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<title>Four-Vectors</title>
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The <code>Vec4</code> class gives a simple implementation of four-vectors.
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The member function names are based on the assumption that these
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represent four-momentum vectors. Thus one can get or set
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<i>p_x, p_y, p_z</i> and <i>e</i>, but not <i>x, y, z</i>
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or <i>t</i>. This is only a matter of naming, however; a
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<code>Vec4</code> can equally well be used to store a space-time
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The <code>Particle</code> object contains a <code>Vec4 p</code> that
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stores the particle four-momentum, and another <code>Vec4 vProd</code>
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for the production vertex. For the latter the input/output method
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names are adapted to the space-time character rather than the normal
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energy-momentum one. Thus a user would not normally access the
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<code>Vec4</code> classes directly, but only via the methods of the
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<code>Particle</code> class,
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see <?php $filepath = $_GET["filepath"];
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echo "<a href='ParticleProperties.php?filepath=".$filepath."' target='page'>";?>Particle Properties</a>.
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Nevertheless you are free to use the PYTHIA four-vectors, e.g. as
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part of some simple analysis code based directly on the PYTHIA output,
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say to define the four-vector sum of a set of particles. But note that
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this class was never set up to allow complete generality, only to
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provide the operations that are of use inside PYTHIA. There is no
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separate class for three-vectors, since such can easily be represented
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by four-vectors where the fourth component is not used.
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Four-vectors have the expected functionality: they can be created,
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copied, added, multiplied, rotated, boosted, and manipulated in other
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ways. Operator overloading is implemented where reasonable. Properties
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can be read out, not only the components themselves but also for derived
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quantities such as absolute momentum and direction angles.
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<h3>Constructors and basic operators</h3>
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A few methods are available to create or copy a four-vector:
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<a name="method1"></a>
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<p/><strong>Vec4::Vec4() </strong> <br/>
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creates a four-vector with all components set to 0.
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<a name="method2"></a>
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<p/><strong>Vec4::Vec4(const Vec4& v) </strong> <br/>
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creates a four-vector copy of the input four-vector.
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<a name="method3"></a>
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<p/><strong>Vec4& Vec4::operator=(const Vec4& v) </strong> <br/>
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copies the input four-vector.
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<a name="method4"></a>
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<p/><strong>Vec4& Vec4::operator=(double value) </strong> <br/>
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gives a four-vector with all components set to <i>value</i>.
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<h3>Member methods for input</h3>
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The values stored in a four-vector can be modified in a few different
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<a name="method5"></a>
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<p/><strong>void Vec4::reset() </strong> <br/>
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sets all components to 0.
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<a name="method6"></a>
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<p/><strong>void Vec4::p(double pxIn, double pyIn, double pzIn, double eIn) </strong> <br/>
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sets all components to their input values.
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<a name="method7"></a>
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<p/><strong>void Vec4::p(Vec4 pIn) </strong> <br/>
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sets all components equal to those of the input four-vector.
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<a name="method8"></a>
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<p/><strong>void Vec4::px(double pxIn) </strong> <br/>
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<strong>void Vec4::py(double pyIn) </strong> <br/>
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<strong>void Vec4::pz(double pzIn) </strong> <br/>
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<strong>void Vec4::e(double eIn) </strong> <br/>
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sets the respective component to the input value.
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<h3>Member methods for output</h3>
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A number of methods provides output of basic or derived quantities:
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<a name="method9"></a>
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<p/><strong>double Vec4::px() </strong> <br/>
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<strong>double Vec4::py() </strong> <br/>
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<strong>double Vec4::pz() </strong> <br/>
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<strong>double Vec4::e() </strong> <br/>
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gets the respective component.
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<a name="method10"></a>
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<p/><strong>double Vec4::mCalc() </strong> <br/>
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<strong>double Vec4::m2Calc() </strong> <br/>
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the (squared) mass, calculated from the four-vectors.
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If <i>m^2 < 0</i> the mass is given with a minus sign,
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<i>-sqrt(-m^2)</i>. Note the possible loss of precision
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in the calculation of <i>E^2 - p^2</i>; for particles the
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correct mass is stored separately to avoid such problems.
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<a name="method11"></a>
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<p/><strong>double Vec4::pT() </strong> <br/>
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<strong>double Vec4::pT2() </strong> <br/>
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the (squared) transverse momentum.
