Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://comphep.sinp.msu.ru/_media/download/hep-ph-0208011.pdf Äàòà èçìåíåíèÿ: Wed Sep 13 17:54:59 2006 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 22:41:30 2016 Êîäèðîâêà: IBM-866

Laboratoire de Physique Th‡orique LAPTH e Chemin de Bellevue, B.P. 110, F-74941 Annecy-le-Vieux, Cedex, France. Laboratory of Particle Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Moscow Region, Russian Federation

LAPTH-926/02

arXiv:hep-ph/0208011 v1 1 Aug 2002

LanHEP--
a package for automatic generation of Feynman rules in field theory Version 2.0

A. V. Semenov

2002


LanHEP -- a package for automatic generation of Feynman rules in field theory. Version 2.0

Abstract. The LanHEP program for Feynman rules generation in momentum representation is presented. It reads the Lagrangian written in a compact form, close to the one used in publications. It means that Lagrangian terms can be written with summation over indices of broken symmetries and using special symbols for complicated expressions, such as covariant derivative and strength tensor for gauge fields. The output is Feynman rules in terms of physical fields and independent parameters. This output can be written in LaTeX format and in the form of CompHEP mo del files, which allows one to start calculations of pro cesses in the new physical mo del. Although this job is rather straightforward and can be done manually, it requires careful calculations and in mo dern theories with many particles and vertices, such as supersymmetric mo dels, can lead to errors and misprints. The program allows one to intro duce into CompHEP new gauge theories as well as various anomalous terms. E-mail: semenov@theory.sinp.msu.ru WWW page: http://theory.sinp.msu.ru/~semenov/lanhep.html


Intro duction
LanHEP has been designed as part of the CompHEP package [1], a software for automatic calculations in high energy physics. CompHEP allows symbolic computation of the matrix element squared of any pro cess with up to 6 incoming and outgoing particles for a given physical mo del (i.e. a mo del defined by a set of Feynman rules as a table of vertices in the momentum representation) and then numerical calculation of cross-sections and various distributions. The main purpose of the new option given by LanHEP is to easily intro duce new mo dels. LanHEP makes possible the generation of Feynman rules for propagators and vertices in momentum representation starting from the Lagrangian defined by a user in some simple format very similar to canonical co ordinate representation. The user should prepare a text file with description of all Lagrangian terms in the format close to the form used in standard publications. Of course, the user has to describe all particles and parameters appearing in Lagrangian terms. The main LanHEP features are: § LanHEP expands expression and combines similar terms; § it performs the Fourier transformation by replacing derivatives with momenta of particles; § it writes Feynman rules in the form of four tables in CompHEP format as well as tables in LaTeX format; § user can define the substitution rules, for example for covariant derivative; § it is possible to define multiplets, and (if necessary) their components; § user can write Lagrangian terms with Lorentz and multiplet indices explicitly or omit indices (all or some of them); § LanHEP performs explicit summation over the indices in the Lagrangian, if the corresponding components for multiplets and matrices are intro duced; § several tests can be applied to check the correctness of the Lagrangian; § supersymmetric theories can be described using superpotential formalism; § BRST invariance of the Lagrangian can be tested, and ghost interaction can be constructed; § LanHEP allows the user to intro duce vertices with 4 fermions or 4 colored particles (such vertices can't be intro duced directly in CompHEP) by means of auxiliary field with constant propagator; The LanHEP software is written in C programming language. The first version [2] was released in August 1996.

1
1.1

Getting started with LanHEP
QED

We start with a simple exercise, illustrating the main ideas and features of LanHEP. The first physical mo del is Quantum Electro dynamics. The QED Lagrangian is L
QE D

1 = - F² F 4

²

+ e ² (i² + ge A² )e - mee ï ï
3


model QED/1. parameter ee=0.31333:'elementary electric charge'. spinor e1/E1:(electron, mass me=0.000511). vector A/A:(photon). let F^mu^nu=deriv^mu*A^nu-deriv^nu*A^mu. lterm -1/4*(F^mu^nu)**2 - 1/2*(deriv^mu*A^mu)**2. lterm E1*(i*gamma*deriv+me)*e1. lterm ee*E1*gamma*A*e1.
Figure 1: LanHEP input file for the generation of QED Feynman rules

and the gauge fixing term in Feynman gauge has the form L
GF

1 = - (² A² )2 . 2

Here e(x) is the spinor electron-positron field, m is the electron mass, A² (x) is the vector photon field, F ² = ² A - A² , and ge is the elementary electric charge. The LanHEP input file to generate the Feynman rules for QED is shown in Fig. 1. First of all, the input file consists of statements. Each statement begins with one of the reserved keywords and ends by a full-stop '.' symbol. First line says that this is a mo del with the name QED and number 1. This information is supplied for CompHEP, the name QED will be displayed in its list of mo dels. In CompHEP package each mo del is described by four files: 'varsN.mdl', 'funcN.mdl', 'prtclsN.mdl', 'lgrngnN.mdl' , where N is the very number specified in the model statement. The model statement stands first in the input file. If this statement is absent, LanHEP do es not generate the four standard CompHEP files, but just builds the mo del and prints a diagnostic, if errors are found. The second line in the input file contains declaration of the mo del parameter, denoting elementary electric charge ge as ee. For each parameter used in the mo del one should declare its numeric value and an optional comment (it is also used in CompHEP menus). The next two lines declare particles. Statement names spinor, vector correspond to the particle spin. So, we declare electron denoted by e1 (the corresponding antiparticle name is E1) and photon denoted by A (with antiparticle name being A, since the antiparticle for photon is identical to particle). After the particle name we give in brackets some options. The first one is the full name of the particle, used in CompHEP; the second option declares the mass of this particle. The let statement in the next line declares the substitution rule for symbol F. Predefined name deriv, which is reserved for the derivative x , will be replaced after the Fourier transformation by the momentum of the particle multiplied by -i. The rest of the lines describe terms in the Lagrangian. Here the reserved name gamma denotes Dirac's -matrices. One can see that the indices are written separated with the caret symbol '^'. Note that in the last two lines we have omitted indices. It means that LanHEP restores omitted indices automatically. Really, one can type the last term in the full format: lterm ee*E1^a*gamma^a^b^mu*A^mu*e1^b. It corresponds to ge ea ab eb A² with all indices written. Note that the order of ob jects in the ï² monomial is important to restore indices automatically.

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1.2

QCD

Now let us consider the case of Quantum Chromo dynamics. The Lagrangian for the gluon fields reads LY where Ga (x) is the gluon field, gs is the strong coupling constant and f abc are purely imaginary ² structure constants of the S U (3) color group. The quark kinetic term and its interaction with the gluon has the form LF = qi ² ² qi + gs aj qi ² qj Gc , ï iï ²
a where ij are Gell-Mann matrices. Gauge fixing terms in Feynman gauge together with the corresponding Faddev-Popov ghost term are 1 - (² G² )2 + igs f abc ca Gb ² cc , ï² a 2 where (c, c) are unphysical ghost fields. ï The corresponding LanHEP input file is shown in Fig. 2. a F² = ² Ga - Ga - gs f ² abc c Gb G , ² M

1 =- F 4

a²

a F² ,

model QCD/2. parameter gg=1.117:'Strong coupling'. spinor q/Q:(quark, mass mq=0.01, color c3). vector G/G:(gluon, color c8, gauge). let F^mu^nu^a = deriv^nu*G^mu^a - deriv^mu*G^nu^a gg*f SU3^a^b^c*G^mu^b*G^nu^c. lterm -F**2/4-(deriv*G)**2/2. lterm Q*(i*gamma*deriv+mq)*q. lterm i*gg*f SU3*ccghost(G)*G*deriv*ghost(G). lterm gg*Q*gamma*lambda*G*q.
Figure 2: Input file for the generation of QCD Feynman rules

