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An Analytical Comparison of Different Rotation Methods of Anomalous U(1) Charges Kara Merfeld, Caleb Smith

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An Analytical Comparison of Different Rotation Methods of Anomalous U(1) Charges Kara Merfeld, Caleb Smith 1 An Analytical Comparison of Different Rotation Methods of Anomalous U(1) Charges Kara Merfeld, Caleb Smith Abstract—Heterotic string theories offer the pos- a string. As the quarks are pulled apart, the string sibility of additional gauge groups arising from the connecting them is stretched and the attractive force symmetry breaking of SO(32) or E8 × E8. These increases. At some point, the string would break arise from compactifying the ten space-time dimen- sions of superstring theory to arrive at the four large and two more quarks would come into existence, space-time dimensions. This study focuses on produc- one at each of the newly created ends[12]. This ing and analyzing string models using the weakly- would account for the absence of individual quarks. coupled free fermionic heterotic string (WCFFHS) When quantum chromodynamics (QCD) came into formalism. Anomalous U(1) gauge groups naturally existence it overshadowed string theory through its arise in models generated by the WCFFHS formalism. Phenomenological analysis and supersymmetry calcu- experimentally confirmed success in modeling the lations are greatly simplified by transforming the U(1) strong force. However, when it was discovered that charges such that only one U(1) is anomalous. Multiple string theory inherently contains gravity, the theory algorithms are derived and presented performing this began to gain support as a grand unifying theory[6]. operation. One of these methods was selected and It would include all of the fundamental particles and implemented into the WCFFHS model building frame- work. Preliminary results of applying this calculation interactions as the result of a single phenomenon: to three generated models are included. different vibrational modes of strings[8]. By the 1980’s there were five distinct string theories, and Index Terms—Anomaly, Rotation Matrices, Normal- 500 ization, Heterotic, WCFFHS formalism there now exist over 10 string models[14]. It is not a question of which of these models is correct, but rather which models most readily reproduce the I. INTRODUCTION SM. These string models include gravity, allow for A. String Theory dark energy, and depend on only one free parameter: The standard model (SM) of particle physics the length of the strings. These attributes make it an is a collection of the fundamental particles that appealing theory, despite its current inability to be combine together to describe all matter and forces. experimentally verified. The SM contains six quarks and six anti-quarks (not including color), six leptons and six anti-leptons, B. Compactification and twelve gauge bosons. There is also a Higgs bo- Of the five string theories, type I, type IIA, type son, which introduces particle masses in the theory IIB, heterotic E8 × E8 and heterotic SO(32), the through electroweak symmetry breaking. However, focus of this study is on heterotic strings. Het- the SM is incomplete, as it does not account for erotic strings include only closed-string models, in dark matter/energy, and offers no description of which the left-moving modes are supersymmetric, gravity[13]. It also gives no explanation for the and the right-moving modes are bosonic[9]. The wide range of particle masses between different right-movers exist in 26 dimensions, while the left- generations (the hierarchy problem)[7]. Finally, it movers exist in 10. The extra 16 dimensions are possesses about 20 free parameters which must be toroidally compactified, making them internal de- experimentally, rather than theoretically, determined. grees of freedom[11]. In order to make contact with An underlying theory is desired. realistic physics, the remaining dimensions of the String theory first came into existence in the string model are also toroidally compactified down 1960’s, as an attempt to explain the strong force to the four dimensions with which the macroscopic within hadrons. It was hypothesized that the strong world is familiar. The compactified dimensions then force between quarks was due to a “strings” con- exist on scales smaller than that which current necting quarks, with one quark at each endpoint of technology can probe. 