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<a name="method12"></a>
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<p/><strong>double Vec4::pAbs() </strong> <br/>
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<strong>double Vec4::pAbs2() </strong> <br/>
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the (squared) absolute momentum.
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<a name="method13"></a>
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<p/><strong>double Vec4::eT() </strong> <br/>
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<strong>double Vec4::eT2() </strong> <br/>
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the (squared) transverse energy,
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<i>eT = e * sin(theta) = e * pT / pAbs</i>.
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<a name="method14"></a>
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<p/><strong>double Vec4::theta() </strong> <br/>
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the polar angle, in the range 0 through
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<a name="method15"></a>
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<p/><strong>double Vec4::phi() </strong> <br/>
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the azimuthal angle, in the range <i>-pi</i> through <i>pi</i>.
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<a name="method16"></a>
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<p/><strong>double Vec4::thetaXZ() </strong> <br/>
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the angle in the <i>xz</i> plane, in the range <i>-pi</i> through
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<i>pi</i>, with 0 along the <i>+z</i> axis.
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<a name="method17"></a>
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<p/><strong>double Vec4::pPos() </strong> <br/>
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<strong>double Vec4::pNeg() </strong> <br/>
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the combinations <i>E+-p_z</i>.
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<h3>Friend methods for output</h3>
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There are also some <code>friend</code> methods that take one, two
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or three four-vectors as argument. Several of them only use the
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three-vector part of the four-vector.
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<a name="method18"></a>
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<p/><strong>friend ostream& operator<<(ostream&, const Vec4& v) </strong> <br/>
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writes out the values of the four components of a <code>Vec4</code> and,
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within brackets, a fifth component being the invariant length of the
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four-vector, as provided by <code>mCalc()</code> above, and it all
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ended with a newline.
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<a name="method19"></a>
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<p/><strong>friend double m(const Vec4& v1, const Vec4& v2) </strong> <br/>
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<strong>friend double m2(const Vec4& v1, const Vec4& v2) </strong> <br/>
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the (squared) invariant mass.
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<a name="method20"></a>
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<p/><strong>friend double dot3(const Vec4& v1, const Vec4& v2) </strong> <br/>
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<a name="method21"></a>
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<p/><strong>friend double cross3(const Vec4& v1, const Vec4& v2) </strong> <br/>
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<a name="method22"></a>
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<p/><strong>friend double theta(const Vec4& v1, const Vec4& v2) </strong> <br/>
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<strong>friend double costheta(const Vec4& v1, const Vec4& v2) </strong> <br/>
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the (cosine) of the opening angle between the vectors,
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in the range 0 through <i>pi</i>.
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<a name="method23"></a>
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<p/><strong>friend double phi(const Vec4& v1, const Vec4& v2) </strong> <br/>
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<strong>friend double cosphi(const Vec4& v1, const Vec4& v2) </strong> <br/>
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the (cosine) of the azimuthal angle between the vectors around the
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<i>z</i> axis, in the range 0 through <i>pi</i>.
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<a name="method24"></a>
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<p/><strong>friend double phi(const Vec4& v1, const Vec4& v2, const Vec4& v3) </strong> <br/>
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<strong>friend double cosphi(const Vec4& v1, const Vec4& v2, const Vec4& v3) </strong> <br/>
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the (cosine) of the azimuthal angle between the first two vectors
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around the direction of the third, in the range 0 through <i>pi</i>.
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<h3>Operations with four-vectors</h3>
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Of course one should be able to add, subtract and scale four-vectors,
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<a name="method25"></a>
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<p/><strong>Vec4 Vec4::operator-() </strong> <br/>
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return a vector with flipped sign for all components, while leaving
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the original vector unchanged.
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<a name="method26"></a>
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<p/><strong>Vec4& Vec4::operator+=(const Vec4& v) </strong> <br/>
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add a four-vector to an existing one.
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<a name="method27"></a>
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<p/><strong>Vec4& Vec4::operator-=(const Vec4& v) </strong> <br/>
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subtract a four-vector from an existing one.
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<a name="method28"></a>
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<p/><strong>Vec4& Vec4::operator*=(double f) </strong> <br/>
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multiply all four-vector components by a real number.
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<a name="method29"></a>
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<p/><strong>Vec4& Vec4::operator/=(double f) </strong> <br/>
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divide all four-vector components by a real number.
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<a name="method30"></a>
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<p/><strong>friend Vec4 operator+(const Vec4& v1, const Vec4& v2) </strong> <br/>
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add two four-vectors.