Table 1: QCD Feynman rules generated by LanHEP in LaTeX output format

Fields in the vertex G²p q ïG G r qap ï G G
²p ²p

qbq G G
q q

G G G

Variational derivative of Lagrangian by fields ² -gs p3 fpqr
² gs ab r pq

²r r r

gs fpqr (p g 3 G
s

²

2 gs (g ² g fpqt frst - g ² g fpqt frst + g ² g fprt fq

- p g 2

²

² - p3 g



+ p g 1

²

+ p² g 2



- p g ² ) 1
st

+g ² g fpst fqrt - g ² g fprt fqst - g ² g fpst fqrt ) Since QCD uses ob jects with color indices, one has to declare the indices of these ob jects. There are three types of color indices supported by LanHEP. These types are referred as color c3 (color triplets), color c3b (color antitriplets), and color c8 (color o ctets). One can see
5


that color c3 index type appears among the options in the quark q declaration, and the color c8 one in the gluon G declaration. Antiquark Q has got color index of type color c3b as antiparticle to quark. LanHEP allows contraction of an index of type color c3 only with another index of type color c3b, and two indices of type color c8. Of course, in Lagrangian terms each index has to be contracted with its partner, since Lagrangian has to be a scalar. LanHEP allows also to use in the Lagrangian terms a predefined symbol lambda with the three indices of types color c3, color c3b, color c8 corresponding to Gell-Mann matrices. Symbol f SU3 denotes the structure constant f abc of color S U (3) group (all three indices have the type color c8). Option gauge in the declaration of G allows to use names ghost(G) and ccghost(G) for the ghost fields c and c in Lagrangian terms and in let statements. ï Table 1 shows Feynman rules generated by LanHEP in LaTeX format after pro cessing the input file presented in Fig. 2. Four gluon vertex rule is indicated in the last line. Note that the output in CompHEP format has no 4-gluon vertex explicitly; it is expressed effectively through 3-leg vertices by a constant propagator of some auxiliary field (see section 9 for more details). 1.3 Higgs sector of the Standard Model

model Higgs/1. parameter EE = 0.31333 : 'Electromagnetic coupling constant', SW = 0.4740 : 'sin of the Weinberg angle (PDG-94)', CW = Sqrt(1-SW**2) : 'cos of the Weinberg angle'. let g=EE/SW, g1=EE/CW. vector A/A: (photon, gauge), Z/Z:('Z boson', mass MZ = 91.187, gauge), 'W+'/'W-': ('W boson', mass MW = MZ*CW, gauge). scalar H/H:(Higgs, mass MH = 200, width wH = 1.461). let B = -SW*Z+CW*A. let W = {'W+', CW*Z+SW*A, 'W-'}. let phi = { -i*gsb('W+'), (vev(2*MW/EE*SW)+H+i*gsb(Z))/Sqrt2 }, Phi = anti(phi). lterm -2*lambda*(phi*anti(phi)-v**2/2)**2 where lambda=(g*MH/MW)**2/16, v=2*MW*SW/EE. let D^a^b^mu =(deriv^mu+i*g1/2*B^mu)*delta(2)^a^b +i*g/2*taupm^a^b^c*W^mu^c, Dc^a^b^mu=(deriv^mu-i*g1/2*B^mu)*delta(2)^a^b -i*g/2*taupm^a^b^c*anti(W)^mu^c. lterm D^a^b^mu*phi^b*Dc^a^c^mu*Phi^c.
Figure 3: Input file for the Higgs sector of the Standard Model

The third example illustrates using the multiplets in the framework of LanHEP. Let us consider the Higgs sector of the Standard Mo del. The Higgs doublet can be defined as = (
2M es
W s

-iWf+ + H + iZf )/ 2

,

where sw is the sinus of the weak angle, H is the Higgs field, Zf and Wf‘ are goldstone bosons corresponding to Z and W ‘ gauge fields. The self-interaction of Higgs field read as LH = -2( - 2/2)2 ,
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Table 2: LanHEP output: Feynman rules for Higgs self-interaction

Fields in the vertex HHH H H H H H
+ WF + WF + WF - WF

Variational derivative of Lagrangian by fields eM 2 - 3 MWHw 2 s - - - - - - - -
1 eM H 2 2 MW s w 1 eM H 2 2 MW s w 3 e2 M H 2 4 MW 2 s w 2 1 e2 M H 2 4 MW 2 s w 2 1 e2 M H 2 4 MW 2 s w 2 1 e2 M H 2 2 MW 2 s w 2 1 e2 M H 2 4 MW 2 s w 2 3 e2 M H 2 4 MW 2 s w 2

Z

F

Z H W Z

F

H H H

H
+ F - WF

F

Z

F - WF

+ WF - WF

- WF

Z Z
F

F

Z Z
F

F

Z

F

Z

F

where = (g MH /MW )2 /16, and the vaccuum expectation value = 2MW /g (here and below g = e/sw , g = e/cw .) The gauge interaction of a Higgs doublet is given by the term (D² )(D ² ) , where D² = ² + ig B² /2 + ig W² . Here B is U (1) singlet and W is S U (2) triplet, B² = -sw Z² + cw A² ,
+ W² a W² = cw Z² + sw A² , - W²





and is the vector ( + , 3 , - ). The above mo del can be represented by the LanHEP co de shown in Table 3. New features in this example include the definition of special symbols for coupling constants g , g , gauge and Higgs fields, and the covariant derivative by means of let statement. Note that in the declaration for the fields we have used dummy indices (vector and isospin). The multiplets are defined by the components in the curly brackets. Note that the option gauge in the declaration of gauge fields allows to use the name gsb(Z) and gsb('W+') for the goldstone bosons.

2

Structure of LanHEP input file

The LanHEP input file is the sequence of statements, each starts with a special identifier (such as parameter, lterm etc) and ends with the full-stop '.' symbol. Statement can o ccupy several lines in the input file. This section is aimed to clarify the syntax of LanHEP input files, i.e. the structure of the statements. 2.1 Constants and identifiers

First of all, each word in any statement is either an identifier or a constant. Indentfiers are the names of particles, parameters etc. Examples of identifiers from the previous section are particle names
7


Table 3: LanHEP output: Feynman rules for Higgs gauge interaction

Fields in the vertex - A² W + WF A² A² H H H H H W W
+ + + WF + WF

Variational derivative of Lagrangian by fields ieMW g ² -ieMW g
²

W W W Z Z
F

-

- WF -

W W Z Z
² ²

+ +

-e(p² - p² ) 2 3

² ²

- WF - ²

+ WF ² ²

eMW ² g sw e - 1 siw (p² - p² 1 3 2 ² ² 1 ie (p 2 - p 1 ) 2 sw eMW ² g cw 2 sw i - 1 c w e w (p ² - 1 2 s ieMW sw ² - cw g 1e (p ² - p ² ) 3 2 2 sw ieMW sw ² g cw 1e (p ² - p ² 1 3 2 sw 1 (1-2sw 2 )e 2 cw sw 2 ²

)

p² ) 3

- WF - WF

Z Z Z Z Z
²

F F

+ WF + WF + WF

W W A H H W

- ² - ²

)

- WF

A² A² A² A² A² A² H H H H W W W
+ + +

+ WF

- WF - WF

-

2e g

(p ² - p ² ) 1 2

W
+

+



1 ie2 ² g 2 sw

+ WF

W
-

-

- WF

- - -

1 ie2 ² g 2 sw 1 e2 2 sw 1 e2 2 sw

Z Z Z


F F

g g

² ²
2

+ WF + WF

W

- WF + ²

(1-2sw 2 )e cw sw 1 e2 ² g 2 sw 2 e2 1 2 cw 2 s
w 2 2

g

²

H H W
+ + WF ² ² ²

W Z
²

W Z


-

²

g

²

- WF

Z Z
- F

- WF

W
-

- ²

-

1 ie 2 cw

g

²

1 ie2 ² g 2 cw 1 e2 ² g 2 sw 2 1 e2 ² g 2 sw 2 1 e2 ² g 2 cw 1 e2 ² g 2 cw 1 (1-2sw 2 )2 e 2 cw 2 sw 2 1 e2 2 cw 2 s
w 2 2

+ WF

W Z Z Z Z
F ²

W

Z Z Z Z
F F

F

- WF

+ WF + WF

W

- ²

- WF

g

²

Z

²

Z



Z

Z

F

g

²

8


e1 E1 A q Q G The first word in each statement is also an identifier, defining the function which this statement performs. The identifiers are usually combinations of letters and digits starting with a letter. If an identifier do es not respect this rule, it should be quoted. For example, the names of W ‘ bosons must be written as 'W+' and 'W-', since they contain '+' and 'í' symbols. Constants can be classified as § integers: they consist of optional sign followed by one or more decimal digits, such as 0 1 -1 123 -98765 Integers can appear in Lagrangian terms, parameter definition and in other expressions. § Floating point numbers include optional sign, several decimal digits of mantissa with an embedded perio d (decimal point) with at least one digit before and after the perio d, and optional exponent. The exponent, if present, consist of letter E or e followed by an optional sign and one or more decimal digits. Valid examples of floating point numbers are 1.0 -1.0 0.000511 5.11e-4 Floating point numbers are used only as parameter values (coupling constants, particle masses etc). They can not be explicitly used in Lagrangian terms. § String constants may include arbitrary symbols. They are used as comments in parameter statements, full particle names in the declaration of a particle, etc. Examples from the previous section are electron photon If a string constant contains any character besides letters and digits or do es not begin with a letter, it should be quoted. For example, the comments in QED and QCD input files (see previous section) contain blank spaces, so they are quoted: 'elementary electric charge' 'Strong coupling' 2.2 Comments

User can include comments into the LanHEP input file in two ways. First, symbol '%' denotes the comment till the end of current line. Second way allows one to comment any number of lines by putting a part of input file between '/*' (begin of comment) and '*/' (end of comment) symbols. 2.3 Including files

LanHEP allows the user to divide the input file into several files. To include the file file, the user should use the statement read file. The standard extension '.mdl' of the file name may be omitted in this statement. Another way to include a file is provided by the use statement as use file. The use statement reads the file only once, next appearances of this statement with the same argument do nothing. This function prevents multiple reading of the same file. This form can be used mainly to include some standard mo dules, such as declaration of Standard Mo del particles to be used for writing some extensions of this mo del.
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2.4