2 While several formalisms exist for constructing C. U(1) Rotation heterotic string models, the focus of this research The U(1) charges qij are defined by the following is on the weakly-coupled free fermionic heterotic two equations. string (WCFFHS) formalism. In the WCFFHS for- malism, the focus is on worldsheet fermions: 8 from Qi j ji = qij j ji (6) the left-moving SUSY, and 32 from fermionizing hQi j ji = qij (7) the 16 compact directions[5]. Notationally, the left- movers and right-movers are represented together In equation (6), Qi is the matrix operator for the th th as: i U(1), while j ji is the j matter state vector. Here qij (the ij element of our U(1) charge matrix) (RLjjRR) (1) is an eigenvalue of the operator Qi and eigenvector where R is twice the rank of the non-abelian Lie j ji. In equation (7), jQii is the basis vector for th gauge group generated, and the subscript indicates the i U(1) (not to be confused with the matrix the direction. For example, in ten large space-time Qi, which is a different basis of matrices). Equation dimensions this is represented by: (7) defines the U(1) charges qij as the inner product of the charge basis vector jQii and the matter state (8jj32): (2) j ji. A matrix of U(1) charges with elements qij is constructed in accordance with equations (6) and The number of compactified directions is the num- (7). It is this matrix of U(1) charges that shall be ber of bosons, so with each compactified dimension, considered in the following sections. the number of large dimensions decreases by 1, The trace of a U(1) charge is the sum of the and the rank of each the left- and right-movers charges of the physical states under that U(1) factor, increases by two, because there are two coupled and is computed as: fermions created for every fermionized boson[1]. m m When compactification occurs repeatedly, it can be X X T r(Q ) = h j Q i = q (8) seen that when D is the number of dimensions[10]: i j i ij j=1 j=1 D = 9 : (10jj34) where equation (7) has been applied and m has been D = 8 : (12jj36) defined as the total number of matter states. When the traces of these U(1) charges are com- . puted, it is easily seen that many of the charges D = 4 : (20jj44): have non-zero traces, making them anomalous. In the heterotic theories, it is possible to modify the In general, the total rank, RT , can be found using: U(1) basis without changing the physics so that all n but one of the U(1) traces are equal to zero[3]. That RT = 26 − D − (3) is, after the manipulation, only one U(1) factor will 2 remain anomalous. Transforming the U(1) charges where n is the total number of left-right pairings. into this form greatly simplifies phenomenological The total rank is also the sum of the ranks of the computations performed with the U(1) charges, in- abelian and non-abelian charges: cluding supersymmetry calculations[2]. The focus of this study is to develop an algorithm to efficiently R = R + R : (4) T A NA perform such a transformation. Note that in the model building process, only the Specifically, the goal is to transform the U(1) non-abelian group factors are constructed on the abelian charges so that: first pass, after which RNA can be determined. A 0 T r(Q1 ) 6= 0 (9) realistic string model must have D = 4 large space- T r(Q 0) = 0 (10) time dimensions[11]. Combining equations (3) and i 0 (4) with D = 4 yields the number of abelian U(1) where Qi is the transformed charge at index i, charges RA: and 1 < i ≤ RA. In order for the physics to be n invariant, it is required that the orthogonal basis R = 22 − − R : (5) A 2 NA formed by the U(1) states be normalized before and after calculation. Therefore, imposing an arbitrary Equation (5) gives the number of abelian U(1) normalization of length squared 2 yields: charges that are explicitly constructed within the model. hQi j Qji = 2δij (11) 3 before the transformation, and: will yield a new basis in which only one U(1) is anomalous. The second requirement (16) follows Q 0 Q 0 = 2δ (12) i j ij from equations (11) and (12) and ensures that we after the transformation, where δij is the normal will preserve the U(1) normalization and orthogo- Kronecker delta. nality. To accomplish this, an orthogonal RA × RA rotation matrix R is defined that, when acting on A. Algorithm I the traces, gives: From the properties of R, if the rotated charges 2 3 2 0 3 are defined as: T r(Q1) T r(Q1 ) 0 0 6 T r(Q2) 7 6 T r(Q2 ) 7 Qi = ai jQii + aj jQji (17) R 6 7 = 6 7 : (13) 6 . 7 6 . 7 0 4 . 5 4 . 5 Qj = aj jQii − ai jQji (18) 0 T r(QR ) T r(QR ) A A where ai; aj 2 ~a, and i and j are the indices of The objective is to find R that will satisfy equations the charges being mixed. The orthogonality between 0 0 (9) and (10) when it is applied to the U(1) traces Qi and Qj is guaranteed[4]. It will be shown and equations (11) and (12) when applied to the that the elimination of all but one anomalous charge U(1) charges.
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