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<a name="method31"></a>
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<p/><strong>friend Vec4 operator-(const Vec4& v1, const Vec4& v2) </strong> <br/>
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subtract two four-vectors.
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<a name="method32"></a>
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<p/><strong>friend Vec4 operator*(double f, const Vec4& v) </strong> <br/>
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<strong>friend Vec4 operator*(const Vec4& v, double f) </strong> <br/>
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multiply a four-vector by a real number.
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<a name="method33"></a>
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<p/><strong>friend Vec4 operator/(const Vec4& v, double f) </strong> <br/>
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divide a four-vector by a real number.
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<a name="method34"></a>
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<p/><strong>friend double operator*(const Vec4& v1, const Vec4 v2) </strong> <br/>
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There are also a few related operations that are normal member methods:
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<a name="method35"></a>
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<p/><strong>void Vec4::rescale3(double f) </strong> <br/>
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multiply the three-vector components by <i>f</i>, but keep the
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fourth component unchanged.
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<a name="method36"></a>
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<p/><strong>void Vec4::rescale4(double f) </strong> <br/>
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multiply all four-vector components by <i>f</i>.
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<a name="method37"></a>
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<p/><strong>void Vec4::flip3() </strong> <br/>
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flip the sign of the three-vector components, but keep the
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fourth component unchanged.
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<a name="method38"></a>
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<p/><strong>void Vec4::flip4() </strong> <br/>
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flip the sign of all four-vector components.
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<h3>Rotations and boosts</h3>
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A common task is to rotate or boost four-vectors. In case only one
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four-vector is affected the operation may be performed directly on it.
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However, in case many particles are affected, the helper class
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<code>RotBstMatrix</code> can be used to speed up operations.
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<a name="method39"></a>
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<p/><strong>void Vec4::rot(double theta, double phi) </strong> <br/>
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rotate the three-momentum with the polar angle <i>theta</i>
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and the azimuthal angle <i>phi</i>.
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<a name="method40"></a>
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<p/><strong>void Vec4::rotaxis(double phi, double nx, double ny, double nz) </strong> <br/>
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rotate the three-momentum with the azimuthal angle <i>phi</i>
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around the direction defined by the <i>(n_x, n_y, n_z)</i>
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<a name="method41"></a>
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<p/><strong>void Vec4::rotaxis(double phi, Vec4& n) </strong> <br/>
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rotate the three-momentum with the azimuthal angle <i>phi</i>
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around the direction defined by the three-vector part of <i>n</i>.
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<a name="method42"></a>
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<p/><strong>void Vec4::bst(double betaX, double betaY, double betaZ) </strong> <br/>
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boost the four-momentum by <i>beta = (beta_x, beta_y, beta_z)</i>.
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<a name="method43"></a>
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<p/><strong>void Vec4::bst(double betaX, double betaY, double betaZ,double gamma) </strong> <br/>
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boost the four-momentum by <i>beta = (beta_x, beta_y, beta_z)</i>,
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where the <i>gamma = 1/sqrt(1 - beta^2)</i> is also input to allow
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better precision when <i>beta</i> is close to unity.
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<a name="method44"></a>
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<p/><strong>void Vec4::bst(const Vec4& p) </strong> <br/>
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boost the four-momentum by <i>beta = (p_x/E, p_y/E, p_z/E)</i>.
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<a name="method45"></a>
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<p/><strong>void Vec4::bst(const Vec4& p, double m) </strong> <br/>
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boost the four-momentum by <i>beta = (p_x/E, p_y/E, p_z/E)</i>,
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where the <i>gamma = E/m</i> is also calculated from input to allow
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better precision when <i>beta</i> is close to unity.
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<a name="method46"></a>
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<p/><strong>void Vec4::bstback(const Vec4& p) </strong> <br/>
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boost the four-momentum by <i>beta = (-p_x/E, -p_y/E, -p_z/E)</i>.
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<a name="method47"></a>
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<p/><strong>void Vec4::bstback(const Vec4& p, double m) </strong> <br/>
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boost the four-momentum by <i>beta = (-p_x/E, -p_y/E, -p_z/E)</i>,
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where the <i>gamma = E/m</i> is also calculated from input to allow
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better precision when <i>beta</i> is close to unity.