Conditional processing of the model

Let us consider the LanHEP input files for the Standard Mo del with both the t'Ho oft-Feynman and unitary gauges. It is clear that these input files differ by several lines only -- the declaration of gauge bosons and Higgs doublet. It is more convenient to have only one mo del definition file. In this case the conditional statements are necessary. LanHEP allows the user to define several keys , and use these keys to branch among several variants of the model. The keys have to be declared by the keys statement: keys name1=value1, name2=value2, ... . Then one can use the conditional statements: do if key==value1. actions1 do else if key==value2. actions2 do else if key==value3. ... do else. default actions end if. The statements do else if and do else are not obligatory. The value of the key is a number or symbolic string. An example of using these statement in the Standard mo del may read: keys Gauge=unitary. do if Gauge==Feynman. vector Z/Z:('Z-boson',mass MZ = 91.187, width wZ = 2.502, gauge). do else if Gauge==unitary. vector Z/Z:('Z-boson',mass MZ = 91.187, width wZ = 2.502). do else. write('Error: key Gauge must be either Feynman or unitary'). quit. end if. Thus, to change the choice of gauge fixing in the generated Feynman rules it is enough to mo dify one word in the input file. There is another way, to set the key value from the command line at the launch of LanHEP: lhep -key Gauge=Feynman filename. If the value of key is set from the command line at the program launch, the value for this key in the keys statement is ignored.

3

Ob jects in the expressions for Lagrangian terms

Each symbol which may appear in algebraic expressions (names of parameters, fields, etc) has a fixed order of indices and their types. If this ob ject is used in any expression, one should write its indices in the same order as they were defined when the ob ject has been declared. Besides the types of indices corresponding to the color S U (3) group: color c3 (color triplet), color c3b (color antitriplet) and color c8 (color o ctet) described in the previous example, there are default types of indices for Lorentz group: vector, spinor and cspinor (antispinor). User can also declare new types of indices corresponding to the symmetries other than color
10


S U (3) group. In this case any ob ject (say, particle) may have indices related to this new group. This possibility will be described in Section 8. If an index appears twice in some monomial of an expression, LanHEP assumes summation over this index. Types of such indices must allow the contraction, i.e. they should be one of the pairs: spinor and cspinor, two vector, color c3 and color c3b, two color c8. In general the following ob jects are available to appear in the expressions for a Lagrangian: integers and identifiers of parameters, particles, specials, let-substitutions and arrays. There are also predefined symbols i, denoting imaginary unit i (i2 = -1) and Sqrt2, which is a parameter with value 2. 3.1 Parameters

Parameters are scalar ob jects (i.e. they have no indices). Parameters denote coupling constants, masses and widths of particles, etc. To intro duce a new parameter one should use the parameter statement, which has the generic form parameter name =value :comment. § name is an identifier of newly created parameter. § value is an integer or floating point number or an expression. One can use previously declared parameters and integers joined by standard arithmetical operators '+', '-', '*', '/', and '**' (power). § comment is an optional comment to clarify the meaning of parameter, it is used in CompHEP help windows. Comment has to be a string constant, so if it contains blank spaces or other special characters, it must be quoted (see Section 3). In the extression for parameters one can use functions sqrt, pow, sin, asin, cos, acos, tan, atan, atan2, fabs (same as in C programming language). One can add new functions through the use of the statement external func(function, arity). Arity is the number of arguments of the function. If this function is absent in the standard C library, it should be written in C and added to the library file usrlib.a in the CompHEP working directory. 3.2 Particles

Particles are ob jects to denote physical particles. They may have indices. It is possible to use three statements to declare a new particle, at the same time the second and the third statements define the corresponding Lorentz index: scalar P/aP:(options). spinor P/aP:(options). vector P/aP:(options). P and aP are identifiers of particle and antiparticle. In the case of truly neutral particles (when antiparticle is identical to the particle itself ) one should use the form P/P with identical names for particle and antiparticle. It is possible to write only the particle name, e.g. scalar P:(options).

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In this case the name of corresponding antiparticle is generated automatically. It satisfies usual CompHEP convention, when the name of antiparticle differs from particle by altering case of the first letter. So for electron name e1 automatically generated antiparticle name be E1. If the name contains symbol '+' it is replaced by 'í' and vice versa. The option is comma-separated list of options for a declared particle, and it may include following items:

the the will the

§ the first element in this list must be the full name of the particle, (e.g. electron and photon in our example.) Full name is a string constant, so it should be quoted if it contains blanks, etc. § mass param=value defines the mass of the particle. Here param is an identifier of a new parameter, which is used to denote the mass; value is its value, it has the same syntax as in the parameter statement, comment for this new parameter being generated automatically. If this option omitted, the mass is assumed to be zero. § width param=value declares the width of the particle. It has the same syntax as for mass option. § itype is a type of index of some symmetry; one can use default index types for color S U (3) group (see QCD example in Section 2). It is possible to use user-defined index types (see Section 8) and Lorentz group indices vector, spinor, cspinor. § left or right says that the massless spinor particle is an eigenstate of (1 - 5 )/2 or (1 + 5 )/2 pro jectors, so this fermion is left-handed or right-handed. § gauge declares the vector particle as a gauge boson. This option generates the corresponding ghosts and goldstone bosons names for the named particle (see below). When a particle name is used in any expression (in Lagrangian terms), one should remember that the first index is either a vector or a spinor (if this particle is not a Lorentz scalar). Then the indices follow in the same order as index types in the options list. So, in the case of quark declaration (see the QCD example) the first index is spinor, and the second one is color triplet. There are several functions taking particle name as an argument which can be used in algebraic expressions. These functions are replaced with auxiliary particle names, which are generated automatically. § Ghost field names in gauge theories are generated by the functions ghost(name) 'name .c' and ccghost(name) 'name .C' (see for instance Table 1). Here and below name is the name of the corresponding gauge boson. § Goldstone boson field name in the t'Ho oft-Feynman gauge is generated by the function gsb(name) 'name .f'. ï § The name for a charge conjugate spinor particle c = C T is generated by the function cc(name ) 'name .c'. Charge conjugate fermion has the same indices types and ordering, however index of spinor type is replaced by the index type cspinor and vice versa. § vev(expr ) is used in the Lagrangian for vacuum expectation values. Function vev ensures that deriv*vev(expr ) is zero. In other words, vev function forces LanHEP to treat expr as a scalar particle which will be replaced by expr in Feynman rules. § The function anti(name ) generates antiparticle name for the particle name.

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3.3

Specials

Besides parameters and particles other indexed such as -matrices, group structure constants, etc may appear in the Lagrangian terms. We refer such ob jects as specials. Predefined specials of the Lorentz group are: § gamma stands for the ² -matrices. It has three indices of spinor, cspinor and vector types. § gamma5 denotes 5 matrix. It has two indices of spinor and cspinor types. § moment has one index of vector type. At the stage of Feynman rules generation this symbol is replaced by the particle moment. § deriv is replaced by -ip² , where p² is the particle moment. It has one vector index. Note that deriv*A*B means ( A)B , not (AB ). § For the derivative of a pro duct, derivp(expr) can be used. derivp(A*B) is equivalent to deriv*A*B+A*deriv*B. Specials of the color S U (3) group are: § lambda denotes Gell-Mann -matrices. It has three indices: color c3, color c3b and color c8. § f SU3 is the S U (3) structure constant. It has three indices of color c8 type. § eps c3 and eps c3b are antisymmetric tensors. The first has 3 indices of the type color c3, and the second 3 color c3b indices. Note that for specials the order of indices types is fixed. Users can declare new specials with the help of a facility defined in Section 8 to intro duce user-defined indices types. 3.4 Let-substitutions

LanHEP allows the user to intro duce new symbols and then substitute them in Lagrangian terms by some expressions. Substitution has the generic form let name =expr. where name is the identifier of newly defined ob ject. The expression has the same structure as those in Lagrangian terms, however here expression may have free (non-contracted) indices. Typical example of using a substitution rule is a definition of the QED covariant derivative as let Deriv^mu=deriv^mu + i*ee*A^mu. corresponding to D² = ² + ige A² . More complicated example is the declaration
²

i( ² - ² )/2 matrices:

let sigma^a^b^mu^nu = i*(gamma^a^c^mu*gamma^c^b^nu - gamma^a^c^nu*gamma^c^b^mu)/2. Note that the order of indices types of new symbol is fixed by the declaration. So, first two indices of sigma after this declaration are spinor and antispinor, third and fourth are vector indices.

