries

��= �� =- ��=

= 2 12,�1

2; 12,�12

12,�1

2;�12,12

∗ /{ 12,�1

2; 12,�12

2 12,�1

2; �12,12

2 1.

Here in fact we have two results. A."Hierarchy relations" Because the number of the dominating amplitudes is smaller than that of independent amplitudes (de scribing all the observable quantities),the hierarchy scenario contains the smaller number of amplitudes describing the same number of observables. So we must have the relations between the observables, hierarchy relations or hierarchy sum rules. �� = �� =- �� B.Numerical values for asymmetry parameters !

�� 1.

In fact, the kinematic hierarchy is a consequences of the kinematic structure of helicity amplitudes and of the assumption that in the considered region

ms

4D1D2∗ 1

4sin2D3D4

∗

or the assumption that and D1 and D2

∗are nonsingular functions in the asymptotic region.

Predictions of perturbative QCD The predictions of perturbative QCD, taking into account the helicity conservation rule gives for high energy 900 scattering in the framework of perturbative QCD [1] S.J.Brodsky, G.P.Lapage, Phys. Rev., D24 (1981) 2848. [2] S.J.Brodsky, C.E.Carlson, H.J.Lipkin, Phys. Rev., D20 (1979) 2278. [3] G.R.Farrar, S.Gottlieb, D.Sivers, G.H.Thomas, Phys. Rev., D20 (1979) 202. �� = �� = ��

13 .

Experiment For experiment we have at 900, for �� (p�

2 GeV 2 ) the following values: E.A.Crosbie et al., Phys. Rev., D23 (1981) 600 [18] A.D.Krish, Phys.Rev. Lett., 63 (1989) 1137

�� 3 81 0 26 �� 4 79 0 52 and �� (5.56)=0.59. So, we see that there is the tendency of rising up the data with p�

2 in agreement with our asymptotic predictions. ---------------

FORMALISM FOR HELICITY AMPLITHDES FOR ANY MASSES AND ARBITRARY SPIN 2 PARTICLE --- 2 JETS 2PARTICLE --- N PARTICLE REACTIONS A lot of diagrams … Helisity approach … Small parameters at asymptotic energies we have in two approaches …

Acknowledgments I would like to acknowledge Prof. Harald Fritzsch, useful discussions and the kind hospitality when I was at Ludwig-Maximilian University in Munchen some times ago and for invitation on this Seminar. I wish to express my appreciation to Prof. Willibald Plessas and to Irina Jurav’lova. Thay halped ma in getting visa. Thanks for technical support to my old friend Sasha Dorochov and to my new friends Martin Volk and Daniel Kupelwieser.