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<a name="method48"></a>
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<p/><strong>void Vec4::rotbst(const RotBstMatrix& M) </strong> <br/>
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perform a combined rotation and boost; see below for a description
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of the <code>RotBstMatrix</code>.
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For a longer sequence of rotations and boosts, and where several
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<code>Vec4</code> are to be rotated and boosted in the same way,
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a more efficient approach is to define a <code>RotBstMatrix</code>,
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which forms a separate auxiliary class. You can build up this
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4-by-4 matrix by successive calls to the methods of the class,
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such that the matrix encodes the full sequence of operations.
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The order in which you do these calls must agree with the imagined
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order in which the rotations/boosts should be applied to a
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four-momentum, since in general the operations do not commute.
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<a name="method49"></a>
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<p/><strong>RotBstMatrix::RotBstMatrix() </strong> <br/>
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creates a diagonal unit matrix, i.e. one that leaves a four-vector
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<a name="method50"></a>
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<p/><strong>RotBstMatrix::RotBstMatrix(const RotBstMatrix& Min) </strong> <br/>
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creates a copy of the input matrix.
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<a name="method51"></a>
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<p/><strong>RotBstMatrix& RotBstMatrix::operator=(const RotBstMatrix4& Min) </strong> <br/>
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copies the input matrix.
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<a name="method52"></a>
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<p/><strong>void RotBstMatrix::rot(double theta = 0., double phi = 0.) </strong> <br/>
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rotate by this polar and azimuthal angle.
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<a name="method53"></a>
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<p/><strong>void RotBstMatrix::rot(const Vec4& p) </strong> <br/>
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rotate so that a vector originally along the <i>+z</i> axis becomes
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parallel with <i>p</i>. More specifically, rotate by <i>-phi</i>,
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<i>theta</i> and <i>phi</i>, with angles defined by <i>p</i>.
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<a name="method54"></a>
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<p/><strong>void RotBstMatrix::bst(double betaX = 0., double betaY = 0., double betaZ = 0.) </strong> <br/>
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boost by this <i>beta</i> vector.
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<a name="method55"></a>
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<p/><strong>void RotBstMatrix::bst(const Vec4&) </strong> <br/>
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<strong>void RotBstMatrix::bstback(const Vec4&) </strong> <br/>
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boost with a <i>beta = p/E</i> or <i>beta = -p/E</i>, respectively.
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<a name="method56"></a>
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<p/><strong>void RotBstMatrix::bst(const Vec4& p1, const Vec4& p2) </strong> <br/>
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boost so that <i>p_1</i> is transformed to <i>p_2</i>. It is assumed
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that the two vectors obey <i>p_1^2 = p_2^2</i>.
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<a name="method57"></a>
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<p/><strong>void RotBstMatrix::toCMframe(const Vec4& p1, const Vec4& p2) </strong> <br/>
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boost and rotate to the rest frame of <i>p_1</i> and <i>p_2</i>,
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with <i>p_1</i> along the <i>+z</i> axis.
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<a name="method58"></a>
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<p/><strong>void RotBstMatrix::fromCMframe(const Vec4& p1, const Vec4& p2) </strong> <br/>
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rotate and boost from the rest frame of <i>p_1</i> and <i>p_2</i>,
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with <i>p_1</i> along the <i>+z</i> axis, to the actual frame of
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<i>p_1</i> and <i>p_2</i>, i.e. the inverse of the above.
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<a name="method59"></a>
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<p/><strong>void RotBstMatrix::rotbst(const RotBstMatrix& Min); </strong> <br/>
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combine the current matrix with another one.
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<a name="method60"></a>
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<p/><strong>void RotBstMatrix::invert() </strong> <br/>
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invert the matrix, which corresponds to an opposite sequence and sign
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of rotations and boosts.
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<a name="method61"></a>
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<p/><strong>void RotBstMatrix::reset() </strong> <br/>
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reset to no rotation/boost; i.e. the default at creation.
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<a name="method62"></a>
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<p/><strong>double RotBstMatrix::deviation() </strong> <br/>
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crude estimate how much a matrix deviates from the unit matrix:
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the sum of the absolute values of all non-diagonal matrix elements
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plus the sum of the absolute deviation of the diagonal matrix
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<a name="method63"></a>
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<p/><strong>friend ostream& operator<<(ostream&, const RotBstMatrix& M) </strong> <br/>
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writes out the values of the sixteen components of a
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<code>RotBstMatrix</code>, on four consecutive lines and
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ended with a newline.
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