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3.5

Arrays

LanHEP allows to define components of indexed ob jects. In this case, contraction of indices will be performed as an explicit sum of pro ducts of the corresponding components. An ob ject with explicit components has to be written as {expr1, expr2 ..., exprN }^i where expressions correspond to components. All indices of components (if present) have to be written at the each component, and the index numbering components has to be written after closing curly bracket. Of course, all the components must have the same types of free (non-contracted) indices. Arrays are usually applied for the definition of multiplets and matrices corresponding to broken symmetries. Typical example of arrays usage is a declaration of electron-neutrino isospin doublet l1 (and antidoublet L1) let l1^a^I = { n1^a, e1^a}^I, L1^a^I = { N1^a, E1^a}^I. Here we suppose that n1 was declared as the spinor particle (neutrino), with the antiparticle name N1. Note that the functions anti and cc can be applied also to the multiplets, so the construction let l1^a^I = { n1^a, e1^a}^I, L1^a^I = anti(l1)^a^I. is possible. Matrices can be represented as arrays which have other arrays as components. However, it is more convenient to declare them with dummy indices, see Section 5.4 (the same is correct for multiplets also). It is possible also to use arrays directly in the Lagrangian, rather than only in the declaration of let-substitution. When LanHEP is launched, it has already declared some frequently used matrices. They are: § tau1, tau2, tau3 are -matrices 1 , 2 , 3 : 1 = 01 10 , 2 = 0 -i i0 , 3 = 1 0 0 -1 ;

§ taupm is a vector ( + , 3 , - );

§ taup and taum are matrices ‘ = ( 1 ‘ i 2 )/ 2;
123

§ tau is a vector = (1 , 2 , 3 );

§ eps is the antisymmetrical tensor (

= 1 );

§ Tau1, Tau2, Tau3 are generators of S U (2) group adjoint representation (3-dimensional analog of -matrices) T 1 , T 2 , T 3 with commutative relations [Ti , Tj ] = -iij k Tk : 0 -1 0 1 0 1 , T 1 = -1 2 0 10 § Taup and Taum corresponds to T
‘





0i 0 1 T 2 = -i 0 -i , 2 0i 0 = (T 1 ‘ iT 2 )/ 2;





10 0 0 ; T3 = 0 0 0 0 -1





§ Taupm is a vector T = (T + , T 3, T - ).
14


3.6

Two-component notation for fermions n using two-component notation for fermions. The confour-component notations is summarized by the following ï , ï = ï
T

It is possible to write the Lagrangia nection between two-component and relations: , c = = ï

,

ï c =

ï

T

.

If the user has declared a spinor particle p (with antiparticle P), the LanHEP notation for its components is: up(p) down(p) ï up(cc(p)) or up(P) ï down(cc(p)) or down(P) To use the four-vector ² one should use the statement ï special sigma:(spinor2,spinor2,vector). Note that the sigma ob ject is not defined by default, thus the above statement is required. It is possible also to use another name instead of sigma (this ob ject can be recognized by LanHEP by the types of its indices). LanHEP uses the following rules to convert the two-component fermions to four-component ones (we use PR,L = (1 ‘ 5 )/2): 1 2 1 2 ïï 1 2 ïï 1 2 1 2 ïï 1 2 1 2 1 2 ïï = = = = = = = = ï 1 PL 2 ï 1 PR 2 ïc PL 2 1 c ï 1 PR 2 c ïc 1 PL 2 c ïc 1 PR 2 ï1 PL c 2 ïc 1 PR 2 up(P1)*up(p2) down(P1)*down(p2) up(p1)*up(p2) down(P1)*down(P2) up(p1)*up(P2) down(p1)*down(P2) up(P1)*up(P2) down(p1)*down(p2) down(P1)*sigma*up(p2) down(p1)*sigma*up(P2) P1*(1-gamma5)/2*p2 P1*(1+gamma5)/2*p2 cc(p1)*(1-gamma5)/2*p2 P1*(1+gamma5)/2*cc(P2) cc(p1)*(1-gamma5)/2*cc(P2) cc(p1)*(1+gamma5)/2*cc(P2) P1*(1-gamma5)/2*cc(P2) cc(p1)*(1+gamma5)/2*p2

ï ï 1 ² 2 = 1 ² PL 2 ² c ïc 1 2 = 1 ² PL 2 ï

P1*gamma*(1-gamma5)/2*p2 cc(p1)*gamma*(1-gamma5)/2*cc(P2)

4

Lagrangian expressions

When all parameters and particles necessary for the intro duction of a (physical) mo del are declared, one can enter Lagrangian terms with the help of the lterm statement: lterm expr. Elementary ob jects of expression are integers, identifiers of parameters, particles, specials, let-substitutions, and arrays. These elementary ob jects can be combined by the usual arithmetic operators as § expr1+expr2 (addition), § expr1íexpr2 (subtraction), § expr1*expr2 (pro duct), § expr1/expr2 (fraction; here expr2 must be a pro duct of integers and parameters),
15


§ expr1**N (Nth power of expr1; N must be an integer). One can use brackets '(' and ')' to force the precedence of operators. Note, that indices can follow only elementary ob jects symbols, i.e. if A1 and A2 were declared as two vector particles then valid expression for their sum is A1^mu+A2^mu, rather than (A1+A2)^mu. 4.1 Where-substitutions

More general form of expressions involves where-substitutions: expr where subst. In the simple form subst is name=repl or several constructions of this type separated by comma ','. In the form of such kind each instance of identifier name in expr is replaced by repl. Note that in contrast to let-substitutions, where-substitution do es not create a new ob ject. LanHEP simply replaces name by repl, and then pro cesses the resulting expression. It means in particular that name can not have indices, although it can denote an ob ject with indices: lterm F**2 where F=deriv^mu*A^nu-deriv^nu*A^mu. is equivalent to lterm (deriv^mu*A^nu-deriv^nu*A^mu)**2. The substitution rule intro duced by the keyword where is active only within the current lterm statement. More general form of where-substitution allows to use several name =repl substitution rules separated by semicolon ';'. In this case expr will be replaced by the sum of expressions; each term in this sum is pro duced by applying one of the substitution rules from semicolon-separated list to the expression expr. This form is useful for writing the Lagrangian where many particles have a similar interaction. For example, if u,d,s,c,b,t are declared as quark names, their interaction with the gluon may read as lterm gg*anti(psi)*gamma*lambda*G*psi where psi=u; psi=d; psi=s; psi=c; psi=b; psi=t. The equivalent form is lterm gg*U*gamma*lambda*G*u + gg*D*gamma*lambda*G*d + gg*C*gamma*lambda*G*c + gg*S*gamma*lambda*G*s + gg*T*gamma*lambda*G*t + gg*B*gamma*lambda*G*b. Where-substitution can also be used in let statement. In this case one should use brackets: let lsub=(expr where wsub=expr1). 4.2 Adding Hermitian conjugate terms

It is possible to generate hermitian conjugate terms automatically by putting the symbol AddHermConj to lterm statement: lterm expr + AddHermConj. Continuing the former example, one can write: lterm a*H*H*h + b*H**3 + AddHermConj.

16


Note, that the symbol AddHermConj adds the hermitian conjugate expressions to all terms in the lterm statement. It means in particular that in the statement lterm expr1 + (expr2 + AddHermConj). the conjugate terms are added to both expr1 and expr2 . Thus, one should not place selfconjugate terms in the lterm statement where AddHermConj present (or one should supply these terms with 1/2 factor). 4.3 Using the superpotential formalism in the MSSM and its extensions particular, in the Minimal Supersymmetric Standard Mo del he superpotential -- a polynomial W depending on scalar fields defined in terms of superfields, we do not consider this case). to the Lagrangian: Yukawa terms in the form - 1 2 2W i j + H.c. Ai Aj

In supersymmetric mo dels (in (MSSM) [3]) one makes use of t Ai (superpotential also can be Then, there is the contribution

and Fi Fi terms, where Fi = W/ Ai (for more details, see [3] and references therein). To use this formalism in LanHEP, one should define first the multiplets of matter fields and then define the superpotential through the let-substitution statement. The example of the MSSM with a single generation may read: keep lets W. let W=eps*(mu*H1*H2+ml*H1*L*R+md*H1*Q*D+mu*H2*Q*U). Here symbols H1, H2, L, R, Q, U, D are defined somewhere else as doublets and singlets in terms of scalar particles. Note that before the definition of W this symbol should appear in the keep lets statement. It is necessary to notify LanHEP that let-substitutions (multiplets) at the definition of W should not be expanded. Without this statement, the representation used by LanHEP for W will not contain symbols of multiplets but only the particles which were used at multiplets definition. Since W was declared in the keep lets statement and contains the symbols of multiplets, one can evaluate the variational derivative of W by one or two multiplets, e.g. df(W,H1) or df(W,H1,L). Thus the Yukawa terms may be written: lterm - df(W,H1,H2)*fH1*fH2 - ... + AddHermConj.

Here fH1, fH2 are fermionic partners of corresponding multiplets. To intro duce the Fi Fi terms one needs to declare the conjugate superpotential, e.g. Wc, and write: lterm - df(W,H1)*df(Wc,H1c) - .... A better way is to use the function dfdfc(W,H1) instead: lterm - dfdfc(W,H1) - .... The function evaluates the variational derivative, multiply it by the conjugate expression and returns the result. Moreover, it can intro duce auxiliary fields to split vertices with 4 color particles (in the case of CompHEP output); see Section 9 for more details.

17


4.4

Generation of counterterms

Ultraviolet divergencies in renormalized quantum field theories are compensated by renormalization of wave functions i i (1 + zi ), masses Mi Mi + Mi , charges ei ei + ei , and may be some other Lagrangian constants. Such transformation of the Lagrangian assumes changes in the Feynman rules. In particular vertices are changed due to the appearence of new terms -- counterterms. It is possible to treat this transformation at 1-lo op level by means of LanHEP package. The statement transform obj -> expr. forces LanHEP to substitute symbol of a parameter or a particle, obj, by the expression expr when this ob ject appears in the Lagrangian expressions of the Lagrangian. For example, if the electric charge e in QED is renormalized than the statement transform e -> e+d e. makes LanHEP substitute each o ccurence of e symbol in further expressions by e+d e. Of course, e and d e should be declared as parameters before this statement. The transform statement is very similar to the let statement and uses the same syntax for expression expr. It has however two advantages. First, one do es not need to intro duce new name for transformed parameter or field. Second, if one has the LanHEP file for some physical mo del, it is enough to add counterterms statements into the input file just after the statements declaring parameters and particles. In 1-lo op approximation renormalization parameters appear in expressions only at first power, i.e. Zi Zj = 0, where Zi means all renormalization parameters ( ei , Mi , zi , ...). This should be done with the help of the following statement infinitesimal d Z1, d Z2, ..., d Zn. It declares that a monomial should be omitted if it contains more than one of parameters d Z1, d Z 2 , ..., d Z n . An example for QED mo del with renormalization reads as: parameter e = 0.3133: 'Electric charge'. vector A/A:photon. spinor e1/E1:(electron, mass me=0.000511). parameter d e, d me, d e1, d A. infinitesimal d e, d me, d e1, d A. transform e -> e*(1+d e), me -> me+d me, A -> A*(1+d A), e1 -> e1 + d e1*e1, E1 -> E1 + d e1*E1. lterm e*E1*(i*gamma*deriv+gamma*A+me)*e1. 4.5 Constructing the ghost Lagrangian

The ghost Lagrangian can be constructed by means of the BRST formalism. To use it, the user should first declare the BRST transformations for fields (see Section 6.4). The gauge-fixing Lagrangian reads as: L
GF

1 1 = G- G+ + | GZ | 2 + | G | 2 , 2 2

18


where Gi (i = ‘, , Z ) are gauge-fixing functions. The ghost Lagrangian ensures the BRST invariance of the entire Lagrangian and can be written as ([7]) L
Gh

= -ci B ï

RS T

Gi + B

RS T

~ L Gh .

~ where BRS T LGh is an overall function, which is BRST-invariant. So, for the photon and G = ( § A), the LanHEP co de for gauge-fixing and ghost terms reads as: let G A = deriv*A. lterm -G A**2/2. lterm -'A.C'*brst(G A). Here the brst function is used to get BRST-transformation of the specified expression. The inverse BRST transformation can also be used. One can declare the transformations for the fields by means of brsti transform. The function brsti(expr) can be used in lterm statements.

5

Omitting indices

Physicists usually do not write all possible indices in the Lagrangian terms. LanHEP also allows a user to omit indices. This feature can simplify the intro duction of the expressions and makes them more readable. Compare two possible forms: lterm E1^a*gamma^a^b^mu*A^mu*e1^b.
² corresponding to ge ea (x)ab eb (x)A² (x), and ï

lterm E1*gamma^mu*e1*A^mu. corresponding to ge e(x) ² e(x)A² (x)). Furthermore, while physicists usually write vector indices ï explicitly in the formulas, in LanHEP vector indices also can be omitted: lterm -i*ee*E1*gamma*e1*A. Generally speaking, when the user omits indices in the expressions, LanHEP faces two problems: which indices were omitted and how to contract indices. 5.1 Restoring the omitted indices

When the indexed ob ject is declared the corresponding set of indices is assumed. Thus, if the quark q is declared as spinor q:('some quark', color c3). its first index is spinor and the second one belongs to the color c3 type. If both indices are omitted in some expression, LanHEP generates them in the correspondence to order (spinor, color c3). However, if only one index is written, for example in the form q^a, LanHEP has to recognize whether the index a is of color c3 or of spinor types. To solve this problem LanHEP lo oks up the list of indices omitting order. By default this list is set to [spinor, color c3, color c8, vector]

19


The algorithm to restore omitted indices is the following. First, LanHEP assumes that user has omitted indices which belong to the first type (and corresponding antitype) from this list. Continuing the consideration of our example with particle q one can see that since this particle is declared having one spinor index (the first type in the list) LanHEP checks whether the number of indices declared for this ob ject without spinor index equals the number of indices written explicitly by user. In our example (when user has written q^a) this is true. In the following LanHEP concludes that the user omitted the spinor index and that explicitly written index is of color c3 type. In other cases, when the supposition fails if the user has omitted indices of the first type in the list of indices omitting order, LanHEP go es to the second step. It assumes that user has omitted indices of first two types from this list. If this assumption also fails, LanHEP assumes that the user has omitted indices of first three types in the list and so on. At each step LanHEP subtracts the number of indices of these types assumed to be omitted from the full number of indices declared for the ob ject, and checks whether this number of resting indices equals the number of explicitly written indices. If LanHEP fails when the list of indices omitting order is completed, error message is returned by the program. Note that if the user would like to omit indices of some type, he must omit all indices of this type (and antitype) as well as the indices of all types which precede in the list of indices omitting order. For example, if ob ject Y is declared with one spinor, two vector and three color c8 indices, than § the form Y^a^b^c^d^e^f means that the user wrote all the indices explicitly; § the form Y^a^b^c^d^e means that the user omitted spinor index and wrote vector and color c8 ones; § the form Y^a^b means that the user omitted spinor and color c8 indices and wrote only two vector ones; § the form Y means that the user omitted all indices; § all other forms, involving different number of written indices, are incorrect. One can say that indices should be omitted in the direct correspondence with abrupting the list of indices omitting order from left to right. One could change the list of indices omitting order types by the statement SetDefIndex. For example, for default setting it lo oks like SetDefIndex(spinor, color c3, color c8, vector). Each argument in the list is a type of index. 5.2 Contraction of restored indices

A dummy index can be contracted only with some another dummy index. LanHEP expands the expression and restores indices in each monomial. LanHEP reads ob jects in the monomial from the left to the right and checks whether the restored indices are present. If such index appears LanHEP seeks for the restored index of the appropriate type at the next ob jects. Note, that the program do es not check whether the ob ject with the first restored index has another restored index of the appropriate type. Thus, if F is declared as let-substitution for the strength tensor of electromagnetic field (with two vector indices) then expression F*F (as well as F**2) after pro cessing dummy indices turns to the implied form F^mu^nu*F^mu^nu rather than F^mu^mu*F^nu^nu.

20


This algorithm makes the contractions to be sensitive to the order of ob jects in the monomial. Let us lo ok again at the QED example. Expression E1*gamma*A*e1 (as well as A*e1*gamma*E1) leads to correct result where the vector index of photon is contracted with the same index of -matrix, spinor index of electron is contracted with antispinor index of -matrix and antispinor index of positron is contracted with spinor index of -matrix. However the expression E1*e1*A*gamma leads to the wrong form E1^a*e1^a*A^mu*gamma^b^c^mu, because the first antispinor index after the electron belongs to positron. Spinor indices of gamma stay free (noncontracted) since no more ob jects with appropriate indices (so, LanHEP will report an error since Lagrangian term is not a scalar). Note, that in the vertex with two -matrices the situation is more ambiguous. Let's lo ok at the term corresponding to the electron anomalous magnetic moment e(x)( ² - ² )e(x)F² . ï The correct LanHEP expression is e1*(gamma^mu*gamma^nu - gamma^nu*gamma^mu)*E1*F^mu^nu Here vector indices can't be omitted, since it lead to the contraction of vector indices of matrices. One can see also that the form e1*E1*(gamma^mu*gamma^nu - gamma^nu*gamma^mu)*F^mu^nu will correspond to the expression e(x)e(x)S p( ² )F² . Here LanHEP has got scalar Lorentzï invariant expression in the Lagrangian term, so it has no reason to report an error. These examples mean that the user should clearly realize how the indices will be restored and contracted, or (s)he has to write all indices explicitly. 5.3 Let-substitutions

Another problem arises when the dummy indices stay free, this is the case for the let statement. LanHEP allows only two ways to avoid ambiguity in the order of indices types: either user specifies all the indices at the name of new symbol and free indices in the corresponding expression, or he should omit all free indices. In the latter case the order of indices types is defined following the order of free dummy indices in the first monomial of the expression. For example if A1 and A2 are vectors and c1 and c2 are spinors, the statement let d=A1*c1+c2*A2. declares a new ob ject d with two indices, the first is a vector index and the second is a spinor index according to their order in the monomial A1*c1. Of course, each monomial in the expression must have the same types of free indices. 5.4 Arrays

The usage of arrays with dummy indices allows us to define matrices conveniently. For example, the declaration of -matrices (defined in Section 3.5) can be written as let tau1 = {{0, 1}, { 1, 0}}. let tau2 = {{0, i}, {-i, 0}}. let tau3 = {{1, 0}, { 0,-1}}. One can see that in such way of declaration a matrix is written "column by column". The declaration of ob jects with three `explicit' indices can be done using the ob jects already defined. For example, when -matrices are defined as before, it is easy to define the vector (1 , 2 , 3 ) as let tau = {tau1, tau2, tau3}.
21


The ob ject tau has three indices, first pair selects the element of the matrix, while the matrix itself is selected by the third index, i.e. tau^i^j^a corresponds to ia . j On the other hand, the declaration of structure constants of a group is more complicated. Declaring such an ob ject one should bear in mind that omitting indices implies that in a sequence of components the second index of an ob ject is changed after the full cycle of the first index, the third index is changed after the full cycle of the second one, etc. For example, a declaration of the antisymmetrical tensor abc can read as let eps = {{{0,0,0}, {0,0,-1}, {0,1,0}}, {{0,0,1}, {0,0,0}, {-1,0,0}}, {{0,-1,0}, {1,0,0}, {0,0,0}}}.
111

One can easily see that the components are listed here in the following order: ,
211

,

311

,

121

,

221

,

321

,

131

, ...

233

,

333

.

The declaration of more complex ob jects such as S U (3) structure constants can be made in the same way.

6
6.1

Checking the correctness of the mo del
Checking electric charge conservation

LanHEP can check whether the intro duced vertices satisfy electric charge conservation law. This option is available, if the user declares some parameter to denote elementary electric charge (say, ee in QED example), and than indicate, which particle is a photon by the statement SetEM(photon, param). So, in example of Section 1 this statement could be SetEM(A,ee). Electric charge of each particle is determined by analyzing its interaction with the photon. LanHEP checks whether the sum of electric charges of particles in each vertex equals zero. 6.2 Testing Hermitian conjugate terms

LanHEP is able to check the correctness of the hermitian conjugate terms in the Lagrangian. To do this, user should use the statement: CheckHerm. For example, let us consider the following (physically meaningless) input file, where a charged scalar field is declared and cubic terms are intro duced: scalar h/H. parameter a,b,c. lterm a*(h*h*H+H*H*h)+b*h*h*H+c*H*H*h + h**3. CheckHerm. The output of LanHEP is the following (here 2 is the symmetry factor): CheckHerm: vertex (h, h, h): conjugate (H, H, H) not found. CheckHerm: inconsistent conjugate vertices: (H, h, h) (H, H, h) 2*a <-> 2*a 2*b <-> (not found) (not found) <-> 2*c
22


If the conjugate vertex is not found, a warning message is printed. If both vertices are present but they are inconsistent, more detailed output is provided. For each couple of the incorrect vertices LanHEP outputs 3 kinds of monomials: 1) those which are found in both vertices (these monomials are correctly conjugated), 2) the monomials found only in first vertex, and 3) the monomials found only in the second vertex. 6.3 Probing the kinetic and mass terms of the Lagrangian

When LanHEP is used to generate CompHEP mo del files, the information about particles propagators is taken from the particle declaration, where particle mass and width (for BreitWigner's propagators) are provided. Thus, the user is not obliged to supply kinetic and mass terms in lterm statements. Even if these terms are written, they do not affect the CompHEP output. LanHEP allows user to examine whether the mass sector of the Lagrangian is consistent. To do this, use the statement CheckMasses. This statement must be put after all lterm statements of the input file. When the CheckMasses statement is used, LanHEP creates the file named masses.chk (in the directory where LanHEP was started). This file contains warning messages if some inconsistencies are found: § missing or incorrect kinetic term; § mass terms with mass different from the value specified at the particle declaration; § off-diagonal mass terms. Note that in more involved mo dels like the MSSM the masses could depend on other parameters and LanHEP is not able to prove that expressions in actual mass matrix and in parameters declared to be the masses of particles are identical. Moreover, it is often impossible to express the masses as formulas written in terms of independent parameters. For this reason LanHEP evaluates the expressions appearing in the mass sector numerically (based on the parameter values specified by the user). It is typical for the MSSM that some fields are rotated by unitary matrices to diagonalize mass terms. In some cases, values of mixing matrices elements can not be expressed by formulas and need to be evaluated numerically. LanHEP can recognize the mixing matrix if the elements of this matrix were used in OrthMatrix statement. LanHEP restores and prints the mass original matrix before fields rotation by mixing matrix. 6.4 Checking BRST invariance

LanHEP can check the BRST invariance (see the reviews in [5, 6]) of the Lagrangian. First, the user should declare the BRST transformations for the fields in the mo del by means of brst transform statement: brst transform field -> expression.
+ - For example, the BRST transformation for the photon A² = ² cA + ie(W² c- - W² c+ ) can be prescribed by the statement:

brst transform A -> deriv*'A.c'+i*EE*('W+'*'W-.c'-'W-'*'W+.c'). The file 'sm brst.mdl' in LanHEP distribution contains the co de for gauge and Higgs fields transformation corresponding to the CompHEP implementation of the Standard mo del. Since the transformations are defined, the statement
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CheckBRST. enables the BRST transf lterm statement). The correct, the output files simplified and remain in 6.5 ormation for Lagrangian terms (so it should be placed before the first output is CompHEP or LaTeX file. Certainly, if the Lagrangian is are empty. However some expressions identical to zero could be not the output.

Extracting vertices

New statement allows to extract a class of vertices into separate file. This feature is useful in checking large mo dels with thousands of vertices, such as the MSSM. The statement has the f o rm SelectVertices(file, vlist ). Here file is a file name to output selected vertices, and vlist determines the class of vertices. The pattern vlist may have two possible formats. In the first format vlist is a list of particles: [P1, P2, P3, ... Pn ] All vertices consisting only of the listed particles P1, P2,... are extracted. For example, the pattern [A, Z, 'W+', 'W-', e1, E1, n1, N1] selects vertices of leptons of first generation interaction with gauge fields and the self-interaction of gauge fields (we used here particles notation of CompHEP). The second format reads [P11/P12../P1n, P21/.../P2m, ... ] Here are several groups of particles, each group is joined by slash '/' symbol. Selected vertices have the number of legs equal to the number of groups in the list, each leg in the vertex corresponds to one group. For example, [e1/E1/n1/N1, e1/E1/n1/N1, A/Z/'W+'/'W-'] selects vertices with two legs corresponding to leptons, third leg is gauge boson. Thus, this pattern extracts vertices of leptons interactions with gauge bosons. Note that the vertices of gauge bosons interaction are not selected in this case. If a group consists of one particle, it must be typed twice to enable second format (e.g. [P1/P1, P2/P2, P3/P3] for the vertex consisting of particles P1, P2, P3. It is possible to specify some options in SelectVertices statement. The WithAnti option adds antiparticle names for each particle in the list (in the first format) or in the each group (in the second format. The Delete option removes selected vertices from the LanHEP's internal vertices list, so these vertices are not written in output file lgrngN.mdl (lgrngN.tex). This could be useful if SelectVertices statements are supposed to distribute all vertices which should be in the mo del, than some 'unexpected' vertices can be found in output lgrngN.mdl file. An example of lepton-gauge vertices selection could read as SelectVertices( 'lept-gauge.mdl', [e1/n1, e1/n1, A/Z/'W+'], WithAnti, Delete).

24


7
7.1

Simplifying the expression for vertices
Orthogonal matrices

If some parameters appear to be the elements of the orthogonal matrix such as quark mixing Cabbibo-Kobayashi-Maskava matrix, one should declare them by the statement OrthMatrix( {{a11 , a12 , a13 }, {a21 , a22 , a23 }, {a31 , a32 , a33 }} ). where aij denote the parameters. Such a declaration permits LanHEP to reduce expressions which contain these parameters by taking into account the properties of the orthogonal matrices. Note that this statement has no relation to the arrays; it just declares that these parameters aij satisfy the correspondent relations. Of course, one can declare further a matrix with these parameters as components by means of the let statement. 7.2 Reducing trigonometric expressions

Some physical mo dels, such as the MSSM and general Two Higgs Doublet Mo del [4], involve large expressions built up of trigonometric functions. In particlular, after the diagonalization of the Lagrangian in the Higgs sector, in the mo dels mentioned above two angles, and , are intro duced, thus the Lagrangian may be written in LanHEP notation using the following definitions: parameter sa=0.5:'sinus alpha', ca=Sqrt(1-sa**2):'cosine alpha'. parameter sb=0.9:'sinus beta', cb=Sqrt(1-sb**2):'cosine beta'. One can find in the output expressions like sa*cb+ca*sb = sin( + ) and much more complicated ones. To simplify the output and make it more readable the user can define new parameters, for example: parameter sapb=sa*cb+ca*sb:'sin(a+b)'. In order to force LanHEP to substitute the expression sa*cb+ca*sb by the parameter sapb, one should use the statement SetAngle(sa*cb+ca*sb=sapb). It is possible also to substitute an expressions by any polynomial involving paramters, for example, SetAngle((sa*cb+ca*sb)**2+(sa*cb-ca*sb)**2=sapb**2+samb**2). where samb should be previously declared as a parameter. Note, that LanHEP expands both expressions before setting the substitution rule. The left-hand side expression can also contain symbols defined by let statement. In the Lagrangian expressions LanHEP expresses powers of cosines, ca**N and cb**N through sines using recursively the formula cosN = cosN -2 (1 - sin2 ) to combine all similar terms. The same pro cedure is performed for the left-hand side expression in SetAngle statement. If the substitution for one angle combination is defined, LanHEP offers to define the other ones for expressions consisting of parameters used in previous SetAngle statements. In our example, for all expressions consisting of parameters ca, cb, sa, sb, the following message is printed (e.g. for cos( + )): Warning: undefined angle combination; use: SetAngle(ca*cb-sa*sb=aa000).
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Here aa000 is an automatically generated parameter. The message is printed only once for each expression. This feature is disabled by default; to enable pro ducing these messages, one should issue the statement option UndefAngleComb=1. 7.3 Heuristics for trigonometric expressions

LanHEP can apply several heuristic algorithms to simplify these expressions. For each angle , the user should declare parameters for sin , cos , sin 2, cos 2, tan . Than one should use the angle statement: angle sin=p1, cos=p2, sin2=p3, cos2=p4, tan=p5, texname=name. Here pN -- parameter identifiers, name -- LaTeX name for angle, it is used to generate automatically LaTeX names for trigonometric functions of this angle if these names are not set explicitly by SetTexName statement. This statement should immediately follow the declaration of the parameters for sin and cos . Only the sin and cos options are obligatory. Other parameters (i.e. sin 2, cos 2, tan ) should be declared if these parameters are defined before sin and cos . If the former parameters will be declared in terms of latter ones, they are recognized automatically and they need not appear in angle statements. For example, the declaration for trigonometric functions of angle in MSSM may read: parameter tb=2.52:'Tangent beta'. parameter sb=tb/Sqrt(1+tb**2):'Sinus beta'. parameter cb=Sqrt(1-sb**2):'Cosine beta'. angle sin=sb, cos=cb, tan=tb, texname='\\beta'. parameter s2b=2*sb*cb:'Sinus 2 beta'. parameter c2b=cb**2-sb**2:'Cosine 2 beta'.

Here the parameters s2b and c2b are recognized automatically by LanHEP as sin 2 and cos 2 since they are declared in terms of sa, ca parameters. For a couple of angles ( , in this example) the user should declare the parameters for sin( + ), cos( + ), sin( - ), and cos( - ) to allow using all implemented heuristics: parameter parameter parameter parameter sapb=sa*cb+ca*sb samb=sa*cb-ca*sb capb=ca*cb-sa*sb camb=ca*cb+sa*sb : : : : 'sin(a+b)'. 'sin(a-b)'. 'cos(a+b)'. 'cos(a-b)'.

These parameters are recognized as trigonometric functions automatically, by analysis of the right-side expressions. It is possible to control the usage of heuristics by the statement: option SmartAngleComb=N. where N is a number: 0 heuristics are switched off; 1 heuristics are switched on (by default); 2 same as 1, prints the generated substitution rule if the simplified expressions consists of more than 3 monomials; 3 same as 1, prints all generated substitution rule.
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4 same as 1, prints all generated substitution rule and some intermediate expressions (debug mo de). The substitution rules are printed as SetAngle statement and can be used for manual improvement of expressions. 7.4 LaTeX output

LanHEP generates LaTeX output instead of CompHEP mo del files if the user set -tex in the command line to start LanHEP. Three files are pro duced: 'varsN .tex', 'prtclsN .tex' and 'lgrngN .tex'. The first file contains names of parameter used in physical model and their values. The second file describes the particles, together with propagators derived from the vertices. The last file lists intro duced vertices. LanHEP uses Greek letters ², , ... for vector indices, letters a, b, c... for spinor ones and p, q , r ... for color indices (and for indices of other groups, if they were defined). It is possible to inscribe names for particles and parameters to use them in LaTeX output. It can be done by the statement SetTexName([ ident=texname, ... ]). Here ident is an identifier of particle or parameter, and texname is string constant containing LaTeX command. Note, that for intro ducing backslash '\' in quoted string constant one should type it twice: '\\'. For example, if one has declared neutrino with name n1 (and name for antineutrino N1) than the statement SetTexName([n1='\\nu^e', N1='\\bar{\\nu}^e']). makes LanHEP to use symbols e and e for neutrino and antineutrino in LaTeX tables. ï

8
8.1

Declaration of new index typ es and indexed ob jects
Declaring new groups

Index type is defined by two keywords1 : group name and representation name. Thus, color triplet index type color c3 has group name color and representation name c3. LanHEP allows user to intro duce new group names by the group statement: group gname. Here gname is a string constant, which becomes the name of newly declared group. Representation names for each group name must be declared by the statement repres gname:(rlist) where rlist is a comma-separated list of representation names declaration for the already declared group name gname. Each such declaration has the form either rname or rname/crname. In the first case index which belongs to the gname rname type can be contracted with another index of the same type; in the second case index of gname rname type can be contracted only with an index of gname crname type. For example, definition for color S U (3) group with fundamental, conjugate fundamental and adjoint representations lo oks like:
1

The exception is Lorentz group, corresponding indices types are defined by a single keyword.

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group color:SU(3). repres color:(c3/c3b,c8). So, three index types can be used: color c3, color c3b, color c8. The contraction of these indices is allowed by pairs (color c3, color c3b) and (color c8, color c8) indices. 8.2 Declaring new specials

Specials with indices of user-defined types can be declared by means of special statement: special name :(ilist). Here name is the name of new symbol, and ilist is a comma-separated list of indices types. For example, Gell-Mann matrices can be defined as (although color group and its indices types are already defined): special lambda:(color c3, color c3b, color c8). To define Dirac's -matrices one can use the command special gamma:(spinor, cspinor, vector). 8.3 Arrays

Array, i.e. the ob ject with explicit components, can also have the user-defined type of index. In this case generic form of such ob ject is { expr1, expr2 ... ,exprN ; itype } where N expressions expr1 ... exprN of N components are separated by comma, and itype is an optional index type. If itype is omitted LanHEP uses default group name wild and index type wild N, where N is a number of components in the array.

9

Splitting the vertices with 4 colored particles

The CompHEP Lagrangian tables do not describe explicitly the color structure of a vertex. If color particles are present in the vertex, the following implicit contractions are assumed (supposing p, q , r are color indices of particles in the vertex): § pq for two color particles A1 , A2 ; p q § §f
r pq pq r

for three particles, which are color triplet, antitriplet and o ctet; for three color o ctets.

Other color structures are forbidden in CompHEP. So, to intro duce the 4-gluon vertex f pqr Gq Gr f pstGs Gt one should split this 4-legs vertex ² ² p q into 3-legs vertices f pqr G² Gr X² :
G G G G G G X G G + G G X G G + G G X G G

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p Here the field X² is a Lorenz tensor and color o ctet, and this field has constant propagator. If gluon name in CompHEP is 'G', the name 'G.t' is used for this tensor particle; its indices denoted as 'm ' and 'M ' (' ' is the number of the particle in table item). The described transformation is performed by LanHEP automatically and transparently for the user. Each vertex containing 4 color particles is split to 2 vertices which are joined by an automatically generated auxiliary field. The same technique is applied in the MSSM where more vertices with 4 color particles appear: vertices with 2 gluons and 2 squarks and vertices with 4 squarks. However, the large amount of vertices with 4 squarks requires many auxiliary fields, which can easily break CompHEP limitations on the particles number. It is possible however to reduce significantly the number of vertices and auxiliary fields if one intro duce auxiliary fields at the level of multiplets. The vertices with 4 squarks come from D D and F F terms. For example, there is the term 1a D D a in the Lagrangian, 2GG a DG = gs (Q a Qi + D a D + D a D ), i

where Q, D , U are squarks multiplets, and is Gell-Mann matrix. Instead of evaluating this expression and that splitting all vertices independently, one can intro duce one color o ctet a auxiliary field a and write this Lagrangian term as DG a . i i Other D D terms contain both color and colorless particles. Thus, the term DA DA with
i DA = g1 (Q T i Q + L T i L + H1 T i H1 + H2 T i H2 ),

can be represented as
2 2 g1 (Q T i Q) i + g1 (Q T i Q)(L T i L + H1 T i H1 + H2 T i H2 ) + g1 (L T i L + H1 T i H1 + H2 T iH2 ) 2

where i is the triplet of auxiliary fields. This terms can also be written in the another form:
2 2 g1 (Q T i Q + L T i L) i + g1 (Q T i Q + L T i L)(H1 T iH1 + H2 T i H2 ) + g1 (H1 T i H1 + H2 T i H2 )2 ,

where all vertices with 4 scalars (except vertices with Higgs particles) are splitted. Although the latter splitting is not obligatory, it can reduce significantly the amount of vertices. The similar technique is applicable to the F F terms, with the transformation Fi Fi Fi i + Fi i . Thus, we distinguish two types of vertices splitting: splitting at multiplet level and splitting at vertices level. Note that splitting the vertices with two gluons and two squarks must be done at vertices level after combining the similar terms, otherwise they would contain the elements of squark mixing matrices. The vertices splitting at multiplet level is implemented in LanHEP mainly for MSSM needs. The first case refers to D D terms. The user should declare several let-substitutions and then put in lterm statement the squared sum: let a1=g*Q*tau*q/2, a2=g*L*tau*l/2, a3=g*H1*tau*h1/2, a4=g*H2*tau*h2/2. lterm - ( a1 + a2 + a3 + a4 ) ** 2 / 2. In this case LanHEP lo oks for the square of the sum of several let-substitution symbols, each containing two color or merely scalar particles. If such expression is found, it is replaced as in the previous formulas. The vertices splitting in F F terms is performed by dfdfc function (see previous section). After taking the variational derivative the monomials with two color or scalar particles (except
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Higgs ones) are multiplied by auxiliary fields, thus mediating the vertices with 4 color (scalar) particles. The multiplet level vertices splitting is controlled by the statement option SplitCol1=N. where N is a number: -1 remove all vertices with 4 color particles from Lagrangian; 0 turn off multiplet level vertices splitting; 1 allows vertices splitting with 4 color multiplets; 2 allows vertices splitting with any 4 scalar multiplets except Higgs ones (more generally, any multiplets containing vev's). The value of this option can be set to different values before executing different lterm statements. The vertices level splitting is performed after combining similar terms of the Lagrangian. This splitting can be controlled by the statement option SplitCol2=N. where N is a number: 0 disable vertex level splitting; 1 enable vertex level splitting (only for vertices with 4 color particles). For CompHEP output, the default value is 2 for SplitCol1 and 1 for SplitCol2. For LaTeX output, default value is 0 for both options.

10

Installation

To install LanHEP on your computer, you should get the archive file from the WWW-page http://theory.sinp.msu.ru/~semenov/lanhep.html. Unpack the archive: gunzip lhep2000.tar.gz tar xf lhep2000.tar The archive contains the directory lanhep with C source files. If you do not have gcc compiler, you should tune the makefile, change 'gcc' to the compiler which you have. To create the executable file (called lhep), go to lanhep directory and type make When LanHEP is complied, remove the source files by typing make clean The archive also contains the subdirectory mdl with startup file and examples for several physical mo dels. Add the directory containing LanHEP to your PATH environment variable. Then LanHEP can be started from any other directory, it can find automatically files in the mdl directory.

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11

Running LanHEP and the command line options

As mentioned above, LanHEP can read the mo del description from the input file prepared by the user. To start LanHEP write the command lhep filename options where the possible options are described in the next section. If the filename is omitted, LanHEP prints it's prompt and waits for the keyboard input. In the last case, user's input is copied into the file lhep.log and can be inspected in the following. To finish the work with LanHEP, type 'quit.' or simply press ^D (or ^Z at MS DOS computers). 11.1 Options

Possible options, which can be used in the command line to start LanHEP are: -OutDir directory Set the directory where output files will be placed. -InDir directory Set the default directory where to search files, which included by read and use statements. -tex LanHEP generates LaTeX files instead of CompHEP mo del tables. -frc If -tex option is set, forces LanHEP to split 4-fermion and 4-color vertices just as it is made for CompHEP files. -texLines num Set number of lines in LaTeX tables to num. After the specified number of lines, LanHEP continues writing current table on the next page of LaTeX the output. Default value is 40. -texLineLength num Controls width of the Lagrangian table. Default value is 35, user can vary table width by changing this parameter. -nocdot removes \cdot commands between the parameters symbols in the LaTeX output. This option is useful when all parameters have prescribed LaTeX names. -c4 pro duces the output files for CompHEP version 4x.xx, which has slightly different format than 3x.xx. -allvrt pro duces CompHEP-like ouput with all vertices of the Lagrangian, including 1leg and 2-leg ones which contribute to the mass matrix. Vertices with more than 4 legs are also in the output (this is not supported by CompHEP now). -evl num For the lengthly expression in a vertex (more then num monomials) an auxiliary variable is generated, and the expression is stored in the parameters table. This allows to increase significantly the speed of symbolic calculation of the matrix element by CompHEP. num=2 is recommended.

Acknowlegements
This work was supported in part by the CERNíINTAS grant 99í0377 and by RFFR grant 01-02-16710. Author thanks A.Belyaev, G.B‡langer, F.Boudjema, F.Cuypers, M.Dubinin, e A.Gladyshev, V.Ilyin for fruitful discussions, valuable proposals, and testing the program.

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References
[1] E. Bo os et al., INP MSUí94í36/358, SNUTP 94-116 (Seoul, 1994); hep-ph/9503280 P. Baikov et al., Physical Results by means of CompHEP, in Proc.of X Workshop on High Energy Physics and Quantum Field Theory (QFTHEP-95), ed.by B.Levtchenko, V.Savrin, Moscow, 1996, p.101 , hep-ph/9701412 A.Pukhov, E.Bo os, M.Dubinin, V.Edneral, V.Ilyin, D.Kovalenko, A.Kryukov, V.Savrin, S.Shichanin, and A.Semenov. "CompHEP - a package for evaluation of Feynman diagrams and integration over multi-particle phase space. User's manual for version 33." Preprint INP MSU 98-41/542; hep-ph/9908288. [2] A. Semenov. LanHEP -- a package for automatic generation of Feynman rules. User's manual. INP MSU 96í24/431, Moscow, 1996; hep-ph/9608488 A. Semenov. Nucl.Inst.&Meth. A393 (1997) p. 293. A. Semenov. LanHEP - a package for automatic generation of Feynman rules from the Lagrangian. Updated version 1.3. INP MSU Preprint 98í2/503. [3] J. Rosiek. Phys. Rev. D41 (1990) p. 3464. [4] J.F. Gunion, H.E. Haber, G. Kane and S. Dawson. The Higgs Hunter Guide, AddisonWesley, 1990. [5] T. Kugo, I. Ojima. Phys. Lett. 73B (1978) 459. [6] L. Baulieu. Phys. Rep. 129 (1985) 1í74. [7] F. Boudjema, private communication.

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Contents
1 Get 1.1 1.2 1.3 2 Str 2.1 2.2 2.3 2.4 3 Ob 3.1 3.2 3.3 3.4 3.5 3.6 ting started QED . . . . QCD . . . . Higgs sector w . . of ith .. .. the LanHEP ................................... ................................... Standard Mo del . . . . . . . . . . . . . . . . . . . . . . . . . file ... ... ... he mo .. .. .. del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . it . . . . . . . . . . . . . . . . . . . . . . . . . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 6 7 7 9 9 10 10 11 11 13 13 14 15 15 16 16 17 18 18 19 19 20 21 21 22 22 22 23 23 24 25 25 25 26 27 27 27 28 28

ucture of LanHEP input Constants and identifiers . Comments . . . . . . . . . Including files . . . . . . . Conditional pro cessing of t jects in the expressions Parameters . . . . . . . Particles . . . . . . . . . Specials . . . . . . . . . Let-substitutions . . . . Arrays . . . . . . . . . . Two-component notation

for .. .. .. .. .. for

Lagrangian ........ ........ ........ ........ ........ fermions . . . . . in . . . . . . . . t . . . . . . . . he . . . . . . . . he . . . . . .

ter .. .. .. .. .. ..

ms .. .. .. .. .. .. . . a . . . . . . . . ng . . . . . .

4 Lagr 4.1 4.2 4.3 4.4 4.5

angian expressions Where-substitutions . . . . . . . . Adding Hermitian conjugate terms Using the superpotential formalism Generation of counterterms . . . . Constructing the ghost Lagrangian . . . . . . . . . . . .

.... .... MSSM .... .... . . . . . . . . . . . . . . g . . . . . . . . . . . . ra . . . . . .

.. .. nd .. .. . . . . . . . . . . n . . . . . .

.... .... extensio .... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.. .. ns .. .. . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Omitting indices 5.1 Restoring the omitted indices 5.2 Contraction of restored indices 5.3 Let-substitutions . . . . . . . 5.4 Arrays . . . . . . . . . . . . .

6 Checking the correctness of the model 6.1 Checking electric charge conservation . 6.2 Testing Hermitian conjugate terms . . 6.3 Probing the kinetic and mass terms of t 6.4 Checking BRST invariance . . . . . . 6.5 Extracting vertices . . . . . . . . . . . 7 Simplifying the expression for vertices 7.1 Orthogonal matrices . . . . . . . . . . 7.2 Reducing trigonometric expressions . . 7.3 Heuristics for trigonometric expressions 7.4 LaTeX output . . . . . . . . . . . . . . 8 Dec 8.1 8.2 8.3 laration of new index Declaring new groups . Declaring new specials Arrays . . . . . . . . . types .... .... .... and ... ... ...

.. .. La .. .. . . . . . . . .

. . ia . . . . . .

indexed ..... ..... .....

ob .. .. ..

jec .. .. ..

ts ................ ................ ................

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9 Splitting the vertices with 4 colored particles 10 Installation

28 30

11 Running LanHEP and the command line options 31 11.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

34